CBSE Class 12 Mathematics 2024 question paper (65/3)

1. A.
   $a_{ij} = \begin{cases} 0, & \text{if } i = j \\ 1, & \text{if } i \neq j \end{cases}$
2. B.
   $a_{ij} = 1, \forall\, i, j$
3. C.
   $a_{ij} = 0, \forall\, i, j$
4. D.
   $a_{ij} = \begin{cases} 0, & \text{if } i \neq j \\ 1, & \text{if } i = j \end{cases}$

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1. A.
   one-one but not onto
2. B.
   onto but not one-one
3. C.
   both one-one and onto
4. D.
   neither one-one nor onto

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1. A.
   $2a$
2. B.
   $2b$
3. C.
   $2c$
4. D.
   0

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1. A.
   discontinuous at $x = 1$ only
2. B.
   discontinuous at $x = 0$ only
3. C.
   discontinuous at $x = 0, 1$
4. D.
   continuous everywhere

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1. A.
   $1{\cdot}5$ cm/s
2. B.
   6 cm/s
3. C.
   3 cm/s
4. D.
   $2{\cdot}25$ cm/s

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1. A.
   $f(-x) = f(x)$
2. B.
   $f(-x) = -f(x)$
3. C.
   $f(a - x) = f(x)$
4. D.
   $f(a - x) = -f(x)$

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1. A.
   variable separable differential equation.
2. B.
   homogeneous differential equation.
3. C.
   first order linear differential equation.
4. D.
   differential equation whose degree is not defined.

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1. A.
   collinear vectors which are not parallel
2. B.
   parallel vectors
3. C.
   perpendicular vectors
4. D.
   unit vectors

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1. A.
   $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$
2. B.
   $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2$
3. C.
   $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1$
4. D.
   $\cos \alpha + \cos \beta + \cos \gamma = 1$

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1. A.
   feasible solutions
2. B.
   constraints
3. C.
   optimal solutions
4. D.
   infeasible solutions

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1. A.
   $0{\cdot}6$
2. B.
   $0{\cdot}4$
3. C.
   $0{\cdot}5$
4. D.
   0

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1. A.
   a skew symmetric matrix
2. B.
   a symmetric matrix
3. C.
   a null matrix
4. D.
   an identity matrix

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1. A.
   2
2. B.
   $-2$
3. C.
   $\pm 2$
4. D.
   $\mp 2$

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1. A.
   $\left(\dfrac{3}{2}\right)^x \dfrac{\log 2}{\log 3}$
2. B.
   $\left(\dfrac{2}{3}\right)^x \dfrac{\log 3}{\log 2}$
3. C.
   $\left(\dfrac{2}{3}\right)^x \dfrac{\log 2}{\log 3}$
4. D.
   $\left(\dfrac{3}{2}\right)^x \dfrac{\log 3}{\log 2}$

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1. A.
   $[-6, 4]$
2. B.
   $[0, 4]$
3. C.
   $[4, 6]$
4. D.
   $[0, 6]$

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1. A.
   0
2. B.
   $\dfrac{\pi}{4}$
3. C.
   $\dfrac{\pi}{2}$
4. D.
   $\pi$

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1. A.
   $x + 2y \le 76$, $2x + y \ge 104$, $x, y \ge 0$
2. B.
   $x + 2y \le 76$, $2x + y \le 104$, $x, y \ge 0$
3. C.
   $x + 2y \ge 76$, $2x + y \le 104$, $x, y \ge 0$
4. D.
   $x + 2y \ge 76$, $2x + y \ge 104$, $x, y \ge 0$

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1. A.
   $\begin{bmatrix} \dfrac{1}{2} & 0 & 0 \\ 0 & \dfrac{1}{3} & 0 \\ 0 & 0 & \dfrac{1}{5} \end{bmatrix}$
2. B.
   $30 \begin{bmatrix} \dfrac{1}{2} & 0 & 0 \\ 0 & \dfrac{1}{3} & 0 \\ 0 & 0 & \dfrac{1}{5} \end{bmatrix}$
3. C.
   $\dfrac{1}{30} \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}$
4. D.
   $\dfrac{1}{30} \begin{bmatrix} \dfrac{1}{2} & 0 & 0 \\ 0 & \dfrac{1}{3} & 0 \\ 0 & 0 & \dfrac{1}{5} \end{bmatrix}$

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1. A.
   Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. B.
   Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. C.
   Assertion (A) is true, but Reason (R) is false.
4. D.
   Assertion (A) is false, but Reason (R) is true.

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1. A.
   Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. B.
   Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. C.
   Assertion (A) is true, but Reason (R) is false.
4. D.
   Assertion (A) is false, but Reason (R) is true.

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1. A.
   2
2. B.
   $-2$
3. C.
   $\pm 2$
4. D.
   $\mp 2$

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1. A.
   $\left(\dfrac{5}{e}\right)^x \dfrac{1}{\log 5}$
2. B.
   $\left(\dfrac{e}{5}\right)^x \dfrac{1}{\log 5}$
3. C.
   $\left(\dfrac{5}{e}\right)^x \log 5$
4. D.
   $\left(\dfrac{e}{5}\right)^x \log 5$

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1. A.
   $[-6, 4]$
2. B.
   $[0, 4]$
3. C.
   $[4, 6]$
4. D.
   $[0, 6]$

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1. A.
   0
2. B.
   $\dfrac{\pi}{4}$
3. C.
   $\dfrac{\pi}{2}$
4. D.
   $\pi$

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1. A.
   $x + 2y \le 76$, $2x + y \ge 104$, $x, y \ge 0$
2. B.
   $x + 2y \le 76$, $2x + y \le 104$, $x, y \ge 0$
3. C.
   $x + 2y \ge 76$, $2x + y \le 104$, $x, y \ge 0$
4. D.
   $x + 2y \ge 76$, $2x + y \ge 104$, $x, y \ge 0$

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1. A.
   $\begin{bmatrix} \dfrac{1}{2} & 0 & 0 \\ 0 & \dfrac{1}{3} & 0 \\ 0 & 0 & \dfrac{1}{5} \end{bmatrix}$
2. B.
   $30 \begin{bmatrix} \dfrac{1}{2} & 0 & 0 \\ 0 & \dfrac{1}{3} & 0 \\ 0 & 0 & \dfrac{1}{5} \end{bmatrix}$
3. C.
   $\dfrac{1}{30} \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}$
4. D.
   $\dfrac{1}{30} \begin{bmatrix} \dfrac{1}{2} & 0 & 0 \\ 0 & \dfrac{1}{3} & 0 \\ 0 & 0 & \dfrac{1}{5} \end{bmatrix}$

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1. A.
   an identity matrix
2. B.
   a null matrix
3. C.
   a skew symmetric matrix
4. D.
   a symmetric matrix

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1. A.
   $(-\infty, \infty)$
2. B.
   $(1, \infty)$
3. C.
   $[1, \infty)$
4. D.
   $[-1, \infty)$

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1. A.
   $2a$
2. B.
   $2b$
3. C.
   $2c$
4. D.
   0

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1. A.
   discontinuous at $x = 1$ only
2. B.
   discontinuous at $x = 0$ only
3. C.
   discontinuous at $x = 0, 1$
4. D.
   continuous everywhere

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1. A.
   $f'(x) = 0$ and $f'(x)$ changes its sign from positive to negative from left to right of that point.
2. B.
   $f'(x) = 0$ and $f'(x)$ changes its sign from negative to positive from left to right of that point.
3. C.
   $f'(x) = 0$ and $f'(x)$ does not change its sign from left to right of that point.
4. D.
   $f'(x) \neq 0$.

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1. A.
   0
2. B.
   $2 \int\limits_{0}^{a} g(x)\, dx$
3. C.
   $\int\limits_{-a}^{a} g(x)\, dx$
4. D.
   $-\int\limits_{-2a}^{0} g(x)\, dx$

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1. A.
   variable separable differential equation.
2. B.
   homogeneous differential equation.
3. C.
   first order linear differential equation.
4. D.
   differential equation whose degree is not defined.

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1. A.
   collinear vectors which are not parallel
2. B.
   parallel vectors
3. C.
   perpendicular vectors
4. D.
   unit vectors

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1. A.
   $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$
2. B.
   $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2$
3. C.
   $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1$
4. D.
   $\cos \alpha + \cos \beta + \cos \gamma = 1$

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1. A.
   feasible solutions
2. B.
   constraints
3. C.
   optimal solutions
4. D.
   infeasible solutions

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1. A.
   $0{\cdot}6$
2. B.
   $0{\cdot}4$
3. C.
   $0{\cdot}5$
4. D.
   0

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1. A.
   $a_{ij} = \begin{cases} 0, & \text{if } i = j \\ 1, & \text{if } i \neq j \end{cases}$
2. B.
   $a_{ij} = 1, \forall\, i, j$
3. C.
   $a_{ij} = 0, \forall\, i, j$
4. D.
   $a_{ij} = \begin{cases} 0, & \text{if } i \neq j \\ 1, & \text{if } i = j \end{cases}$

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1. A.
   Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. B.
   Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. C.
   Assertion (A) is true, but Reason (R) is false.
4. D.
   Assertion (A) is false, but Reason (R) is true.

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1. A.
   Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. B.
   Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. C.
   Assertion (A) is true, but Reason (R) is false.
4. D.
   Assertion (A) is false, but Reason (R) is true.

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