Assertion-Reason MCQs for Sub-Topics of Topic 10: Calculus

Question 1. Assertion (A): The concept of a limit describes the behavior of a function as the input approaches a particular value, without necessarily reaching it.

Reason (R): The value of $\lim_{x \to a} f(x)$ depends only on the value of $f(a)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): For a function $f(x)$, the right hand limit at $x=a$, denoted as $\lim_{x \to a^+} f(x)$, considers the values of $f(x)$ as $x$ approaches $a$ from values greater than $a$.

Reason (R): The right hand limit is only concerned with values of $x$ strictly less than $a$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The limit $\lim_{x \to a} f(x)$ exists if and only if the left hand limit and the right hand limit at $x=a$ are equal and finite.

Reason (R): If $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = L$, where $L$ is a finite number, then $\lim_{x \to a} f(x) = L$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The limit of the function $f(x) = \frac{1}{x-2}$ as $x \to 2$ does not exist.

Reason (R): The left hand limit $\lim_{x \to 2^-} \frac{1}{x-2} = -\infty$ and the right hand limit $\lim_{x \to 2^+} \frac{1}{x-2} = +\infty$. Since LHL $\neq$ RHL and are infinite, the overall limit does not exist as a finite number.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The limit of the function $f(x) = \begin{cases} x^2+1 & , & x < 1 \\ 2x & , & x \geq 1 \end{cases}$ exists at $x=1$.

Reason (R): The left hand limit $\lim_{x \to 1^-} (x^2+1) = 1^2+1 = 2$ and the right hand limit $\lim_{x \to 1^+} (2x) = 2(1) = 2$. Since LHL = RHL = 2, the limit exists and is 2.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): To evaluate $\lim_{x \to 3} (x^2 - 5x + 6)$, we can substitute $x=3$ directly into the function.

Reason (R): The function $f(x) = x^2 - 5x + 6$ is a polynomial, and the limit of a polynomial as $x \to a$ is equal to the value of the polynomial at $x=a$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): Direct substitution of $x=2$ in $\lim_{x \to 2} \frac{x-2}{x^2-4}$ results in the indeterminate form $\frac{0}{0}$.

Reason (R): Both the numerator and the denominator become zero when $x=2$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): To evaluate $\lim_{x \to 3} \frac{x^2 - 5x + 6}{x - 3}$, factorization is a suitable method.

Reason (R): The numerator $x^2 - 5x + 6$ can be factored as $(x-2)(x-3)$, allowing cancellation of the $(x-3)$ term when $x \neq 3$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The limit $\lim_{x \to 0} \frac{\sqrt{1+x} - 1}{x}$ can be evaluated by rationalizing the numerator.

Reason (R): Rationalizing involves multiplying the numerator and denominator by the conjugate of the numerator, which is $\sqrt{1+x} + 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): $\lim_{x \to 0} \frac{\sin(2x)}{x} = 2$.

Reason (R): This limit can be evaluated using the standard result $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ by setting $\theta = 2x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 11. Assertion (A): The value of $\lim_{x \to 4} (x^2 - 3x + 1)$ is obtained by direct substitution.

Reason (R): Direct substitution is applicable if the function is defined and finite at the limit point.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 12. Assertion (A): The left hand limit of $f(x) = \frac{|x|}{x}$ at $x=0$ is $-1$.

Reason (R): For $x < 0$, $|x| = -x$, so $\frac{|x|}{x} = \frac{-x}{x} = -1$. As $x \to 0^-$ from the left, the function value is constantly $-1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 13. Assertion (A): The existence of $\lim_{x \to a} f(x)$ guarantees that $f(a)$ is defined.

Reason (R): The definition of a limit does not require the function to be defined at the point $x=a$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 14. Assertion (A): To evaluate $\lim_{x \to \infty} \frac{x^2+x+1}{2x^2-x+5}$, we can divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^2$.

Reason (R): Dividing by the highest power of $x$ transforms the expression into a form where limits of terms like $\frac{c}{x^n}$ as $x \to \infty$ are known (they approach 0).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 15. Assertion (A): $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$ is a standard limit result.

Reason (R): This limit arises from the derivative of $e^x$ at $x=0$ using the definition from first principles.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 16. Assertion (A): The limit of $\frac{\ln(1+x)}{x}$ as $x \to 0$ is $1$.

Reason (R): This is a standard limit result related to the derivative of $\ln x$ at $x=1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 17. Assertion (A): The limit $\lim_{x \to a} [f(x) \cdot g(x)]$ is equal to $[\lim_{x \to a} f(x)] \cdot [\lim_{x \to a} g(x)]$ if $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist.

Reason (R): The product rule for limits states that the limit of a product is the product of the limits, provided the individual limits exist.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 18. Assertion (A): The Squeeze Play Theorem (Sandwich Theorem) is used to find the limit of a function that is 'squeezed' between two other functions whose limits are known and equal at the point of interest.

Reason (R): If $g(x) \leq f(x) \leq h(x)$ for all $x$ in an interval containing $a$ (except possibly at $a$), and $\lim_{x \to a} g(x) = L = \lim_{x \to a} h(x)$, then $\lim_{x \to a} f(x) = L$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 19. Assertion (A): $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

Reason (R): This is a fundamental trigonometric limit.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 20. Assertion (A): $\lim_{x \to 0} \frac{\tan x}{x} = 1$.

Reason (R): This can be shown using the fact that $\tan x = \frac{\sin x}{\cos x}$ and the standard limits $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \cos x = 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 21. Assertion (A): $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$ for $a > 0, a \neq 1$.

Reason (R): This is a standard limit result for exponential functions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 22. Assertion (A): $\lim_{x \to \infty} (1 + \frac{1}{x})^x = e$.

Reason (R): This limit is the definition of the mathematical constant $e$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 23. Assertion (A): If $\lim_{x \to a} f(x)$ exists and is equal to $L$, then the limit of $k \cdot f(x)$ as $x \to a$ is $kL$, where $k$ is a constant.

Reason (R): The constant multiple rule for limits states that the limit of a constant times a function is the constant times the limit of the function.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 24. Assertion (A): $\lim_{x \to 0} \frac{\sin(\alpha x)}{\sin(\beta x)} = \frac{\alpha}{\beta}$, where $\beta \neq 0$.

Reason (R): This can be derived using the standard limit $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ by multiplying and dividing the numerator and denominator by $\alpha x$ and $\beta x$ respectively.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 25. Assertion (A): $\lim_{x \to 0} x \sin(\frac{1}{x}) = 0$.

Reason (R): This limit can be evaluated using the Squeeze Play Theorem, as $-|x| \leq x \sin(\frac{1}{x}) \leq |x|$ and $\lim_{x \to 0} -|x| = 0 = \lim_{x \to 0} |x|$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M \neq 0$.

Reason (R): The quotient rule for limits states that the limit of a quotient is the quotient of the limits, provided the limit of the denominator is non-zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): $\lim_{x \to a} c = c$, where $c$ is a constant.

Reason (R): The limit of a constant function is the constant itself, regardless of the value $x$ approaches.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If $f(x) \leq g(x)$ for all $x$ near $a$ (except possibly at $a$), and $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then $L \leq M$.

Reason (R): This property of limits is often used in conjunction with the Squeeze Play Theorem.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): $\lim_{x \to 0} \frac{\sin x}{x^2}$ does not exist.

Reason (R): $\lim_{x \to 0} \frac{\sin x}{x} = 1$, so $\lim_{x \to 0} \frac{\sin x}{x^2} = \lim_{x \to 0} (\frac{\sin x}{x} \cdot \frac{1}{x}) = 1 \cdot \lim_{x \to 0} \frac{1}{x}$. Since $\lim_{x \to 0} \frac{1}{x}$ does not exist (LHL is $-\infty$, RHL is $+\infty$), the product limit does not exist.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$.

Reason (R): This is a standard trigonometric limit derived using trigonometric identities or L'Hopital's rule.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): $\lim_{x \to 0} \frac{\tan (\alpha x)}{x} = \alpha$ for $\alpha \neq 0$.

Reason (R): Using $\lim_{\theta \to 0} \frac{\tan \theta}{\theta} = 1$, we have $\lim_{x \to 0} \frac{\tan (\alpha x)}{x} = \lim_{x \to 0} \alpha \frac{\tan (\alpha x)}{\alpha x} = \alpha \lim_{x \to 0} \frac{\tan (\alpha x)}{\alpha x} = \alpha \cdot 1 = \alpha$ (assuming $\alpha x \to 0$ as $x \to 0$).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): $\lim_{x \to \infty} (1 + \frac{k}{x})^x = e^k$ for any real number $k$.

Reason (R): This is a generalization of the standard limit for $e$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): $\lim_{x \to 0} \frac{\ln(1+ax)}{bx} = \frac{a}{b}$, where $b \neq 0$.

Reason (R): This can be derived using the standard limit $\lim_{\theta \to 0} \frac{\ln(1+\theta)}{\theta} = 1$ by manipulating the expression.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): If $\lim_{x \to a} f(x) = L$, then $\lim_{x \to a} |f(x)| = |L|$.

Reason (R): The absolute value function is continuous.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$, provided the individual limits exist.

Reason (R): The difference rule for limits allows us to subtract the limits of two functions if their individual limits exist.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 11. Assertion (A): $\lim_{x \to \infty} \sin x$ does not exist.

Reason (R): As $x \to \infty$, the value of $\sin x$ oscillates between $-1$ and $1$ and does not approach a single finite value.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 12. Assertion (A): $\lim_{x \to 0} \frac{\sin^{-1} x}{x} = 1$.

Reason (R): Let $y = \sin^{-1} x$. As $x \to 0$, $y \to 0$. The limit becomes $\lim_{y \to 0} \frac{y}{\sin y} = \lim_{y \to 0} \frac{1}{\sin y / y} = \frac{1}{1} = 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 13. Assertion (A): $\lim_{x \to a} x^n = a^n$ for any real number $n$, provided $a^n$ is defined.

Reason (R): This is a standard algebraic limit result.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 14. Assertion (A): $\lim_{x \to 0} \frac{(1+x)^n - 1}{x} = n$ for any real number $n$.

Reason (R): This is a standard limit result that can be proven using binomial expansion or L'Hopital's rule.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 15. Assertion (A): $\lim_{x \to 0} \frac{e^{ax} - e^{bx}}{x} = a-b$.

Reason (R): $\lim_{x \to 0} \frac{e^{ax} - e^{bx}}{x} = \lim_{x \to 0} \frac{(e^{ax}-1) - (e^{bx}-1)}{x} = \lim_{x \to 0} \frac{e^{ax}-1}{x} - \lim_{x \to 0} \frac{e^{bx}-1}{x} = a \lim_{x \to 0} \frac{e^{ax}-1}{ax} - b \lim_{x \to 0} \frac{e^{bx}-1}{bx} = a(1) - b(1) = a-b$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 16. Assertion (A): $\lim_{x \to 0} \frac{\cos x - 1}{x} = 0$.

Reason (R): This is a standard trigonometric limit derived from $\lim_{x \to 0} \frac{1-\cos x}{x^2}$ and $\lim_{x \to 0} x = 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 17. Assertion (A): $\lim_{x \to 0} \frac{\sin x - x}{x^3} = -\frac{1}{6}$.

Reason (R): This limit can be evaluated using series expansions of $\sin x$ or L'Hopital's Rule repeatedly.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 18. Assertion (A): $\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$ for any rational number $n$.

Reason (R): This is a standard algebraic limit result and forms the basis for the power rule of differentiation.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 19. Assertion (A): The limit of $\frac{1-\cos(ax)}{x^2}$ as $x \to 0$ is $\frac{a^2}{2}$.

Reason (R): This is derived from the standard limit $\lim_{\theta \to 0} \frac{1-\cos \theta}{\theta^2} = \frac{1}{2}$ by substituting $\theta = ax$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 20. Assertion (A): $\lim_{x \to 0} \frac{e^{kx} - 1}{x} = k$ for any real number $k$.

Reason (R): This is obtained from the standard limit $\lim_{\theta \to 0} \frac{e^\theta - 1}{\theta} = 1$ by substituting $\theta = kx$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): A function $f(x)$ is continuous at a point $x=a$ if $\lim_{x \to a} f(x) = f(a)$.

Reason (R): This is the formal definition of continuity at a point, requiring the limit to exist, the function to be defined at the point, and the limit value to equal the function value.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The function $f(x) = |x|$ is continuous at $x=0$.

Reason (R): $\lim_{x \to 0^-} |x| = 0$, $\lim_{x \to 0^+} |x| = 0$, and $f(0) = |0| = 0$. Since $\lim_{x \to 0} |x| = 0 = f(0)$, the function is continuous at $x=0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): A function $f(x)$ is continuous on an open interval $(a, b)$ if it is continuous at every point in the interval.

Reason (R): Continuity on an open interval does not involve checking limits or function values at the endpoints $a$ and $b$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): A function $f(x)$ is continuous on a closed interval $[a, b]$ if it is continuous on $(a,b)$, continuous from the right at $a$, and continuous from the left at $b$.

Reason (R): Continuity from the right at $a$ means $\lim_{x \to a^+} f(x) = f(a)$, and continuity from the left at $b$ means $\lim_{x \to b^-} f(x) = f(b)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The function $f(x) = \frac{1}{x-1}$ has a jump discontinuity at $x=1$.

Reason (R): A jump discontinuity occurs when the left and right hand limits exist but are not equal.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The function $f(x) = \frac{x^2-1}{x-1}$ has a removable discontinuity at $x=1$.

Reason (R): A removable discontinuity occurs at $x=a$ if $\lim_{x \to a} f(x)$ exists but $f(a)$ is undefined or $\lim_{x \to a} f(x) \neq f(a)$. Here, $\lim_{x \to 1} \frac{x^2-1}{x-1} = \lim_{x \to 1} (x+1) = 2$, but $f(1)$ is undefined.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): If $f$ and $g$ are continuous functions at $x=a$, then the product function $f \cdot g$ is also continuous at $x=a$.

Reason (R): The product of two continuous functions is always continuous.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): If $f$ is continuous and $g$ is continuous, then the composite function $(f \circ g)(x) = f(g(x))$ is continuous.

Reason (R): The composition of continuous functions is continuous, provided the range of $g$ is in the domain of $f$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The function $f(x) = \sin(|x|)$ is continuous for all real numbers.

Reason (R): $f(x)$ is a composition of the continuous function $\sin x$ and the continuous function $|x|$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): If $f(x)$ is continuous at $x=a$, then $\lim_{x \to a} f(x)$ must exist and be equal to $f(a)$.

Reason (R): This is the definition of continuity at a point, which implies the left and right limits must be equal and finite, and equal to the function value.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 11. Assertion (A): The function $f(x) = \begin{cases} 1 & , & x \in \mathbb{Q} \\ 0 & , & x \notin \mathbb{Q} \end{cases}$ (Dirichlet function) is discontinuous everywhere.

Reason (R): In any interval, however small, there are both rational and irrational numbers. Thus, $\lim_{x \to a} f(x)$ does not exist for any $a$, so the function is not continuous anywhere.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 12. Assertion (A): A function with an infinite discontinuity at $x=a$ has at least one of its one-sided limits equal to $\infty$ or $-\infty$ at $x=a$.

Reason (R): Infinite discontinuities are characterized by the function value tending towards infinity near the point of discontinuity, often seen in functions like $1/x$ at $x=0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 13. Assertion (A): If $f(x)$ is continuous on $[a, b]$, then it attains its maximum and minimum values on $[a, b]$.

Reason (R): This is the Extreme Value Theorem, which applies to continuous functions on closed intervals.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 14. Assertion (A): If a function is discontinuous at a point, its limit at that point cannot exist.

Reason (R): If the limit exists and is finite, and the function is defined at the point, the function is continuous if the limit equals the function value. Discontinuity means this condition is not met, implying either the limit doesn't exist or it doesn't match the function value.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 15. Assertion (A): All polynomial functions are continuous for all real numbers.

Reason (R): Polynomial functions are sums and products of basic continuous functions like $x$ and constants, and the algebra of continuous functions states that sums and products of continuous functions are continuous.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 16. Assertion (A): The function $f(x) = \text{cosec } x$ is continuous on its domain.

Reason (R): $\text{cosec } x$ is defined as $\frac{1}{\sin x}$ and is discontinuous where $\sin x = 0$, i.e., at $x = n\pi$ for integer $n$. Its domain is all real numbers except $n\pi$. On this domain, it is a quotient of continuous functions where the denominator is non-zero, hence it is continuous.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 17. Assertion (A): The function $f(x) = \begin{cases} x+1 & , & x \leq 0 \\ 1 & , & x > 0 \end{cases}$ is continuous at $x=0$.

Reason (R): $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x+1) = 0+1 = 1$. $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} 1 = 1$. $f(0) = 0+1 = 1$. Since LHL = RHL = $f(0)$, the function is continuous at $x=0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 18. Assertion (A): A function with a jump discontinuity cannot be made continuous by redefining the function at a single point.

Reason (R): A jump discontinuity implies that the left and right limits are different. Redefining the function value at the point of discontinuity cannot make these two different limits equal.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 19. Assertion (A): If $f$ is continuous and non-zero on an interval, then $\frac{1}{f}$ is also continuous on that interval.

Reason (R): The reciprocal of a continuous function is continuous wherever the original function is non-zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 20. Assertion (A): The function $f(x) = x \sin(\frac{1}{x})$ for $x \neq 0$ and $f(0)=0$ is continuous at $x=0$.

Reason (R): $\lim_{x \to 0} x \sin(\frac{1}{x}) = 0$ (using Squeeze Theorem) and $f(0)=0$. Since the limit equals the function value, it is continuous at $x=0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The derivative of a function $f(x)$ at a point $x=a$ is defined as $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$, provided the limit exists.

Reason (R): This definition is based on the concept of the instantaneous rate of change of the function at $x=a$, represented by the slope of the tangent line.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): A function is differentiable at a point if and only if the left hand derivative and the right hand derivative at that point are equal and finite.

Reason (R): The existence of the limit $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ implies that the limits as $h \to 0^+$ (RHD) and $h \to 0^-$ (LHD) are equal and finite.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If a function $f(x)$ is differentiable at a point $x=a$, then it must be continuous at $x=a$.

Reason (R): Differentiability implies continuity, but the converse is not true.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The function $f(x) = |x-2|$ is not differentiable at $x=2$.

Reason (R): At $x=2$, the graph of $f(x)$ has a sharp corner (a cusp), where the tangent line is not uniquely defined, and the left and right derivatives are $-1$ and $1$, respectively.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The derivative of $f(x) = \cos x$ from first principles is $-\sin x$.

Reason (R): Using $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{\cos(x+h) - \cos x}{h}$, applying trigonometric identities and standard limits leads to $-\sin x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): A function can be continuous at a point but not differentiable at that point.

Reason (R): The function $f(x) = |x|$ at $x=0$ is an example of a function that is continuous but not differentiable at a point.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): Differentiability of a function $f(x)$ on an open interval $(a, b)$ means $f(x)$ is differentiable at every point in $(a, b)$.

Reason (R): Differentiability on an open interval $(a, b)$ does not require checking the derivative at the endpoints $a$ and $b$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): If a function is differentiable at every point in its domain, it is called a differentiable function.

Reason (R): The set of points where a function is differentiable is called its domain of differentiability.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The derivative process finds the slope of the tangent line to the function's graph at a given point.

Reason (R): The derivative measures the instantaneous rate of change of the function.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): The definition of the derivative involves a limit.

Reason (R): The instantaneous rate of change is obtained by taking the limit of the average rate of change as the interval shrinks to zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The derivative of a constant function $f(x) = c$ is $f'(x) = 0$.

Reason (R): The rate of change of a quantity that does not change is zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The derivative of $f(x) = x^7$ is $f'(x) = 7x^6$.

Reason (R): The power rule for differentiation states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$ for any real number $n$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If $f(x) = u(x) + v(x)$, then $f'(x) = u'(x) + v'(x)$.

Reason (R): The sum rule for derivatives states that the derivative of a sum is the sum of the derivatives.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The derivative of $f(x) = 3x^4 - 2x^2 + 5x - 1$ is $f'(x) = 12x^3 - 4x + 5$.

Reason (R): This is found by applying the power rule, constant multiple rule, sum/difference rules, and the derivative of a constant.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The derivative of $f(x) = \sin x$ is $f'(x) = \cos x$, and the derivative of $g(x) = \cos x$ is $g'(x) = \sin x$.

Reason (R): The derivatives of $\sin x$ and $\cos x$ are $\cos x$ and $-\sin x$, respectively.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The derivative of $f(x) = \tan x$ is $f'(x) = \text{sec}^2 x$.

Reason (R): This is a standard derivative formula for trigonometric functions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): The derivative of $f(x) = e^x$ is $f'(x) = e^x$.

Reason (R): The exponential function $e^x$ is its own derivative.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): The derivative of $f(x) = a^x$ is $f'(x) = a^x \log_a e$.

Reason (R): The derivative of $a^x$ is $a^x \ln a$. Since $\ln a = \frac{\log_a a}{\log_a e} = \frac{1}{\log_a e}$, the formula is correct.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The derivative of $f(x) = \ln x$ is $f'(x) = \frac{1}{x}$ for $x > 0$.

Reason (R): This is a standard derivative formula for the natural logarithmic function.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): The derivative of $f(x) = \log_a x$ is $f'(x) = \frac{1}{x \ln a}$ for $x > 0, a > 0, a \neq 1$.

Reason (R): Using the change of base formula, $\log_a x = \frac{\ln x}{\ln a}$. Differentiating with respect to $x$, $\frac{d}{dx}(\frac{\ln x}{\ln a}) = \frac{1}{\ln a} \frac{d}{dx}(\ln x) = \frac{1}{\ln a} \cdot \frac{1}{x} = \frac{1}{x \ln a}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 11. Assertion (A): If $f(x) = u(x) \cdot v(x)$, then $f'(x) = u'(x) v'(x)$.

Reason (R): The product rule for derivatives states that $(uv)' = u'v + uv'$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 12. Assertion (A): If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$, provided $v(x) \neq 0$.

Reason (R): This is the quotient rule for derivatives.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 13. Assertion (A): The derivative of $\sec x$ is $\sec x \tan x$.

Reason (R): $\sec x = \frac{1}{\cos x}$. Using the quotient rule, $\frac{d}{dx}(\frac{1}{\cos x}) = \frac{0 \cdot \cos x - 1 \cdot (-\sin x)}{\cos^2 x} = \frac{\sin x}{\cos^2 x} = \frac{\sin x}{\cos x} \cdot \frac{1}{\cos x} = \tan x \sec x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 14. Assertion (A): The derivative of $\text{cosec } x$ is $-\text{cosec } x \text{cot } x$.

Reason (R): This is a standard derivative formula for trigonometric functions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 15. Assertion (A): The derivative of $f(x) = c \cdot u(x)$ is $f'(x) = c \cdot u'(x)$, where $c$ is a constant.

Reason (R): The constant multiple rule allows us to factor out a constant before differentiating.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): A composite function is a function of a function, like $f(g(x))$.

Reason (R): In $f(g(x))$, the function $g(x)$ is the input to the function $f$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The chain rule is used to find the derivative of composite functions.

Reason (R): If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The derivative of $(2x+3)^5$ is $5(2x+3)^4 \cdot 2 = 10(2x+3)^4$ using the chain rule.

Reason (R): Let $y = u^5$ and $u = 2x+3$. Then $\frac{dy}{du} = 5u^4$ and $\frac{du}{dx} = 2$. By the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^4 \cdot 2 = 10(2x+3)^4$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The derivative of $\sin(x^2)$ is $\cos(x^2)$.

Reason (R): The chain rule states that $\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$. Here $f(u) = \sin u$ and $g(x) = x^2$. So $f'(u) = \cos u$ and $g'(x) = 2x$. The derivative is $\cos(x^2) \cdot 2x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The derivative of $e^{\tan x}$ is $e^{\tan x} \text{sec}^2 x$.

Reason (R): Using the chain rule with $y = e^u$ and $u = \tan x$, we get $\frac{dy}{du} = e^u$ and $\frac{du}{dx} = \text{sec}^2 x$. $\frac{dy}{dx} = e^u \cdot \text{sec}^2 x = e^{\tan x} \text{sec}^2 x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The chain rule can be extended for more than two functions, like $f(g(h(x)))$.

Reason (R): If $y = f(u)$, $u = g(v)$, and $v = h(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): The derivative of $\sqrt{x^2+1}$ is $\frac{x}{\sqrt{x^2+1}}$ using the chain rule.

Reason (R): Let $y = \sqrt{u}$ and $u = x^2+1$. Then $\frac{dy}{du} = \frac{1}{2\sqrt{u}}$ and $\frac{du}{dx} = 2x$. By the chain rule, $\frac{dy}{dx} = \frac{1}{2\sqrt{x^2+1}} \cdot 2x = \frac{x}{\sqrt{x^2+1}}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): The chain rule is essential for differentiating composite functions involving trigonometric, exponential, or logarithmic functions applied to algebraic expressions.

Reason (R): The chain rule connects the derivative of the outer function (evaluated at the inner function) with the derivative of the inner function.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The derivative of $\sin(\sin x)$ is $\cos(\sin x) \cos x$.

Reason (R): Using the chain rule with outer function $f(u) = \sin u$ and inner function $g(x) = \sin x$. $f'(u) = \cos u$ and $g'(x) = \cos x$. $\frac{d}{dx} f(g(x)) = f'(g(x))g'(x) = \cos(\sin x) \cos x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): The derivative of $\ln(x^2+4)$ is $\frac{2x}{x^2+4}$.

Reason (R): Using the chain rule with $y = \ln u$ and $u = x^2+4$. $\frac{dy}{du} = \frac{1}{u}$ and $\frac{du}{dx} = 2x$. $\frac{dy}{dx} = \frac{1}{u} \cdot 2x = \frac{2x}{x^2+4}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): To find $\frac{dy}{dx}$ for the equation $x^3 + y^3 = 3xy$, we use implicit differentiation.

Reason (R): The equation defines $y$ as a function of $x$ implicitly, and it is difficult or impossible to express $y$ explicitly in terms of $x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): Differentiating $y^2$ with respect to $x$ using implicit differentiation gives $2y \frac{dy}{dx}$.

Reason (R): We treat $y$ as a function of $x$ and apply the chain rule: $\frac{d}{dx}(y^2) = \frac{d}{dy}(y^2) \cdot \frac{dy}{dx} = 2y \frac{dy}{dx}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If $y = f(x)$ and $x = g(y)$ is its inverse function, then $\frac{dy}{dx} = \frac{1}{g'(y)}$.

Reason (R): By the chain rule, $\frac{dy}{dx} \cdot \frac{dx}{dy} = \frac{dy}{dy} = 1$. So $\frac{dy}{dx} = \frac{1}{dx/dy} = \frac{1}{g'(y)}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The derivative of $\cos^{-1} x$ is $-\frac{1}{\sqrt{1-x^2}}$ for $|x| < 1$.

Reason (R): This is a standard derivative formula for inverse trigonometric functions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The derivative of $\tan^{-1} (\frac{2x}{1-x^2})$ is $\frac{2}{1+x^2}$.

Reason (R): Using the substitution $x = \tan \theta$, we get $\tan^{-1}(\frac{2 \tan \theta}{1-\tan^2 \theta}) = \tan^{-1}(\tan 2\theta) = 2\theta$. Since $\theta = \tan^{-1} x$, the expression becomes $2 \tan^{-1} x$. The derivative of $2 \tan^{-1} x$ is $2 \cdot \frac{1}{1+x^2} = \frac{2}{1+x^2}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): If $x = \sin y$, then $\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}$.

Reason (R): Differentiating $x = \sin y$ with respect to $y$ gives $\frac{dx}{dy} = \cos y$. Since $\cos y = \sqrt{1-\sin^2 y} = \sqrt{1-x^2}$ for a suitable range of $y$, $\frac{dy}{dx} = \frac{1}{dx/dy} = \frac{1}{\cos y} = \frac{1}{\sqrt{1-x^2}}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): Implicit differentiation requires using the chain rule whenever a term involving $y$ is differentiated with respect to $x$.

Reason (R): This is because $y$ is treated as a function of $x$ in implicit differentiation.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): The derivative of $\sec^{-1} x$ is $\frac{1}{|x|\sqrt{x^2-1}}$ for $|x|>1$.

Reason (R): This is a standard derivative formula for inverse trigonometric functions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): For $x^2 - y^2 = 1$, $\frac{dy}{dx} = \frac{x}{y}$ using implicit differentiation.

Reason (R): Differentiating $x^2 - y^2 = 1$ implicitly with respect to $x$ gives $2x - 2y \frac{dy}{dx} = 0$. Solving for $\frac{dy}{dx}$ yields $\frac{dy}{dx} = \frac{2x}{2y} = \frac{x}{y}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): If $y = \text{cosec}^{-1} x$, then $\frac{dy}{dx} = -\frac{1}{x\sqrt{x^2-1}}$.

Reason (R): The derivative of $\text{cosec}^{-1} x$ is $-\frac{1}{|x|\sqrt{x^2-1}}$. For $x>1$, $|x|=x$, so the statement is correct for $x>1$. For $x<-1$, $|x|=-x$, and the derivative is $\frac{1}{x\sqrt{x^2-1}}$. The provided Reason is partially correct but lacks the absolute value for the general formula.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): Logarithmic differentiation simplifies the process of finding the derivative of a function that is a product, quotient, or power of other functions.

Reason (R): Taking the logarithm of such a function converts products into sums, quotients into differences, and powers into products, which are easier to differentiate.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): To differentiate $y = (\sin x)^x$, logarithmic differentiation is the most suitable method.

Reason (R): The function is in the form $f(x)^{g(x)}$, which is best handled by taking the natural logarithm of both sides before differentiating.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If $x = f(t)$ and $y = g(t)$ are parametric equations defining $y$ as a function of $x$, then $\frac{dy}{dx} = \frac{g'(t)}{f'(t)}$, provided $f'(t) \neq 0$.

Reason (R): This formula for the derivative of a function defined parametrically is obtained by applying the chain rule $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): If $y = \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)}}$, taking the natural logarithm before differentiating simplifies the process.

Reason (R): $\ln y = \frac{1}{2} [\ln(x-1) + \ln(x-2) - \ln(x-3) - \ln(x-4)]$, which can be differentiated term by term.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): If $x = t^2$ and $y = t^3$, then $\frac{dy}{dx} = \frac{3}{2}t$.

Reason (R): $\frac{dx}{dt} = 2t$ and $\frac{dy}{dt} = 3t^2$. $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3t^2}{2t} = \frac{3}{2}t$ for $t \neq 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): Logarithmic differentiation is necessary for differentiating functions like $y = x^x$.

Reason (R): The power rule applies to $x^n$ (constant power) and the exponential rule applies to $a^x$ (constant base). For a variable base and variable power, the derivative cannot be found using just these basic rules.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): The derivative of $y = x^x$ is $y'(1 + \ln x)$.

Reason (R): Taking $\ln y = x \ln x$, differentiating implicitly gives $\frac{1}{y}\frac{dy}{dx} = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1$. So $\frac{dy}{dx} = y(1 + \ln x) = x^x (1 + \ln x)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): Functions in parametric form express both the independent and dependent variables in terms of a third variable, the parameter.

Reason (R): This form is particularly useful when the direct relationship between $x$ and $y$ is complicated or when describing curves where $y$ is not a single-valued function of $x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): To find the derivative of $y = (\cos x)^{\sin x}$, logarithmic differentiation is used.

Reason (R): This function is of the form $f(x)^{g(x)}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): If $x = \sin t$ and $y = \cos (2t)$, then $\frac{dy}{dx} = -4 \sin t$.

Reason (R): $\frac{dx}{dt} = \cos t$. $\frac{dy}{dt} = - \sin(2t) \cdot 2 = -2 \sin(2t) = -2(2 \sin t \cos t) = -4 \sin t \cos t$. $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-4 \sin t \cos t}{\cos t} = -4 \sin t$ (for $\cos t \neq 0$).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The second order derivative of a function $f(x)$ is the derivative of $f'(x)$ with respect to $x$.

Reason (R): This is represented as $\frac{d}{dx} \left(\frac{dy}{dx}\right)$ or $\frac{d^2 y}{dx^2}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If $y = x^4$, then the second derivative $\frac{d^2 y}{dx^2} = 12x^2$.

Reason (R): The first derivative $\frac{dy}{dx} = 4x^3$. The second derivative is the derivative of $4x^3$, which is $4 \cdot 3x^2 = 12x^2$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If $y = \sin x$, then $\frac{d^2 y}{dx^2} = -\sin x$.

Reason (R): $\frac{dy}{dx} = \cos x$. $\frac{d^2 y}{dx^2} = \frac{d}{dx}(\cos x) = -\sin x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): If $y = e^{ax}$, the $n$-th order derivative $\frac{d^n y}{dx^n} = a^n e^{ax}$.

Reason (R): Each differentiation with respect to $x$ brings down a factor of $a$ from the exponent.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): If $y = \ln x$, then $\frac{d^2 y}{dx^2} = \frac{1}{x^2}$.

Reason (R): $\frac{dy}{dx} = \frac{1}{x}$. $\frac{d^2 y}{dx^2} = \frac{d}{dx}(\frac{1}{x}) = \frac{d}{dx}(x^{-1}) = -1 \cdot x^{-2} = -\frac{1}{x^2}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The second derivative of a function $y=f(x)$ defined parametrically by $x=g(t)$ and $y=h(t)$ is given by $\frac{d^2 y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$.

Reason (R): We first find $\frac{dy}{dx} = \frac{h'(t)}{g'(t)}$, which is a function of $t$. Then we differentiate this function with respect to $t$ and divide by $\frac{dx}{dt}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): Higher order derivatives provide information about the rate of change of the rate of change of a function.

Reason (R): The second derivative, for example, describes the acceleration if the first derivative describes velocity.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): If $y = \sin (ax+b)$, then $\frac{d^2 y}{dx^2} = -a^2 \sin (ax+b)$.

Reason (R): $\frac{dy}{dx} = a \cos (ax+b)$. $\frac{d^2 y}{dx^2} = a \cdot (-a \sin (ax+b)) = -a^2 \sin (ax+b)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): If $y = e^x \cos x$, then $\frac{d^2 y}{dx^2} = -2e^x \sin x$.

Reason (R): $\frac{dy}{dx} = e^x \cos x - e^x \sin x = e^x(\cos x - \sin x)$. $\frac{d^2 y}{dx^2} = e^x(\cos x - \sin x) + e^x(-\sin x - \cos x) = e^x \cos x - e^x \sin x - e^x \sin x - e^x \cos x = -2e^x \sin x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): The process of finding higher order derivatives involves successive differentiation.

Reason (R): To find the $n$-th derivative, we differentiate the function $n$ times consecutively.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): Rolle's Theorem requires a function $f(x)$ to be continuous on $[a,b]$ and differentiable on $(a,b)$, with $f(a) = f(b)$.

Reason (R): These conditions are necessary for the existence of at least one point $c \in (a,b)$ where $f'(c) = 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If a function satisfies the conditions of Rolle's Theorem, then there is a point where the tangent to the curve is parallel to the x-axis.

Reason (R): Rolle's Theorem states that there exists $c \in (a,b)$ such that $f'(c) = 0$, and $f'(c)$ is the slope of the tangent line at $x=c$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): Lagrange's Mean Value Theorem is applicable to a function $f(x)$ if it is continuous on $[a,b]$ and differentiable on $(a,b)$.

Reason (R): These are the only conditions required for the existence of a point $c \in (a,b)$ satisfying the conclusion of the theorem.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The geometric interpretation of the Mean Value Theorem is that there exists a point on the curve where the tangent is parallel to the secant line connecting the endpoints of the interval.

Reason (R): The slope of the tangent at $x=c$ is $f'(c)$, and the slope of the secant line connecting $(a, f(a))$ and $(b, f(b))$ is $\frac{f(b) - f(a)}{b - a}$. The theorem states $f'(c) = \frac{f(b) - f(a)}{b - a}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): Rolle's Theorem can be seen as a special case of the Mean Value Theorem where $f(a) = f(b)$.

Reason (R): When $f(a) = f(b)$ in the MVT conclusion $f'(c) = \frac{f(b) - f(a)}{b - a}$, it simplifies to $f'(c) = 0$, which is the conclusion of Rolle's Theorem.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): For the function $f(x) = x^2$ on $[0, 2]$, the value of $c$ guaranteed by the Mean Value Theorem is $1$.

Reason (R): $f(0)=0, f(2)=4$. $\frac{f(2)-f(0)}{2-0} = \frac{4-0}{2} = 2$. $f'(x) = 2x$. Setting $f'(c) = 2c = 2$, we get $c=1$. Since $1 \in (0,2)$, the theorem is verified and $c=1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): If a function is not differentiable at a point in the open interval $(a,b)$, Rolle's Theorem might not apply even if other conditions are met.

Reason (R): Differentiability on $(a,b)$ is a necessary condition for Rolle's Theorem.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): The Mean Value Theorem can be used to prove that if $f'(x) = 0$ for all $x$ in an interval, then $f(x)$ is a constant function on that interval.

Reason (R): If $f'(x) = 0$ on $(a,b)$, then for any $x_1, x_2 \in (a,b)$ with $x_1 < x_2$, by MVT on $[x_1, x_2]$, there exists $c \in (x_1, x_2)$ such that $\frac{f(x_2)-f(x_1)}{x_2-x_1} = f'(c)$. Since $f'(c) = 0$, $f(x_2) - f(x_1) = 0$, so $f(x_1) = f(x_2)$. Thus $f(x)$ is constant.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): If $f(x)$ is a polynomial, it satisfies the continuity and differentiability conditions for both Rolle's and Mean Value Theorems on any closed interval.

Reason (R): Polynomials are continuous and differentiable everywhere.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): For the function $f(x) = (x-1)(x-2)(x-3)$ on $[1, 3]$, there exists $c \in (1, 3)$ such that $f'(c) = 0$.

Reason (R): $f(x)$ is a polynomial, so it's continuous on $[1,3]$ and differentiable on $(1,3)$. Also, $f(1) = 0$, $f(2)=0$, $f(3)=0$. Since $f(1) = f(3)$, Rolle's Theorem applies.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): If the volume of a sphere is increasing at a certain rate, its radius is also increasing at a related rate.

Reason (R): The rate of change of volume with respect to time is related to the rate of change of radius with respect to time through differentiation.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If the edge of a cube is increasing at a rate of $3$ cm/s, the volume is increasing at a rate of $9a^2$ cm$^3$/s, where $a$ is the current edge length.

Reason (R): Volume $V = a^3$. $\frac{dV}{dt} = \frac{d}{dt}(a^3) = 3a^2 \frac{da}{dt}$. Given $\frac{da}{dt} = 3$, so $\frac{dV}{dt} = 3a^2(3) = 9a^2$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): In economics, marginal cost is the rate of change of total cost with respect to the quantity produced.

Reason (R): If $C(x)$ is the total cost function for producing $x$ units, the marginal cost is $C'(x)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): If the total revenue from selling $x$ units is $R(x) = 100x - x^2$, the marginal revenue when $x=10$ units is calculated by finding $R'(10)$.

Reason (R): Marginal revenue is the derivative of the total revenue function, and evaluating it at a specific quantity gives the instantaneous rate of change of revenue at that quantity.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): Related rates problems involve finding the rate of change of one variable when the rate of change of another related variable is known.

Reason (R): These problems are solved by finding a relationship between the variables, differentiating implicitly with respect to time, and substituting the known values.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The rate of change of the surface area of a sphere with respect to its radius is equal to the circumference of a great circle of the sphere.

Reason (R): Surface Area $S = 4\pi r^2$. $\frac{dS}{dr} = 8\pi r$. The circumference of a great circle is $2\pi r$. So Assertion is false.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): If the total cost function is $C(x) = 5x^2 + 20x + 100$, the marginal cost is $10x + 20$.

Reason (R): Marginal cost is $C'(x)$, and the derivative of $5x^2 + 20x + 100$ is $10x + 20$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): The rate of change of the area of a circle with respect to its radius at $r=5$ cm, if the radius is increasing at $1$ cm/s, is $10\pi$ cm$^2$/s.

Reason (R): $A = \pi r^2$. $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$. At $r=5$ and $\frac{dr}{dt}=1$, $\frac{dA}{dt} = 2\pi (5)(1) = 10\pi$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): Marginal profit is the rate of change of total profit with respect to the number of units produced or sold.

Reason (R): If $P(x)$ is the total profit function, then marginal profit is $P'(x)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): In a related rates problem involving a conical pile of sand, if the height is always equal to the radius, the volume $V = \frac{1}{3}\pi h^3$.

Reason (R): The volume of a cone is $V = \frac{1}{3}\pi r^2 h$. If $h=r$, substitute $r$ with $h$ to get $V = \frac{1}{3}\pi h^2 h = \frac{1}{3}\pi h^3$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The equation of the tangent to the curve $y=f(x)$ at the point $(x_0, y_0)$ is $y - y_0 = f'(x_0)(x - x_0)$.

Reason (R): The slope of the tangent line at $(x_0, y_0)$ is $f'(x_0)$, and the point-slope form of a line is $y - y_0 = m(x - x_0)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The equation of the normal to the curve $y=f(x)$ at $(x_0, y_0)$ is $y - y_0 = -\frac{1}{f'(x_0)}(x - x_0)$, provided $f'(x_0) \neq 0$.

Reason (R): The normal line is perpendicular to the tangent line at the point of tangency. The slope of the normal is the negative reciprocal of the slope of the tangent.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): Differentials can be used to find the approximate value of $f(x + \Delta x)$.

Reason (R): The approximation is given by $f(x + \Delta x) \approx f(x) + dy = f(x) + f'(x) \Delta x$, where $dy = f'(x) dx$ and $dx = \Delta x$ is small.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): If $\Delta x$ is a small error in the measurement of $x$, then the approximate error $\Delta y$ in the function $y=f(x)$ is given by $dy = f'(x) dx$, where $dx = \Delta x$.

Reason (R): The differential $dy$ is defined as $f'(x) dx$, and for small $\Delta x$, $dy$ is a good approximation for $\Delta y = f(x+\Delta x) - f(x)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): If the radius of a sphere is measured with a relative error of $1\%$, the relative error in its volume is approximately $3\%$.

Reason (R): Volume $V = \frac{4}{3}\pi r^3$. $\frac{dV}{dr} = 4\pi r^2$. So $dV = 4\pi r^2 dr$. The relative error in volume is $\frac{dV}{V} = \frac{4\pi r^2 dr}{\frac{4}{3}\pi r^3} = \frac{3dr}{r} = 3 (\frac{dr}{r})$. If $\frac{dr}{r} = 0.01$ (1%), then $\frac{dV}{V} = 3(0.01) = 0.03$ (3%).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The approximate value of $(81)^{1/4}$ is exactly $3$.

Reason (R): Differentials provide approximations, not exact values, unless the function is linear.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): The equation of the tangent to the curve $y^2 = 4ax$ at the point $(x_0, y_0)$ is $yy_0 = 2a(x+x_0)$.

Reason (R): Differentiating implicitly, $2y \frac{dy}{dx} = 4a$, so $\frac{dy}{dx} = \frac{2a}{y}$. At $(x_0, y_0)$, the slope is $\frac{2a}{y_0}$. The equation of the tangent is $y - y_0 = \frac{2a}{y_0}(x - x_0) \implies y y_0 - y_0^2 = 2ax - 2ax_0$. Since $(x_0, y_0)$ is on the curve, $y_0^2 = 4ax_0$. So $y y_0 - 4ax_0 = 2ax - 2ax_0 \implies y y_0 = 2ax + 2ax_0 = 2a(x+x_0)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): The percentage error in the measurement of the area of a circle is double the percentage error in the measurement of its radius.

Reason (R): If $A = \pi r^2$, then $\frac{dA}{A} = 2 \frac{dr}{r}$. Multiplying by 100 gives the percentage error relationship.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The approximation $f(x + \Delta x) \approx f(x) + f'(x) \Delta x$ is based on the idea that the tangent line is a good approximation of the curve near the point of tangency.

Reason (R): The tangent line is the linear approximation of the function at a point.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): The normal to the curve $y = x^2$ at $(1, 1)$ is $x + 2y - 3 = 0$.

Reason (R): The slope of the tangent at $(1,1)$ is $\frac{dy}{dx}|_{x=1} = 2x|_{x=1} = 2$. The slope of the normal is $-\frac{1}{2}$. The equation of the normal is $y - 1 = -\frac{1}{2}(x - 1) \implies 2(y-1) = -(x-1) \implies 2y - 2 = -x + 1 \implies x + 2y - 3 = 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): A function $f(x)$ is strictly increasing on an interval if for any $x_1 < x_2$ in the interval, $f(x_1) < f(x_2)$.

Reason (R): This is the definition of a strictly increasing function.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If $f'(x) > 0$ for all $x$ in an interval, then $f(x)$ is strictly increasing on that interval.

Reason (R): A positive derivative indicates that the tangent line has a positive slope, meaning the function is rising.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If $f'(x) \geq 0$ for all $x$ in an interval, then $f(x)$ is increasing on that interval.

Reason (R): A non-negative derivative indicates that the function is either rising or staying constant.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The function $f(x) = x^3 - 3x$ is strictly increasing on $(-\infty, -1)$ and $(1, \infty)$, and strictly decreasing on $(-1, 1)$.

Reason (R): $f'(x) = 3x^2 - 3 = 3(x^2-1) = 3(x-1)(x+1)$. $f'(x) > 0$ for $x < -1$ or $x > 1$. $f'(x) < 0$ for $-1 < x < 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): To find the intervals where a function is increasing or decreasing, we analyze the sign of its first derivative.

Reason (R): The sign of the first derivative tells us whether the function is increasing or decreasing at a point.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The function $f(x) = e^x$ is strictly increasing for all real numbers.

Reason (R): The derivative $f'(x) = e^x$, and $e^x > 0$ for all real $x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): The function $f(x) = \cos x$ is strictly decreasing on $(0, \pi)$.

Reason (R): The derivative $f'(x) = -\sin x$, and $-\sin x < 0$ for $x \in (0, \pi)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): A function is monotonic on an interval if it is either entirely increasing or entirely decreasing on that interval.

Reason (R): Monotonicity describes the overall trend of the function's values over an interval.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): If a function is strictly increasing, its derivative must be strictly positive.

Reason (R): A zero derivative at a point allows the function to flatten out momentarily, violating strict increase. (Consider $f(x)=x^3$ at $x=0$).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): Applied problems, like maximizing revenue or minimizing cost, often involve finding intervals of increasing or decreasing functions (e.g., profit increasing with production up to a point).

Reason (R): The behavior of the derivative helps determine the direction of change in the dependent variable as the independent variable changes.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): Local extrema (maxima or minima) of a differentiable function can only occur at critical points where the derivative is zero.

Reason (R): At a local extremum where the derivative exists, the tangent line is horizontal, meaning its slope is zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If $f'(c) = 0$, then $f(x)$ must have a local extremum at $x=c$.

Reason (R): A critical point where $f'(c)=0$ could be a local maximum, a local minimum, or a point of inflection (like $f(x)=x^3$ at $x=0$).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The First Derivative Test checks the sign of the derivative around a critical point $c$ to determine if it is a local extremum.

Reason (R): If $f'(x)$ changes sign from positive to negative as $x$ increases through $c$, there is a local maximum at $c$. If it changes from negative to positive, there is a local minimum.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): If $f''(c) > 0$ at a critical point $c$ where $f'(c) = 0$, then $f(x)$ has a local minimum at $x=c$ by the Second Derivative Test.

Reason (R): A positive second derivative indicates concavity upwards, which corresponds to a local minimum at a stationary point.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The absolute maximum and minimum values of a continuous function on a closed interval $[a,b]$ are found by comparing the function values at critical points in $(a,b)$ and the endpoints $a$ and $b$.

Reason (R): The Extreme Value Theorem guarantees that a continuous function on a closed interval attains its absolute extrema within that interval, and these occur at critical points or endpoints.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): Optimization word problems often require setting up a function to be maximized or minimized and then using calculus techniques to find the extrema.

Reason (R): The first derivative is used to find critical points, and the first or second derivative tests are used to classify these points as local maxima or minima.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): If a function $f(x)$ has a local maximum at $x=c$, then $f'(c)$ must be zero.

Reason (R): This is true only if the function is differentiable at $x=c$. A local maximum can occur at a point where the derivative is undefined (e.g., $f(x) = -|x|$ at $x=0$).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): A critical point is a point in the domain of the function where the derivative is zero or undefined.

Reason (R): Local extrema can only occur at critical points.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): If the Second Derivative Test is inconclusive (i.e., $f'(c)=0$ and $f''(c)=0$), the point $c$ is necessarily a point of inflection.

Reason (R): While $f''(c)=0$ is a condition for a possible point of inflection, the second derivative test being inconclusive does not definitively mean it's an inflection point. It could still be a local extremum (e.g., $f(x)=x^4$ at $x=0$).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): Applied optimization problems in business and economics, such as finding maximum profit or minimum cost, directly use the concepts of extrema of functions.

Reason (R): Calculus provides the tools (derivatives) to find these optimal values efficiently.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): Integration is the reverse process of differentiation.

Reason (R): Finding the indefinite integral of a function is equivalent to finding its antiderivative.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The indefinite integral of $f(x)$ is denoted by $\int f(x) dx$.

Reason (R): The symbol $\int$ is the integral sign, and $dx$ indicates that the integration is with respect to the variable $x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The indefinite integral $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$, where $C$ is the constant of integration.

Reason (R): Differentiating $\frac{x^{n+1}}{n+1} + C$ with respect to $x$ gives $\frac{d}{dx}(\frac{x^{n+1}}{n+1} + C) = \frac{n+1}{n+1} x^{(n+1)-1} + 0 = x^n$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The indefinite integral of a function is a unique function.

Reason (R): If $F(x)$ is an antiderivative of $f(x)$, then $F(x) + C$ is also an antiderivative for any constant $C$. Thus, there is a family of antiderivatives, not a unique one.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): $\int \sin x dx = -\cos x + C$.

Reason (R): The derivative of $-\cos x + C$ is $\frac{d}{dx}(-\cos x + C) = -(-\sin x) + 0 = \sin x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The collection of all antiderivatives of a function $f(x)$ is called the indefinite integral of $f(x)$.

Reason (R): The term "indefinite" refers to the presence of the arbitrary constant of integration $C$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): If $F(x)$ is an antiderivative of $f(x)$, then $\int k f(x) dx = k F(x) + C$ for any constant $k$.

Reason (R): The constant multiple property of indefinite integrals allows factoring out a constant from the integral sign.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): $\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$.

Reason (R): The integral of a sum is the sum of the integrals.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): $\int \frac{1}{x} dx = \ln |x| + C$ for $x \neq 0$.

Reason (R): The derivative of $\ln x$ is $\frac{1}{x}$ for $x>0$, and the derivative of $\ln(-x)$ is $\frac{1}{-x}(-1) = \frac{1}{x}$ for $x<0$. Combining these gives the result for $x \neq 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): Geometrically, the indefinite integral represents a family of curves that are vertical translations of each other.

Reason (R): The constant of integration $C$ corresponds to the vertical shift of the antiderivative graph. Different values of $C$ give different curves with the same slope function $f(x)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The method of integration by substitution is based on the chain rule of differentiation.

Reason (R): If $\int f(g(x)) g'(x) dx$, letting $u = g(x)$, so $du = g'(x) dx$, transforms the integral into $\int f(u) du$, which mirrors the chain rule in reverse: $\frac{d}{dx} F(g(x)) = F'(g(x)) g'(x) = f(g(x)) g'(x)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): To evaluate $\int 2x (x^2+1)^3 dx$, a suitable substitution is $u = x^2+1$.

Reason (R): If $u = x^2+1$, then $du = 2x dx$. The integral becomes $\int u^3 du$, which is easily integrated.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The formula for integration by parts is $\int u dv = uv - \int v du$.

Reason (R): This formula is derived from the product rule of differentiation $(uv)' = u'v + uv'$, which can be rewritten as $uv' = (uv)' - u'v$ and then integrated.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): To evaluate $\int x \sin x dx$, integration by parts is a suitable technique.

Reason (R): We can choose $u=x$ and $dv = \sin x dx$, so $du = dx$ and $v = -\cos x$. Applying the formula gives $\int x \sin x dx = -x \cos x - \int (-\cos x) dx = -x \cos x + \int \cos x dx = -x \cos x + \sin x + C$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The integral $\int e^x (\sin x + \cos x) dx$ can be evaluated using the standard result $\int e^x (f(x) + f'(x)) dx = e^x f(x) + C$.

Reason (R): If we take $f(x) = \sin x$, then $f'(x) = \cos x$. The integral matches the form $\int e^x (f(x) + f'(x)) dx$, so the result is $e^x \sin x + C$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The substitution method is effective when the integrand contains a function and its derivative (or a constant multiple of its derivative).

Reason (R): The differential $du$ absorbs the derivative term, simplifying the integrand.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): When using integration by parts $\int u dv = uv - \int v du$, the choice of $u$ and $dv$ is arbitrary and does not affect the ease of integration.

Reason (R): A good choice of $u$ simplifies when differentiated, and a good choice of $dv$ is easily integrable, ideally making the integral $\int v du$ simpler than the original integral.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): $\int \ln x dx = x \ln x - x + C$.

Reason (R): Using integration by parts with $u = \ln x$ and $dv = dx$, we get $du = \frac{1}{x} dx$ and $v=x$. $\int \ln x dx = x \ln x - \int x \cdot \frac{1}{x} dx = x \ln x - \int 1 dx = x \ln x - x + C$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The integral $\int x^2 e^x dx$ can be evaluated using integration by parts twice.

Reason (R): We can choose $u=x^2$ and $dv = e^x dx$. This reduces the power of $x$ in the $\int v du$ term, and repeating the process eliminates the polynomial part.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): Integration by substitution is suitable for integrals of the form $\int \frac{f'(x)}{f(x)} dx$.

Reason (R): Let $u = f(x)$. Then $du = f'(x) dx$. The integral becomes $\int \frac{du}{u} = \ln |u| + C = \ln |f(x)| + C$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The method of partial fractions is used to integrate rational functions where the degree of the numerator is less than the degree of the denominator.

Reason (R): Partial fraction decomposition breaks down a complex rational function into simpler rational functions that are easier to integrate.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): To integrate $\frac{x}{(x-1)(x+2)} dx$, we decompose the integrand into partial fractions of the form $\frac{A}{x-1} + \frac{B}{x+2}$.

Reason (R): The denominator is a product of distinct linear factors.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): To integrate rational functions of $\sin x$ and $\cos x$, the substitution $t = \tan(\frac{x}{2})$ is often effective.

Reason (R): This substitution transforms $\sin x$ into $\frac{2t}{1+t^2}$, $\cos x$ into $\frac{1-t^2}{1+t^2}$, and $dx$ into $\frac{2 dt}{1+t^2}$, converting the integrand into a rational function of $t$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The integral $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}(\frac{x}{a}) + C$.

Reason (R): This is a standard integral formula.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The integral $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$.

Reason (R): This is a standard integral formula derived using trigonometric substitution $x = a \tan \theta$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): If the degree of the numerator of a rational function is greater than or equal to the degree of the denominator, we must perform polynomial long division before applying partial fraction decomposition.

Reason (R): Partial fraction decomposition is only applicable to proper rational functions (degree of numerator less than degree of denominator).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): For a repeated linear factor $(x-a)^n$ in the denominator, the partial fraction decomposition includes terms $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \dots + \frac{A_n}{(x-a)^n}$.

Reason (R): This ensures that all possible terms arising from the repeated factor are accounted for in the decomposition.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): The integral $\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}(\frac{x}{a}) + C$.

Reason (R): This is a standard integral form, often derived using trigonometric substitution $x = a \sin \theta$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The integral $\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln |x + \sqrt{x^2 - a^2}| + C$.

Reason (R): This is a standard integral formula.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): To integrate $\frac{x}{x^2+1} dx$, partial fractions is the required method.

Reason (R): The denominator $x^2+1$ has no real roots and is an irreducible quadratic factor.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The definite integral $\int_{a}^{b} f(x) dx$ can be defined as the limit of a sum.

Reason (R): The limit of a Riemann sum as the number of subintervals approaches infinity and the width of the largest subinterval approaches zero gives the definite integral.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The Fundamental Theorem of Integral Calculus part 2 states that if $F(x)$ is an antiderivative of $f(x)$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$.

Reason (R): This theorem provides a method to evaluate definite integrals without having to calculate the limit of a sum.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The definite integral $\int_{a}^{b} f(x) dx$ represents the area under the curve $y=f(x)$ from $x=a$ to $x=b$ if $f(x) \geq 0$ on $[a,b]$.

Reason (R): The definition of the definite integral as the limit of a sum of areas of rectangles approximates the area under the curve.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): $\int_{1}^{2} x^2 dx = \frac{7}{3}$.

Reason (R): An antiderivative of $x^2$ is $\frac{x^3}{3}$. By the Fundamental Theorem, $\int_{1}^{2} x^2 dx = [\frac{x^3}{3}]_1^2 = \frac{2^3}{3} - \frac{1^3}{3} = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The Fundamental Theorem of Integral Calculus part 1 states that if $F(x) = \int_{a}^{x} f(t) dt$, then $F'(x) = f(x)$.

Reason (R): This theorem shows that differentiation and integration are inverse processes.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The constant of integration $C$ is not included when evaluating definite integrals using the Fundamental Theorem.

Reason (R): The constant $C$ cancels out when evaluating $F(b) - F(a)$, as $(F(b)+C) - (F(a)+C) = F(b) - F(a)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): If $f(x)$ is continuous on $[a,b]$, then $\int_{a}^{b} f(x) dx$ always exists.

Reason (R): The definition of the definite integral as the limit of a sum applies to continuous functions on a closed interval, and the limit exists for such functions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): The value of $\int_{a}^{a} f(x) dx = 0$.

Reason (R): By the Fundamental Theorem, $\int_{a}^{a} f(x) dx = F(a) - F(a) = 0$. Geometrically, the area under a curve from a point to itself is zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): $\int_{b}^{a} f(x) dx = - \int_{a}^{b} f(x) dx$.

Reason (R): By definition or Fundamental Theorem, $\int_{b}^{a} f(x) dx = F(a) - F(b) = -(F(b) - F(a)) = -\int_{a}^{b} f(x) dx$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): If $f(x) = c$ (a constant), then $\int_{a}^{b} c dx = c(b-a)$.

Reason (R): The area under the constant function $y=c$ from $a$ to $b$ is a rectangle with width $(b-a)$ and height $c$, and its area is $c(b-a)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): To evaluate $\int_{0}^{\pi/2} \sin x dx$, we find the antiderivative of $\sin x$, which is $-\cos x$, and evaluate it from $0$ to $\pi/2$.

Reason (R): By the Fundamental Theorem of Calculus, $\int_{0}^{\pi/2} \sin x dx = [-\cos x]_0^{\pi/2} = -\cos(\pi/2) - (-\cos 0) = -0 - (-1) = 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): To evaluate $\int_{0}^{1} x(x^2+1)^3 dx$, substitution can be used.

Reason (R): Let $u = x^2+1$. When $x=0$, $u=1$. When $x=1$, $u=2$. $du = 2x dx$, so $x dx = \frac{1}{2} du$. The integral becomes $\int_{1}^{2} u^3 \frac{1}{2} du = \frac{1}{2} [\frac{u^4}{4}]_1^2 = \frac{1}{8}(2^4 - 1^4) = \frac{1}{8}(16-1) = \frac{15}{8}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): One property of definite integrals is $\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$ for any $c$ between $a$ and $b$, assuming $f$ is integrable.

Reason (R): This property, the interval splitting property, allows breaking down an integral over a larger interval into a sum of integrals over subintervals.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$.

Reason (R): This is a standard property of definite integrals often used to simplify integrals, particularly those involving trigonometric functions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): $\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx$ if $f(x)$ is an even function.

Reason (R): If $f(x)$ is an even function, its graph is symmetric about the y-axis. The area from $-a$ to $a$ is twice the area from $0$ to $a$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): $\int_{-1}^{1} x^3 dx = 0$.

Reason (R): $f(x) = x^3$ is an odd function, since $f(-x) = (-x)^3 = -x^3 = -f(x)$. For an odd function, $\int_{-a}^{a} f(x) dx = 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): To evaluate $\int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$, we can use the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$.

Reason (R): Applying the property, the integral becomes $\int_{0}^{\pi/2} \frac{\sin (\pi/2 - x)}{\sin (\pi/2 - x) + \cos (\pi/2 - x)} dx = \int_{0}^{\pi/2} \frac{\cos x}{\cos x + \sin x} dx$. Adding the original integral and this new integral results in $\int_{0}^{\pi/2} \frac{\sin x + \cos x}{\sin x + \cos x} dx = \int_{0}^{\pi/2} 1 dx = [x]_0^{\pi/2} = \pi/2$. Thus, the original integral is $\frac{\pi/2}{2} = \pi/4$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): If $f(x)$ is integrable on $[a,b]$ and $f(x) \geq 0$ for all $x \in [a,b]$, then $\int_{a}^{b} f(x) dx \geq 0$.

Reason (R): The definite integral represents the area under the curve. If the function is non-negative, the area is non-negative.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): $\int_{a}^{b} k f(x) dx = k \int_{a}^{b} f(x) dx$ for any constant $k$.

Reason (R): The constant multiple property holds for definite integrals just as it does for indefinite integrals.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): Evaluating definite integrals by substitution requires changing the limits of integration according to the substitution.

Reason (R): If the substitution is $u = g(x)$ and the original limits are $a$ and $b$ for $x$, the new limits for $u$ become $g(a)$ and $g(b)$. This avoids substituting back to $x$ after integration.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The area of the region bounded by the curve $y=f(x)$, the x-axis, and the lines $x=a$ and $x=b$ (where $a < b$) is given by $\int_{a}^{b} f(x) dx$ if $f(x) \geq 0$ on $[a,b]$.

Reason (R): The definite integral represents the signed area between the curve and the x-axis.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If $f(x) \leq 0$ on $[a,b]$, the area bounded by the curve $y=f(x)$, the x-axis, $x=a$, and $x=b$ is given by $|\int_{a}^{b} f(x) dx|$ or $\int_{a}^{b} |f(x)| dx$.

Reason (R): The integral $\int_{a}^{b} f(x) dx$ will be negative if the function is below the x-axis. Area is a positive quantity, so we take the absolute value.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The area between two curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$, where $f(x) \geq g(x)$ on $[a,b]$, is given by $\int_{a}^{b} [f(x) - g(x)] dx$.

Reason (R): We integrate the difference between the upper function and the lower function over the interval.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The area of the region bounded by the parabola $y = x^2$, the x-axis, and the line $x=2$ is $\frac{8}{3}$ square units.

Reason (R): The area is given by $\int_{0}^{2} x^2 dx$. $\int_{0}^{2} x^2 dx = [\frac{x^3}{3}]_0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The area bounded by the curves $y=x$ and $y=x^2$ is $\frac{1}{6}$ square units.

Reason (R): The curves intersect when $x = x^2$, i.e., $x^2-x=0 \implies x(x-1)=0$, so $x=0$ and $x=1$. On $[0,1]$, $x \geq x^2$. The area is $\int_{0}^{1} (x - x^2) dx = [\frac{x^2}{2} - \frac{x^3}{3}]_0^1 = (\frac{1}{2} - \frac{1}{3}) - (0-0) = \frac{3-2}{6} = \frac{1}{6}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): If a region is bounded by a curve $x=f(y)$, the y-axis, and horizontal lines $y=c$ and $y=d$, the area is given by $\int_{c}^{d} f(y) dy$ if $f(y) \geq 0$ on $[c,d]$.

Reason (R): Integration with respect to $y$ calculates the area by summing up horizontal rectangular strips.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): The area between two curves $x=f(y)$ and $x=g(y)$ from $y=c$ to $y=d$, where $f(y) \geq g(y)$ on $[c,d]$, is given by $\int_{c}^{d} [f(y) - g(y)] dy$.

Reason (R): We integrate the difference between the rightmost function and the leftmost function with respect to $y$ over the interval $[c,d]$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): If the function $f(x)$ is positive on $[a, c)$ and negative on $(c, b]$, the total area bounded by the curve, the x-axis, $x=a$, and $x=b$ is $\int_{a}^{c} f(x) dx + |\int_{c}^{b} f(x) dx|$.

Reason (R): To find the total area, we integrate the absolute value of the function over the entire interval, or sum the absolute values of the integrals over subintervals where the function's sign is constant.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The area of the region bounded by the circle $x^2+y^2=a^2$ is $\pi a^2$.

Reason (R): The area can be found by evaluating $4 \int_{0}^{a} \sqrt{a^2 - x^2} dx$, which is $4 \cdot [\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}(\frac{x}{a})]_0^a = 4 \cdot [(0 + \frac{a^2}{2} \sin^{-1}(1)) - (0+0)] = 4 \cdot \frac{a^2}{2} \cdot \frac{\pi}{2} = \pi a^2$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): Applications of integration are not limited to finding areas but include concepts like volume, arc length, and work done.

Reason (R): Integration is essentially a summation process that can be applied to various quantities that can be approximated by sums of small parts.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): A differential equation is an equation involving an independent variable, a dependent variable, and derivatives of the dependent variable with respect to the independent variable.

Reason (R): $\frac{dy}{dx} = x+y$ is an example of a differential equation.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The order of the differential equation $(\frac{d^2 y}{dx^2})^3 + (\frac{dy}{dx})^2 + y = x$ is $3$.

Reason (R): The order of a differential equation is the highest order of the derivative involved in the equation. The highest order derivative is $\frac{d^2 y}{dx^2}$, which is of order 2. The exponent of this term is 3, but that determines the degree, not the order.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The degree of the differential equation $\frac{d^2 y}{dx^2} = \sqrt{\frac{dy}{dx} + y}$ is $2$.

Reason (R): To find the degree, we first clear the equation of radicals involving derivatives by squaring both sides: $(\frac{d^2 y}{dx^2})^2 = \frac{dy}{dx} + y$. The highest order derivative is $\frac{d^2 y}{dx^2}$, and its power is 2. The equation is a polynomial in terms of derivatives.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): A general solution of a differential equation contains arbitrary constants equal to the order of the differential equation.

Reason (R): This is because finding the general solution typically involves integrating the equation, and each integration step introduces an arbitrary constant.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): A particular solution of a differential equation is obtained from the general solution by assigning specific values to the arbitrary constants.

Reason (R): These specific values are determined by using initial or boundary conditions provided with the differential equation.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The differential equation $\frac{dy}{dx} = f(x) g(y)$ can be solved by the variable separable method.

Reason (R): The variable separable method involves rearranging the equation to $\frac{dy}{g(y)} = f(x) dx$ and then integrating both sides.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The differential equation $(x^2+y^2) dx - 2xy dy = 0$ is a homogeneous differential equation.

Reason (R): A differential equation $M(x,y) dx + N(x,y) dy = 0$ is homogeneous if $M(x,y)$ and $N(x,y)$ are homogeneous functions of the same degree. Here, $M(x,y) = x^2+y^2$ (degree 2) and $N(x,y) = -2xy$ (degree 2).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): To solve a homogeneous differential equation $\frac{dy}{dx} = F(\frac{y}{x})$, we use the substitution $y = vx$.

Reason (R): Substituting $y=vx$ implies $\frac{dy}{dx} = v + x \frac{dv}{dx}$, which transforms the equation into a variable separable form in terms of $v$ and $x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The general solution of $\frac{dy}{dx} = e^{x-y}$ is $e^y = e^x + C$.

Reason (R): The equation can be written as $\frac{dy}{dx} = e^x e^{-y}$, which is separable: $e^y dy = e^x dx$. Integrating both sides gives $\int e^y dy = \int e^x dx$, so $e^y = e^x + C$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The equation $\frac{dy}{dx} = \frac{x+y+1}{x+y+2}$ can be reduced to a homogeneous form.

Reason (R): Equations of the form $\frac{dy}{dx} = \frac{ax+by+c}{Ax+By+C}$ where $aB - Ab \neq 0$ can be reduced to homogeneous form by substituting $x = X+h, y = Y+k$ and choosing $h, k$ to make the linear terms zero. In this case, $a=1, b=1, A=1, B=1$, so $aB-Ab = 1(1) - 1(1) = 0$. The lines $x+y+1=0$ and $x+y+2=0$ are parallel. A different substitution is needed for this case.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The general solution of the differential equation $\frac{dy}{dx} = \frac{y}{x}$ is $y = Cx$.

Reason (R): This is a homogeneous equation. Using $y=vx$, we get $v+x\frac{dv}{dx} = v$, so $x\frac{dv}{dx} = 0$. This implies $\frac{dv}{dx}=0$ (assuming $x \neq 0$), so $v = C$. Substituting back $y=vx$ gives $y=Cx$. Alternatively, it's separable: $\frac{dy}{y} = \frac{dx}{x}$, which integrates to $\ln|y| = \ln|x| + \ln|C|$, so $|y| = |Cx|$, which gives $y=Cx$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): The order of the differential equation representing the family of curves $y = ax^2 + bx + c$ is 3.

Reason (R): The differential equation is formed by eliminating the arbitrary constants. Here, there are three constants ($a, b, c$), so the order should be 3. $y' = 2ax+b$, $y'' = 2a$, $y''' = 0$. The resulting differential equation is $y''' = 0$, which is of order 3.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): The equation $(x+y) dx + x dy = 0$ is a homogeneous differential equation.

Reason (R): $M(x,y) = x+y$ is homogeneous of degree 1, and $N(x,y) = x$ is homogeneous of degree 1. Since the degrees are the same, the equation is homogeneous.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The differential equation $\frac{dy}{dx} = \sin(x+y)$ can be solved by the variable separable method.

Reason (R): The expression $\sin(x+y)$ cannot be factored into a product of a function of $x$ only and a function of $y$ only.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): A singular solution of a differential equation is a solution that cannot be obtained from the general solution by specializing the constants.

Reason (R): Singular solutions may exist and are not covered by the family of curves represented by the general solution.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The differential equation $x \frac{dy}{dx} + 2y = x^2$ is a first order linear differential equation.

Reason (R): A first order linear differential equation in $y$ has the standard form $\frac{dy}{dx} + P(x)y = Q(x)$. Dividing the given equation by $x$ gives $\frac{dy}{dx} + \frac{2}{x}y = x$, which is of the standard form with $P(x) = \frac{2}{x}$ and $Q(x) = x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): For the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$, the integrating factor (IF) is $e^{\int P(x) dx}$.

Reason (R): Multiplying the standard form of the linear DE by the integrating factor makes the left-hand side the derivative of the product of $y$ and the integrating factor.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): The integrating factor for $\frac{dy}{dx} + \frac{y}{x} = x^2$ is $x$.

Reason (R): Here $P(x) = \frac{1}{x}$. The integrating factor is $e^{\int \frac{1}{x} dx} = e^{\ln|x|} = |x|$. Since we usually consider the positive case, IF $= x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The general solution of $\frac{dy}{dx} + P(x)y = Q(x)$ is given by $y \cdot (\text{IF}) = \int Q(x) \cdot (\text{IF}) dx + C$.

Reason (R): Multiplying the original equation by IF gives $\frac{d}{dx}(y \cdot \text{IF}) = Q(x) \cdot \text{IF}$. Integrating both sides with respect to $x$ yields $y \cdot \text{IF} = \int Q(x) \cdot \text{IF} dx + C$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The differential equation $(x^2+1)\frac{dy}{dx} + 2xy = 4x^2$ is linear.

Reason (R): The equation is of the form $\frac{dy}{dx} + P(x)y = Q(x)$ where $P(x) = \frac{2x}{x^2+1}$ and $Q(x) = \frac{4x^2}{x^2+1}$. The dependent variable $y$ and its derivative $\frac{dy}{dx}$ appear only to the first power and are not multiplied together or involved in non-linear functions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The integrating factor for the linear DE $\frac{dx}{dy} + P(y)x = Q(y)$ is $e^{\int P(y) dy}$.

Reason (R): This is the integrating factor when the independent variable is $y$ and the dependent variable is $x$, making the LHS the derivative of $x \cdot \text{IF}$ with respect to $y$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): The general solution of $\frac{dy}{dx} + y \cot x = \text{cosec } x$ is $y \sin x = x + C$.

Reason (R): $P(x) = \cot x$, $\text{IF} = e^{\int \cot x dx} = e^{\ln|\sin x|} = \sin x$. $Q(x) = \text{cosec } x$. Solution is $y \sin x = \int \text{cosec } x \cdot \sin x dx + C = \int 1 dx + C = x + C$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): The differential equation $(\frac{dy}{dx})^2 + y = x$ is a first order linear differential equation.

Reason (R): In a linear differential equation, the dependent variable and its derivatives must appear only in the first power and should not be multiplied together.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): The method of integrating factor is applicable only to first order linear differential equations.

Reason (R): Higher order or non-linear differential equations require different solution techniques.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): The integrating factor for the equation $y' + y = 1$ is $e^x$.

Reason (R): $P(x) = 1$, so IF $= e^{\int 1 dx} = e^x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): Differential equations are powerful tools for modeling real-world phenomena that involve rates of change.

Reason (R): Many physical, biological, and economic processes can be described by equations relating a quantity to its derivative(s) with respect to time or another variable.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The differential equation $\frac{dP}{dt} = kP$ models exponential population growth or decay.

Reason (R): This equation states that the rate of change of population $P$ with respect to time $t$ is directly proportional to the current population $P$. The solution is $P(t) = P_0 e^{kt}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): Newton's Law of Cooling can be formulated as a differential equation.

Reason (R): The law states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. If $T(t)$ is the object's temperature and $T_m$ is the ambient temperature, the DE is $\frac{dT}{dt} = k(T - T_m)$ for some constant $k < 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): Solving a differential equation obtained from a real-world problem often involves finding a particular solution.

Reason (R): The initial conditions or boundary conditions provided in the problem are used to determine the specific values of the arbitrary constants in the general solution.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The differential equation describing radioactive decay is $\frac{dN}{dt} = -\lambda N$, where $N$ is the amount of radioactive substance and $\lambda$ is a positive constant.

Reason (R): This equation indicates that the rate of decay is proportional to the current amount of the substance, and the negative sign signifies that the amount is decreasing over time.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. Assertion (A): The velocity of an object falling under gravity and air resistance proportional to velocity can be modeled by a first-order differential equation.

Reason (R): The net force (mass times acceleration) is equal to the difference between the gravitational force and the air resistance force, and acceleration is the derivative of velocity.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 7. Assertion (A): In banking, the continuous compounding of interest can be modeled by a differential equation.

Reason (R): If $A(t)$ is the amount at time $t$ and $r$ is the annual interest rate, the rate of change of the amount is proportional to the current amount, i.e., $\frac{dA}{dt} = rA$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 8. Assertion (A): Formulating a differential equation from a word problem requires translating the description of the rate of change into mathematical terms involving derivatives.

Reason (R): The phrase "rate of change of Y with respect to X" translates to $\frac{dY}{dX}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 9. Assertion (A): A first-order differential equation can model phenomena where the future state depends only on the current state and the rate of change at that state.

Reason (R): First-order derivatives represent instantaneous rates of change depending on the current values of the variables.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 10. Assertion (A): Solving an applied problem modeled by a differential equation involves finding the general solution first, and then using given conditions to find the particular solution relevant to the specific problem.

Reason (R): The general solution represents a family of potential outcomes, while the particular solution pinpoints the single outcome that matches the given scenario.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.