Assertion-Reason MCQs for Sub-Topics of Topic 2: Algebra

Question 1. Assertion (A): In the algebraic expression $5x^2 - 3x + 7$, the term $7$ is a constant.

Reason (R): A constant term in an algebraic expression has a fixed numerical value and does not contain any variables.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): In the term $-4ab$, the coefficient of $ab$ is $-4$.

Reason (R): The coefficient of a term is the numerical factor that multiplies the variable part.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The expression $p/q$ is an algebraic expression.

Reason (R): An algebraic expression is formed by combining variables and constants using the operations of addition, subtraction, multiplication, and division.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): The value of the expression $2x + 1$ when $x=5$ is $11$.

Reason (R): To find the value of an expression for a given value of the variable, substitute the value into the expression and perform the calculations.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): $3a^2b$ and $5ab^2$ are like terms.

Reason (R): Like terms have the same variables raised to the same corresponding powers.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The sum of $4x - y$ and $x + 2y$ is $5x + y$.

Reason (R): When adding algebraic expressions, we combine the coefficients of like terms.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The product of $(x+1)$ and $(x-1)$ is $x^2 - 1$.

Reason (R): The difference of squares identity is $(a+b)(a-b) = a^2 - b^2$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The result of dividing $12a^3b^2$ by $4ab$ is $3a^2b$.

Reason (R): When dividing monomials, we divide the coefficients and subtract the exponents of corresponding variables.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): Subtraction of algebraic expressions is commutative.

Reason (R): $A - B = B - A$ for any two algebraic expressions A and B.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): To multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial.

Reason (R): The distributive property of multiplication over addition (and subtraction) applies to algebraic expressions.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The expression $\sqrt{x} + 5$ is not a polynomial.

Reason (R): The powers of the variable in a polynomial must be non-negative integers.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The degree of the polynomial $3x^5 - 7x^2 + 10$ is 5.

Reason (R): The degree of a polynomial is the highest power of the variable in the polynomial.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): A polynomial with exactly two terms is called a binomial.

Reason (R): 'Bi' means two, indicating two terms in the polynomial.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): If $P(2) = 0$ for a polynomial $P(x)$, then $x=2$ is a zero of the polynomial.

Reason (R): A zero of a polynomial $P(x)$ is a value of $x$ for which $P(x) = 0$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The graph of a quadratic polynomial $y = ax^2+bx+c$ with $a \neq 0$ is a straight line.

Reason (R): The graph of a quadratic polynomial is a parabola.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): When the polynomial $P(x)$ is divided by $(x-a)$, the remainder is equal to $P(a)$.

Reason (R): This statement is known as the Remainder Theorem.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): If $P(a) = 0$ for a polynomial $P(x)$, then $(x-a)$ is a factor of $P(x)$.

Reason (R): This statement is known as the Factor Theorem.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): When $x^2 - 5x + 6$ is divided by $(x-2)$, the remainder is 0.

Reason (R): By the Remainder Theorem, the remainder is $P(2) = 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): $(x+1)$ is a factor of $x^3 + 1$.

Reason (R): By the Factor Theorem, if $(x+1)$ is a factor, then $P(-1) = 0$. $P(-1) = (-1)^3 + 1 = -1 + 1 = 0$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): In the Division Algorithm $P(x) = Q(x) D(x) + R(x)$, the degree of the remainder $R(x)$ is always strictly less than the degree of the divisor $D(x)$.

Reason (R): If the degree of $R(x)$ were greater than or equal to the degree of $D(x)$, the division process could continue.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): $(a+b)^2 = a^2 + b^2 + 2ab$ is an algebraic identity.

Reason (R): An algebraic identity is an equality that is true for all values of the variables involved.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The expression $4x^2 - 9y^2$ can be factorised using the identity $a^2 - b^2 = (a-b)(a+b)$.

Reason (R): $4x^2 = (2x)^2$ and $9y^2 = (3y)^2$, so the expression is in the form of a difference of squares.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The value of $55^2 - 45^2$ is 1000.

Reason (R): Using the identity $a^2 - b^2 = (a-b)(a+b)$, $55^2 - 45^2 = (55-45)(55+45) = 10 \times 100 = 1000$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): $(x-y)^3 = x^3 - y^3 - 3xy(x-y)$.

Reason (R): The identity for $(a-b)^3$ is $a^3 - b^3 - 3a^2b + 3ab^2$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): If $a+b+c=0$, then $a^3+b^3+c^3 = 3abc$.

Reason (R): The identity $a^3+b^3+c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ is true for all values of $a, b, c$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The expression $5x^2 - 10xy$ can be factorised as $5x(x - 2y)$.

Reason (R): Factorisation by taking out a common factor involves identifying the greatest common factor of the terms and writing the expression as a product.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The expression $ab + ac + bd + cd$ can be factorised by grouping as $(a+d)(b+c)$.

Reason (R): Factorisation by grouping involves arranging terms and factoring common factors from groups to get a common binomial factor.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The polynomial $x^2 - 7x + 12$ can be factorised by splitting the middle term into $-3x$ and $-4x$.

Reason (R): We look for two numbers whose product is the constant term (12) and whose sum is the coefficient of the middle term (-7).

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): The polynomial $x^3 - 8$ can be factorised as $(x-2)(x^2+4x+4)$.

Reason (R): The identity for the difference of cubes is $a^3 - b^3 = (a-b)(a^2+ab+b^2)$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): If $(x-a)$ is a factor of a polynomial $P(x)$, then $x=a$ is a root of the equation $P(x)=0$.

Reason (R): A root of an equation is a value of the variable that satisfies the equation.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The equation $5x - 1 = 9$ is a linear equation in one variable.

Reason (R): A linear equation in one variable can be written in the form $ax+b=0$, where $a$ and $b$ are constants and $a \neq 0$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The solution of the equation $2x+3 = 7$ is $x=2$.

Reason (R): Substituting $x=2$ into the equation gives $2(2)+3 = 4+3 = 7$, which is a true statement.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): Using transposition, $3x = 10 - 4$ is obtained from $3x + 4 = 10$.

Reason (R): Transposing a term from one side of the equation to the other requires changing its sign.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): Multiplying both sides of the equation $\frac{x}{2} = 5$ by 2 gives $x=10$, and the solution remains the same.

Reason (R): Performing the same non-zero arithmetic operation on both sides of a linear equation maintains the equality and the solution.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): A word problem involving finding a single unknown quantity often leads to a linear equation in one variable.

Reason (R): Variables in algebra are used to represent unknown quantities.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): $x + y = 7$ is a linear equation in two variables.

Reason (R): A linear equation in two variables can be written in the form $Ax + By + C = 0$, where $A$ and $B$ are not both zero.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The equation $x = 5$ represents a straight line parallel to the y-axis.

Reason (R): In the equation $x=k$, the x-coordinate of every point on the line is $k$, while the y-coordinate can be any real number, resulting in a vertical line.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): A linear equation in two variables has infinitely many solutions.

Reason (R): The graph of a linear equation in two variables is a straight line, and every point on the line represents a solution.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): The point $(2, 3)$ is a solution to the equation $x + y = 5$.

Reason (R): Substituting $x=2$ and $y=3$ into the equation gives $2+3=5$, which is a true statement.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The equation $y = 0$ represents the x-axis.

Reason (R): On the x-axis, the y-coordinate of every point is zero.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): A system of two linear equations in two variables can have a unique solution, no solution, or infinitely many solutions.

Reason (R): Geometrically, two lines in a plane can intersect at one point, be parallel and distinct, or be coincident.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): A consistent system of linear equations has at least one solution.

Reason (R): If the lines represented by the equations are intersecting or coincident, the system is consistent.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The system $\begin{cases} 2x + 3y = 5 \\ 4x + 6y = 10 \end{cases}$ has infinitely many solutions.

Reason (R): The ratio of coefficients $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ for this system ($2/4 = 3/6 = 5/10 = 1/2$).

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): The system $\begin{cases} x + y = 1 \\ x + y = 2 \end{cases}$ is inconsistent.

Reason (R): Parallel lines represent an inconsistent system.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The graphical method involves finding the intersection point(s) of the lines.

Reason (R): A solution to a system of equations is a point that satisfies all equations in the system.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The equation $x^2 - 5x + 6 = 0$ is a quadratic equation.

Reason (R): A quadratic equation is a polynomial equation of degree 2.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The sum of the roots of $2x^2 - 8x + 6 = 0$ is 4.

Reason (R): For a quadratic equation $ax^2+bx+c=0$, the sum of the roots is $-b/a$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The roots of $x^2 - 9 = 0$ are $x=3$ and $x=-3$.

Reason (R): The equation can be factored as $(x-3)(x+3) = 0$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): The discriminant of $x^2 + x + 1 = 0$ is $-3$.

Reason (R): The discriminant $D = b^2 - 4ac$ for $ax^2+bx+c=0$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): If the discriminant of a quadratic equation with real coefficients is negative, the roots are real and distinct.

Reason (R): A negative discriminant means the roots are non-real (complex).

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): $i^4 = 1$.

Reason (R): $i^2 = -1$, so $i^4 = (i^2)^2 = (-1)^2 = 1$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The sum of $(3+2i)$ and $(1-4i)$ is $4-2i$.

Reason (R): Complex numbers are added by summing their real parts and summing their imaginary parts separately.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The product of $i$ and $-i$ is 1.

Reason (R): $i \times (-i) = -i^2 = -(-1) = 1$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): The reciprocal of $i$ is $-i$.

Reason (R): The reciprocal of a complex number $z$ is $1/z$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The real part of the complex number $5i$ is 5.

Reason (R): A purely imaginary number has a real part equal to zero.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The complex number $2 - 3i$ is represented by the point $(2, -3)$ in the Argand plane.

Reason (R): In the Argand plane, the real part of $a+bi$ is the x-coordinate and the imaginary part is the y-coordinate.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The modulus of the complex number $-1 - i$ is $\sqrt{2}$.

Reason (R): The modulus $|a+bi| = \sqrt{a^2+b^2}$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The conjugate of the complex number $5+2i$ is $5-2i$.

Reason (R): The conjugate of $a+bi$ is obtained by changing the sign of the real part.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): If $z$ is a complex number such that $z = -\bar{z}$, then $z$ must be purely imaginary.

Reason (R): If $z=a+bi$, then $-\bar{z} = -(a-bi) = -a+bi$. If $a+bi = -a+bi$, then $a=-a$, which means $2a=0$, so $a=0$. Thus $z=bi$, which is purely imaginary.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The polar form $r(\cos \theta + i \sin \theta)$ of a complex number is unique.

Reason (R): The argument $\theta$ is unique up to a multiple of $2\pi$, but the principal argument in $(-\pi, \pi]$ is unique.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The quadratic equation $x^2 + x + 1 = 0$ has complex roots.

Reason (R): The discriminant of the equation is $D = 1^2 - 4(1)(1) = -3$, which is negative.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): If a quadratic equation with real coefficients has $2+i$ as one root, then $2-i$ must be the other root.

Reason (R): Complex roots of a quadratic equation with real coefficients always occur in conjugate pairs.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The roots of the equation $x^2 - 4x + 8 = 0$ are $2 \pm 2i$.

Reason (R): The quadratic formula for the roots of $ax^2+bx+c=0$ is $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): The quadratic equation with roots $3i$ and $-3i$ is $x^2 + 9 = 0$.

Reason (R): The sum of roots is $0$ and the product of roots is $9$, leading to the equation $x^2 - (sum)x + (product) = 0$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The equation $x^2 + kx + k = 0$ has complex roots if $0 < k < 4$.

Reason (R): The roots are complex if the discriminant $b^2 - 4ac < 0$. For this equation, $k^2 - 4k < 0 \implies k(k-4) < 0$, which is true when $0 < k < 4$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): $3x + 5y \ge 10$ is a linear inequality in two variables.

Reason (R): It involves linear expressions and an inequality sign.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): When solving $-4x > 8$, we divide by $-4$ and change the inequality sign to get $x < -2$.

Reason (R): Dividing an inequality by a negative number reverses the direction of the inequality sign.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The solution set of $x \ge 2$ on a number line includes the point $x=2$.

Reason (R): The symbol '$\ge$' means 'greater than or equal to'.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): The graph of $y > 1$ in the coordinate plane is the region above the line $y=1$, excluding the line itself.

Reason (R): The '>' symbol indicates that the boundary line is not included in the solution set.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The feasible region for a system of linear inequalities in two variables is the area where all inequalities are satisfied simultaneously.

Reason (R): The feasible region is the intersection of the half-planes represented by each linear inequality.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The sequence $5, 10, 15, 20, \dots$ is an Arithmetic Progression (AP).

Reason (R): In an AP, the difference between any term and its preceding term is constant.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The sequence $3, 9, 27, 81, \dots$ is a Geometric Progression (GP).

Reason (R): In a GP, the ratio of any term to its preceding term is constant.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The $n$-th term of the AP $2, 5, 8, \dots$ is $3n-1$.

Reason (R): The formula for the $n$-th term of an AP is $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): For any two distinct positive numbers, their Arithmetic Mean (AM) is strictly greater than their Geometric Mean (GM).

Reason (R): The relationship between AM and GM is $AM \ge GM$, and equality holds only when the two numbers are equal.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The sum of the first $n$ natural numbers is given by the formula $\frac{n(n+1)}{2}$.

Reason (R): The sequence of natural numbers $1, 2, 3, \dots$ is an AP with first term 1 and common difference 1.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The Principle of Mathematical Induction is a method used to prove statements for all positive integers.

Reason (R): The principle consists of proving a base case and an inductive step that shows the property holds for $k+1$ if it holds for $k$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): In proving $P(n): 1+2+\dots+n = \frac{n(n+1)}{2}$ for $n \ge 1$, the base case is $P(1)$.

Reason (R): The statement needs to be proven for the smallest positive integer, which is 1.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): If we successfully prove the base case $P(n_0)$ and the inductive step $P(k) \implies P(k+1)$ for all $k \ge n_0$, then we have proven $P(n)$ for all integers $n \ge n_0$.

Reason (R): The Principle of Mathematical Induction guarantees that if a property holds for a starting value and propagates, it holds for all subsequent integers.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): To prove $P(n): n^2 > 2n$ for $n \ge 3$, the base case is $P(3)$.

Reason (R): The statement is specified to be true for integers greater than or equal to 3, so 3 is the smallest value for which the statement is claimed to be true.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): Strong induction is a more powerful method than standard induction.

Reason (R): In strong induction, the inductive hypothesis assumes $P(k)$ is true, while in standard induction, it assumes $P(j)$ is true for all $n_0 \le j \le k$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The number of ways to arrange 3 distinct objects is $3! = 6$.

Reason (R): $n!$ represents the number of permutations of $n$ distinct objects.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The number of ways to choose a committee of 2 from 5 people is $\text{C}(5, 2) = 10$.

Reason (R): A committee selection is a combination problem because the order of selection does not matter.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): $\text{P}(n, r) = \text{C}(n, r) \times r!$.

Reason (R): A permutation is a combination followed by an arrangement (of the selected items).

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): $\text{C}(10, 3) = \text{C}(10, 7)$.

Reason (R): The property $\text{C}(n, r) = \text{C}(n, n-r)$ states that choosing $r$ items from $n$ is the same as choosing $n-r$ items to leave behind.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The number of distinct permutations of the letters in the word 'INDIA' is 60.

Reason (R): The formula for permutations with repeated letters is $\frac{n!}{p_1! p_2! \dots p_k!}$. In 'INDIA', there are 5 letters with 'I' repeated twice, so the number of permutations is $\frac{5!}{2!} = 60$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The expansion of $(x+y)^n$ for a positive integer $n$ has $n+1$ terms.

Reason (R): The terms in the expansion correspond to the values of $r$ from 0 to $n$ in the general term $\text{C}(n, r) x^{n-r} y^r$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The 4th term in the expansion of $(a+b)^6$ is $\text{C}(6, 3) a^3 b^3$.

Reason (R): The general term is $T_{r+1} = \text{C}(n, r) a^{n-r} b^r$. For the 4th term, $r+1=4$, so $r=3$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The sum of the coefficients in the expansion of $(2x - 3y)^5$ is $-1$.

Reason (R): The sum of coefficients in $(a+b)^n$ is $2^n$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): The middle term in the expansion of $(x+y)^{10}$ is the 5th term.

Reason (R): If the index $n$ is even, the middle term is the $(n/2 + 1)$-th term.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The coefficient of the term independent of $x$ in the expansion of $(x + 1/x)^4$ is 6.

Reason (R): The general term is $\text{C}(4, r) x^{4-r} (1/x)^r = \text{C}(4, r) x^{4-2r}$. The term independent of $x$ occurs when $4-2r=0$, so $r=2$. The coefficient is $\text{C}(4, 2) = 6$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): Two matrices can be added only if they have the same order.

Reason (R): Matrix addition is performed by adding corresponding elements, which requires the matrices to have the same dimensions.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): Matrix multiplication is commutative.

Reason (R): For any two matrices A and B, $AB = BA$ is always true.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): If matrix A is of order $2 \times 3$ and matrix B is of order $3 \times 2$, then the product matrix AB is of order $2 \times 2$.

Reason (R): If A is of order $m \times n$ and B is of order $n \times p$, the product AB is of order $m \times p$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): A square matrix is a matrix where the number of rows is equal to the number of columns.

Reason (R): The identity matrix is a type of square matrix.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The product of a zero matrix and any other matrix (for which the product is defined) is always the zero matrix.

Reason (R): Multiplying any number by zero results in zero.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The transpose of a matrix A is obtained by interchanging its rows and columns.

Reason (R): If $A = (a_{ij})$, then $A' = (a_{ji})$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): If $A$ is a square matrix such that $A = A'$, then A is a symmetric matrix.

Reason (R): This is the definition of a symmetric matrix.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The diagonal elements of a skew-symmetric matrix are always zero.

Reason (R): For a skew-symmetric matrix, $A' = -A$, which implies $a_{ii} = -a_{ii}$ for diagonal elements.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): For any square matrix A, $A+A'$ is a symmetric matrix.

Reason (R): $(A+A')' = A' + (A')' = A' + A = A+A'$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): An invertible matrix is also called a non-singular matrix.

Reason (R): A non-singular matrix is a square matrix whose determinant is non-zero, and its inverse exists.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): The determinant of the matrix $\begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix}$ is 0.

Reason (R): In the given matrix, the first column is twice the second column, a linear dependency makes the determinant zero.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): If A and B are square matrices of the same order, then $\det(AB) = \det(A) \det(B)$.

Reason (R): The determinant of a product of matrices is the product of their determinants.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): The area of a triangle with vertices $(1, 1), (2, 2), (3, 3)$ is 0.

Reason (R): If the three vertices of a triangle are collinear, the area of the triangle is zero.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): The adjoint of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.

Reason (R): The adjoint of a matrix is the transpose of its cofactor matrix.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): If A is a square matrix, then $A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| I$.

Reason (R): This is a fundamental property relating a matrix, its adjoint, and its determinant.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): A square matrix A has an inverse if and only if its determinant is non-zero.

Reason (R): If $|A| \neq 0$, the matrix is singular and its inverse exists.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): The solution to a system of linear equations $AX=B$, where A is the coefficient matrix and $|A| \neq 0$, is given by $X = A^{-1}B$.

Reason (R): Multiplying the matrix equation $AX=B$ by $A^{-1}$ on the left side gives $A^{-1}(AX) = A^{-1}B$, which simplifies to $X = A^{-1}B$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): Cramer's Rule is applicable to solve any system of linear equations $AX=B$.

Reason (R): Cramer's Rule requires the coefficient matrix A to be square and its determinant $|A|$ to be non-zero.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): If $|A|=0$ for a system $AX=B$, the system cannot have a unique solution.

Reason (R): If $|A|=0$, the matrix A is singular, and the equation $AX=B$ cannot be solved by multiplying by $A^{-1}$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): The homogeneous system $AX=O$ ($B=O$) has a non-trivial solution if $|A|=0$.

Reason (R): If $|A|=0$, the matrix A is singular, and there are linear dependencies among the rows/columns, leading to infinite solutions (including non-trivial ones for $B=O$).

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 1. Assertion (A): A problem involving finding the speeds of a boat in still water and a stream, given upstream and downstream times/distances, can be solved using a system of linear equations.

Reason (R): Upstream and downstream speeds are linear combinations of the boat's speed and the stream's speed ($u-v$ and $u+v$).

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 2. Assertion (A): A word problem about finding a two-digit number where the digits are reversed after adding a value often leads to a linear equation in one variable.

Reason (R): If the digits are $t$ and $u$, the number is $10t+u$. The reversed number is $10u+t$. Relationships between these are typically linear equations in $t$ and $u$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 3. Assertion (A): A problem involving the time taken by pipes to fill or empty a tank can often be solved using equations based on their filling/emptying rates.

Reason (R): If a pipe fills a tank in $T$ units of time, its rate of filling is $1/T$ of the tank per unit of time.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 4. Assertion (A): A word problem about finding two positive numbers given the sum of their squares and the difference of their squares can be solved using a system of two equations involving squares of variables.

Reason (R): Let the numbers be $x$ and $y$. The given conditions translate to equations like $x^2+y^2 = S$ and $x^2-y^2 = D$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.

Question 5. Assertion (A): Problems involving price, quantity, and total cost, where one variable changes causing another to change, can often lead to quadratic equations.

Reason (R): If total cost $= \text{price} \times \text{quantity}$, and both price and quantity are expressed in terms of one variable (e.g., initial price $p$, new price $p \pm \Delta p$, initial quantity $Q = Cost/p$, new quantity $Q \mp \Delta Q = Cost/(p \pm \Delta p)$), substituting these relationships can result in a quadratic equation in $p$.

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

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

(C) A is true, but R is false.

(D) A is false, but R is true.