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In a simple word, it is the duplicate copy of question paper. Model papers - It gives you an idea of exam pattern, questions and marking scheme. Here, the no. of Sample Papers is 12

Board Class 12 Sample Paper - 1

Academic Session - 2024-2025

Subject : Mathematics

Time : 3 Hours

Full Marks : 80

Answered question numbers => 0

1)

A) $D_f=R$, $R_f={ \{ y \in R: y \gt 1 \} }$ B) $D_f=R^+$, $R_f={ \{ y \in R: y \ge 1 \} }$ C) $D_f=R$, $R_f={ \{ y \in R: y \le1 \} }$ D) $D_f=R$, $R_f={ \{ y \in R: y \ge -1 \} }$

2)

A) $A^{-1}B^{-1}$ B) $B^{-1}A^{-1}$ C) $BA^{-1}$ D) $AB^{-1}$

3)

A) $\pm \sqrt{5}$ B) $\pm \sqrt{3}$ C) $\pm 2 \sqrt{3}$ D) $\pm 3 \sqrt{3}$

4)

A) \begin{bmatrix} -cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix} B) \begin{bmatrix} cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix} C) \begin{bmatrix} cos 2 \alpha & sin 2 \alpha \\ sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix} D) \begin{bmatrix} cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & -cos 2 \alpha \\ \end{bmatrix}

5)

A) $\frac{16}{3}$ B) $- \frac{33}{2}$ C) $\frac{33}{2}$ D) $\frac{25}{2}$

6)

A) 1 B) $\frac{2}{3}$ C) $\frac{1}{2}$ D) $\frac{1}{3}$

7)

A) 0 B) 1 C) -1 D) none of these

8)

A) R - {3} B) R C) $(0, \infty)$ D) none of these

9)

A) 1 B) 4 C) 0 D) -3

10)

A) $\frac{5}{9}$ B) $\frac{4}{9}$ C) $\frac{1}{3}$ D) none of these

11)

A) $\frac{e}{2} -1$ B) e-1 C) e+1 D) none of these

12)

A) $\frac{2x}{x^2 - 1} - \frac{2 ln(x^2 + 1) }{(2x + 1)^2} $ B) $\frac{2x}{x^2 + 1} - \frac{2 ln(x^2 + 1) }{(2x + 1)^2} $ C) $\frac{2x}{x^2 + 1} - \frac{2 ln(x^2 + 1) }{(2x - 1)^2} $ D) $\frac{2x}{x^2 + 1} - \frac{2 ln(x^2 - 1) }{(2x + 1)^2} $

13)

A) $- \frac{2}{3x}$ B) $\frac{2}{x}$ C) $\frac{1}{3x}$ D) $\frac{2}{3x}$

14)

A) x tan y = c B) x cos y = c C) x sec y = c D) x cot y = c

15)

A) $- \frac{1}{2} (log \, tan \, x)^2$ B) $\frac{1}{2} (log \, cot \, x)^2$ C) $ (log \, tan \, x)^2 + c$ D) $\frac{1}{2} (log \, tan \, x)^2 + c$

16)

A) $\frac{19}{6}$ B) $\frac{17}{6}$ C) $\frac{13}{6}$ D) $\frac{23}{6}$

17)

A) $\frac{5}{\sqrt{7}}$ B) $\frac{5}{\sqrt{3}}$ C) $\frac{3}{\sqrt{6}}$ D) $\frac{5}{\sqrt{6}}$

18)

A) $\frac{10}{7}$ B) $-\frac{10}{7}$ C) $-\frac{5}{7}$ D) $\frac{5}{7}$

19)

A) $]\frac{2}{3}, 3[$ B) $]\frac{5}{3}, 3[$ C) $]\frac{5}{3}, 2[$ D) $]\frac{1}{3}, 3[$

20)

A) $\frac{3}{5}$ B) $\frac{2}{3}$ C) $\frac{2}{5}$ D) $\frac{4}{5}$

21)

A) $-2 \lt x \lt 6$ B) $2 \lt x \lt 6$ C) $3 \lt x \lt 6$ D) none of these

22)

A) $cos^{-1}x + \sqrt{1-x^2}+c$ B) $tan^{-1}x + \sqrt{1-x^2}+c$ C) $sin^{-1}x + \sqrt{1-x^2}+c$ D) $sin^{-1}x - \sqrt{1-x^2}+c$

23)

A) $\frac{4}{15}x \sqrt{x} (5-3x) + c$ B) $\frac{2}{15}x \sqrt{x} (5+3x) + c$ C) $\frac{2}{15}x \sqrt{x} (5-3x) + c$ D) none of these

24)

A) $\frac{15}{5-\sqrt{3}}$m B) $\frac{16}{6-\sqrt{3}}$m C) $\frac{14}{4-\sqrt{3}}$m D) $\frac{15}{6-\sqrt{3}}$m

25)

A) (2, -7) B) (2, -9) C) (2, -5) D) (2, 9)

26)

A) 10 B) 6 C) 15 D) 25

27)

A) $\frac{9}{8}$ sq. units B) $\frac{10}{7}$ sq. units C) $\frac{13}{8}$ sq. units D) none of these

28)

A) either a=1 and b=0 or a=1 and $b \, \in \, R$ B) either a=0 and b=1 or a=-1 and $b \, \in \, R$ C) either a=1 and b=1 or a=-1 and $b \, \in \, R$ D) either a=1 and b=0 or a=-1 and $b \, \in \, R$

29)

A) x = -3, y = 2, z = -1 B) x = 3, y = 2, z = -1 C) x = -3, y = 2, z = 1 D) x = -3, y = 2, z = -2

30)

A) (1, 0, 7) B) (1, 1, 7) C) (-1, 0, 7) D) (1, 0, 6)

Result

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Hide Show Answers

1) $D_f=R$, $R_f={ \{ y \in R: y \ge 1 \} }$ , 2) $B^{-1}A^{-1}$, 3) $\pm 2 \sqrt{3}$, 4) \begin{bmatrix} cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix}, 5) $\frac{33}{2}$, 6) $\frac{1}{2}$, 7) 1, 8) R - {3}, 9) -3, 10) $\frac{5}{9}$, 11) $\frac{e}{2} -1$, 12) $\frac{2x}{x^2 + 1} - \frac{2 ln(x^2 + 1) }{(2x + 1)^2} $, 13) $\frac{2}{3x}$ , 14) x cos y = c, 15) $\frac{1}{2} (log \, tan \, x)^2 + c$, 16) $\frac{17}{6}$, 17) $\frac{5}{\sqrt{6}}$, 18) $-\frac{10}{7}$, 19) $]\frac{5}{3}, 3[$, 20) $\frac{2}{5}$, 21) $2 \lt x \lt 6$, 22) $sin^{-1}x + \sqrt{1-x^2}+c$, 23) $\frac{2}{15}x \sqrt{x} (5-3x) + c$, 24) $\frac{16}{6-\sqrt{3}}$m, 25) (2, -9), 26) 15, 27) $\frac{9}{8}$ sq. units, 28) either a=1 and b=0 or a=-1 and $b \, \in \, R$, 29) x = -3, y = 2, z = -1, 30) (1, 0, 7),

ShowHide solution of some questions.

1
Find domain and range of the function f : R $\rightarrow $ R, f(x)=$x^2-1$

2
If A and B are invertible matrices of the same order, then $(AB)^{-1} = ?$

3
Find the value of x, \begin{equation} \begin{vmatrix} 2 & x \\ x & 1 \\ \end{vmatrix} = \begin{vmatrix} 2 & 3 \\ 4 & 1 \\ \end{vmatrix} \end{equation}

4
If \begin{equation} A = \begin{bmatrix} sin \alpha & cos \alpha \\ -cos \alpha & sin \alpha \\ \end{bmatrix} \end{equation}, then $A^2$

5
If \begin{equation} A= \begin{bmatrix} 1 & k & 3 \\ 3 & k & -2 \\ 2 & 3 & -4 \\ \end{bmatrix} \end{equation} is singular, then k = ?

6
Find the value of \begin{equation} \begin{vmatrix} cos \, 50^o & sin \, 10^o \\ sin \, 50^o & cos \, 10^o \\ \end{vmatrix} \end{equation}

7
\begin{equation} f(x) = \lvert x \rvert = \left\{ \begin{array}{ll} -x & \quad x = 0 \\ kx & \quad x \geq 0 \end{array} \right. \end{equation}, if f(x) is continuous at x = 0, then the value of k is

8
The set of points where the f(x) is defined by f(x) = |x - 3| cos x is differentiable, is

9
Evaluate $\,\, \lim_{x \to 0} \frac{x^2 - 3x}{x}$
Solution

10
The probability of A, B and C of solving a problem are $\frac{1}{6}$, $\frac{1}{5}$ and $\frac{1}{3}$ respectively. What is the probability that the problem is solved ?

11
Evaluate $\int_0^1 \, \frac{xe^x}{(1+x)^2} \, dx$ .

12
Calculate the derivative of $g(x) = \frac{ln(x^2 + 1)}{2x + 1} .$

13
Derivative of $x^2$ w.r.t. $x^3$

14
The solution of differential equation $x \frac{dy}{dx} = cot \, y$.
Solution

15
Evaluate $\int \frac{log \, tan \, x }{sin \, x \, cos \, x} \, dx$ .

16
The value of tan{$cos^{-1}\frac{4}{5} + tan^{-1}\frac{2}{3}$}

17
What is the projection of $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ on $\vec{b}=\hat{i}- 2\hat{j}+\hat{k}$ ?

18
The lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-5}$ are perpendicular to each other, then k = ?

19
Find the intervals on which the following function $f(x) = (x-1) (x-3)^2$, is decreasing.

20
A die is thrown twice and the sum of the numbers is observed to be 8. What is the conditional probability that the number 5 has appeared at least once ?

21
The intervals in which the function $f(x)=x^3-12 x^2+36x +17$ is decreasing

22
Evaluate $\int \, \sqrt{\frac{1-x}{1+x}}$ .
Solution

23
Evaluate $\int \, (1-x)\sqrt{x} \, dx$ .

24
A window is in the form of rectangle surmounted by an equilateral triangle. Given that the perimeter is 16 meters, find the width of the window in order that the maximum amount of light may be admitted.

25
Find the point on the curve $y=x^3-11x+5$ at which the equation of the tangent is y=x-11
Solution

26
The maximum value of P=5x+3y subject to constraints $5x+2y \le 10$, $x \ge 0$ and $y \ge 0$ is

27
Find the area bounded by curves$x^2=4y$ and the line x = 4y - 2

28
Let f : R $\rightarrow$ R be a function given by f(x)=ax+b for all $x \, \in \, R$. Find the constants a and b such that $f \, o \, f \, = \, I_R$.

29
Solve the following system of equations using matrix method : 2x + 8y + 5z = 5 x + y + z = -2 x + 2y - z = 2

30
Find the image of the point (1, 6, 3) in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$.
Solution

Choose a question from set no1: