Single Best Answer MCQs for Sub-Topics of Topic 10: Calculus

Question 1. What is the concept of a limit of a function $f(x)$ as $x$ approaches $a$?

(A) The exact value of $f(a)$

(B) The value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$, but not necessarily equal to $a$

(C) The largest value of $f(x)$ in the neighbourhood of $a$

(D) The smallest value of $f(x)$ in the neighbourhood of $a$

Question 2. The limit $\lim\limits_{x \to 3} (2x + 5)$ can be evaluated by direct substitution because:

(A) The function $f(x) = 2x+5$ is a polynomial.

(B) The function is defined at $x=3$.

(C) Direct substitution does not result in an indeterminate form.

(D) All of the above.

Question 3. Evaluate $\lim\limits_{x \to 2} \frac{x^2 - 4}{x - 2}$.

(A) 0

(B) 4

(C) $\infty$

(D) Undefined

Question 4. What is the value of the left-hand limit of $f(x) = \begin{cases} x+1 & , & x < 1 \\ x^2 - 1 & , & x \geq 1 \end{cases}$ at $x=1$?

(A) 0

(B) 1

(C) 2

(D) Does not exist

Question 5. For the limit of a function to exist at a point, which condition must be met?

(A) The left-hand limit must exist.

(B) The right-hand limit must exist.

(C) The left-hand limit must be equal to the right-hand limit and finite.

(D) The function must be defined at that point.

Question 6. Evaluate $\lim\limits_{x \to 0} \frac{\sqrt{1+x} - 1}{x}$.

(A) 0

(B) 1/2

(C) 1

(D) 2

Question 7. If $\lim\limits_{x \to a^-} f(x) = L_1$ and $\lim\limits_{x \to a^+} f(x) = L_2$, then $\lim\limits_{x \to a} f(x)$ exists if and only if:

(A) $L_1$ and $L_2$ exist.

(B) $L_1 = L_2$.

(C) $L_1 = L_2$ and are finite.

(D) $f(a)$ exists and $f(a) = L_1 = L_2$.

Question 8. Which of the following is an indeterminate form?

(A) $0/5$

(B) $5/0$

(C) $0/0$

(D) $\infty/5$

Question 9. To evaluate $\lim\limits_{x \to 1} \frac{x^3 - 1}{x^2 - 1}$, which method is suitable?

(A) Direct Substitution

(B) Rationalization

(C) Factorization

(D) Squeeze Theorem

Question 10. Evaluate $\lim\limits_{x \to 3} \frac{x^2 - 5x + 6}{x - 3}$.

(A) 0

(B) 1

(C) 3

(D) Undefined

Question 11. What is the significance of the left-hand limit?

(A) It is the value the function approaches as $x$ approaches from values greater than $a$.

(B) It is the value the function approaches as $x$ approaches from values less than $a$.

(C) It is the value of the function at $a$.

(D) It determines if the function is continuous at $a$.

Question 12. For the function $f(x) = \frac{1}{x-2}$, what is $\lim\limits_{x \to 2^+} f(x)$?

(A) 0

(B) 1

(C) $\infty$

(D) $-\infty$

Question 13. If $\lim\limits_{x \to a} f(x)$ exists, does $f(a)$ necessarily exist?

(A) Yes, always.

(B) No, not necessarily.

(C) Yes, if the limit is finite.

(D) Yes, if the limit is non-zero.

Question 14. Evaluate $\lim\limits_{x \to 0} \frac{\sqrt{4+x} - 2}{x}$ using rationalization.

(A) 1/4

(B) 1/2

(C) 1

(D) 0

Question 15. What is the primary reason for using factorization or rationalization when evaluating limits?

(A) To simplify the expression.

(B) To remove the indeterminate form.

(C) To find the value of the function at the limit point.

(D) To check for continuity.

Question 16. Evaluate $\lim\limits_{h \to 0} \frac{(x+h)^2 - x^2}{h}$.

(A) 0

(B) $x^2$

(C) $2x$

(D) $2x+h$

Question 17. For the function $f(x) = \begin{cases} 2x & , & x < 2 \\ 4 & , & x = 2 \\ x^2 & , & x > 2 \end{cases}$, what is $\lim\limits_{x \to 2} f(x)$?

(A) 4

(B) 2

(C) 8

(D) Does not exist

Question 18. Evaluate $\lim\limits_{x \to -1} \frac{x^2 + 3x + 2}{x^2 - 1}$.

(A) -1/2

(B) 1/2

(C) 0

(D) Undefined

Question 19. If a function $f(x)$ is such that $g(x) \leq f(x) \leq h(x)$ for all $x$ in an open interval containing $a$, and $\lim\limits_{x \to a} g(x) = L$ and $\lim\limits_{x \to a} h(x) = L$, then $\lim\limits_{x \to a} f(x) = L$. This statement is part of which theorem?

(A) Factor Theorem

(B) Rationalization Principle

(C) Squeeze Play Theorem (Sandwich Theorem)

(D) Fundamental Theorem of Calculus

Question 20. Which of the following limits can be evaluated using direct substitution?

(A) $\lim\limits_{x \to 0} \frac{\sin x}{x}$

(B) $\lim\limits_{x \to 1} \frac{x^2 - 1}{x - 1}$

(C) $\lim\limits_{x \to 2} (x^3 + 1)$

(D) $\lim\limits_{x \to \infty} \frac{1}{x}$

Question 21. What is the right-hand limit of $f(x) = \begin{cases} 2x+3 & , & x < 0 \\ 3x-1 & , & x \geq 0 \end{cases}$ at $x=0$?

(A) 3

(B) -1

(C) 2

(D) Does not exist

Question 22. If $\lim\limits_{x \to 5} f(x) = 7$, which statement is definitely true?

(A) $f(5) = 7$

(B) $f(x)$ is continuous at $x=5$

(C) As $x$ gets close to 5, $f(x)$ gets close to 7.

(D) The function $f(x)$ is defined at $x=5$.

Question 1. If $\lim\limits_{x \to a} f(x) = L$ and $\lim\limits_{x \to a} g(x) = M$, then by the sum rule for limits, $\lim\limits_{x \to a} [f(x) + g(x)]$ is:

(A) $L+M$

(B) $L-M$

(C) $L \times M$

(D) $L/M$, provided $M \neq 0$

Question 2. Evaluate $\lim\limits_{x \to 2} (3x^2 - 5x + 2)$.

(A) 4

(B) 0

(C) 6

(D) 8

Question 3. What is the value of $\lim\limits_{x \to a} c$, where $c$ is a constant?

(A) $a$

(B) $c$

(C) 0

(D) Undefined

Question 4. Evaluate $\lim\limits_{x \to 0} \frac{\sin(5x)}{x}$.

(A) 1

(B) 5

(C) 1/5

(D) 0

Question 5. According to the Squeeze Play Theorem, if $g(x) \leq f(x) \leq h(x)$ and $\lim\limits_{x \to a} g(x) = L$, what must be true about $\lim\limits_{x \to a} h(x)$ for the theorem to apply to $f(x)$?

(A) $\lim\limits_{x \to a} h(x)$ must be greater than $L$.

(B) $\lim\limits_{x \to a} h(x)$ must be less than $L$.

(C) $\lim\limits_{x \to a} h(x)$ must be equal to $L$.

(D) $\lim\limits_{x \to a} h(x)$ must not exist.

Question 6. Evaluate $\lim\limits_{x \to 0} \frac{\tan x}{x}$.

(A) 0

(B) 1

(C) $\infty$

(D) -1

Question 7. Evaluate $\lim\limits_{x \to 0} \frac{e^{2x} - 1}{x}$.

(A) 1

(B) 2

(C) 1/2

(D) 0

Question 8. Evaluate $\lim\limits_{x \to 0} (1 + 3x)^{1/x}$.

(A) $e$

(B) $e^3$

(C) $e^{1/3}$

(D) 1

Question 9. What is the value of $\lim\limits_{x \to 0} \frac{\log_e(1+4x)}{x}$?

(A) 0

(B) 1

(C) 4

(D) 1/4

Question 10. Evaluate $\lim\limits_{x \to a} \frac{x^5 - a^5}{x - a}$.

(A) $a^4$

(B) $5a^4$

(C) $a^5$

(D) $5a^5$

Question 11. If $\lim\limits_{x \to a} f(x) = L \neq 0$ and $\lim\limits_{x \to a} g(x) = M \neq 0$, then $\lim\limits_{x \to a} \frac{f(x)}{g(x)}$ is:

(A) $L/M$

(B) $M/L$

(C) $L-M$

(D) $L+M$

Question 12. Evaluate $\lim\limits_{x \to 0} \frac{1 - \cos(2x)}{x^2}$.

(A) 1

(B) 2

(C) 1/2

(D) 0

Question 13. The standard limit $\lim\limits_{x \to 0} \frac{a^x - 1}{x}$ is equal to:

(A) 1

(B) $\log_e a$

(C) $a$

(D) 0

Question 14. Evaluate $\lim\limits_{x \to 0} x \cdot \text{cosec } x$.

(A) 0

(B) 1

(C) $\infty$

(D) -1

Question 15. Evaluate $\lim\limits_{x \to \infty} (1 + \frac{1}{x})^x$.

(A) 1

(B) $e$

(C) $\infty$

(D) 0

Question 16. Evaluate $\lim\limits_{x \to 1} \frac{\log_e x}{x-1}$.

(A) 0

(B) 1

(C) $e$

(D) -1

Question 17. If $\lim\limits_{x \to a} f(x)$ exists, which of the following statements is incorrect?

(A) $\lim\limits_{x \to a} |f(x)|$ exists.

(B) $\lim\limits_{x \to a} \sqrt{f(x)}$ exists if $\lim\limits_{x \to a} f(x) \geq 0$.

(C) $\lim\limits_{x \to a} \frac{1}{f(x)}$ exists.

(D) $\lim\limits_{x \to a} (c \cdot f(x))$ exists for any constant $c$.

Question 18. Evaluate $\lim\limits_{x \to 0} \frac{\sin ax}{\sin bx}$ where $a, b \neq 0$.

(A) 1

(B) a/b

(C) b/a

(D) 0

Question 19. Evaluate $\lim\limits_{x \to 0} \frac{(1+x)^n - 1}{x}$.

(A) 1

(B) n

(C) $n-1$

(D) 0

Question 20. For the limit $\lim\limits_{x \to 0} \frac{\sin x}{x}$, the Squeeze Theorem can be applied using the inequality $\cos x < \frac{\sin x}{x} < 1$ for $x \in (-\pi/2, \pi/2)$ and $x \neq 0$. What is $\lim\limits_{x \to 0} \cos x$?

(A) 0

(B) 1

(C) $\infty$

(D) -1

Question 21. Evaluate $\lim\limits_{x \to 0} \frac{\log_a(1+x)}{x}$ for $a > 0, a \neq 1$.

(A) 1

(B) $\log_e a$

(C) $\frac{1}{\log_e a}$

(D) $a$

Question 1. A function $f(x)$ is said to be continuous at a point $x=a$ if:

(A) $\lim\limits_{x \to a} f(x)$ exists.

(B) $f(a)$ is defined.

(C) $\lim\limits_{x \to a} f(x) = f(a)$.

(D) The left-hand limit equals the right-hand limit.

Question 2. For the function $f(x) = \begin{cases} x+k & , & x \leq 1 \\ 3x-2 & , & x > 1 \end{cases}$ to be continuous at $x=1$, the value of $k$ must be:

(A) 0

(B) 1

(C) 2

(D) -1

Question 3. A function $f(x)$ is continuous in an open interval $(a, b)$ if:

(A) It is defined for all $x \in (a, b)$.

(B) The limit exists for all $x \in (a, b)$.

(C) It is continuous at every point $c \in (a, b)$.

(D) It is continuous at $a$ and $b$.

Question 4. Which type of discontinuity occurs when $\lim\limits_{x \to a^-} f(x) = \lim\limits_{x \to a^+} f(x) \neq f(a)$ (if $f(a)$ is defined), or $f(a)$ is undefined?

(A) Jump Discontinuity

(B) Infinite Discontinuity

(C) Removable Discontinuity

(D) Essential Discontinuity

Question 5. If $f$ and $g$ are continuous functions at $x=a$, then which of the following is NOT necessarily continuous at $x=a$?

(A) $f+g$

(B) $f-g$

(C) $f/g$

(D) $f \cdot g$

Question 6. If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then the composite function $(f \circ g)(x) = f(g(x))$ is continuous at:

(A) $g(a)$

(B) $f(a)$

(C) $a$

(D) $f(g(a))$

Question 7. Identify the type of discontinuity for the function $f(x) = \frac{|x|}{x}$ at $x=0$.

(A) Removable Discontinuity

(B) Jump Discontinuity

(C) Infinite Discontinuity

(D) No discontinuity

Question 8. Is the function $f(x) = \sin x$ continuous for all real values of $x$?

(A) Yes

(B) No

(C) Only for $x \geq 0$

(D) Only for $x \leq 0$

Question 9. Which of the following functions has a removable discontinuity at $x=2$?

(A) $f(x) = \frac{1}{x-2}$

(B) $f(x) = |x-2|$

(C) $f(x) = \frac{x^2 - 4}{x - 2}$

(D) $f(x) = \begin{cases} x & , & x \neq 2 \\ 5 & , & x = 2 \end{cases}$

Question 10. If a function is continuous on a closed interval $[a, b]$, does it attain its maximum and minimum values in that interval?

(A) Yes, by Extreme Value Theorem.

(B) No, not necessarily.

(C) Only if the function is also differentiable.

(D) Only if the function is monotonic.

Question 11. The function $f(x) = \lfloor x \rfloor$ (floor function) has discontinuity at:

(A) All integers

(B) All non-integers

(C) Only at $x=0$

(D) No discontinuity

Question 12. If $f$ is continuous at $a$, and $f(a) > 0$, then there exists an interval around $a$ where $f(x) > 0$. This is a consequence of:

(A) Intermediate Value Theorem

(B) Extreme Value Theorem

(C) Definition of Continuity

(D) Sign preserving property of continuous functions

Question 13. For what value of $c$ is the function $f(x) = \begin{cases} cx+1 & , & x \leq 3 \\ cx^2-1 & , & x > 3 \end{cases}$ continuous at $x=3$?

(A) 1/3

(B) 2/3

(C) 1/4

(D) -1/4

Question 14. If $f(x)$ is continuous at $x=a$, then $\lim\limits_{h \to 0} f(a+h)$ is equal to:

(A) $f(a)$

(B) $\lim\limits_{x \to a} f(x)$

(C) Both (A) and (B)

(D) Neither (A) nor (B)

Question 15. The function $f(x) = \frac{1}{x^2}$ has an infinite discontinuity at $x=0$. What are the left and right-hand limits at $x=0$?

(A) $\lim\limits_{x \to 0^-} f(x) = -\infty, \lim\limits_{x \to 0^+} f(x) = \infty$

(B) $\lim\limits_{x \to 0^-} f(x) = \infty, \lim\limits_{x \to 0^+} f(x) = -\infty$

(C) $\lim\limits_{x \to 0^-} f(x) = \infty, \lim\limits_{x \to 0^+} f(x) = \infty$

(D) $\lim\limits_{x \to 0^-} f(x) = -\infty, \lim\limits_{x \to 0^+} f(x) = -\infty$

Question 16. Which of the following functions is continuous everywhere on the real number line?

(A) $f(x) = \tan x$

(B) $f(x) = \frac{1}{x}$

(C) $f(x) = x^3 - 5x + 2$

(D) $f(x) = \text{cosec } x$

Question 17. If $f(x)$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs, then by the Intermediate Value Theorem, there exists at least one root of $f(x) = 0$ in:

(A) $[a, b]$

(B) $(a, b)$

(C) $(-\infty, \infty)$

(D) $[f(a), f(b)]$ or $[f(b), f(a)]$

Question 18. Let $f(x) = \begin{cases} \sin x & , & x \neq 0 \\ k & , & x = 0 \end{cases}$. For $f(x)$ to be continuous at $x=0$, the value of $k$ should be:

(A) 0

(B) 1

(C) -1

(D) Any real number

Question 19. The function $f(x) = |x-1|$ is continuous at:

(A) $x=1$ only

(B) All real numbers

(C) All real numbers except $x=1$

(D) Only for $x \geq 1$

Question 20. If $f(x)$ is a continuous function, is $|f(x)|$ also continuous?

(A) Yes

(B) No, not always.

(C) Only if $f(x) \geq 0$.

(D) Only if $f(x) \leq 0$.

Question 21. The number of points of discontinuity for the function $f(x) = \frac{x^2 - 9}{x - 3}$ is:

(A) 0

(B) 1

(C) 2

(D) Infinite

Question 22. A function $f(x)$ is said to be continuous on a closed interval $[a, b]$ if it is continuous on $(a, b)$, and:

(A) $\lim\limits_{x \to a^+} f(x) = f(a)$ and $\lim\limits_{x \to b^-} f(x) = f(b)$.

(B) $\lim\limits_{x \to a^-} f(x) = f(a)$ and $\lim\limits_{x \to b^+} f(x) = f(b)$.

(C) $\lim\limits_{x \to a} f(x) = f(a)$ and $\lim\limits_{x \to b} f(x) = f(b)$.

(D) $\lim\limits_{x \to a^+} f(x) = \lim\limits_{x \to a^-} f(x)$ and $\lim\limits_{x \to b^+} f(x) = \lim\limits_{x \to b^-} f(x)$.

Question 1. The derivative of a function $f(x)$ at a point $x=a$, denoted by $f'(a)$, is defined as:

(A) $\lim\limits_{h \to 0} \frac{f(a+h) + f(a)}{h}$

(B) $\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$, provided the limit exists and is finite.

(C) $\lim\limits_{h \to 0} \frac{f(a) - f(a-h)}{h}$

(D) $\frac{f(a) - f(0)}{a - 0}$

Question 2. For a function $f(x)$ to be differentiable at a point $x=a$, the left-hand derivative (LHD) and the right-hand derivative (RHD) at $x=a$ must:

(A) Both exist.

(B) Both be equal.

(C) Both exist and be equal and finite.

(D) Both be zero.

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

(A) Continuous at $x=a$.

(B) Discontinuous at $x=a$.

(C) Defined at $x=a$, but not necessarily continuous.

(D) Neither continuous nor discontinuous.

Question 4. Which of the following functions is continuous at $x=0$ but NOT differentiable at $x=0$?

(A) $f(x) = x^2$

(B) $f(x) = \sin x$

(C) $f(x) = |x|$

(D) $f(x) = e^x$

Question 5. The left-hand derivative of $f(x)$ at $x=a$ is given by:

(A) $\lim\limits_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$

(B) $\lim\limits_{h \to 0^-} \frac{f(a+h) - f(a)}{h}$

(C) $\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$

(D) $\lim\limits_{h \to 0^-} \frac{f(a) - f(a-h)}{h}$

Question 6. If a function is differentiable in an open interval $(a, b)$, then it is:

(A) Continuous in $(a, b)$ and at $a$ and $b$.

(B) Continuous only in $(a, b)$.

(C) Not necessarily continuous in $(a, b)$.

(D) Continuous at $a$ and $b$ but not necessarily in $(a, b)$.

Question 7. The process of finding the derivative of a function is called:

(A) Integration

(B) Differentiation

(C) Limiting

(D) Factoring

Question 8. What is the geometric interpretation of the derivative of a function at a point?

(A) The area under the curve.

(B) The slope of the tangent line to the curve at that point.

(C) The length of the curve.

(D) The value of the function at that point.

Question 9. Can a function be differentiable at a point where it is discontinuous?

(A) Yes, always.

(B) Yes, sometimes.

(C) No, never.

(D) Only if the discontinuity is removable.

Question 10. Calculate the derivative of $f(x) = x^2$ from the first principle at $x=1$.

(A) 1

(B) 2

(C) 3

(D) 0

Question 11. If $f(x) = |x-3|$, is $f(x)$ differentiable at $x=3$? Why or why not?

(A) Yes, because it is continuous at $x=3$.

(B) Yes, because LHD = RHD = 0 at $x=3$.

(C) No, because the graph has a sharp corner at $x=3$, meaning LHD $\neq$ RHD.

(D) No, because it is discontinuous at $x=3$.

Question 12. Which of the following is a correct statement?

(A) If a function is continuous, it is differentiable.

(B) If a function is differentiable, it is continuous.

(C) Continuity and differentiability are unrelated.

(D) If a function is differentiable at a point, it is also defined at that point.

Question 13. The right-hand derivative of $f(x)$ at $x=a$ is given by:

(A) $\lim\limits_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$

(B) $\lim\limits_{h \to 0^-} \frac{f(a+h) - f(a)}{h}$

(C) $\lim\limits_{h \to 0^+} \frac{f(a) - f(a-h)}{h}$

(D) $\lim\limits_{h \to 0^-} \frac{f(a) - f(a-h)}{h}$

Question 14. If a function is differentiable on a closed interval $[a, b]$, then it is also differentiable on the open interval $(a, b)$ and the limits of the difference quotient exist at the endpoints $a$ and $b$ (one-sided derivatives). This statement is:

(A) True

(B) False

(C) Depends on the function

(D) Only true for polynomials

Question 15. The derivative of $f(x) = c$ (constant) from the first principle is:

(A) 1

(B) $c$

(C) 0

(D) $c+h$

Question 16. Consider the function $f(x) = \begin{cases} x^2 & , & x \leq 0 \\ x & , & x > 0 \end{cases}$. Is it differentiable at $x=0$?

(A) Yes

(B) No

(C) Cannot be determined

(D) Differentiable only from the right

Question 17. What is the slope of the tangent to the curve $y = x^3$ at $x=2$?

(A) 6

(B) 8

(C) 12

(D) 4

Question 18. If a function is differentiable at a point, it means the function's graph has a well-defined, non-vertical tangent line at that point. This is a key idea in:

(A) Applied Mathematics perspective of differentiation.

(B) Limit calculation.

(C) Integration theory.

(D) Probability theory.

Question 19. The function $f(x) = \sin(|x|)$ is differentiable at:

(A) All real numbers

(B) All real numbers except $x=0$

(C) Only positive real numbers

(D) Only non-zero real numbers

Question 20. What is the LHD of $f(x) = |x|$ at $x=0$?

(A) 1

(B) -1

(C) 0

(D) Does not exist

Question 21. What is the RHD of $f(x) = |x|$ at $x=0$?

(A) 1

(B) -1

(C) 0

(D) Does not exist

Question 22. If a function is differentiable at $x=a$, then the limit $\lim\limits_{h \to 0} \frac{f(a+h) - f(a-h)}{2h}$ is equal to:

(A) $f'(a)$

(B) $2f'(a)$

(C) $\frac{1}{2} f'(a)$

(D) 0

Question 1. If $y = x^n$, then $\frac{dy}{dx}$ is given by the power rule as:

(A) $nx^{n-1}$

(B) $x^{n-1}$

(C) $nx^{n+1}$

(D) $n x^n \log x$

Question 2. Find the derivative of $y = 5x^3 - 2x^2 + 7x - 1$.

(A) $15x^2 - 4x + 7$

(B) $5x^2 - 2x + 7$

(C) $15x^3 - 4x^2 + 7x$

(D) $3x^2 - 2x + 7$

Question 3. If $u$ and $v$ are differentiable functions of $x$, then the product rule states that $\frac{d}{dx}(uv) =$

(A) $u'v'$

(B) $uv' + vu'$

(C) $uv' - vu'$

(D) $u'v + uv'$

Question 4. Find the derivative of $y = \sin x$.

(A) $\cos x$

(B) $-\cos x$

(C) $\tan x$

(D) $\sec x$

Question 5. Find the derivative of $y = e^x$.

(A) $xe^{x-1}$

(B) $e^x$

(C) $e^x \log x$

(D) $\frac{e^x}{x}$

Question 6. Find the derivative of $y = \log_e x$.

(A) $x$

(B) 1

(C) $1/x$

(D) $\log_e x$

Question 7. Find the derivative of $y = \text{cosec } x$.

(A) $-\text{cosec } x \cot x$

(B) $\text{cosec } x \cot x$

(C) $-\cot^2 x$

(D) $\cot^2 x$

Question 8. Find the derivative of $y = \sqrt{x}$.

(A) $1/\sqrt{x}$

(B) $2\sqrt{x}$

(C) $1/(2\sqrt{x})$

(D) $1/x$

Question 9. If $y = uv$, where $u$ and $v$ are functions of $x$, then $\frac{d}{dx}(\frac{u}{v}) =$

(A) $\frac{u'v - uv'}{v^2}$

(B) $\frac{uv' - u'v}{v^2}$

(C) $\frac{u'v + uv'}{v^2}$

(D) $\frac{u'v'}{v^2}$

Question 10. Find the derivative of $y = a^x$, where $a$ is a constant $(a>0, a \neq 1)$.

(A) $x a^{x-1}$

(B) $a^x$

(C) $a^x \log_e a$

(D) $a^x \log_a x$

Question 11. Find the derivative of $y = \log_a x$, where $a$ is a constant $(a>0, a \neq 1)$.

(A) $1/x$

(B) $\frac{1}{x \log_e a}$

(C) $\frac{\log_e x}{a}$

(D) $a^x$

Question 12. The derivative of $y = \sec x$ is:

(A) $\sec x \tan x$

(B) $\tan^2 x$

(C) $\sec^2 x$

(D) $-\sec x \tan x$

Question 13. If $y = c \cdot f(x)$, where $c$ is a constant and $f(x)$ is differentiable, then $\frac{dy}{dx} = $

(A) $c \cdot f'(x)$

(B) $c + f'(x)$

(C) $f'(x)$

(D) $c \cdot f(x)$

Question 14. Find the derivative of $y = x^{100}$.

(A) $100x^{99}$

(B) $99x^{100}$

(C) $100x^{101}$

(D) $x^{99}$

Question 15. Find the derivative of $y = \cos x$.

(A) $\sin x$

(B) $-\sin x$

(C) $\cot x$

(D) $-\cot x$

Question 16. Find the derivative of $y = \frac{1}{x^2}$.

(A) $-\frac{2}{x^3}$

(B) $\frac{2}{x^3}$

(C) $-\frac{1}{x}$

(D) $\frac{1}{x}$

Question 17. Find the derivative of $y = x \sin x$.

(A) $\sin x + x \cos x$

(B) $\cos x + x \sin x$

(C) $\sin x - x \cos x$

(D) $x \cos x$

Question 18. The derivative of $y = 5$ is:

(A) 5

(B) 1

(C) $5x$

(D) 0

Question 19. Find the derivative of $y = \tan x$.

(A) $\cot x$

(B) $\sec^2 x$

(C) $\text{cosec}^2 x$

(D) $\tan x \sec x$

Question 20. If $y = \frac{\sin x}{x}$, then $\frac{dy}{dx} =$

(A) $\frac{x \cos x - \sin x}{x^2}$

(B) $\frac{\cos x - x \sin x}{x^2}$

(C) $\frac{x \sin x - \cos x}{x^2}$

(D) $\frac{\cos x}{x}$

Question 21. Find the derivative of $y = x \log_e x - x$.

(A) $\log_e x$

(B) $1 + \log_e x$

(C) $\log_e x - 1$

(D) $1/x$

Question 22. The derivative of $y = \cot x$ is:

(A) $\sec^2 x$

(B) $-\sec^2 x$

(C) $\text{cosec}^2 x$

(D) $-\text{cosec}^2 x$

Question 1. A composite function is a function where one function is inside another, like $f(g(x))$. If $y=f(u)$ and $u=g(x)$, the chain rule states that $\frac{dy}{dx} =$

(A) $\frac{dy}{du} + \frac{du}{dx}$

(B) $\frac{dy}{du} \cdot \frac{du}{dx}$

(C) $\frac{dy}{dx} \cdot \frac{du}{dy}$

(D) $\frac{du}{dy} \cdot \frac{dy}{dx}$

Question 2. Find the derivative of $y = \sin(x^2)$.

(A) $\cos(x^2)$

(B) $2x \cos(x^2)$

(C) $\sin(2x)$

(D) $x^2 \cos(x^2)$

Question 3. Find the derivative of $y = (2x+3)^5$.

(A) $5(2x+3)^4$

(B) $10(2x+3)^4$

(C) $(2x+3)^4$

(D) $5(2x+3)^6$

Question 4. Find the derivative of $y = e^{\sin x}$.

(A) $e^{\sin x}$

(B) $\cos x \cdot e^{\sin x}$

(C) $\sin x \cdot e^{\sin x}$

(D) $e^{\cos x}$

Question 5. Find the derivative of $y = \log_e(\cos x)$.

(A) $\tan x$

(B) $-\tan x$

(C) $\frac{1}{\cos x}$

(D) $-\cot x$

Question 6. If the cost function for producing $x$ units is $C(x) = \sqrt{x^2 + 100}$, find the marginal cost when $x=6$. (Applied Maths perspective)

(A) $\frac{6}{\sqrt{136}}$

(B) $\frac{6}{10}$

(C) $\frac{12}{\sqrt{136}}$

(D) $\frac{x}{\sqrt{x^2 + 100}}$

Question 7. Find the derivative of $y = \tan(\sqrt{x})$.

(A) $\sec^2(\sqrt{x})$

(B) $\frac{\sec^2(\sqrt{x})}{2\sqrt{x}}$

(C) $\frac{\sec(\sqrt{x})\tan(\sqrt{x})}{2\sqrt{x}}$

(D) $\sec^2(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}$

Question 8. If $y = f(u)$, $u = g(v)$, and $v = h(x)$, then $\frac{dy}{dx} =$

(A) $\frac{dy}{du} + \frac{du}{dv} + \frac{dv}{dx}$

(B) $\frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx}$

(C) $\frac{dy}{dx} \cdot \frac{dx}{dv} \cdot \frac{dv}{du}$

(D) $\frac{du}{dy} \cdot \frac{dv}{du} \cdot \frac{dx}{dv}$

Question 9. Find the derivative of $y = \sin(\cos x)$.

(A) $\cos(\cos x)$

(B) $-\sin x \cos(\cos x)$

(C) $\cos(\cos x) \sin x$

(D) $-\cos x \sin(\cos x)$

Question 10. Find the derivative of $y = \log_e(x^2 + 5)$.

(A) $\frac{1}{x^2+5}$

(B) $\frac{2x}{x^2+5}$

(C) $2x \log_e(x^2+5)$

(D) $\frac{1}{2x}$

Question 11. Find the derivative of $y = \sqrt{\sin x}$.

(A) $\frac{\cos x}{2\sqrt{\sin x}}$

(B) $\frac{\sin x}{2\sqrt{\cos x}}$

(C) $\frac{1}{2\sqrt{\sin x}}$

(D) $\cos x \sqrt{\sin x}$

Question 12. If the revenue function is $R(x) = 100x - x^2$, and $x$ is related to time $t$ by $x(t) = 5t+1$, find the rate of change of revenue with respect to time $\frac{dR}{dt}$. (Applied Maths)

(A) $100 - 2x$

(B) $5(100 - 2x)$

(C) $5(100 - 10t - 2)$

(D) $100 - 10t - 2$

Question 13. Find the derivative of $y = e^{ax+b}$.

(A) $e^{ax+b}$

(B) $a e^{ax+b}$

(C) $b e^{ax+b}$

(D) $(ax+b) e^{ax+b - 1}$

Question 14. Find the derivative of $y = \tan^2 x$.

(A) $2 \tan x$

(B) $2 \tan x \sec^2 x$

(C) $\sec^4 x$

(D) $2 \tan x \cot x$

Question 15. If $y = \log_e(\sec x + \tan x)$, find $\frac{dy}{dx}$.

(A) $\sec x$

(B) $\tan x$

(C) $\frac{1}{\sec x + \tan x}$

(D) $\sec^2 x + \tan x \sec x$

Question 16. Find the derivative of $y = \cos(x^3 - 5x)$.

(A) $-\sin(x^3 - 5x)$

(B) $(3x^2 - 5) \sin(x^3 - 5x)$

(C) $(-(3x^2 - 5)) \sin(x^3 - 5x)$

(D) $\sin(3x^2 - 5)$

Question 17. If $y = \sqrt{e^x}$, find $\frac{dy}{dx}$.

(A) $\sqrt{e^x}$

(B) $\frac{1}{2\sqrt{e^x}}$

(C) $\frac{e^x}{2\sqrt{e^x}}$

(D) $2\sqrt{e^x}$

Question 18. If the demand for a product is $D(p) = (1000 - 5p)^2$, where $p$ is the price in $\textsf{₹}$, find the rate of change of demand with respect to price. (Applied Maths)

(A) $2(1000 - 5p)$

(B) $10(1000 - 5p)$

(C) $-10(1000 - 5p)$

(D) $-5(1000 - 5p)^2$

Question 19. Find the derivative of $y = \sin^3(x)$.

(A) $3 \sin^2 x \cos x$

(B) $3 \cos^2 x \sin x$

(C) $3 \sin^2 x$

(D) $3 \cos^2 x$

Question 20. Find the derivative of $y = \log_e(\log_e x)$.

(A) $1/x$

(B) $\frac{1}{\log_e x}$

(C) $\frac{1}{x \log_e x}$

(D) $\frac{\log_e x}{x}$

Question 21. Find the derivative of $y = e^{x^2+2x+1}$.

(A) $(x^2+2x+1)e^{x^2+2x+1}$

(B) $(2x+2)e^{x^2+2x+1}$

(C) $e^{2x+2}$

(D) $e^{(x+1)^2}$

Question 22. Find the derivative of $y = \cos^2(\sin x)$.

(A) $2 \cos(\sin x) \sin(\sin x) \cos x$

(B) $-2 \cos(\sin x) \sin(\sin x) \cos x$

(C) $-2 \cos(\sin x) \sin(\sin x)$

(D) $2 \cos(\sin x) \cos x$

Question 1. If $x^2 + y^2 = 25$, find $\frac{dy}{dx}$ using implicit differentiation.

(A) $-x/y$

(B) $x/y$

(C) $-y/x$

(D) $y/x$

Question 2. If $y = \sin^{-1} x$, find $\frac{dy}{dx}$.

(A) $\frac{1}{\sqrt{1-x^2}}$

(B) $\frac{1}{\sqrt{x^2-1}}$

(C) $-\frac{1}{\sqrt{1-x^2}}$

(D) $\frac{1}{1+x^2}$

Question 3. If $y = \tan^{-1} x$, find $\frac{dy}{dx}$.

(A) $\frac{1}{\sqrt{1-x^2}}$

(B) $\frac{1}{1+x^2}$

(C) $-\frac{1}{1+x^2}$

(D) $\frac{1}{\sqrt{x^2-1}}$

Question 4. Find $\frac{dy}{dx}$ if $xy = 5$.

(A) $5/x^2$

(B) $-5/x^2$

(C) $-y/x$

(D) $y/x$

Question 5. If $y = \cos^{-1} x$, find $\frac{dy}{dx}$.

(A) $\frac{1}{\sqrt{1-x^2}}$

(B) $-\frac{1}{\sqrt{1-x^2}}$

(C) $\frac{1}{\sqrt{x^2-1}}$

(D) $\frac{1}{1+x^2}$

Question 6. If $y = \sec^{-1} x$, find $\frac{dy}{dx}$.

(A) $\frac{1}{|x|\sqrt{x^2-1}}$

(B) $\frac{1}{\sqrt{x^2-1}}$

(C) $-\frac{1}{|x|\sqrt{x^2-1}}$

(D) $\frac{1}{1+x^2}$

Question 7. Find $\frac{dy}{dx}$ if $\sin y = x$.

(A) $\cos y$

(B) $1/\cos y$

(C) $-\cos y$

(D) $-1/\cos y$

Question 8. If $y = \cot^{-1} x$, find $\frac{dy}{dx}$.

(A) $\frac{1}{1+x^2}$

(B) $-\frac{1}{1+x^2}$

(C) $\frac{1}{\sqrt{1-x^2}}$

(D) $-\frac{1}{\sqrt{1-x^2}}$

Question 9. If $f(x)$ is a differentiable function and $f^{-1}(x)$ exists and is differentiable, then the derivative of $f^{-1}(x)$ at $y=f(x)$ is given by:

(A) $f'(x)$

(B) $1/f'(x)$

(C) $-f'(x)$

(D) $1/f(x)$

Question 10. Find $\frac{dy}{dx}$ if $x^3 + y^3 = 3axy$.

(A) $\frac{3ay - 3x^2}{3y^2 - 3ax}$

(B) $\frac{ay - x^2}{y^2 - ax}$

(C) $\frac{ax - y^2}{x^2 - ay}$

(D) $\frac{x^2 - ay}{y^2 - ax}$

Question 11. If $y = \text{cosec}^{-1} x$, find $\frac{dy}{dx}$.

(A) $\frac{1}{|x|\sqrt{x^2-1}}$

(B) $-\frac{1}{|x|\sqrt{x^2-1}}$

(C) $\frac{1}{\sqrt{1-x^2}}$

(D) $-\frac{1}{1+x^2}$

Question 12. If $y = \sin^{-1}(2x)$, find $\frac{dy}{dx}$.

(A) $\frac{1}{\sqrt{1-4x^2}}$

(B) $\frac{2}{\sqrt{1-4x^2}}$

(C) $\frac{1}{\sqrt{1-x^2}}$

(D) $\frac{2}{\sqrt{1-x^2}}$

Question 13. Find $\frac{dy}{dx}$ if $\cos(xy) = c$.

(A) $-\frac{y \sin(xy)}{x \sin(xy)}$

(B) $\frac{y}{x}$

(C) $-\frac{y}{x}$

(D) $\frac{x}{y}$

Question 14. If $y = \tan^{-1}(\frac{2x}{1-x^2})$, find $\frac{dy}{dx}$.

(A) $\frac{1}{1+x^2}$

(B) $\frac{2}{1+x^2}$

(C) $\frac{2}{\sqrt{1-x^2}}$

(D) $\frac{1}{\sqrt{1-x^2}}$

Question 15. Find $\frac{dy}{dx}$ if $e^{x+y} = xy$.

(A) $\frac{y - e^{x+y}}{e^{x+y} - x}$

(B) $\frac{y - xy}{xy - x}$

(C) $\frac{y - x}{y - x}$

(D) $\frac{y e^{x+y} - 1}{1 - x e^{x+y}}$

Question 16. If $y = \sin^{-1}(\cos x)$, find $\frac{dy}{dx}$.

(A) $\frac{-\sin x}{\sqrt{1-\cos^2 x}}$

(B) $\frac{\cos x}{\sqrt{1-\sin^2 x}}$

(C) -1

(D) 1

Question 17. The derivative of $y = \sin^{-1} x + \cos^{-1} x$ is:

(A) $\frac{2}{\sqrt{1-x^2}}$

(B) 0

(C) 1

(D) $\frac{1}{\sqrt{1-x^2}}$

Question 18. If $y = \tan^{-1}(\frac{x}{\sqrt{1-x^2}})$, find $\frac{dy}{dx}$.

(A) $\frac{1}{\sqrt{1-x^2}}$

(B) $\frac{-1}{\sqrt{1-x^2}}$

(C) $\frac{1}{1+x^2}$

(D) $-\frac{1}{1+x^2}$

Question 19. Find $\frac{dy}{dx}$ if $\sqrt{x} + \sqrt{y} = \sqrt{a}$.

(A) $-\sqrt{y/x}$

(B) $\sqrt{y/x}$

(C) $-\sqrt{x/y}$

(D) $\sqrt{x/y}$

Question 20. If $y = \sec^{-1}(\frac{1}{2x^2 - 1})$, find $\frac{dy}{dx}$ for $0 < x < 1/\sqrt{2}$.

(A) $\frac{-2}{\sqrt{1-x^2}}$

(B) $\frac{2}{\sqrt{1-x^2}}$

(C) -2

(D) 2

Question 21. If $y = \cos^{-1}(\frac{1-x^2}{1+x^2})$, find $\frac{dy}{dx}$.

(A) $\frac{1}{1+x^2}$

(B) $\frac{2}{1+x^2}$

(C) $\frac{-2}{1+x^2}$

(D) $\frac{2x}{1+x^2}$

Question 1. Logarithmic differentiation is particularly useful for functions of the form:

(A) $f(x) = c^x$

(B) $f(x) = u(x)^{v(x)}$

(C) $f(x) = \sin(ax+b)$

(D) $f(x) = x^n$

Question 2. Find the derivative of $y = x^x$ using logarithmic differentiation.

(A) $x^x$

(B) $x^x (1 + \log_e x)$

(C) $x \cdot x^{x-1}$

(D) $\log_e x$

Question 3. If $x = a \cos \theta$ and $y = a \sin \theta$, find $\frac{dy}{dx}$ in terms of $\theta$.

(A) $\tan \theta$

(B) $-\cot \theta$

(C) $\cot \theta$

(D) $-\tan \theta$

Question 4. Functions defined by equations like $x = f(t), y = g(t)$, where $t$ is a parameter, are called:

(A) Explicit functions

(B) Implicit functions

(C) Parametric functions

(D) Inverse functions

Question 5. If $y = (x+1)^{x+1}$, find $\frac{dy}{dx}$.

(A) $(x+1)^{x+1} [1 + \log_e(x+1)]$

(B) $(x+1)^{x+1} \log_e(x+1)$

(C) $(x+1) \log_e(x+1)$

(D) $(x+1)^x$

Question 6. If $x = at^2$ and $y = 2at$, find $\frac{dy}{dx}$.

(A) $t$

(B) $1/t$

(C) $2a$

(D) $2at$

Question 7. Find the derivative of $y = (\sin x)^{\cos x}$.

(A) $(\sin x)^{\cos x} [\sin x \log_e(\sin x) + \cos x \cot x]$

(B) $(\sin x)^{\cos x} [-\sin x \log_e(\sin x) + \cos x \cot x]$

(C) $(\sin x)^{\cos x} [\cos x \log_e(\sin x) + \sin x \cot x]$

(D) $(\sin x)^{\cos x} [-\cos x \log_e(\sin x) - \sin x \cot x]$

Question 8. If $x = a(\theta + \sin \theta)$ and $y = a(1 - \cos \theta)$, find $\frac{dy}{dx}$.

(A) $\tan(\theta/2)$

(B) $\cot(\theta/2)$

(C) $-\tan(\theta/2)$

(D) $-\cot(\theta/2)$

Question 9. Logarithmic differentiation simplifies finding the derivative of products and quotients of many functions by converting them into:

(A) Powers

(B) Sums and Differences

(C) Trigonometric functions

(D) Exponentials

Question 10. If $x = \sin t$ and $y = \cos t$, find $\frac{dy}{dx}$.

(A) $\tan t$

(B) $-\tan t$

(C) $\cot t$

(D) $-\cot t$

Question 11. Find the derivative of $y = \frac{(x-1)(x-2)}{(x-3)(x-4)}$ using logarithmic differentiation.

(A) $y \left[ \frac{1}{x-1} + \frac{1}{x-2} - \frac{1}{x-3} - \frac{1}{x-4} \right]$

(B) $y \left[ \frac{1}{x-1} + \frac{1}{x-2} + \frac{1}{x-3} + \frac{1}{x-4} \right]$

(C) $\frac{1}{(x-3)(x-4)} - \frac{(x-1)(x-2)}{((x-3)(x-4))^2} (x-3+x-4)$

(D) $y \left[ \log_e(x-1) + \log_e(x-2) - \log_e(x-3) - \log_e(x-4) \right]$

Question 12. If $x = \sin^3 t$ and $y = \cos^3 t$, find $\frac{dy}{dx}$ at $t=\pi/4$.

(A) 1

(B) -1

(C) 0

(D) Undefined

Question 13. Find the derivative of $y = (\log_e x)^x$.

(A) $(\log_e x)^x \left[ \frac{1}{\log_e x} + \log_e(\log_e x) \right]$

(B) $(\log_e x)^x \left[ \frac{1}{x \log_e x} + \log_e(\log_e x) \right]$

(C) $x (\log_e x)^{x-1}$

(D) $\log_e(\log_e x)$

Question 14. If $x = \frac{1-t^2}{1+t^2}$ and $y = \frac{2t}{1+t^2}$, then $\frac{dy}{dx}$ is equal to:

(A) $y/x$

(B) $-y/x$

(C) $x/y$

(D) $-x/y$

Question 15. Find the derivative of $y = x^{\sin x}$.

(A) $x^{\sin x} [\cos x \log_e x + \frac{\sin x}{x}]$

(B) $\sin x \cdot x^{\sin x - 1}$

(C) $x^{\sin x} \cos x \log_e x$

(D) $x^{\sin x} \frac{\sin x}{x}$

Question 16. If $x = e^t \sin t$ and $y = e^t \cos t$, find $\frac{dy}{dx}$.

(A) $\frac{\cos t - \sin t}{\cos t + \sin t}$

(B) $\frac{\cos t + \sin t}{\cos t - \sin t}$

(C) $\tan t$

(D) $-\cot t$

Question 17. Find the derivative of $y = (\sin x)^{\sin x}$.

(A) $(\sin x)^{\sin x} (1 + \log_e \sin x)$

(B) $(\sin x)^{\sin x} \cos x (1 + \log_e \sin x)$

(C) $\sin x (\sin x)^{\sin x - 1}$

(D) $(\sin x)^{\sin x} \log_e \sin x$

Question 18. If $x = ct$ and $y = c/t$, find $\frac{dy}{dx}$.

(A) $1/t^2$

(B) $-1/t^2$

(C) $t^2$

(D) $-t^2$

Question 19. Find the derivative of $y = (1+x^2)^{x^2}$.

(A) $(1+x^2)^{x^2} [2x \log_e(1+x^2) + \frac{2x^3}{1+x^2}]$

(B) $x^2 (1+x^2)^{x^2-1} (2x)$

(C) $(1+x^2)^{x^2} [\log_e(1+x^2) + \frac{x^2}{1+x^2}]$

(D) $(1+x^2)^{x^2} [2x + \frac{2x^3}{1+x^2}]$

Question 20. If $x = 2 \cos t - \cos 2t$ and $y = 2 \sin t - \sin 2t$, find $\frac{dy}{dx}$.

(A) $\tan(3t/2)$

(B) $-\cot(3t/2)$

(C) $\cot(3t/2)$

(D) $-\tan(3t/2)$

Question 21. If $y = x^{\log_e x}$, find $\frac{dy}{dx}$.

(A) $x^{\log_e x} \frac{2 \log_e x}{x}$

(B) $x^{\log_e x} \log_e x$

(C) $(\log_e x) x^{\log_e x - 1}$

(D) $2 (\log_e x) x^{\log_e x - 1}$

Question 1. If $y = x^4$, find the second order derivative $\frac{d^2 y}{dx^2}$.

(A) $4x^3$

(B) $12x^2$

(C) $12x$

(D) $x^2$

Question 2. If $y = \sin x$, find $\frac{d^2 y}{dx^2}$.

(A) $\cos x$

(B) $-\sin x$

(C) $\sin x$

(D) $-\cos x$

Question 3. If $y = e^{2x}$, find $\frac{d^2 y}{dx^2}$.

(A) $2e^{2x}$

(B) $4e^{2x}$

(C) $e^{2x}$

(D) $2e^{2x} \log 2$

Question 4. If $y = \log_e x$, find $\frac{d^2 y}{dx^2}$.

(A) $1/x$

(B) $-1/x^2$

(C) $1/x^2$

(D) $-\log_e x$

Question 5. If $y = x^2 + 3x + 2$, find $\frac{d^2 y}{dx^2}$.

(A) $2x+3$

(B) 2

(C) 3

(D) 0

Question 6. If $y = x \sin x$, find $\frac{d^2 y}{dx^2}$.

(A) $\cos x + \sin x$

(B) $\cos x - \sin x$

(C) $2 \cos x - x \sin x$

(D) $2 \sin x - x \cos x$

Question 7. If $x = a \cos t$, $y = b \sin t$, find $\frac{d^2 y}{dx^2}$ at $t=\pi/2$.

(A) $b/a$

(B) $-b/a^2$

(C) $-b/a^2 \sin^3 t$

(D) $-b/a^2$

Question 8. If $y = A e^{mx} + B e^{nx}$, where $A, B, m, n$ are constants, then $\frac{d^2 y}{dx^2} - (m+n)\frac{dy}{dx} + mn y$ is equal to:

(A) 0

(B) 1

(C) $A e^{mx} + B e^{nx}$

(D) $(m+n)y$

Question 9. If $y = \cos(\log_e x)$, find $\frac{d^2 y}{dx^2}$.

(A) $-\sin(\log_e x) - \cos(\log_e x)$

(B) $\frac{-\sin(\log_e x)}{x}$

(C) $\frac{-\cos(\log_e x)}{x^2}$

(D) $\frac{- \sin(\log_e x) - \cos(\log_e x)}{x^2}$

Question 10. If $y = \sin(2x)$, find the third derivative $\frac{d^3 y}{dx^3}$.

(A) $-4 \cos(2x)$

(B) $-8 \cos(2x)$

(C) $8 \sin(2x)$

(D) $-8 \sin(2x)$

Question 11. If $y = x^3 \log_e x$, find $\frac{d^2 y}{dx^2}$.

(A) $3x^2 \log_e x + x^2$

(B) $6x \log_e x + 5x$

(C) $6x \log_e x + 3x$

(D) $6x \log_e x + x$

Question 12. If $y = \tan^{-1} x$, find $\frac{d^2 y}{dx^2}$.

(A) $\frac{1}{1+x^2}$

(B) $\frac{-2x}{(1+x^2)^2}$

(C) $\frac{2x}{(1+x^2)^2}$

(D) $-\frac{1}{(1+x^2)^2}$

Question 13. If $y = \log_e(\sin x)$, find $\frac{d^2 y}{dx^2}$.

(A) $\cot x$

(B) $-\text{cosec}^2 x$

(C) $\text{cosec}^2 x$

(D) $-\cot^2 x$

Question 14. If $x = \cos \theta$, $y = \sin^3 \theta$, find $\frac{d^2 y}{dx^2}$ at $\theta = \pi/2$.

(A) 0

(B) 3

(C) -3

(D) 6

Question 15. If $y = x^n$, what is the $n$-th order derivative, $\frac{d^n y}{dx^n}$?

(A) $n!$

(B) $n! x$

(C) 0

(D) $n x^n$

Question 16. If $y = e^{ax}$, what is the $n$-th order derivative, $\frac{d^n y}{dx^n}$?

(A) $a^n e^{ax}$

(B) $n! e^{ax}$

(C) $a^n x e^{ax}$

(D) $n e^{ax}$

Question 17. If $y = \frac{1}{x}$, find $\frac{d^2 y}{dx^2}$.

(A) $-\frac{1}{x^2}$

(B) $\frac{2}{x^3}$

(C) $-\frac{2}{x^3}$

(D) $\frac{1}{x^2}$

Question 18. If $y = \sin(ax+b)$, what is the second order derivative $\frac{d^2 y}{dx^2}$?

(A) $-a^2 \sin(ax+b)$

(B) $a^2 \cos(ax+b)$

(C) $-a \sin(ax+b)$

(D) $-a^2 \cos(ax+b)$

Question 19. If $y = \sin^2 x$, find $\frac{d^2 y}{dx^2}$.

(A) $2 \sin x \cos x$

(B) $2 \cos 2x$

(C) $-2 \sin 2x$

(D) $2 \cos x$

Question 20. If $y = \log_e(1+x)$, find $\frac{d^2 y}{dx^2}$.

(A) $\frac{1}{1+x}$

(B) $-\frac{1}{(1+x)^2}$

(C) $\frac{1}{(1+x)^2}$

(D) $-\log_e(1+x)$

Question 21. If $x = ct, y = c/t^2$, find $\frac{d^2 y}{dx^2}$ at $t=1$.

(A) $-2/c$

(B) $6/c^2$

(C) $-6/c^2 t^4$

(D) $6/c^2$

Question 1. Which of the following is NOT a condition for Rolle's Theorem to apply to a function $f(x)$ on $[a, b]$?

(A) $f(x)$ is continuous on $[a, b]$.

(B) $f(x)$ is differentiable on $(a, b)$.

(C) $f(a) = f(b)$.

(D) $f'(c) = 0$ for some $c \in (a, b)$.

Question 2. Rolle's Theorem states that if $f(x)$ satisfies the conditions on $[a, b]$, then there exists at least one value $c$ in $(a, b)$ such that:

(A) $f(c) = 0$

(B) $f'(c) = 0$

(C) $f(c) = f(a)$

(D) $f'(c) = \frac{f(b)-f(a)}{b-a}$

Question 3. What is the geometric interpretation of Rolle's Theorem?

(A) The tangent to the curve is parallel to the x-axis at some point between $a$ and $b$, provided $f(a)=f(b)$.

(B) The tangent to the curve is parallel to the chord joining $(a, f(a))$ and $(b, f(b))$.

(C) The function crosses the x-axis at some point between $a$ and $b$.

(D) The function reaches its maximum or minimum in $(a, b)$.

Question 4. Lagrange's Mean Value Theorem states that if $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one value $c$ in $(a, b)$ such that:

(A) $f'(c) = 0$

(B) $f(c) = \frac{f(a)+f(b)}{2}$

(C) $f'(c) = \frac{f(b)-f(a)}{b-a}$

(D) $f(c) = f(a)$

Question 5. What is the geometric interpretation of Lagrange's Mean Value Theorem?

(A) The tangent to the curve is horizontal at some point.

(B) The tangent to the curve is parallel to the chord joining $(a, f(a))$ and $(b, f(b))$ at some point between $a$ and $b$.

(C) The function has a local extremum in $(a, b)$.

(D) The area under the curve equals the area of a rectangle.

Question 6. For the function $f(x) = x^2 - 4x + 3$ on $[1, 3]$, verify Rolle's Theorem. What is the value of $c$?

(A) 1

(B) 2

(C) 3

(D) 0

Question 7. Does Rolle's Theorem apply to $f(x) = |x|$ on $[-1, 1]$? Why?

(A) Yes, because $f(-1) = f(1)$.

(B) Yes, because it is continuous on $[-1, 1]$.

(C) No, because it is not differentiable at $x=0$ in $(-1, 1)$.

(D) No, because $f(-1) \neq f(1)$.

Question 8. For the function $f(x) = x^2$ on $[1, 4]$, find the value of $c$ that satisfies Lagrange's Mean Value Theorem.

(A) 2.5

(B) 2

(C) 3

(D) 3.5

Question 9. Rolle's Theorem is a special case of Lagrange's Mean Value Theorem when:

(A) $a=b$

(B) $f(a) = f(b)$

(C) $f'(c) = 0$

(D) The function is a polynomial

Question 10. For the function $f(x) = \frac{1}{x}$ on [-1, 1], does Rolle's Theorem apply? Why?

(A) Yes, $f(-1) = f(1)$.

(B) Yes, it is differentiable on (-1, 1).

(C) No, it is not continuous on [-1, 1].

(D) No, $f'(x) \neq 0$ for any $x \in (-1, 1)$.

Question 11. For the function $f(x) = x^{2/3}$ on $[-1, 1]$, does Rolle's Theorem apply? Why?

(A) Yes, $f(-1) = f(1)$.

(B) Yes, it is continuous on $[-1, 1]$.

(C) No, it is not differentiable at $x=0$ in $(-1, 1)$.

(D) Both (A) and (B).

Question 12. If a car travels 100 km in 2 hours, Lagrange's Mean Value Theorem implies that at some point during the journey, the car's instantaneous speed was:

(A) Less than 50 km/hr.

(B) Exactly 50 km/hr.

(C) More than 50 km/hr.

(D) Constant throughout the journey.

Question 13. For the function $f(x) = (x-1)(x-2)(x-3)$ on $[1, 3]$, find the number of values of $c$ in $(1, 3)$ such that $f'(c) = 0$.

(A) 0

(B) 1

(C) 2

(D) 3

Question 14. Does Lagrange's Mean Value Theorem apply to $f(x) = |x|$ on [-1, 1]? Why?

(A) Yes, because it is continuous on [-1, 1].

(B) Yes, because $\frac{f(1)-f(-1)}{1-(-1)} = 0$.

(C) No, because it is not differentiable on (-1, 1).

(D) Both (A) and (B).

Question 15. For the function $f(x) = e^x$ on $[0, 1]$, find the value of $c$ that satisfies Lagrange's Mean Value Theorem.

(A) $\log_e (e-1)$

(B) $\log_e e$

(C) $e-1$

(D) 1

Question 16. The minimum number of real roots of $f'(x)=0$ in $(a,b)$ when Rolle's theorem is applicable to $f(x)$ in $[a,b]$ is:

(A) 0

(B) 1

(C) 2

(D) More than 2

Question 17. If $f(x)$ is a polynomial and $f(a) = f(b)$, then by Rolle's Theorem, there exists $c \in (a,b)$ such that $f'(c)=0$. This property is always true for polynomials because they are:

(A) Continuous everywhere.

(B) Differentiable everywhere.

(C) Both continuous and differentiable everywhere.

(D) Have no critical points.

Question 18. For the function $f(x) = \tan x$ on $[0, \pi]$, does Rolle's Theorem apply?

(A) Yes

(B) No, because $f(0) \neq f(\pi)$.

(C) No, because it is not continuous on $[0, \pi]$.

(D) No, because it is not differentiable on $(0, \pi)$.

Question 19. If $f'(x) = 0$ for all $x$ in an interval $(a, b)$, then $f(x)$ is constant in $(a, b)$. This can be deduced from:

(A) Rolle's Theorem

(B) Lagrange's Mean Value Theorem

(C) Extreme Value Theorem

(D) Intermediate Value Theorem

Question 20. For $f(x) = x^3 - 6x^2 + 11x - 6$ on $[1, 3]$, find the number of values of $c$ in $(1, 3)$ such that $f'(c) = 0$.

(A) 0

(B) 1

(C) 2

(D) 3

Question 21. If $f(x) = \log_e x$ on $[1, e]$, find the value of $c$ that satisfies Lagrange's Mean Value Theorem.

(A) $e-1$

(B) $1/(e-1)$

(C) $e$

(D) $1$

Question 22. If a function $f(x)$ is differentiable on $(a, b)$ and $f'(x) > 0$ for all $x \in (a, b)$, then $f(x)$ is increasing on $(a, b)$. This can be shown using:

(A) Rolle's Theorem

(B) Lagrange's Mean Value Theorem

(C) Definition of Derivative

(D) Implicit Differentiation

Question 1. The rate of change of the area of a circle with respect to its radius $r$ when $r=5\ \text{cm}$ is:

(A) $5\pi\ \text{cm}^2/\text{cm}$

(B) $10\pi\ \text{cm}^2/\text{cm}$

(C) $25\pi\ \text{cm}^2/\text{cm}$

(D) $10\ \text{cm}^2/\text{cm}$

Question 2. If the volume of a cube is increasing at a rate of $9\ \text{cm}^3/\text{sec}$, how fast is the surface area increasing when the length of an edge is $10\ \text{cm}$?

(A) $1.8\ \text{cm}^2/\text{sec}$

(B) $3.6\ \text{cm}^2/\text{sec}$

(C) $5.4\ \text{cm}^2/\text{sec}$

(D) $7.2\ \text{cm}^2/\text{sec}$

Question 3. If $C(x)$ is the cost function for producing $x$ units, the marginal cost is defined as:

(A) $C(x)/x$

(B) $C'(x)$

(C) $\int C(x) dx$

(D) $C(x+1) - C(x)$ (approximately)

Question 4. The radius of a sphere is increasing at the rate of $0.2\ \text{cm}/\text{sec}$. The rate of increase of its volume when the radius is $10\ \text{cm}$ is:

(A) $80\pi\ \text{cm}^3/\text{sec}$

(B) $40\pi\ \text{cm}^3/\text{sec}$

(C) $80\ \text{cm}^3/\text{sec}$

(D) $40\ \text{cm}^3/\text{sec}$

Question 5. If $R(x)$ is the revenue function for selling $x$ units, the marginal revenue is defined as:

(A) $R(x)/x$

(B) $R'(x)$

(C) $\int R(x) dx$

(D) $R(x+1) - R(x)$ (approximately)

Question 6. The length $x$ of a rectangle is decreasing at the rate of $3\ \text{cm}/\text{minute}$ and the width $y$ is increasing at the rate of $2\ \text{cm}/\text{minute}$. When $x=10\ \text{cm}$ and $y=6\ \text{cm}$, the rate of change of the perimeter is:

(A) $-1\ \text{cm}/\text{minute}$

(B) $1\ \text{cm}/\text{minute}$

(C) $-2\ \text{cm}/\text{minute}$

(D) $2\ \text{cm}/\text{minute}$

Question 7. The total cost $C(x)$ in $\textsf{₹}$ associated with producing $x$ units of an item is given by $C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000$. The marginal cost when $x=100$ is:

(A) $\textsf{₹ }30$

(B) $\textsf{₹ }50.6$

(C) $\textsf{₹ }40.6$

(D) $\textsf{₹ }52.6$

Question 8. The rate of change of the area of an equilateral triangle with respect to its side length $s$ is proportional to:

(A) $s$

(B) $s^2$

(C) $\sqrt{s}$

(D) $1/s$

Question 9. A ladder $5\ \text{m}$ long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2\ \text{cm}/\text{sec}$. How fast is the height on the wall decreasing when the foot of the ladder is $4\ \text{m}$ away from the wall?

(A) $1.5\ \text{cm}/\text{sec}$

(B) $1.33\ \text{cm}/\text{sec}$ (approx)

(C) $2.5\ \text{cm}/\text{sec}$

(D) $3\ \text{cm}/\text{sec}$

Question 10. The total revenue $R(x)$ in $\textsf{₹}$ received from the sale of $x$ units of a product is given by $R(x) = 3x^2 + 36x + 5$. The marginal revenue when $x=5$ is:

(A) $\textsf{₹ }36$

(B) $\textsf{₹ }66$

(C) $\textsf{₹ }30$

(D) $\textsf{₹ }5$

Question 11. Sand is pouring from a pipe at the rate of $12\ \text{cm}^3/\text{sec}$. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is $4\ \text{cm}$?

(A) $1/(48\pi)\ \text{cm}/\text{sec}$

(B) $1/(36\pi)\ \text{cm}/\text{sec}$

(C) $1/(24\pi)\ \text{cm}/\text{sec}$

(D) $1/(12\pi)\ \text{cm}/\text{sec}$

Question 12. If the cost function is $C(x) = 0.007x^3 - 0.003x^2 + 15x + 4000$, the marginal cost when 17 units are produced is (approximately):

(A) $\textsf{₹ }21.96$

(B) $\textsf{₹ }15$

(C) $\textsf{₹ }20$

(D) $\textsf{₹ }18.5$

Question 13. The rate of change of the area of a square with respect to its side length $a$ when $a=4\ \text{cm}$ is:

(A) $4\ \text{cm}^2/\text{cm}$

(B) $8\ \text{cm}^2/\text{cm}$

(C) $16\ \text{cm}^2/\text{cm}$

(D) $2\ \text{cm}^2/\text{cm}$

Question 14. A stone is dropped into a quiet lake and waves move in circles at a speed of $4\ \text{cm}/\text{sec}$. At the instant when the radius of the circular wave is $10\ \text{cm}$, how fast is the enclosed area increasing?

(A) $8\pi\ \text{cm}^2/\text{sec}$

(B) $16\pi\ \text{cm}^2/\text{sec}$

(C) $40\pi\ \text{cm}^2/\text{sec}$

(D) $80\pi\ \text{cm}^2/\text{sec}$

Question 15. The total revenue in $\textsf{₹}$ received from the sale of $x$ units of a product is given by $R(x) = 13x^2 + 26x + 15$. The marginal revenue when $x=7$ is:

(A) $\textsf{₹ }26x + 26$

(B) $\textsf{₹ }182 + 26$

(C) $\textsf{₹ }208$

(D) $\textsf{₹ }13(49) + 26(7) + 15$

Question 16. The marginal cost represents:

(A) The average cost per unit.

(B) The cost of producing the next unit.

(C) The total cost at a certain production level.

(D) The fixed cost.

Question 17. An edge of a variable cube is increasing at the rate of $3\ \text{cm}/\text{sec}$. How fast is the volume of the cube increasing when the edge is $10\ \text{cm}$ long?

(A) $30\ \text{cm}^3/\text{sec}$

(B) $300\ \text{cm}^3/\text{sec}$

(C) $900\ \text{cm}^3/\text{sec}$

(D) $2700\ \text{cm}^3/\text{sec}$

Question 18. A spherical balloon is being inflated by pumping $900$ cubic cm of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is $15\ \text{cm}$.

(A) $\frac{1}{10\pi}\ \text{cm}/\text{sec}$

(B) $\frac{1}{20\pi}\ \text{cm}/\text{sec}$

(C) $\frac{1}{30\pi}\ \text{cm}/\text{sec}$

(D) $\frac{1}{40\pi}\ \text{cm}/\text{sec}$

Question 19. If the total cost function is $C(x)$ is given, the marginal cost is calculated as:

(A) $\frac{dC}{dx}$

(B) $\frac{C(x)}{x}$

(C) $\int C(x) dx$

(D) $C(x+1)$

Question 20. The marginal revenue is the rate of change of total revenue with respect to:

(A) Price

(B) Time

(C) Quantity sold

(D) Cost

Question 21. If the total cost function is $C(x) = x^2 + 2x + 500$, the marginal cost at $x=10$ is:

(A) $\textsf{₹ }22$

(B) $\textsf{₹ }120$

(C) $\textsf{₹ }10$

(D) $\textsf{₹ }20$

Question 1. The slope of the tangent to the curve $y = x^2 - 3x + 2$ at the point where $x=2$ is:

(A) 1

(B) 2

(C) -1

(D) 0

Question 2. The equation of the tangent to the curve $y = x^3$ at $(1, 1)$ is:

(A) $y - 1 = 3(x - 1)$

(B) $y - 1 = 1(x - 1)$

(C) $y - 1 = -1(x - 1)$

(D) $y - 1 = 0(x - 1)$

Question 3. The equation of the normal to the curve $y = x^2$ at $(1, 1)$ is:

(A) $y - 1 = 2(x - 1)$

(B) $y - 1 = -1/2(x - 1)$

(C) $y - 1 = 1/2(x - 1)$

(D) $y - 1 = -2(x - 1)$

Question 4. Using differentials, the approximate value of $\sqrt{25.3}$ is:

(A) 5.03

(B) 5.02

(C) 5.01

(D) 5.003

Question 5. If the radius of a sphere is measured as $10\ \text{cm}$ with an error of $0.02\ \text{cm}$, the approximate error in calculating its volume is:

(A) $4\pi (10)^2 (0.02)\ \text{cm}^3$

(B) $\frac{4}{3}\pi (10)^3\ \text{cm}^3$

(C) $3 \cdot 4\pi (10)^2 (0.02)\ \text{cm}^3$

(D) $4\pi (10)^3 (0.02)\ \text{cm}^3$

Question 6. The percentage error in the volume of a cube if there is a $1\%$ error in measuring the length of its edge is:

(A) $1\%$

(B) $2\%$

(C) $3\%$

(D) $6\%$

Question 7. The angle made by the tangent to the curve $y^2 = 4ax$ at $(at^2, 2at)$ with the x-axis is:

(A) $\tan^{-1}(t)$

(B) $\tan^{-1}(1/t)$

(C) $\tan^{-1}(2t)$

(D) $\tan^{-1}(2/t)$

Question 8. Using differentials, the approximate value of $(0.009)^{1/3}$ is:

(A) 0.2

(B) 0.21

(C) 0.208

(D) 0.198

Question 9. The absolute error in $y = x^3$ when $x=2$ and the error in $x$ is $0.01$ is approximately:

(A) $0.03$

(B) $0.12$

(C) $0.01$

(D) $0.06$

Question 10. Find the point on the curve $y = x^2 - 2x + 3$ where the tangent is parallel to the x-axis.

(A) $(1, 2)$

(B) $(2, 3)$

(C) $(0, 3)$

(D) $(-1, 6)$

Question 11. The relative error in $y = x^2$ when $x=3$ and $\Delta x = 0.02$ is approximately:

(A) $0.04/3$

(B) $0.04/9$

(C) $0.01$

(D) $0.0067$ (approx)

Question 12. The equation of the tangent to the curve $y = \sin x$ at $(0, 0)$ is:

(A) $y=x$

(B) $y=-x$

(C) $y=0$

(D) $x=0$

Question 13. The angle between the curves $y^2 = x$ and $x^2 = y$ at the point $(1, 1)$ is:

(A) $\tan^{-1}(1/3)$

(B) $\tan^{-1}(3)$

(C) $\pi/2$

(D) $\pi/4$

Question 14. Using differentials, the approximate value of $\tan(46^\circ)$ is (use $1^\circ \approx 0.01745$ radians):

(A) $1 + 0.01745$

(B) $1 - 0.01745$

(C) $1$

(D) $0.01745$

Question 15. If the measurement of the side of a square is subject to an error of $0.1\%$, the error in the area is approximately:

(A) $0.1\%$

(B) $0.2\%$

(C) $0.01\%$

(D) $0.02\%$

Question 16. The points on the curve $y = x^3 - 3x^2 - 9x + 7$ where the tangents are parallel to the x-axis are:

(A) $(3, -20)$ and $(-1, 12)$

(B) $(-3, -20)$ and $(1, 12)$

(C) $(3, 12)$ and $(-1, -20)$

(D) $(3, -20)$ and $(1, -4)$

Question 17. The approximate value of $(82)^{1/4}$ using differentials is:

(A) $3.009$

(B) $3.008$

(C) $3.007$

(D) $3.006$

Question 18. If the percentage error in the radius of a sphere is $p\%$, the percentage error in its surface area is:

(A) $p\%$

(B) $2p\%$

(C) $3p\%$

(D) $p^2 \%$

Question 19. The equation of the normal to the curve $y = \sin x$ at $(0, 0)$ is:

(A) $y=-x$

(B) $y=x$

(C) $x=0$

(D) $y=0$

Question 20. The total error $\Delta y$ in estimating $f(x+\Delta x) - f(x)$ is approximated by $dy = f'(x) \Delta x$. Here $dy$ is called the:

(A) Absolute error

(B) Relative error

(C) Percentage error

(D) Differential of $y$

Question 21. The curve $y = x^{1/3}$ has a vertical tangent at:

(A) $x=0$

(B) $x=1$

(C) $x=-1$

(D) No point

Question 1. A function $f(x)$ is strictly increasing on an interval $(a, b)$ if for any $x_1, x_2 \in (a, b)$ with $x_1 < x_2$, we have:

(A) $f(x_1) \leq f(x_2)$

(B) $f(x_1) < f(x_2)$

(C) $f(x_1) \geq f(x_2)$

(D) $f(x_1) > f(x_2)$

Question 2. A function $f(x)$ is decreasing on an interval $[a, b]$ if for any $x_1, x_2 \in [a, b]$ with $x_1 < x_2$, we have:

(A) $f(x_1) < f(x_2)$

(B) $f(x_1) \leq f(x_2)$

(C) $f(x_1) > f(x_2)$

(D) $f(x_1) \geq f(x_2)$

Question 3. If $f'(x) > 0$ for all $x$ in an interval $(a, b)$, then $f(x)$ is:

(A) Strictly increasing on $(a, b)$.

(B) Increasing on $(a, b)$.

(C) Strictly decreasing on $(a, b)$.

(D) Decreasing on $(a, b)$.

Question 4. Find the interval where $f(x) = x^2 - 4x + 6$ is strictly increasing.

(A) $(-\infty, 2)$

(B) $(2, \infty)$

(C) $(-\infty, \infty)$

(D) $(0, \infty)$

Question 5. Find the interval where $f(x) = 3x - x^3$ is strictly decreasing.

(A) $(-1, 1)$

(B) $(-\infty, -1) \cup (1, \infty)$

(C) $(-\infty, -1)$

(D) $(1, \infty)$

Question 6. If $f'(x) < 0$ for all $x$ in an interval $(a, b)$, then $f(x)$ is:

(A) Strictly increasing on $(a, b)$.

(B) Increasing on $(a, b)$.

(C) Strictly decreasing on $(a, b)$.

(D) Decreasing on $(a, b)$.

Question 7. The function $f(x) = e^x$ is:

(A) Strictly increasing for all $x \in \mathbb{R}$.

(B) Strictly decreasing for all $x \in \mathbb{R}$.

(C) Neither increasing nor decreasing.

(D) Increasing only for $x > 0$.

Question 8. Find the interval where $f(x) = \sin x$ is strictly increasing.

(A) $(0, \pi/2)$

(B) $(-\pi/2, \pi/2)$

(C) $(\pi/2, \pi)$

(D) $(0, \pi)$

Question 9. The function $f(x) = \cos x$ is strictly decreasing in the interval:

(A) $(0, \pi/2)$

(B) $(0, \pi)$

(C) $(\pi/2, \pi)$

(D) $(0, 2\pi)$

Question 10. For a function to be monotonic on an interval, it must be either always increasing or always decreasing on that interval. This statement is:

(A) True

(B) False

(C) True, only if the function is continuous

(D) True, only if the function is differentiable

Question 11. Find the interval where $f(x) = x^3 - 3x^2 + 3x - 1$ is strictly increasing.

(A) $(-\infty, \infty)$

(B) $(-\infty, 1)$

(C) $(1, \infty)$

(D) $(-\infty, 0)$

Question 12. If $f'(x) = 0$ for all $x$ in an interval, the function $f(x)$ is:

(A) Increasing

(B) Decreasing

(C) Constant

(D) Cannot be determined

Question 13. The function $f(x) = |x|$ is:

(A) Strictly increasing on $\mathbb{R}$.

(B) Strictly decreasing on $\mathbb{R}$.

(C) Increasing on $[0, \infty)$ and decreasing on $(-\infty, 0]$.

(D) Neither increasing nor decreasing on $\mathbb{R}$.

Question 14. Find the intervals where $f(x) = 2x^3 - 3x^2 - 36x + 7$ is strictly increasing.

(A) $(-2, 3)$

(B) $(-\infty, -2) \cup (3, \infty)$

(C) $(-\infty, -2)$

(D) $(3, \infty)$

Question 15. If $f'(x) \geq 0$ for all $x$ in an interval $(a, b)$ and $f'(x) = 0$ only at isolated points, then $f(x)$ is:

(A) Increasing

(B) Strictly increasing

(C) Decreasing

(D) Strictly decreasing

Question 16. Find the interval where $f(x) = \log_e x$ is strictly increasing.

(A) $(0, \infty)$

(B) $(1, \infty)$

(C) $(-\infty, \infty)$

(D) $(e, \infty)$

Question 17. The function $f(x) = \tan x$ is strictly increasing in which interval?

(A) $(-\pi/2, \pi/2)$

(B) $(0, \pi)$

(C) $(-\infty, \infty)$

(D) $(-\pi/2, \pi/2) \cup (\pi/2, 3\pi/2)$

Question 18. If the profit function $P(x)$ for producing $x$ units is strictly increasing, it means:

(A) Profit is constant.

(B) Profit decreases as production increases.

(C) Profit increases as production increases.

(D) Marginal profit is zero.

Question 19. Find the interval where $f(x) = \frac{x}{x^2 + 1}$ is strictly increasing.

(A) $(-1, 1)$

(B) $(-\infty, -1) \cup (1, \infty)$

(C) $(-\infty, \infty)$

(D) $(0, \infty)$

Question 20. The function $f(x) = x^3$ is strictly increasing on:

(A) $(-\infty, 0)$

(B) $(0, \infty)$

(C) $(-\infty, \infty)$

(D) $(-\infty, 0] \cup [0, \infty)$

Question 21. If a function is strictly monotonic on an interval, is it necessarily differentiable on that interval?

(A) Yes

(B) No

(C) Only if it is also continuous

(D) Only if the interval is open

Question 1. A function $f(x)$ has a local maximum at $x=c$ if there exists an interval $(c-\delta, c+\delta)$ such that for all $x$ in this interval (except possibly $c$):

(A) $f(x) \geq f(c)$

(B) $f(x) \leq f(c)$

(C) $f(x) = f(c)$

(D) $f'(c) = 0$

Question 2. A critical point of a function $f(x)$ is a point $x=c$ in the domain of $f$ where:

(A) $f'(c) = 0$

(B) $f'(c)$ is undefined

(C) Either $f'(c) = 0$ or $f'(c)$ is undefined.

(D) $f(c) = 0$

Question 3. According to the First Derivative Test, if $f'(x)$ changes from positive to negative as $x$ increases through $c$, then $f(x)$ has a ________ at $x=c$.

(A) Local Minimum

(B) Local Maximum

(C) Inflection Point

(D) No extremum

Question 4. Find the local maximum value of $f(x) = x^2 - 4x + 5$.

(A) 1

(B) 2

(C) 5

(D) Does not exist

Question 5. According to the Second Derivative Test, if $f'(c) = 0$ and $f''(c) > 0$, then $f(x)$ has a ________ at $x=c$.

(A) Local Minimum

(B) Local Maximum

(C) Inflection Point

(D) No extremum

Question 6. Find the critical points of $f(x) = x^{1/3}$.

(A) $x=0$

(B) $x=1$

(C) $x=-1$

(D) No critical points

Question 7. For a function continuous on a closed interval $[a, b]$, the absolute maximum and minimum values occur either at critical points in $(a, b)$ or at:

(A) Endpoints $a$ and $b$.

(B) Points where $f'(x) = 0$.

(C) Points where $f''(x) = 0$.

(D) Inflection points.

Question 8. Find the local minimum value of $f(x) = x^3 - 3x + 2$.

(A) 0

(B) 4

(C) -1

(D) 1

Question 9. A rectangular park is to be designed whose breadth is 3 less than its length. Its area is to be more than 4 square meters. Find the minimum area. (Applied Maths)

(A) 4 square meters

(B) 4.5 square meters

(C) 2.25 square meters

(D) 0 square meters

Question 10. Find the absolute maximum value of $f(x) = x^3$ on $[-2, 1]$.

(A) 1

(B) 8

(C) -8

(D) 0

Question 11. Find two positive numbers whose sum is 16 and the sum of whose squares is minimum.

(A) 8 and 8

(B) 4 and 12

(C) 6 and 10

(D) 7 and 9

Question 12. The function $f(x) = |x|$ has a local minimum at $x=0$. At this point, $f'(x)$ is:

(A) 0

(B) Undefined

(C) Positive

(D) Negative

Question 13. Find the maximum profit that a company can make, if the profit function is given by $P(x) = 41 + 24x - 18x^2$. (Applied Maths)

(A) 41

(B) 24

(C) 49

(D) 50

Question 14. If $f'(c) = 0$ and $f''(c) = 0$, the Second Derivative Test is inconclusive. This means:

(A) There is definitely a local extremum at $x=c$.

(B) There is definitely no local extremum at $x=c$.

(C) We need to use the First Derivative Test to determine if there is an extremum.

(D) $x=c$ is an inflection point.

Question 15. Find the absolute minimum value of $f(x) = x^2$ on $[-2, 1]$.

(A) 4

(B) 1

(C) 0

(D) -2

Question 16. Find the dimensions of the rectangle of perimeter 40 cm that has the maximum area.

(A) 10 cm by 10 cm

(B) 8 cm by 12 cm

(C) 5 cm by 15 cm

(D) 2 cm by 18 cm

Question 17. The points where local maxima or minima occur are called:

(A) Inflection points

(B) Critical points

(C) Endpoints

(D) Roots

Question 18. A manufacturer can sell $x$ items at a price of $\textsf{₹ }(5 - x/100)$ each. The cost of production is $\textsf{₹ }(x/5 + 500)$. Find the number of items to be sold to get maximum profit. (Applied Maths)

(A) 200

(B) 400

(C) 500

(D) 240

Question 19. Find the local maximum value of $f(x) = \sin x + \cos x$ in $[0, \pi/2]$.

(A) 1

(B) $\sqrt{2}$

(C) 2

(D) 0

Question 20. The function $f(x) = x^3$ has a critical point at $x=0$. Does it have a local extremum at $x=0$?

(A) Yes, a local minimum.

(B) Yes, a local maximum.

(C) No, it is an inflection point.

(D) Cannot be determined.

Question 21. Find two positive numbers $x$ and $y$ such that $x+y=60$ and $xy^3$ is maximum.

(A) $x=15, y=45$

(B) $x=30, y=30$

(C) $x=20, y=40$

(D) $x=45, y=15$

Question 22. Find the minimum cost of production if the cost function is $C(x) = 2x^2 - 40x + 2000$. (Applied Maths)

(A) 2000

(B) 1600

(C) 1800

(D) 400

Question 1. An antiderivative of a function $f(x)$ is a function $F(x)$ such that:

(A) $F(x) = f'(x)$

(B) $F'(x) = f(x)$

(C) $\int F(x) dx = f(x)$

(D) $F(x) = \frac{1}{f(x)}$

Question 2. The indefinite integral of a function $f(x)$, denoted by $\int f(x) dx$, represents:

(A) A specific antiderivative of $f(x)$.

(B) The area under the curve of $f(x)$.

(C) The family of all antiderivatives of $f(x)$.

(D) The rate of change of $f(x)$.

Question 3. If $\frac{d}{dx} F(x) = f(x)$, then $\int f(x) dx =$

(A) $F(x)$

(B) $F(x) + C$, where $C$ is the constant of integration.

(C) $F'(x)$

(D) $f'(x)$

Question 4. Evaluate $\int x^n dx$ for $n \neq -1$.

(A) $nx^{n-1} + C$

(B) $\frac{x^{n+1}}{n+1} + C$

(C) $x^{n+1} + C$

(D) $\frac{x^n}{n} + C$

Question 5. Evaluate $\int \cos x dx$.

(A) $\sin x + C$

(B) $-\sin x + C$

(C) $\cos x + C$

(D) $-\cos x + C$

Question 6. Evaluate $\int e^x dx$.

(A) $xe^{x-1} + C$

(B) $e^x + C$

(C) $e^{x+1} + C$

(D) $e^x \log x + C$

Question 7. Evaluate $\int \frac{1}{x} dx$ for $x \neq 0$.

(A) $\log_e x + C$

(B) $-\frac{1}{x^2} + C$

(C) $\log_e |x| + C$

(D) $x^{-1} + C$

Question 8. If the marginal cost function is $MC(x) = 2x + 5$, find the total cost function $C(x)$, given that the fixed cost is $\textsf{₹ }1000$. (Applied Maths)

(A) $x^2 + 5x$

(B) $x^2 + 5x + 1000$

(C) $2x^2/2 + 5x + C$

(D) $x^2 + 5x + C$

Question 9. Evaluate $\int (ax+b)^n dx$ for $n \neq -1$.

(A) $\frac{(ax+b)^{n+1}}{n+1} + C$

(B) $a \frac{(ax+b)^{n+1}}{n+1} + C$

(C) $\frac{1}{a} \frac{(ax+b)^{n+1}}{n+1} + C$

(D) $n(ax+b)^{n-1}a + C$

Question 10. If $\int f(x) dx = F(x) + C$, then $\int k f(x) dx = $

(A) $k F(x) + C$

(B) $F(kx) + C$

(C) $k (F(x) + C)$

(D) $k F(x) + kC$ (where $kC$ is just a new constant)

Question 11. Evaluate $\int (\sin x + \cos x) dx$.

(A) $\cos x - \sin x + C$

(B) $-\cos x + \sin x + C$

(C) $\sin x + \cos x + C$

(D) $-\sin x - \cos x + C$

Question 12. The integral $\int dx$ is equal to:

(A) $x + C$

(B) $1 + C$

(C) $0 + C$

(D) $dx + C$

Question 13. Evaluate $\int \sec^2 x dx$.

(A) $\tan x + C$

(B) $-\cot x + C$

(C) $\sec x + C$

(D) $2 \sec x \tan x + C$

Question 14. If the marginal revenue function is $MR(x) = 6 - 2x$, find the total revenue function $R(x)$, assuming $R(0)=0$. (Applied Maths)

(A) $6x - x^2$

(B) $6x - x^2 + C$

(C) $6x - x^2 + 0$

(D) $6 - 2x + C$

Question 15. Evaluate $\int \frac{1}{\sqrt{1-x^2}} dx$.

(A) $\sin^{-1} x + C$

(B) $\cos^{-1} x + C$

(C) $\tan^{-1} x + C$

(D) $\log_e |\sqrt{1-x^2} + x| + C$

Question 16. The graph of an indefinite integral represents:

(A) A single curve.

(B) A family of parallel curves.

(C) A family of curves intersecting at a single point.

(D) A straight line.

Question 17. Evaluate $\int (x + 1/x)^2 dx$.

(A) $\frac{x^3}{3} + 2x + \frac{1}{x} + C$

(B) $\frac{x^3}{3} + 2 \log_e |x| - \frac{1}{x} + C$

(C) $2(x+1/x)(1-1/x^2) + C$

(D) $\frac{x^3}{3} + 2 \log_e |x| + \frac{1}{x} + C$

Question 18. Evaluate $\int \tan^2 x dx$.

(A) $\tan x - x + C$

(B) $\sec^2 x - 1 + C$

(C) $\sec x + C$

(D) $\tan x + x + C$

Question 19. If $F(x)$ and $G(x)$ are two antiderivatives of the same function $f(x)$, then $F(x)$ and $G(x)$ must differ by:

(A) A constant.

(B) A variable $x$.

(C) A function of $x$.

(D) Zero.

Question 20. Evaluate $\int a^x dx$ where $a > 0, a \neq 1$.

(A) $a^x \log_e a + C$

(B) $\frac{a^x}{\log_e a} + C$

(C) $x a^{x-1} + C$

(D) $a^x + C$

Question 21. If $\int f(x) dx = \sin x + C$, then $f(x)$ is:

(A) $\sin x$

(B) $\cos x$

(C) $-\cos x$

(D) $-\sin x$

Question 1. The method of integration by substitution is based on the:

(A) Product Rule of Differentiation

(B) Quotient Rule of Differentiation

(C) Chain Rule of Differentiation

(D) Sum Rule of Differentiation

Question 2. Evaluate $\int 2x \cos(x^2) dx$ using substitution.

(A) $\sin(x^2) + C$

(B) $-\sin(x^2) + C$

(C) $\cos(x^2) + C$

(D) $x^2 \sin(x^2) + C$

Question 3. The integration by parts formula is $\int u dv = uv - \int v du$. Here, $u$ and $v$ are functions of:

(A) The integration variable.

(B) A constant.

(C) Time.

(D) Another function.

Question 4. Evaluate $\int x e^x dx$ using integration by parts.

(A) $xe^x - e^x + C$

(B) $xe^x + e^x + C$

(C) $x e^x + C$

(D) $\frac{x^2}{2} e^x + C$

Question 5. Evaluate $\int \tan x dx$.

(A) $\log_e |\sec x| + C$

(B) $-\log_e |\cos x| + C$

(C) $\log_e |\sin x| + C$

(D) Both (A) and (B)

Question 6. Evaluate $\int x \sin x dx$ using integration by parts.

(A) $-x \cos x + \sin x + C$

(B) $x \cos x - \sin x + C$

(C) $-x \cos x - \sin x + C$

(D) $x \sin x + \cos x + C$

Question 7. Which function should typically be chosen as $u$ when using integration by parts with $\int x^n e^{ax} dx$?

(A) $e^{ax}$

(B) $x^n$

(C) $dx$

(D) $e^{ax} dx$

Question 8. Evaluate $\int \frac{e^{\tan^{-1} x}}{1+x^2} dx$ using substitution.

(A) $e^{\tan^{-1} x} + C$

(B) $\log_e |1+x^2| + C$

(C) $\tan^{-1} x + C$

(D) $\frac{e^{\tan^{-1} x}}{2x} + C$

Question 9. Evaluate $\int \log_e x dx$ using integration by parts.

(A) $x \log_e x + x + C$

(B) $x \log_e x - x + C$

(C) $\frac{1}{x} + C$

(D) $(\log_e x)^2/2 + C$

Question 10. Evaluate $\int \frac{\sin(\log_e x)}{x} dx$ using substitution.

(A) $\cos(\log_e x) + C$

(B) $-\cos(\log_e x) + C$

(C) $\sin(\log_e x) + C$

(D) $\frac{\cos(\log_e x)}{x} + C$

Question 11. The integral $\int e^x (f(x) + f'(x)) dx$ is equal to:

(A) $e^x f'(x) + C$

(B) $e^x f(x) + C$

(C) $e^x + f(x) + C$

(D) $e^x f(x) + e^x f'(x) + C$

Question 12. Evaluate $\int \sin^3 x \cos x dx$ using substitution.

(A) $3 \sin^2 x \cos x + C$

(B) $\frac{\sin^4 x}{4} + C$

(C) $\frac{\cos^4 x}{4} + C$

(D) $\sin^4 x + C$

Question 13. Evaluate $\int \tan^{-1} x dx$ using integration by parts.

(A) $x \tan^{-1} x - \frac{1}{2} \log_e |1+x^2| + C$

(B) $x \tan^{-1} x + \frac{1}{2} \log_e |1+x^2| + C$

(C) $\frac{1}{1+x^2} + C$

(D) $\tan^{-1} x + \frac{x}{1+x^2} + C$

Question 14. To integrate $\int \frac{f'(x)}{f(x)} dx$, the substitution $u = f(x)$ is typically used. The result is:

(A) $f(x) + C$

(B) $\log_e |f(x)| + C$

(C) $f'(x) + C$

(D) $\frac{1}{f(x)} + C$

Question 15. Evaluate $\int x \log_e x dx$ using integration by parts.

(A) $\frac{x^2}{2} \log_e x - \frac{x^2}{4} + C$

(B) $\frac{x^2}{2} \log_e x + \frac{x^2}{4} + C$

(C) $x \log_e x - x + C$

(D) $\frac{x^2}{2} (\log_e x)^2 + C$

Question 16. Evaluate $\int e^x (\tan x + \sec^2 x) dx$.

(A) $e^x \tan x + C$

(B) $e^x \sec^2 x + C$

(C) $e^x \tan x + e^x \sec^2 x + C$

(D) $e^x (\tan x + \sec^2 x) + C$

Question 17. Evaluate $\int \frac{x}{x^2+1} dx$.

(A) $\log_e |x^2+1| + C$

(B) $\frac{1}{2} \log_e |x^2+1| + C$

(C) $\tan^{-1} x + C$

(D) $2 \log_e |x^2+1| + C$

Question 18. Evaluate $\int \sin^{-1} x dx$.

(A) $x \sin^{-1} x + \sqrt{1-x^2} + C$

(B) $x \sin^{-1} x - \sqrt{1-x^2} + C$

(C) $\frac{1}{\sqrt{1-x^2}} + C$

(D) $x \sin^{-1} x + C$

Question 19. When using integration by parts $\int u dv$, how is $dv$ related to $v$?

(A) $v = \int dv$

(B) $dv = \int v$

(C) $v = dv/dx$

(D) $dv = v dx$

Question 20. Evaluate $\int \frac{x}{\sqrt{x^2+1}} dx$ using substitution.

(A) $\sqrt{x^2+1} + C$

(B) $2\sqrt{x^2+1} + C$

(C) $\log_e |\sqrt{x^2+1}| + C$

(D) $\frac{1}{2\sqrt{x^2+1}} + C$

Question 21. Evaluate $\int x^2 e^x dx$.

(A) $x^2 e^x - 2x e^x + 2e^x + C$

(B) $x^2 e^x + 2x e^x + 2e^x + C$

(C) $\frac{x^3}{3} e^x + C$

(D) $x^2 e^x - 2e^x + C$

Question 1. The method of partial fractions is used for integrating:

(A) Polynomial functions

(B) Rational functions

(C) Trigonometric functions

(D) Exponential functions

Question 2. Evaluate $\int \frac{1}{(x+1)(x+2)} dx$ using partial fractions.

(A) $\log_e |\frac{x+1}{x+2}| + C$

(B) $\log_e |\frac{x+2}{x+1}| + C$

(C) $\log_e |(x+1)(x+2)| + C$

(D) $\tan^{-1}(\frac{x+1}{x+2}) + C$

Question 3. Evaluate $\int \frac{dx}{\sqrt{a^2 - x^2}}$.

(A) $\sin^{-1}(x/a) + C$

(B) $\cos^{-1}(x/a) + C$

(C) $\tan^{-1}(x/a) + C$

(D) $\log_e |x + \sqrt{a^2 - x^2}| + C$

Question 4. Evaluate $\int \frac{dx}{x^2 + a^2}$.

(A) $\tan^{-1}(x/a) + C$

(B) $\frac{1}{a} \tan^{-1}(x/a) + C$

(C) $\frac{1}{2a} \log_e |\frac{a+x}{a-x}| + C$

(D) $\sin^{-1}(x/a) + C$

Question 5. Evaluate $\int \frac{dx}{\sqrt{x^2 + a^2}}$.

(A) $\log_e |x + \sqrt{x^2 + a^2}| + C$

(B) $\sinh^{-1}(x/a) + C$

(C) $\log_e |x + \sqrt{x^2 - a^2}| + C$

(D) Both (A) and (B)

Question 6. To integrate a rational function $\frac{P(x)}{Q(x)}$, if the degree of $P(x)$ is greater than or equal to the degree of $Q(x)$, the first step is typically:

(A) Partial fraction decomposition.

(B) Long division.

(C) Substitution.

(D) Integration by parts.

Question 7. Evaluate $\int \frac{x}{x^2 - 3x + 2} dx$.

(A) $2 \log_e |x-2| - \log_e |x-1| + C$

(B) $\log_e |x-1| - 2 \log_e |x-2| + C$

(C) $\log_e |(x-1)(x-2)| + C$

(D) $\frac{1}{2} \log_e |x^2 - 3x + 2| + C$

Question 8. To integrate rational functions of $\sin x$ and $\cos x$, a common substitution is $t = \tan(x/2)$. In this case, $dx = \frac{2dt}{1+t^2}$, $\sin x = \frac{2t}{1+t^2}$, and $\cos x =$

(A) $\frac{1-t^2}{1+t^2}$

(B) $\frac{1+t^2}{1-t^2}$

(C) $\frac{2t}{1+t^2}$

(D) $\frac{2t}{1-t^2}$

Question 9. Evaluate $\int \frac{dx}{\sqrt{x^2 - a^2}}$.

(A) $\log_e |x + \sqrt{x^2 - a^2}| + C$

(B) $\cosh^{-1}(x/a) + C$

(C) $\log_e |x + \sqrt{a^2 - x^2}| + C$

(D) Both (A) and (B) for $x>a$.

Question 10. Evaluate $\int \sqrt{a^2 - x^2} dx$.

(A) $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}(x/a) + C$

(B) $\frac{x}{2}\sqrt{a^2 - x^2} - \frac{a^2}{2} \sin^{-1}(x/a) + C$

(C) $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \log_e |x + \sqrt{a^2 - x^2}| + C$

(D) $\sqrt{a^2 - x^2} + C$

Question 11. Evaluate $\int \frac{dx}{a^2 - x^2}$ for $a>0$.

(A) $\frac{1}{a} \tan^{-1}(x/a) + C$

(B) $\frac{1}{2a} \log_e |\frac{a+x}{a-x}| + C$

(C) $\frac{1}{2a} \log_e |\frac{x-a}{x+a}| + C$

(D) $\frac{1}{a} \log_e |\frac{a+x}{a-x}| + C$

Question 12. Evaluate $\int \frac{dx}{x^2 - a^2}$ for $a>0$.

(A) $\frac{1}{2a} \log_e |\frac{x-a}{x+a}| + C$

(B) $\frac{1}{2a} \log_e |\frac{a+x}{a-x}| + C$

(C) $\frac{1}{a} \tan^{-1}(x/a) + C$

(D) $\log_e |x^2 - a^2| + C$

Question 13. Evaluate $\int \sqrt{x^2 - a^2} dx$.

(A) $\frac{x}{2}\sqrt{x^2 - a^2} + \frac{a^2}{2} \log_e |x + \sqrt{x^2 - a^2}| + C$

(B) $\frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2} \log_e |x + \sqrt{x^2 - a^2}| + C$

(C) $\frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2} \sin^{-1}(x/a) + C$

(D) $\frac{x}{2}\sqrt{x^2 - a^2} + \frac{a^2}{2} \sin^{-1}(x/a) + C$

Question 14. To use partial fractions for $\frac{P(x)}{Q(x)}$, where $Q(x)$ has a repeated linear factor $(ax+b)^n$, the corresponding terms in the partial fraction decomposition should be:

(A) $\frac{A}{(ax+b)^n}$

(B) $\frac{A}{ax+b} + \frac{B}{(ax+b)^2} + \dots + \frac{N}{(ax+b)^n}$

(C) $\frac{A}{ax+b}$

(D) $\frac{Ax+B}{(ax+b)^n}$

Question 15. Evaluate $\int \frac{dx}{1+\cos x}$.

(A) $\tan x + C$

(B) $\tan(x/2) + C$

(C) $-\cot(x/2) + C$

(D) $x + \sin x + C$

Question 16. Evaluate $\int \sqrt{x^2 + a^2} dx$.

(A) $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2} \log_e |x + \sqrt{x^2 + a^2}| + C$

(B) $\frac{x}{2}\sqrt{x^2 + a^2} - \frac{a^2}{2} \log_e |x + \sqrt{x^2 + a^2}| + C$

(C) $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2} \sin^{-1}(x/a) + C$

(D) $\sqrt{x^2 + a^2} + C$

Question 17. Evaluate $\int \frac{x+1}{x^2 + 4} dx$.

(A) $\frac{1}{2} \log_e |x^2+4| + \frac{1}{2} \tan^{-1}(x/2) + C$

(B) $\log_e |x^2+4| + \tan^{-1}(x/2) + C$

(C) $\frac{1}{2} \log_e |x^2+4| + \tan^{-1}(x/2) + C$

(D) $\log_e |x^2+4| + \frac{1}{2} \tan^{-1}(x/2) + C$

Question 18. To integrate rational functions of $\sin x$ and $\cos x$ using the substitution $t = \tan(x/2)$, the range of $t$ is usually considered as:

(A) $(-\infty, \infty)$

(B) $(-\pi/2, \pi/2)$

(C) $(0, \pi)$

(D) $[-\pi, \pi]$

Question 19. Evaluate $\int \frac{dx}{\sqrt{(x-a)(x-b)}}$.

(A) $2 \log_e |\sqrt{x-a} + \sqrt{x-b}| + C$

(B) $\log_e |x - \frac{a+b}{2} + \sqrt{(x-a)(x-b)}| + C$

(C) $\sin^{-1}(\frac{2x-a-b}{b-a}) + C$

(D) $\cosh^{-1}(\frac{2x-a-b}{b-a}) + C$ (for $x > b$)

Question 20. Evaluate $\int \frac{dx}{\sqrt{x^2 + 2x + 2}}$.

(A) $\log_e |x+1 + \sqrt{x^2+2x+2}| + C$

(B) $\sinh^{-1}(x+1) + C$

(C) $\log_e |x+1 + \sqrt{(x+1)^2+1}| + C$

(D) All of the above.

Question 21. Evaluate $\int \frac{dx}{x^2 + 4x + 5}$.

(A) $\tan^{-1}(x+2) + C$

(B) $\frac{1}{2} \tan^{-1}(x+2) + C$

(C) $\log_e |x^2+4x+5| + C$

(D) $\frac{1}{\sqrt{5}} \tan^{-1}(\frac{x+2}{\sqrt{5}}) + C$

Question 22. Evaluate $\int \frac{x+2}{x^2 + 4x + 5} dx$.

(A) $\frac{1}{2} \log_e |x^2+4x+5| + C$

(B) $\log_e |x^2+4x+5| + C$

(C) $\frac{1}{2} \tan^{-1}(x+2) + C$

(D) $\frac{1}{2} \int \frac{2x+4}{x^2+4x+5} dx + C$

Question 23. Evaluate $\int \frac{dx}{a^2 - x^2}$ for $|x| < a$.

(A) $\frac{1}{2a} \log_e |\frac{a+x}{a-x}| + C$

(B) $\frac{1}{2a} \log_e |\frac{x-a}{x+a}| + C$

(C) $\frac{1}{a} \tan^{-1}(x/a) + C$

(D) $\sin^{-1}(x/a) + C$

Question 24. Evaluate $\int \frac{dx}{x^2 - a^2}$ for $|x| > a$.

(A) $\frac{1}{2a} \log_e |\frac{a+x}{a-x}| + C$

(B) $\frac{1}{2a} \log_e |\frac{x-a}{x+a}| + C$

(C) $\frac{1}{a} \tan^{-1}(x/a) + C$

(D) $\log_e |x^2 - a^2| + C$

Question 25. Evaluate $\int \frac{\cos x}{\sin^2 x + 4} dx$.

(A) $\frac{1}{2} \tan^{-1}(\frac{\sin x}{2}) + C$

(B) $\frac{1}{2} \tan^{-1}(\sin x) + C$

(C) $\tan^{-1}(\frac{\sin x}{2}) + C$

(D) $\log_e |\sin^2 x + 4| + C$

Question 26. Evaluate $\int \frac{\sin x}{\cos^2 x + 2\cos x + 1} dx$.

(A) $\frac{1}{\cos x + 1} + C$

(B) $-\frac{1}{\cos x + 1} + C$

(C) $-\frac{1}{(\cos x + 1)^2} + C$

(D) $\log_e |\cos x + 1| + C$

Question 27. Evaluate $\int \frac{dx}{5 + 4 \cos x}$.

(A) $\frac{2}{3} \tan^{-1}(\frac{1}{3} \tan(x/2)) + C$

(B) $\frac{1}{3} \tan^{-1}(\frac{2}{3} \tan(x/2)) + C$

(C) $\frac{1}{3} \tan^{-1}(\tan(x/2)) + C$

(D) $\frac{2}{3} \tan^{-1}(\tan(x/2)) + C$

Question 28. Evaluate $\int \text{cosec } x dx$.

(A) $\log_e |\text{cosec } x + \cot x| + C$

(B) $\log_e |\tan(x/2)| + C$

(C) $-\log_e |\text{cosec } x - \cot x| + C$

(D) Both (B) and (C)

Question 29. Evaluate $\int \frac{dx}{\sqrt{4-x^2}}$.

(A) $\sin^{-1}(x/2) + C$

(B) $\cos^{-1}(x/2) + C$

(C) $\frac{1}{2} \tan^{-1}(x/2) + C$

(D) $\frac{1}{2} \log_e |\frac{2+x}{2-x}| + C$

Question 30. Evaluate $\int \frac{dx}{\sqrt{x^2+9}}$.

(A) $\log_e |x + \sqrt{x^2+9}| + C$

(B) $\sinh^{-1}(x/3) + C$

(C) $\frac{1}{3} \tan^{-1}(x/3) + C$

(D) Both (A) and (B)

Question 31. Evaluate $\int \frac{dx}{\sqrt{x^2-16}}$.

(A) $\log_e |x + \sqrt{x^2-16}| + C$

(B) $\cosh^{-1}(x/4) + C$

(C) $\sin^{-1}(x/4) + C$

(D) Both (A) and (B) for $x>4$

Question 32. Evaluate $\int \sqrt{4-x^2} dx$.

(A) $\frac{x}{2}\sqrt{4-x^2} + 2 \sin^{-1}(x/2) + C$

(B) $\frac{x}{2}\sqrt{4-x^2} - 2 \sin^{-1}(x/2) + C$

(C) $\frac{x}{2}\sqrt{4-x^2} + \frac{1}{2} \sin^{-1}(x/2) + C$

(D) $\frac{x}{2}\sqrt{4-x^2} + 2 \log_e |x + \sqrt{4-x^2}| + C$

Question 33. Evaluate $\int \sqrt{x^2+9} dx$.

(A) $\frac{x}{2}\sqrt{x^2+9} + \frac{9}{2} \log_e |x + \sqrt{x^2+9}| + C$

(B) $\frac{x}{2}\sqrt{x^2+9} - \frac{9}{2} \log_e |x + \sqrt{x^2+9}| + C$

(C) $\frac{x}{2}\sqrt{x^2+9} + \frac{9}{2} \sin^{-1}(x/3) + C$

(D) $\frac{x}{2}\sqrt{x^2+9} + \frac{3}{2} \log_e |x + \sqrt{x^2+9}| + C$

Question 34. Evaluate $\int \sqrt{x^2-16} dx$.

(A) $\frac{x}{2}\sqrt{x^2-16} + 8 \log_e |x + \sqrt{x^2-16}| + C$

(B) $\frac{x}{2}\sqrt{x^2-16} - 8 \log_e |x + \sqrt{x^2-16}| + C$

(C) $\frac{x}{2}\sqrt{x^2-16} - 16 \log_e |x + \sqrt{x^2-16}| + C$

(D) $\frac{x}{2}\sqrt{x^2-16} + 16 \log_e |x + \sqrt{x^2-16}| + C$

Question 35. Evaluate $\int (x+1)\sqrt{x^2+2x+3} dx$.

(A) $\frac{1}{3} (x^2+2x+3)^{3/2} + C$

(B) $\frac{2}{3} (x^2+2x+3)^{3/2} + C$

(C) $\frac{1}{2} (x^2+2x+3)^{1/2} + C$

(D) $\frac{(x+1)^2}{2} \sqrt{x^2+2x+3} + C$

Question 36. Evaluate $\int (2x+3)\sqrt{x^2+3x+1} dx$.

(A) $(x^2+3x+1)^{3/2} + C$

(B) $\frac{2}{3} (x^2+3x+1)^{3/2} + C$

(C) $\frac{3}{2} (x^2+3x+1)^{3/2} + C$

(D) $\frac{(2x+3)^2}{2} \sqrt{x^2+3x+1} + C$

Question 37. Evaluate $\int \frac{2x+5}{\sqrt{x^2+5x+6}} dx$.

(A) $\sqrt{x^2+5x+6} + C$

(B) $2\sqrt{x^2+5x+6} + C$

(C) $\log_e |\sqrt{x^2+5x+6}| + C$

(D) $\frac{1}{2\sqrt{x^2+5x+6}} + C$

Question 38. Evaluate $\int \frac{dx}{x(x^n+1)}$.

(A) $\frac{1}{n} \log_e |\frac{x^n}{x^n+1}| + C$

(B) $\log_e |\frac{x^n}{x^n+1}| + C$

(C) $\frac{1}{n} \log_e |x^n+1| + C$

(D) $\log_e |x(x^n+1)| + C$

Question 39. Evaluate $\int \frac{dx}{1+3e^x}$.

(A) $x - \frac{1}{3} \log_e |1+3e^x| + C$

(B) $x - \log_e |1+3e^x| + C$

(C) $\log_e |1+3e^x| + C$

(D) $\frac{1}{3} \log_e |1+3e^x| + C$

Question 40. Evaluate $\int \frac{x^2}{x^2+1} dx$.

(A) $x - \tan^{-1} x + C$

(B) $x + \tan^{-1} x + C$

(C) $\frac{x^3}{3(x^2+1)} + C$

(D) $\frac{1}{2} \log_e |x^2+1| + C$

Question 41. Evaluate $\int \frac{1}{x(x^2+1)} dx$.

(A) $\log_e |x| - \frac{1}{2} \log_e |x^2+1| + C$

(B) $\frac{1}{2} \log_e |x^2+1| - \log_e |x| + C$

(C) $\log_e |x| + \log_e |x^2+1| + C$

(D) $\log_e |\frac{x}{x^2+1}| + C$

Question 1. The definite integral $\int_a^b f(x) dx$ is defined as the limit of a sum given by:

(A) $\lim\limits_{n \to \infty} \frac{b-a}{n} \sum\limits_{i=1}^n f(a + i \frac{b-a}{n})$

(B) $\lim\limits_{n \to \infty} \frac{b-a}{n} \sum\limits_{i=0}^n f(a + i \frac{b-a}{n})$

(C) $\lim\limits_{n \to \infty} \frac{b-a}{n} \sum\limits_{i=0}^{n-1} f(a + i \frac{b-a}{n})$

(D) $\lim\limits_{n \to \infty} \sum\limits_{i=0}^{n-1} f(a + i \frac{b-a}{n})$

Question 2. The First Fundamental Theorem of Calculus states that if $F(x) = \int_a^x f(t) dt$, where $f$ is continuous on $[a, b]$, then $F'(x) =$

(A) $f(x)$

(B) $f(x) - f(a)$

(C) $\int_a^x f'(t) dt$

(D) $f(t)$

Question 3. The Second Fundamental Theorem of Calculus (Evaluation Theorem) states that if $F$ is an antiderivative of a continuous function $f$ on $[a, b]$, then $\int_a^b f(x) dx =$

(A) $F(b) + F(a)$

(B) $F'(b) - F'(a)$

(C) $F(b) - F(a)$

(D) $f(b) - f(a)$

Question 4. Evaluate $\int_1^2 x^2 dx$ using the Fundamental Theorem.

(A) $7/3$

(B) $8/3$

(C) $3$

(D) $7$

Question 5. Evaluate $\int_0^{\pi/2} \sin x dx$.

(A) 0

(B) 1

(C) -1

(D) $\pi/2$

Question 6. Evaluate $\int_1^e \frac{1}{x} dx$.

(A) 0

(B) 1

(C) $e$

(D) $1/e$

Question 7. What does $\int_a^b f(x) dx$ represent geometrically if $f(x) \geq 0$ on $[a, b]$?

(A) The length of the curve.

(B) The volume under the curve.

(C) The area bounded by the curve $y=f(x)$, the x-axis, and the lines $x=a, x=b$.

(D) The slope of the curve.

Question 8. Evaluate $\int_0^1 e^x dx$.

(A) $e$

(B) $e-1$

(C) 1

(D) $e+1$

Question 9. Evaluate $\int_{-1}^1 x^3 dx$.

(A) 0

(B) 1/4

(C) 1/2

(D) 1

Question 10. If $\int_a^b f(x) dx = 0$ for $a \neq b$, which statement is necessarily true?

(A) $f(x) = 0$ for all $x \in [a, b]$.

(B) $f(x)$ must change sign in $[a, b]$ if it is continuous and not identically zero.

(C) $f(a) = f(b)$.

(D) The maximum value of $f(x)$ is 0.

Question 11. Find the area under the curve $y = x$ from $x=0$ to $x=4$.

(A) 4 square units

(B) 8 square units

(C) 16 square units

(D) 2 square units

Question 12. For the limit of a sum $\lim\limits_{n \to \infty} \frac{b-a}{n} \sum\limits_{i=1}^n f(a + i h)$, what does $h$ represent?

(A) The number of subintervals.

(B) The height of each rectangle.

(C) The width of each subinterval.

(D) The number of points.

Question 13. Evaluate $\int_0^2 (x+1) dx$ as the limit of a sum.

(A) 2

(B) 3

(C) 4

(D) 5

Question 14. Evaluate $\int_1^2 \frac{1}{x^2} dx$.

(A) 1/2

(B) 1/4

(C) -1/2

(D) 1

Question 15. If $F(x)$ is an antiderivative of $f(x)$, and $G(x)$ is another antiderivative, then $\int_a^b f(x) dx$ is equal to:

(A) $G(b) - G(a)$

(B) $F(b) - F(a)$

(C) Both (A) and (B)

(D) $F(b) - G(a)$

Question 16. Find the area under the curve $y = e^x$ from $x=0$ to $x=1$.

(A) 1

(B) $e$

(C) $e-1$

(D) $e+1$

Question 17. Evaluate $\int_0^{\pi} \cos x dx$.

(A) 0

(B) 1

(C) 2

(D) $\pi$

Question 18. If $\int_1^k \frac{1}{x} dx = \log_e 5$, find the value of $k$.

(A) 1

(B) $e$

(C) 5

(D) $1/5$

Question 19. Find the area bounded by $y = x^2$, the x-axis, and the lines $x=0, x=2$.

(A) 2/3 square units

(B) 4/3 square units

(C) 8/3 square units

(D) 1 square unit

Question 20. What is the value of $\int_a^a f(x) dx$?

(A) $f(a)$

(B) 1

(C) 0

(D) Undefined

Question 21. Evaluate $\int_0^1 (x^2+x) dx$.

(A) 1/3

(B) 1/2

(C) 5/6

(D) 7/6

Question 22. Evaluate $\int_{-2}^2 |x| dx$.

(A) 0

(B) 2

(C) 4

(D) 8

Question 23. Evaluate $\int_0^1 \frac{1}{1+x^2} dx$.

(A) $\pi/4$

(B) $\pi/2$

(C) $\pi$

(D) 0

Question 24. If $F(x) = \int_a^x f(t) dt$, then $F'(c)$ for $a < c < b$ is given by:

(A) $\int_a^c f(t) dt$

(B) $f(c)$

(C) $f(c) - f(a)$

(D) $F(c)$

Question 25. If $f(x)$ takes both positive and negative values on $[a, b]$, $\int_a^b f(x) dx$ represents:

(A) The total area bounded by the curve, above and below the x-axis.

(B) The net signed area (area above x-axis minus area below x-axis).

(C) The area above the x-axis only.

(D) The area below the x-axis only.

Question 1. Evaluate $\int_0^1 \frac{2x}{1+x^2} dx$ using substitution.

(A) $\log_e 2$

(B) $\log_e 1$

(C) $\tan^{-1} 1 - \tan^{-1} 0$

(D) 1

Question 2. Evaluate $\int_0^{\pi/2} \sin^2 x dx$.

(A) $\pi/4$

(B) $\pi/2$

(C) 0

(D) 1

Question 3. Evaluate $\int_0^{\pi/2} \cos^2 x dx$.

(A) $\pi/4$

(B) $\pi/2$

(C) 0

(D) 1

Question 4. The property $\int_a^b f(x) dx = \int_a^b f(t) dt$ means:

(A) The value of the definite integral depends only on the function and the limits, not the variable of integration.

(B) The variable of integration can be changed to any other variable.

(C) $x$ and $t$ represent the same value.

(D) The area under the curve is independent of the chosen variable.

Question 5. The property $\int_a^b f(x) dx = -\int_b^a f(x) dx$ implies that:

(A) Reversing the limits of integration changes the sign of the integral.

(B) The function must be odd.

(C) The integral is always negative.

(D) The integral is always zero.

Question 6. The property $\int_a^c f(x) dx + \int_c^b f(x) dx = \int_a^b f(x) dx$ is valid if:

(A) $a < c < b$ and $f(x)$ is continuous on $[a, b]$.

(B) $c$ is any real number.

(C) $f(x)$ is differentiable on $[a, b]$.

(D) $f(x)$ is a polynomial.

Question 7. Evaluate $\int_{-2}^2 |x-1| dx$.

(A) 5

(B) 2

(C) 4

(D) 8

Question 8. Evaluate $\int_0^2 x \sqrt{x^2+1} dx$ using substitution.

(A) $\frac{1}{3} (5\sqrt{5} - 1)$

(B) $\frac{2}{3} (5\sqrt{5} - 1)$

(C) $\frac{1}{2} (5\sqrt{5} - 1)$

(D) $\frac{1}{3} (5^{3/2})$

Question 9. Evaluate $\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$ using the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$.

(A) 0

(B) $\pi/4$

(C) $\pi/2$

(D) $\pi/8$

Question 10. Evaluate $\int_{-\pi/2}^{\pi/2} \sin^2 x dx$.

(A) $\pi/4$

(B) $\pi/2$

(C) $\pi$

(D) 0

Question 11. Evaluate $\int_{-1}^1 x \cos x dx$.

(A) 0

(B) 1

(C) $\sin 1 - \cos 1$

(D) $2(\sin 1 - \cos 1)$

Question 12. Evaluate $\int_0^1 x e^{x^2} dx$ using substitution.

(A) $e - 1$

(B) $\frac{1}{2} (e - 1)$

(C) $e$

(D) $e/2$

Question 13. Evaluate $\int_1^e \log_e x dx$.

(A) 1

(B) $e-1$

(C) $e$

(D) 0

Question 14. Evaluate $\int_0^{\pi/2} \sqrt{\sin x} \cos x dx$ using substitution.

(A) 1

(B) $2/3$

(C) $3/2$

(D) 0

Question 15. Evaluate $\int_0^1 x(1-x)^n dx$.

(A) $\frac{1}{(n+1)(n+2)}$

(B) $\frac{1}{n+1}$

(C) $\frac{1}{n+2}$

(D) $\frac{n!}{(n+2)!}$

Question 16. Evaluate $\int_0^{\pi/2} \frac{\cos x}{1+\sin x} dx$ using substitution.

(A) $\log_e 2$

(B) $\log_e 1$

(C) 0

(D) 1

Question 17. Evaluate $\int_0^1 \sin^{-1} x dx$.

(A) $\pi/2 - 1$

(B) $\pi/2$

(C) 1

(D) $\pi/2 + 1$

Question 18. Evaluate $\int_0^{2\pi} |\sin x| dx$.

(A) 0

(B) 2

(C) 4

(D) $\pi$

Question 19. Evaluate $\int_0^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} dx$ using the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$.

(A) 0

(B) $\pi/4$

(C) $\pi/2$

(D) 1

Question 20. Evaluate $\int_0^1 x \sqrt{1-x^2} dx$ using substitution.

(A) 1

(B) 1/2

(C) 1/3

(D) 2/3

Question 21. Evaluate $\int_0^{\pi} \sin^3 x dx$.

(A) 0

(B) $2/3$

(C) $4/3$

(D) $\pi/2$

Question 22. Evaluate $\int_0^1 \frac{\tan^{-1} x}{1+x^2} dx$.

(A) $\pi/4$

(B) $(\pi/4)^2$

(C) $(\pi/4)^2 / 2$

(D) $\pi/2$

Question 23. Evaluate $\int_0^{\pi/2} \log_e(\tan x) dx$.

(A) $\pi/2 \log_e 1$

(B) $\pi/4 \log_e 2$

(C) 0

(D) $\log_e 1$

Question 24. If a continuous function $f(x)$ satisfies $\int_0^a f(x) dx = 2 \int_0^{a/2} f(x) dx$, it means:

(A) $f(x)$ is symmetric about $x=a/2$. (i.e. $f(x) = f(a-x)$)

(B) $f(x)$ is periodic with period $a/2$.

(C) $f(x)$ is an even function.

(D) $f(x) = f(x+a/2)$

Question 1. Find the area under the curve $y = x^2$ from $x=0$ to $x=2$.

(A) 4/3 square units

(B) 8/3 square units

(C) 2 square units

(D) 3 square units

Question 2. Find the area bounded by the curve $y = \sin x$, the x-axis, from $x=0$ to $x=\pi$.

(A) 1 square unit

(B) 2 square units

(C) $\pi$ square units

(D) 0 square units

Question 3. Find the area bounded by the parabola $y^2 = 4ax$ and the line $x=a$.

(A) $8a^2/3$ square units

(B) $4a^2/3$ square units

(C) $16a^2/3$ square units

(D) $a^2$ square units

Question 4. Find the area bounded by the curve $y = x^3$, the x-axis, and the lines $x=-1$ to $x=1$.

(A) 0 square units

(B) 1/4 square unit

(C) 1/2 square unit

(D) 1 square unit

Question 5. Find the area of the region bounded by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

(A) $\pi a^2$ square units

(B) $\pi b^2$ square units

(C) $\pi ab$ square units

(D) $a b$ square units

Question 6. Find the area between the curves $y = x^2$ and $y = x$.

(A) 1/6 square unit

(B) 1/3 square unit

(C) 1/2 square unit

(D) 1 square unit

Question 7. Find the area between the curves $y = x$ and $y = x^3$ in the first quadrant.

(A) 1/4 square unit

(B) 1/2 square unit

(C) 1/3 square unit

(D) 1 square unit

Question 8. Find the area bounded by the parabola $y = x^2$ and the line $y=4$.

(A) 8/3 square units

(B) 16/3 square units

(C) 32/3 square units

(D) 4 square units

Question 9. Find the area bounded by the curve $x^2 + y^2 = a^2$.

(A) $\pi a^2$ square units

(B) $2\pi a^2$ square units

(C) $a^2$ square units

(D) $\pi a^2/2$ square units

Question 10. Find the area of the region bounded by the curve $y = \sqrt{x}$, the x-axis, and the line $x=4$.

(A) 4/3 square units

(B) 8/3 square units

(C) 16/3 square units

(D) 2 square units

Question 11. Find the area between the curves $y = \sin x$ and $y = \cos x$ from $x=0$ to $x=\pi/2$.

(A) $\sqrt{2}-1$ square units

(B) $2(\sqrt{2}-1)$ square units

(C) $\sqrt{2}$ square units

(D) 1 square unit

Question 12. Find the area bounded by the curve $y = |x|$, the x-axis, from $x=-2$ to $x=2$.

(A) 2 square units

(B) 4 square units

(C) 8 square units

(D) 0 square units

Question 13. Find the area between the curves $y = 2x - x^2$ and $y = -x$.

(A) 9/2 square units

(B) 9 square units

(C) 27/2 square units

(D) 27 square units

Question 14. Find the area bounded by the curve $y = e^x$, the x-axis, and the lines $x=0, x=1$.

(A) $e$ square units

(B) $e-1$ square units

(C) 1 square unit

(D) $e+1$ square units

Question 15. Find the area bounded by the parabola $y = x^2$ and the line $y=2x$.

(A) 2/3 square units

(B) 4/3 square units

(C) 8/3 square units

(D) 1 square unit

Question 16. Find the area bounded by the lines $y=x$, $y=2x$ and $x=1$.

(A) 1/2 square unit

(B) 1 square unit

(C) 3/2 square units

(D) 2 square units

Question 17. Find the area of the region bounded by $y^2 = 4x$ and the line $x=1$.

(A) 8/3 square units

(B) 4/3 square units

(C) 2/3 square units

(D) 1 square unit

Question 18. Find the area bounded by the curve $y = \cos x$, the x-axis, from $x=0$ to $x=\pi$.

(A) 0 square units

(B) 1 square unit

(C) 2 square units

(D) $\pi$ square units

Question 19. Find the total area bounded by the curve $y = x(x-1)(x-2)$, the x-axis, from $x=0$ to $x=2$.

(A) 1/2 square unit

(B) 1 square unit

(C) 2 square units

(D) 4 square units

Question 20. In a business scenario, if the marginal cost function is $C'(x)$ and the marginal revenue function is $R'(x)$, the increase in profit from producing $x_1$ to $x_2$ units ($x_1 < x_2$) is given by:

(A) $\int_{x_1}^{x_2} (R'(x) + C'(x)) dx$

(B) $\int_{x_1}^{x_2} (R'(x) - C'(x)) dx$

(C) $\int_{x_1}^{x_2} R'(x) dx$

(D) $\int_{x_1}^{x_2} C'(x) dx$

Question 21. The total production from time $t_1$ to $t_2$ given a rate of production $r(t)$ (where $r(t)$ is the rate of change of total production with respect to time) is:

(A) $r(t_2) - r(t_1)$

(B) $r'(t_2) - r'(t_1)$

(C) $\int_{t_1}^{t_2} r(t) dt$

(D) $r(t_1) + r(t_2)$

Question 22. If the velocity of an object moving along a straight line is $v(t)$, the displacement from time $t_1$ to $t_2$ is given by:

(A) $\int_{t_1}^{t_2} v(t) dt$

(B) $\int_{t_1}^{t_2} |v(t)| dt$

(C) $v(t_2) - v(t_1)$

(D) $v'(t_2) - v'(t_1)$

Question 23. 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:

(A) $\int_a^b f(x) dx - \int_a^b g(x) dx$

(B) $\int_a^b (g(x) - f(x)) dx$

(C) $\int_a^b f(x) g(x) dx$

(D) $\int_a^b (f(x) + g(x)) dx$

Question 1. A differential equation is an equation involving:

(A) Variables and constants.

(B) Functions and their derivatives.

(C) Algebraic expressions.

(D) Inequalities.

Question 2. What is the order of the differential equation $\frac{d^2 y}{dx^2} + (\frac{dy}{dx})^3 + y = 0$?

(A) 1

(B) 2

(C) 3

(D) 5

Question 3. What is the degree of the differential equation $\frac{d^2 y}{dx^2} + (\frac{dy}{dx})^3 + y = 0$?

(A) 1

(B) 2

(C) 3

(D) 5

Question 4. What is the order of the differential equation $y' + y'' + (y''')^2 = 0$?

(A) 1

(B) 2

(C) 3

(D) 2

Question 5. What is the degree of the differential equation $y' + y'' + (y''')^2 = 0$?

(A) 1

(B) 2

(C) 3

(D) Undefined

Question 6. What is the order of the differential equation $\frac{d^3 y}{dx^3} + x^2 (\frac{d^2 y}{dx^2})^5 = 0$?

(A) 1

(B) 2

(C) 3

(D) 5

Question 7. What is the degree of the differential equation $\frac{d^3 y}{dx^3} + x^2 (\frac{d^2 y}{dx^2})^5 = 0$?

(A) 1

(B) 2

(C) 3

(D) 5

Question 8. A solution that contains arbitrary constants equal to the order of the differential equation is called a:

(A) Particular Solution

(B) General Solution

(C) Singular Solution

(D) Trivial Solution

Question 9. A solution obtained from the general solution by giving particular values to the arbitrary constants is called a:

(A) Particular Solution

(B) General Solution

(C) Singular Solution

(D) Complete Solution

Question 10. Form the differential equation of the family of curves $y = ax$, where $a$ is an arbitrary constant.

(A) $\frac{dy}{dx} = a$

(B) $y = x \frac{dy}{dx}$

(C) $x = y \frac{dy}{dx}$

(D) $\frac{dy}{dx} = y$

Question 11. Form the differential equation of the family of circles $x^2 + y^2 = a^2$, where $a$ is an arbitrary constant.

(A) $x^2 + y^2 = (\frac{dy}{dx})^2$

(B) $x + y \frac{dy}{dx} = 0$

(C) $2x + 2y \frac{dy}{dx} = 0$

(D) $x \frac{dy}{dx} - y = 0$

Question 12. The equation $y = c_1 e^x + c_2 e^{-x}$ is the general solution of which differential equation?

(A) $y'' - y = 0$

(B) $y'' + y = 0$

(C) $y' - y = 0$

(D) $y' + y = 0$

Question 13. A solution that cannot be obtained from the general solution by assigning values to the arbitrary constants is called a:

(A) Particular Solution

(B) General Solution

(C) Singular Solution

(D) Complete Solution

Question 14. What is the order of the differential equation $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2 y}{dx^2}$?

(A) 1

(B) 2

(C) 3

(D) 1

Question 15. What is the degree of the differential equation $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2 y}{dx^2}$?

(A) 1

(B) 2

(C) 3

(D) Undefined

Question 16. Form the differential equation of the family of lines $y = mx + c$, where $m$ and $c$ are arbitrary constants.

(A) $\frac{dy}{dx} = m$

(B) $\frac{d^2 y}{dx^2} = 0$

(C) $y = x \frac{dy}{dx} + c$

(D) $y = mx + \frac{dy}{dx}$

Question 17. Form the differential equation of the family of parabolas $y^2 = 4ax$, where $a$ is an arbitrary constant.

(A) $2y \frac{dy}{dx} = 4a$

(B) $y = 2x \frac{dy}{dx}$

(C) $y \frac{dy}{dx} = 2ax$

(D) $y \frac{d^2 y}{dx^2} + (\frac{dy}{dx})^2 = 0$

Question 18. Which of the following is a solution to the differential equation $y' = y$?

(A) $y = x$

(B) $y = e^x$

(C) $y = \sin x$

(D) $y = x^2$

Question 19. What does the order of a differential equation indicate?

(A) The power of the highest derivative.

(B) The number of arbitrary constants in the general solution.

(C) The number of independent variables.

(D) The type of functions involved.

Question 20. What does the degree of a differential equation indicate?

(A) The order of the highest derivative.

(B) The number of terms in the equation.

(C) The highest power of the highest order derivative after the equation is made free from radicals and fractions of derivatives.

(D) The number of variables.

Question 21. Form the differential equation of the family of straight lines passing through the origin.

(A) $y = mx$

(B) $y = x \frac{dy}{dx}$

(C) $\frac{dy}{dx} = y$

(D) $\frac{dy}{dx} = x$

Question 22. Form the differential equation of the family of circles having their centres on the x-axis and radius $a$.

(A) $y^2 + (\frac{dy}{dx})^2 = a^2$

(B) $y \frac{dy}{dx} + x - c = 0$

(C) $y y'' + (\frac{dy}{dx})^2 + 1 = 0$

(D) $(x-c)^2 + y^2 = a^2$

Question 23. In Applied Maths, a differential equation often represents:

(A) An algebraic relationship between variables.

(B) A relationship between a quantity and its rate of change.

(C) A constant value.

(D) A geometric shape.

Question 24. A differential equation of the form $\frac{dy}{dx} = f(x)g(y)$ is a:

(A) Linear differential equation.

(B) Homogeneous differential equation.

(C) Variable separable differential equation.

(D) Exact differential equation.

Question 25. The general solution of a differential equation of order $n$ typically contains:

(A) $n$ specific constants.

(B) $n$ arbitrary constants.

(C) $n+1$ arbitrary constants.

(D) No arbitrary constants.

Question 26. The order of the differential equation $\left(\frac{d^2 y}{dx^2}\right)^2 + x \frac{dy}{dx} + y = \cos x$ is:

(A) 1

(B) 2

(C) 3

(D) 5

Question 27. The degree of the differential equation $\left(\frac{d^2 y}{dx^2}\right)^2 + x \frac{dy}{dx} + y = \cos x$ is:

(A) 1

(B) 2

(C) 3

(D) 5

Question 1. The general approach to solving a first-order differential equation involves:

(A) Reducing it to an algebraic equation.

(B) Integrating the equation to find the dependent variable as a function of the independent variable.

(C) Differentiating the equation to simplify it.

(D) Finding its order and degree.

Question 2. Solve the differential equation $\frac{dy}{dx} = e^{x+y}$.

(A) $e^x + e^{-y} = C$

(B) $e^{-x} + e^{-y} = C$

(C) $e^x + e^y = C$

(D) $e^{-x} + e^y = C$

Question 3. Solve the differential equation $\frac{dy}{dx} = \frac{x}{y}$.

(A) $x^2 - y^2 = C$

(B) $x^2 + y^2 = C$

(C) $y = cx^2$

(D) $y = cx$

Question 4. A first-order differential equation is homogeneous if it can be written in the form $\frac{dy}{dx} = f(\frac{y}{x})$. What substitution is typically used to solve such equations?

(A) $y = vx$

(B) $x = vy$

(C) $y = v+x$

(D) $v = xy$

Question 5. Solve the differential equation $\frac{dy}{dx} = \frac{x+y}{x}$.

(A) $y = x \log_e |x| + Cx$

(B) $y = x \log_e |x| + C$

(C) $y = \log_e |x| + Cx$

(D) $y = \log_e |x| + C$

Question 6. Solve the differential equation $\frac{dy}{dx} = y$.

(A) $y = x + C$

(B) $y = Cx$

(C) $y = C e^x$

(D) $y = C \log_e x$

Question 7. Solve the differential equation $(1+x^2) dy - (1+y^2) dx = 0$.

(A) $\tan^{-1} y - \tan^{-1} x = C$

(B) $\tan^{-1} x - \tan^{-1} y = C$

(C) $\log_e |1+x^2| - \log_e |1+y^2| = C$

(D) $\log_e |1+x^2| + \log_e |1+y^2| = C$

Question 8. Solve the differential equation $\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$.

(A) $y^2 = x^2 (\log_e x^2 + C)$

(B) $y^2 = x^2 (2 \log_e x + C)$

(C) $y = x (2 \log_e x + C)$

(D) $\log_e y = \log_e x + C$

Question 9. Solve the differential equation $\frac{dy}{dx} = \frac{y}{x} + \sin(\frac{y}{x})$.

(A) $\tan(y/x) = \log_e |Cx|$

(B) $\tan(y/2x) = Cx$

(C) $\tan(y/2x) = \log_e |Cx|$

(D) $\cot(y/2x) = Cx$

Question 10. The substitution $y = vx$ in a homogeneous differential equation $\frac{dy}{dx} = f(\frac{y}{x})$ transforms it into a variable separable equation in terms of which variables?

(A) $x$ and $y$ only.

(B) $x$ and $v$ only.

(C) $y$ and $v$ only.

(D) $x$, $y$, and $v$.

Question 11. Solve the differential equation $\frac{dy}{dx} = 1 + x + y + xy$.

(A) $\log_e |1+y| = x + x^2/2 + C$

(B) $\log_e |1+y| = \log_e |1+x| + C$

(C) $\log_e |1+y| = x + C$

(D) $\log_e |1+y| = \log_e |1+x| + x + C$

Question 12. Solve the differential equation $\frac{dy}{dx} = \frac{x-y+1}{x-y-1}$.

(A) $(x-y) + \log_e |x-y| = 2x + C$

(B) $(x-y) - \log_e |x-y| = 2y + C$

(C) $(x-y) + \log_e |x-y-1| = 2x + C$

(D) $(x-y) - \log_e |x-y-1| = 2y + C$

Question 13. To solve the equation $\frac{dy}{dx} = \frac{ax+by+c}{Ax+By+C}$, where $aB \neq Ab$, what substitution is used?

(A) $y = vx$

(B) $v = ax+by$

(C) $x = X+h, y = Y+k$, where $(h, k)$ is the intersection of $ax+by+c=0$ and $Ax+By+C=0$.

(D) $v = y/x$

Question 14. Solve the differential equation $\frac{dy}{dx} = \sqrt{\frac{1-y^2}{1-x^2}}$.

(A) $\sin^{-1} y = \sin^{-1} x + C$

(B) $\sin^{-1} x + \sin^{-1} y = C$

(C) $\sqrt{1-y^2} = \sqrt{1-x^2} + C$

(D) $\cos^{-1} y = \cos^{-1} x + C$

Question 15. Solve the differential equation $y dx - x dy = 0$.

(A) $y = x + C$

(B) $y = Cx$

(C) $\log_e |y| - \log_e |x| = C$

(D) Both (B) and (C)

Question 16. Solve the differential equation $\frac{dy}{dx} = \frac{x+2y-3}{2x+4y-6}$.

(A) $x+2y = C$

(B) $x+2y - 3 = C e^{2x}$

(C) $\frac{1}{2}(x+2y) - \frac{3}{2} \log_e |x+2y-3| = x + C$

(D) $(x+2y)^2 = C x$

Question 17. The variable separable method is applicable when the first-order differential equation can be written as:

(A) $g(y) dy = f(x) dx$

(B) $\frac{dy}{dx} = f(x) + g(y)$

(C) $\frac{dy}{dx} = f(x)/g(y)$

(D) Both (A) and (C) after rearrangement.

Question 18. Solve the differential equation $x \frac{dy}{dx} = y \log_e (\frac{y}{x})$.

(A) $\log_e |\log_e (y/x)| = \log_e |Cx|$

(B) $\log_e |\log_e (y/x)| = Cx$

(C) $\log_e (y/x) = C/x$

(D) $\log_e (y/x) = Cx$

Question 19. Solve the differential equation $(x+y)^2 \frac{dy}{dx} = a^2$ using substitution $v = x+y$.

(A) $v + \log_e |v-a| - \log_e |v+a| = x + C$

(B) $v - \frac{a}{2} \log_e |\frac{v-a}{v+a}| = x + C$

(C) $v + \frac{a}{2} \log_e |\frac{v-a}{v+a}| = x + C$

(D) $v + a \log_e |\frac{v-a}{v+a}| = x + C$

Question 20. To solve $\frac{dy}{dx} = \frac{ax+by+c}{Ax+By+C}$ when $\frac{a}{A} = \frac{b}{B} = k$, what substitution is typically used?

(A) $y = vx$

(B) $v = ax+by$ or $v = Ax+By$

(C) $x=X+h, y=Y+k$

(D) $v = y/x$

Question 21. Solve the differential equation $\frac{dy}{dx} = \frac{x-y+5}{x-y-7}$.

(A) $x-y + \log_e |x-y-7| = 2x + C$

(B) $x-y - \log_e |x-y-7| = 2y + C$

(C) $x-y + \log_e |x-y-7| = -12 y + C$

(D) $x-y - 6 \log_e |x-y-7| = -12 y + C$

Question 22. Solve the differential equation $\frac{dy}{dx} = \cos(x+y)$.

(A) $\tan(\frac{x+y}{2}) = x + C$

(B) $\tan(\frac{x+y}{2}) = y + C$

(C) $\tan(x+y) = x + C$

(D) $\tan(x+y) = y + C$

Question 23. Solve the differential equation $e^x \tan y dx + (1-e^x) \sec^2 y dy = 0$.

(A) $\tan y = C (1-e^x)$

(B) $\tan y = C e^x$

(C) $\log_e |\tan y| = \log_e |1-e^x| + C$

(D) $\log_e |e^x| + \log_e |\tan y| = C$

Question 24. Solve the differential equation $(x^2 - y^2) dx + 2xy dy = 0$.

(A) $x^2 + y^2 = Cx$

(B) $x^2 + y^2 = Cy$

(C) $x^2 + y^2 = C$

(D) $x + y^2 = C x^2$

Question 25. A differential equation $\frac{dy}{dx} = f(x,y)$ is homogeneous if $f(tx, ty) = f(x,y)$ for some $t$. This means $f(x,y)$ is a homogeneous function of degree:

(A) 0

(B) 1

(C) $t$

(D) Any real number

Question 26. The general solution of $\frac{dy}{dx} = \frac{y}{x}$ is:

(A) $y = cx$

(B) $y = c/x$

(C) $y = c \log_e |x|$

(D) $y = c e^x$

Question 27. Solve the differential equation $\frac{dy}{dx} = \frac{y}{x} + 1$.

(A) $y = x \log_e |x| + Cx$

(B) $y = x \log_e |x| + C$

(C) $y = \log_e |x| + C$

(D) $y = x(\log_e |x| + C)$

Question 28. Solve the differential equation $\sec^2 x \tan y dx + \sec^2 y \tan x dy = 0$.

(A) $\tan x + \tan y = C$

(B) $\tan x \cdot \tan y = C$

(C) $\log_e |\tan x| + \log_e |\tan y| = C$

(D) $\log_e |\sec x| + \log_e |\sec y| = C$

Question 29. Solve the differential equation $(x+y+1)^2 \frac{dy}{dx} = 1$ using substitution $v = x+y+1$.

(A) $x+y+1 = \tan(x+C)$

(B) $x+y+1 = \tan(y+C)$

(C) $\tan(x+y+1) = x+C$

(D) $\tan(x+y+1) = y+C$

Question 30. The form $\frac{dy}{dx} = \frac{ax+by+c}{Ax+By+C}$ where $aB \neq Ab$ can be reduced to homogeneous form by shifting the origin using $x=X+h, y=Y+k$. The new origin $(h, k)$ is found by solving:

(A) The system of algebraic equations $ax+by=0$ and $Ax+By=0$.

(B) The system of algebraic equations $ax+by+c=0$ and $Ax+By+C=0$.

(C) $h$ and $k$ from the differential equation itself.

(D) $h=k=0$ always works.

Question 1. The standard form of a first-order linear differential equation in $y$ is:

(A) $\frac{dy}{dx} + P(x)y = Q(x)$

(B) $\frac{dy}{dx} = f(x)g(y)$

(C) $M(x,y) dx + N(x,y) dy = 0$

(D) $\frac{dy}{dx} = f(\frac{y}{x})$

Question 2. The integrating factor (IF) for the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is:

(A) $e^{\int Q(x) dx}$

(B) $e^{\int P(x) dx}$

(C) $\int P(x) dx$

(D) $\int Q(x) dx$

Question 3. The general solution of the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is given by:

(A) $y \cdot (\text{IF}) = \int Q(x) dx + C$

(B) $y \cdot (\text{IF}) = \int Q(x) \cdot (\text{IF}) dx + C$

(C) $y = \int Q(x) \cdot (\text{IF}) dx + C$

(D) $x \cdot (\text{IF}) = \int Q(y) \cdot (\text{IF}) dy + C$

Question 4. Find the integrating factor for the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^2$.

(A) $x$

(B) $e^x$

(C) $\log_e x$

(D) $1/x$

Question 5. Solve the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^2$.

(A) $xy = \frac{x^4}{4} + C$

(B) $y/x = \frac{x^3}{3} + C$

(C) $y = x^3 + C$

(D) $xy = \frac{x^3}{3} + C$

Question 6. Find the integrating factor for the differential equation $x \frac{dy}{dx} + 2y = x^2 \log_e x$.

(A) $x$

(B) $x^2$

(C) $e^x$

(D) $\log_e x$

Question 7. Solve the differential equation $x \frac{dy}{dx} + 2y = x^2 \log_e x$.

(A) $yx^2 = \int x^2 \log_e x dx + C$

(B) $yx^2 = \int x^3 \log_e x dx + C$

(C) $y/x^2 = \int \log_e x dx + C$

(D) $yx^2 = \int x^2 \log_e x \cdot x^2 dx + C$

Question 8. A first-order linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$ is linear in which variable(s)?

(A) Only $x$

(B) Only $y$

(C) $x$ and $\frac{dy}{dx}$

(D) $y$ and $\frac{dy}{dx}$

Question 9. The equation $\frac{dx}{dy} + P(y)x = Q(y)$ is a linear differential equation in which variable?

(A) $x$

(B) $y$

(C) $x$ and $y$ are both dependent variables.

(D) $\frac{dx}{dy}$ is the independent variable.

Question 10. Find the integrating factor for $\frac{dx}{dy} + \frac{x}{y} = y^2$.

(A) $y$

(B) $e^y$

(C) $\log_e y$

(D) $1/y$

Question 11. Solve the differential equation $\frac{dy}{dx} + y \cot x = 2x \cot x$.

(A) $y \sin x = x^2 \cot x + C$

(B) $y \sin x = x^2 \sin x + C$

(C) $y \sin x = \int 2x \cos x dx + C$

(D) $y \cot x = 2x \cot x + C$

Question 12. Find the integrating factor for $(x+2y^3) \frac{dy}{dx} = y$.

(A) $e^{\int 1/y dy} = y$

(B) $e^{\int -1/y dy} = 1/y$

(C) $e^{\int 2y^2 dy}$

(D) $e^{\int 1/(x+2y^3) dx}$

Question 13. Solve the differential equation $(x+2y^3) \frac{dy}{dx} = y$.

(A) $x \cdot (1/y) = \int 2y^2 (1/y) dy + C$

(B) $y \cdot x = \int y \cdot y^2 dy + C$

(C) $x/y = y^2 + C$

(D) $x/y = \int 2y^2 dy + C$

Question 14. The general solution of $\frac{dy}{dx} + y = 1$ is:

(A) $y = 1 + C e^{-x}$

(B) $y = 1 + C e^x$

(C) $y = C e^{-x}$

(D) $y = C e^{-x} + x$

Question 15. Find the integrating factor for $\frac{dy}{dx} - y = \cos x$.

(A) $e^{\int -1 dx} = e^{-x}$

(B) $e^{\int 1 dx} = e^x$

(C) $e^{\int \cos x dx} = e^{\sin x}$

(D) $e^{\int -\cos x dx} = e^{-\sin x}$

Question 16. Solve the differential equation $\frac{dy}{dx} - y = \cos x$.

(A) $y e^{-x} = \int e^{-x} \cos x dx + C$

(B) $y e^{-x} = \sin x + C$

(C) $y e^x = \int e^x \cos x dx + C$

(D) $y = e^x \int e^{-x} \cos x dx + C$

Question 17. Solve the differential equation $\sin x \frac{dy}{dx} + y \cos x = \sin x \cos x$.

(A) $y \sin x = \frac{\sin^2 x}{2} + C$

(B) $y \cos x = \frac{\sin^2 x}{2} + C$

(C) $y \sin x = \frac{\cos^2 x}{2} + C$

(D) $y \cos x = \frac{\cos^2 x}{2} + C$

Question 18. Find the integrating factor for $\frac{dy}{dx} + y \sec x = \tan x$.

(A) $|\sec x + \tan x|$

(B) $|\sec x|$

(C) $e^{\sec^2 x}$

(D) $e^{\tan x}$

Question 19. Solve the differential equation $\frac{dy}{dx} + y \sec x = \tan x$.

(A) $y |\sec x + \tan x| = \sec x + C$

(B) $y |\sec x + \tan x| = \tan x + C$

(C) $y |\sec x + \tan x| = \int \tan x |\sec x + \tan x| dx + C$

(D) $y |\sec x| = \tan x + C$

Question 20. Which of the following is a linear differential equation of the first order?

(A) $(\frac{dy}{dx})^2 + y = x$

(B) $y \frac{dy}{dx} + x = 0$

(C) $\frac{dy}{dx} + xy = \sin x$

(D) $\frac{dy}{dx} = \frac{x+y}{x-y}$ (Homogeneous)

Question 21. Find the integrating factor for $\frac{dy}{dx} + \frac{y}{1+x^2} = \frac{e^{\tan^{-1} x}}{1+x^2}$.

(A) $e^{\tan^{-1} x}$

(B) $\tan^{-1} x$

(C) $e^{1+x^2}$

(D) $\frac{1}{1+x^2}$

Question 22. Solve the differential equation $\frac{dy}{dx} + \frac{y}{1+x^2} = \frac{e^{\tan^{-1} x}}{1+x^2}$.

(A) $y e^{\tan^{-1} x} = \int \frac{e^{\tan^{-1} x}}{1+x^2} \cdot e^{\tan^{-1} x} dx + C$

(B) $y e^{\tan^{-1} x} = \int \frac{e^{2 \tan^{-1} x}}{1+x^2} dx + C$

(C) $y e^{\tan^{-1} x} = \int \frac{1}{1+x^2} dx + C$

(D) $y e^{\tan^{-1} x} = \frac{1}{2} e^{2 \tan^{-1} x} + C$

Question 23. The equation $\frac{dy}{dx} + y^2 = x$ is:

(A) Linear

(B) Non-linear

(C) Homogeneous

(D) Variable Separable

Question 24. Find the integrating factor for $\frac{dy}{dx} + \frac{1}{x \log_e x} y = \frac{1}{x}$.

(A) $x$

(B) $\log_e x$

(C) $e^{\log_e x}$

(D) $e^x$

Question 25. Solve the differential equation $\frac{dy}{dx} + \frac{1}{x \log_e x} y = \frac{1}{x}$.

(A) $y \log_e x = \log_e |\log_e x| + C$

(B) $y \log_e x = \int \frac{1}{x} \log_e x dx + C$

(C) $y x \log_e x = \int 1 dx + C$

(D) $y \log_e x = \frac{(\log_e x)^2}{2} + C$

Question 1. The rate of decay of a radioactive substance is proportional to the amount of the substance present at that time $t$. If $A(t)$ is the amount present, this can be modeled by the differential equation:

(A) $\frac{dA}{dt} = k A$

(B) $\frac{dA}{dt} = -k A$

(C) $\frac{dA}{dt} = k t$

(D) $\frac{dA}{dt} = -k t$

Question 2. The population growth in a certain region is proportional to the population at any time $t$. If $P(t)$ is the population, the differential equation is:

(A) $\frac{dP}{dt} = k t$

(B) $\frac{dP}{dt} = k P$

(C) $\frac{dP}{dt} = k/P$

(D) $\frac{dP}{dt} = k/t$

Question 3. A bank account earns interest at a rate $r$ compounded continuously. If $A(t)$ is the amount in the account at time $t$, the differential equation describing this is:

(A) $\frac{dA}{dt} = r$

(B) $\frac{dA}{dt} = r t$

(C) $\frac{dA}{dt} = r A$

(D) $\frac{dA}{dt} = A/r$

Question 4. If a substance cools at a rate proportional to the difference between its temperature $T(t)$ and the ambient temperature $T_a$, this is described by: (Newton's Law of Cooling)

(A) $\frac{dT}{dt} = k (T_a - T)$

(B) $\frac{dT}{dt} = k (T - T_a)$

(C) $\frac{dT}{dt} = k T$

(D) $\frac{dT}{dt} = k T_a$

Question 5. The velocity $v$ of a falling object with air resistance proportional to velocity is modeled by $\frac{dv}{dt} = g - kv$. This is a:

(A) Variable separable DE

(B) Homogeneous DE

(C) Linear DE

(D) Non-linear DE

Question 6. If the growth rate of a population $P$ is given by $\frac{dP}{dt} = kP(M-P)$, this is known as a:

(A) Linear DE

(B) Logistic DE

(C) Homogeneous DE

(D) First order linear DE

Question 7. Solve the population growth differential equation $\frac{dP}{dt} = k P$ with initial population $P(0) = P_0$.

(A) $P(t) = P_0 + kt$

(B) $P(t) = P_0 e^{kt}$

(C) $P(t) = k P_0 t$

(D) $P(t) = P_0 / k$

Question 8. If the rate of decay of a substance is modeled by $\frac{dA}{dt} = -0.05 A$, where $t$ is in years, what is the half-life of the substance?

(A) $\log_e(2) / 0.05$ years

(B) $0.05 / \log_e(2)$ years

(C) $1/0.05$ years

(D) $2 \cdot 0.05$ years

Question 9. The current $I(t)$ in an electrical circuit with resistance $R$ and inductance $L$, driven by a voltage $E(t)$, is described by $L \frac{dI}{dt} + RI = E(t)$. If $R, L$ are constants and $E(t)$ is a function of $t$, this is a:

(A) Variable separable DE

(B) Homogeneous DE

(C) Linear DE

(D) Non-linear DE

Question 10. If a differential equation is used to model the spread of a disease, and $\frac{dS}{dt} = -k S I$, where $S$ is susceptible population and $I$ is infected population, $S$ and $I$ are functions of time. What do the derivatives typically represent? (Applied Maths)

(A) The total number of people.

(B) The rate of change of the number of susceptible and infected people over time.

(C) The initial conditions of the disease.

(D) The recovery rate.

Question 11. In financial modeling, if the rate of change of investment value is proportional to the value itself, and there is a constant withdrawal rate, the model is $\frac{dA}{dt} = rA - w$. This is a: (Applied Maths)

(A) Linear, homogeneous DE

(B) Linear, non-homogeneous DE

(C) Non-linear DE

(D) Second order DE

Question 12. A cup of tea cools from $90^\circ$C to $80^\circ$C in 10 minutes, in a room temperature of $20^\circ$C. What is the temperature of the tea after 20 minutes? (Assume Newton's Law of Cooling, $\frac{dT}{dt} = k(T-T_a)$)

(A) $70^\circ$C

(B) $60^\circ$C

(C) $65^\circ$C

(D) $50^\circ$C

Question 13. If the rate of increase of bacteria in a culture is proportional to the number of bacteria present, and the number triples in 5 hours, how many times the initial number will be present after 10 hours?

(A) 6

(B) 9

(C) 12

(D) 27

Question 14. The marginal revenue function is given by $MR(x) = 10 - 3x$. If the revenue from selling 0 units is $\textsf{₹ }0$, find the total revenue function $R(x)$. (Applied Maths)

(A) $10x - 3x^2/2$

(B) $10 - 3x$

(C) $10x - 3x^2/2 + C$

(D) $10x - 3x^2$

Question 15. A company's sales $S$ are growing at a rate proportional to the current sales. If sales doubled in 5 years, in how many years will they be three times the initial sales? (Applied Maths)

(A) $5 \log_e(3)/\log_e(2)$ years

(B) $5 \log_e(2)/\log_e(3)$ years

(C) 15 years

(D) 10 years

Question 16. A tank initially contains 100 litres of water with 10 kg of salt dissolved in it. Pure water is poured into the tank at the rate of 5 litres/minute. The mixture is kept stirred and flows out at the same rate. If $A(t)$ is the amount of salt in the tank at time $t$, the differential equation governing the amount of salt is:

(A) $\frac{dA}{dt} = 5 - \frac{A}{100} \cdot 5$

(B) $\frac{dA}{dt} = 0 - \frac{A}{100} \cdot 5$

(C) $\frac{dA}{dt} = 10 - \frac{A}{100} \cdot 5$

(D) $\frac{dA}{dt} = 10 - \frac{A}{100}$

Question 17. The velocity $v(t)$ of an object falling under gravity (neglecting air resistance) satisfies the differential equation $\frac{dv}{dt} = g$. If $v(0)=0$, find $v(t)$.

(A) $v(t) = gt$

(B) $v(t) = gt + C$

(C) $v(t) = g$

(D) $v(t) = g/t$

Question 18. In economics, the capital growth $K$ over time $t$ is often modeled by a differential equation. If the rate of increase of capital is proportional to the capital, the model is: (Applied Maths)

(A) $\frac{dK}{dt} = r K + I_0$ (with constant investment $I_0$)

(B) $\frac{dK}{dt} = r K$

(C) $\frac{dK}{dt} = r$

(D) $\frac{dK}{dt} = r/K$

Question 19. A population of insects increases at a rate proportional to its size. If the population doubles in 2 days, how many times its initial size will it be in 6 days?

(A) 4

(B) 6

(C) 8

(D) 16

Question 20. The half-life of Carbon-14 is approximately 5730 years. If a sample initially contains $N_0$ amount, the differential equation for the amount $N(t)$ at time $t$ is $\frac{dN}{dt} = -k N$. The value of $k$ is:

(A) $\log_e(2) / 5730$

(B) $5730 / \log_e(2)$

(C) $1 / 5730$

(D) $2 / 5730$

Question 21. The total cost of producing $x$ units is $C(x)$. The marginal cost is $C'(x)$. If the cost of producing 0 units is $C_0$, then $C(x)$ is given by: (Applied Maths)

(A) $\int C'(x) dx$

(B) $\int C'(x) dx + C_0$

(C) $C'(x) - C_0$

(D) $C_0 \int C'(x) dx$

Question 22. The velocity $v(t)$ of an object satisfies the differential equation $m \frac{dv}{dt} = F - kv$, where $F$ is a constant force and $k$ is a resistance constant. If $v(0)=0$, what is the limiting velocity as $t \to \infty$?

(A) $F/k$

(B) $k/F$

(C) $Fm/k$

(D) $0$

Question 23. The rate of sales $S$ of a new product is proportional to the number of potential customers $M$ who have not yet purchased the product. If $N(t)$ is the number of people who have purchased the product at time $t$, the model is $\frac{dN}{dt} = k(M - N)$. This is a:

(A) Linear DE

(B) Non-linear DE

(C) Homogeneous DE

(D) Bernoulli DE

Question 24. A sum of $\textsf{₹ }10,000$ is invested at an annual interest rate of 8% compounded continuously. The differential equation for the amount $A(t)$ after $t$ years is $\frac{dA}{dt} = 0.08 A$. The amount after $t$ years is:

(A) $10000 e^{0.08t}$

(B) $10000 (1 + 0.08)^t$

(C) $10000 + 0.08 t$

(D) $10000 + 0.08 A$

Question 25. A resource is consumed at a rate proportional to the square root of the remaining amount. If $R(t)$ is the remaining amount at time $t$, the differential equation is:

(A) $\frac{dR}{dt} = k \sqrt{R}$

(B) $\frac{dR}{dt} = -k \sqrt{R}$

(C) $\frac{dR}{dt} = k R^2$

(D) $\frac{dR}{dt} = -k R^2$

Question 26. The rate of change of atmospheric pressure $P$ with respect to altitude $h$ is proportional to the pressure. If $P_0$ is the pressure at sea level $(h=0)$, the differential equation is:

(A) $\frac{dP}{dh} = k P$

(B) $\frac{dP}{dh} = -k P$

(C) $\frac{dP}{dh} = k/P$

(D) $\frac{dP}{dh} = k h$

Question 27. The solution to the differential equation in Q564 with $P(0) = P_0$ is:

(A) $P(h) = P_0 e^{kh}$

(B) $P(h) = P_0 e^{-kh}$

(C) $P(h) = P_0 (1+kh)$

(D) $P(h) = P_0 - kh$