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Mathematics - XII

The Model Paper - 1

Year : 2025 - 26

Time - 3 Hours 0 minutes

Full Marks - 80

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Cur Set 1

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Not Answered 30 Answered 0

In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 1

Instructions

Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 1

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1 ) The product of the matrices $[1 \,\,\, 2 \,\,\, 3 ]^T$ and $[2 \,\,\, 3 \,\,\, 4 ] $ is

2 ) if A is a 2-rowed square matrix and |A| = 6, then A . Adj A = ?

3 ) The area of $\triangle ABC$ having vertices A(2, -6), B(5, 4) and C(k, 4) is 35 sq units. Find the value of k.

4 ) If A and B are symmetric square matrices of the same order, then (AB - BA) is always

5 ) Find the value of \begin{equation} \begin{vmatrix} 0 & b & -c \\ -b & 0 & a \\ c & -a & 0 \\ \end{vmatrix} \end{equation}

6 ) If \begin{equation*} g(x)=\begin{cases} \frac{sin^{-1}}{x} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, find k.

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

8 ) Evaluate $\,\, \lim_{x \to 0} \frac{sec x - 1}{x}$

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

10 ) If $x=t^2$ and $y=t^3$, then $\frac{d^2y}{dx^2}=?$

11 ) $tan^{-1}(1+x) + tan^{-1}(1 - x) = \frac{\pi}{2}$

12 ) Evaluate $\int \frac{x^4 + 1}{x^2 + 1} \, dx$ .

13 ) Evaluate $\int_{-a}^a \, f(x) \, dx$ .

14 ) If $y=x^4 + 10$ and if x changes from 2 to 1.99, what is the approximate change in y.

15 ) The solution of differential equation $\frac{dy}{dx} = \frac{-2xy}{1+x^2}$ is.

16 ) If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$, then

17 ) If the planes 2x - 4y +3z = 5 and $x + 2y + \lambda z = 12$ are perpendicular to each other, then $\lambda= ?$

18 ) In a class, 60% of the students read mathematics, 25% biology and 15% both mathematics and biology. One student selected at random. What is the probability that he reads mathematics, if it is known that he reads biology ?

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

20 ) Evaluate $\int \, \frac{x^2+1}{x^4+1} \, dx$ .

21 ) Evaluate $\int \, e^x \frac{sin \, 4x - 4}{1- cos \, 4x} \, dx$ .

22 ) The function $f(x)=cos^{-1}(sin \, x +cos \, x)$ strictly decreasing function in the interval -

23 ) If the curves $x=y^2$ and $xy=k$ cut at right angles, then

24 ) An open box with a square base is to be made out of a given cardboard of area $c^2$ square units. The maximum volume of the box is

25 ) A bag contains 1 white and 6 red balls. Another bag contains 4 white and 3 red balls. One of the two bag is selected at random and a ball is drawn from it, which is found to be white. Find the probability that the ball drawn is from the bag A.

26 ) The constraints $-x+y \le 1$, $-x+3y \le 9$ and $x \ge 0$, $y \ge 0$ defines on

27 ) Let $f : R \rightarrow (-1, \, 1)$ ,is given by $f(x)=\frac{10^x-10^{-x}}{10^x+10^{-x}}$ is invertible. Find $f^{-1}(x)$.

28 ) Solve the following system of equations using matrix method : 2x - 3y + 5z = 11 3x + 2y - 4z = -5 x + y - 2z = -3

29 ) Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane $\vec{r}.(-2\hat{j}+4\hat{k})+5=0$.

30 ) Find the area bounded by curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x}{a}+\frac{y}{b}=1$.

Close

1 )

The product of the matrices $[1 \,\,\, 2 \,\,\, 3 ]^T$ and $[2 \,\,\, 3 \,\,\, 4 ] $ is

1) \begin{bmatrix} 2 & 3 & 6 \\ 4 & 6 & 8 \\ 6 & 9 & 10 \\ \end{bmatrix}

2) \begin{bmatrix} 2 & 3 & 6 \\ 4 & 5 & 8 \\ 6 & 9 & 12 \\ \end{bmatrix}

3) \begin{bmatrix} 2 & 3 & 6 \\ 4 & 6 & 8 \\ 4 & 9 & 12 \\ \end{bmatrix}

4) \begin{bmatrix} 2 & 3 & 6 \\ 4 & 6 & 8 \\ 6 & 9 & 12 \\ \end{bmatrix}

2 )

if A is a 2-rowed square matrix and |A| = 6, then A . Adj A = ?

1) \begin{bmatrix} 6 & 0 \\ 0 & 6 \\ \end{bmatrix}

2) \begin{bmatrix} 3 & 0 \\ 0 & 3 \\ \end{bmatrix}

3) \begin{bmatrix} 3 & 0 \\ 0 & 6 \\ \end{bmatrix}

4) none of these

3 )

The area of $\triangle ABC$ having vertices A(2, -6), B(5, 4) and C(k, 4) is 35 sq units. Find the value of k.

1) -12, 2

2) -12, -2

3) 12, 2

4) 12, -2

4 )

If A and B are symmetric square matrices of the same order, then (AB - BA) is always

1) a symmetric square matrix

2) zero square matrix

3) a skew symmetric square matrix

4) an identity matrix

5 )

Find the value of \begin{equation} \begin{vmatrix} 0 & b & -c \\ -b & 0 & a \\ c & -a & 0 \\ \end{vmatrix} \end{equation}

1) 1

2) 0

3) 2

4) -1

6 )

If \begin{equation*} g(x)=\begin{cases} \frac{sin^{-1}}{x} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, find k.

1) 1

2) 0

3) -1

4) none of these

7 )

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

1) R - {3}

2) R

3) $(0, \infty)$

4) none of these

8 )

Evaluate $\,\, \lim_{x \to 0} \frac{sec x - 1}{x}$

1) 0

2) 1

3) -1

4) none of these

9 )

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

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

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

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

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

10 )

If $x=t^2$ and $y=t^3$, then $\frac{d^2y}{dx^2}=?$

1) $\frac{3}{2t}$

2) $\frac{3}{t}$

3) $\frac{3}{4t}$

4) none of these

11 )

$tan^{-1}(1+x) + tan^{-1}(1 - x) = \frac{\pi}{2}$

1) 0

2) 3

3) 2

4) -1

12 )

Evaluate $\int \frac{x^4 + 1}{x^2 + 1} \, dx$ .

1) $\frac{x^2}{3} - x - 2 tan^{-1}x + c$

2) $\frac{x^2}{3} + x - 2 tan^{-1}x + c$

3) $\frac{x^2}{3} - x + 2 tan^{-1}x + c$

4) $\frac{x^2}{3} - x - tan^{-1}x + c$

13 )

Evaluate $\int_{-a}^a \, f(x) \, dx$ .

1) $\int_{0}^a \, [f(x)+f(-x)] \, dx$

2) $2 \int_{0}^a \, [f(x)+f(-x)] \, dx$

3) $2 \int_{0}^a \, [f(x)+f(-x)] \, dx$

4) none of these

14 )

If $y=x^4 + 10$ and if x changes from 2 to 1.99, what is the approximate change in y.

1) -0.32

2) -0.3

3) 0.32

4) -0.12

15 )

The solution of differential equation $\frac{dy}{dx} = \frac{-2xy}{1+x^2}$ is.

1) $y(x^2 - 1)=c$

2) $y^2(x^2 + 1)=c$

3) $y(x^2 + 1)=c$

4) $y^2(x + 1)=c$

16 )

If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$, then

1) $\vec{a} \parallel \vec{b}$

2) $|\vec{a}| = |\vec{b}|$

3) $\vec{a} \perp \vec{b}$

4) none of these

17 )

If the planes 2x - 4y +3z = 5 and $x + 2y + \lambda z = 12$ are perpendicular to each other, then $\lambda= ?$

1) 2

2) -2

3) 3

4) -3

18 )

In a class, 60% of the students read mathematics, 25% biology and 15% both mathematics and biology. One student selected at random. What is the probability that he reads mathematics, if it is known that he reads biology ?

1) $\frac{5}{8}$

2) $\frac{2}{5}$

3) $\frac{3}{5}$

4) $\frac{3}{8}$

19 )

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

1) $2 \frac{\pi}{8} log \, 3$

2) $\frac{\pi}{6} log \, 2$

3) $\frac{\pi}{8} log \, 3$

4) $\frac{\pi}{8} log \, 2$

20 )

Evaluate $\int \, \frac{x^2+1}{x^4+1} \, dx$ .

1) $\frac{1}{\sqrt{2}}cot^{-1}[\frac{1}{\sqrt{2}}(x-\frac{1}{x})]+c$

2) $\frac{1}{\sqrt{2}}tan^{-1}[\frac{1}{\sqrt{2}}(x+\frac{1}{x})]+c$

3) $\frac{1}{\sqrt{3}}tan^{-1}[\frac{1}{\sqrt{3}}(x-\frac{1}{x})]+c$

4) $\frac{1}{\sqrt{2}}tan^{-1}[\frac{1}{\sqrt{2}}(x-\frac{1}{x})]+c$

21 )

Evaluate $\int \, e^x \frac{sin \, 4x - 4}{1- cos \, 4x} \, dx$ .

1) $e^x cot \, 2x+c$

2) $-e^x cot \, 2x+c$

3) $e^x tan \, 2x+c$

4) $-e^x tan \, 2x+c$

22 )

The function $f(x)=cos^{-1}(sin \, x +cos \, x)$ strictly decreasing function in the interval -

1) $[0, \frac{\pi}{4}]$

2) $(0, \frac{\pi}{4}]$

3) $[0, \frac{\pi}{4})$

4) $(0, \frac{\pi}{4})$

23 )

If the curves $x=y^2$ and $xy=k$ cut at right angles, then

1) $8k^2=-1$

2) $6k^2=1$

3) $10k^2=1$

4) $8k^2=1$

24 )

An open box with a square base is to be made out of a given cardboard of area $c^2$ square units. The maximum volume of the box is

1) $\frac{c^3}{6\sqrt{2}}$

2) $\frac{c^3}{6\sqrt{3}}$

3) $\frac{5c^3}{6\sqrt{3}}$

4) $\frac{5c^2}{6\sqrt{3}}$

25 )

A bag contains 1 white and 6 red balls. Another bag contains 4 white and 3 red balls. One of the two bag is selected at random and a ball is drawn from it, which is found to be white. Find the probability that the ball drawn is from the bag A.

1) $\frac{2}{3}$

2) $\frac{3}{5}$

3) $\frac{2}{5}$

4) $\frac{1}{5}$

26 )

The constraints $-x+y \le 1$, $-x+3y \le 9$ and $x \ge 0$, $y \ge 0$ defines on

1) both bounded and unbounded space

2) bounded feasible space

3) unbounded feasible space

4) none of these

27 )

Let $f : R \rightarrow (-1, \, 1)$ ,is given by $f(x)=\frac{10^x-10^{-x}}{10^x+10^{-x}}$ is invertible. Find $f^{-1}(x)$.

1) $\frac{1}{2} log_{10} \frac{1-x}{1+x}$

2) $\frac{1}{2} log_{10} \frac{1+x}{1-x}$

3) $\frac{1}{2} log_{10} \frac{1+2x}{1-2x}$

4) $\frac{1}{2} log_{10} \frac{1-2x}{1+2x}$

28 )

Solve the following system of equations using matrix method : 2x - 3y + 5z = 11 3x + 2y - 4z = -5 x + y - 2z = -3

1) x = -1, y = 2, z = 3

2) x = 1, y = 2, z = -3

3) x = 1, y = 2, z = 3

4) x = 1, y = -2, z = 3

29 )

Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane $\vec{r}.(-2\hat{j}+4\hat{k})+5=0$.

1) $\frac{13}{14} \sqrt{6}$, $(-\frac{1}{12}, \frac{25}{12},- \frac{2}{12})$

2) $\frac{13}{12} \sqrt{6}$, $(-\frac{1}{12}, \frac{25}{12},-\frac{2}{12})$

3) $\frac{13}{12} \sqrt{6}$, $(\frac{1}{12}, \frac{25}{12},- \frac{2}{12})$

4) $\frac{13}{12} \sqrt{6}$, $(-\frac{1}{12}, \frac{25}{12},- \frac{2}{12})$

30 )

Find the area bounded by curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x}{a}+\frac{y}{b}=1$.

1) $(\frac{\pi ab}{3} -\frac{ab}{2})$ sq. units

2) $(\frac{\pi ab}{4} -\frac{ab}{2})$ sq. units

3) $(\frac{\pi ab}{4} +\frac{ab}{2})$ sq. units

4) $(\frac{\pi ab}{4} -\frac{ab}{3})$ sq. units

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1 ) \begin{bmatrix} 2 & 3 & 6 \\ 4 & 6 & 8 \\ 6 & 9 & 12 \\ \end{bmatrix} 2 ) \begin{bmatrix} 6 & 0 \\ 0 & 6 \\ \end{bmatrix} 3 ) 12, -2 4 ) a skew symmetric square matrix 5 ) 0 6 ) 1 7 ) R - {3} 8 ) 0 9 ) $\frac{2x}{x^2 + 1} - \frac{2 ln(x^2 + 1) }{(2x + 1)^2} $ 10 ) $\frac{3}{4t}$ 11 ) 0 12 ) $\frac{x^2}{3} - x - 2 tan^{-1}x + c$ 13 ) $\int_{0}^a \, [f(x)+f(-x)] \, dx$ 14 ) -0.32 15 ) $y(x^2 + 1)=c$ 16 ) $\vec{a} \perp \vec{b}$ 17 ) 2 18 ) $\frac{3}{5}$ 19 ) $\frac{\pi}{8} log \, 2$ 20 ) $\frac{1}{\sqrt{2}}tan^{-1}[\frac{1}{\sqrt{2}}(x-\frac{1}{x})]+c$ 21 ) $e^x cot \, 2x+c$ 22 ) $[0, \frac{\pi}{4}]$ 23 ) $8k^2=1$ 24 ) $\frac{c^3}{6\sqrt{3}}$ 25 ) $\frac{1}{5}$ 26 ) both bounded and unbounded space 27 ) $\frac{1}{2} log_{10} \frac{1+x}{1-x}$ 28 ) x = 1, y = 2, z = 3 29 ) $\frac{13}{12} \sqrt{6}$, $(-\frac{1}{12}, \frac{25}{12},- \frac{2}{12})$ 30 ) $(\frac{\pi ab}{4} -\frac{ab}{2})$ sq. units

Mathematics - XII

The Model Paper - 2

Year : 2025 - 26

Time - 3 marks 0 minutes

Full Marks - 80

Previous Set

Cur Set 2

Next Set

Pages=>

Not Answered 30 Answered 0

In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 2

Instructions

Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 2

Close

×

1 ) Find a 2 x 2 matrix $\, A=[a_{ij} \,] $, $\, a_{ij} \, =\, 2i + 3j - 6$.

2 ) If A is an invertible square matrix then $A^{-1} = ?$

3 ) Find the value of \begin{vmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \\ \end{vmatrix}

4 ) If A is 3-rowed square matrix and |A| = 5, then |adj A| = ?

5 ) Find the value of \begin{equation} \begin{vmatrix} cos \, 70^o & sin \, 20^o \\ sin \, 70^o & cos \, 20^o \\ \end{vmatrix} \end{equation}

6 ) \begin{equation*} g(x)=\begin{cases} x \quad &\text{if} \, x \in \mathbb{Q} \\ -x \quad &\text{if} \, x \notin \mathbb{Q} \\ \end{cases} \end{equation*}

7 ) The function $\, f(x) = e^{-|x|} \,$ is

8 ) Evaluate $\,\, \lim_{x \to 1} \frac{x^2 - 1}{x - 1}$

9 ) Determine the derivative of h(x) = sin(2x) + cos(3x).

10 ) In a curve $y=t^2+2t$ and $x=t^3$, find the slope of $\frac{dy}{dx}$ at x=5.

11 ) if $tan^{-1}x = \frac{\pi}{4} - tan^{-1} \frac{1}{3}$, then x = ?

12 ) Evaluate $\int \sqrt{e^x - 1} \, dx$ .

13 ) Evaluate $\int_1^2 \, |x^2 -3x + 2| \, dx$ .

14 ) A circular metal plate expands upon heating so that its radius increases by 3%. Find the approximate increase in the area of the plate, it being given that the radius of the plate was 10 cm before heating.

15 ) The solution of differential equation $\frac{dy}{dx} = 2^{x+y}$.

16 ) If $|\vec{a}|=\sqrt{26}$, $|\vec{b}|=7$ and $|\vec{a} \, \times \, \vec{b}| =35 $, then $\vec{a} \,. \, \vec{b}$ is

17 ) 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 = ?

18 ) 8 coins are tossed simultaneously. The probability of getting 6 heads is

19 ) Evaluate $\int_0^2 \, x\sqrt{2-x} \, dx$ .

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

21 ) Evaluate $\int \, \frac{3x+1}{(x+2)(x-2)^2} \, dx$ .

22 ) The function $f(x)=-\frac{x}{2}+sin\, x$ is strictly increasing in

23 ) Find the angle of intersection of the curves $y^2=2a$ and $x^2+y^2=8$

24 ) If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, the area of the triangle is maximum when angle between is

25 ) Suppose 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

26 ) The minimum value of P=x+3y subject to constraints $2x+y \le 20$, $x+2y \le 20$ and $x \ge 0$, $y \ge 0$ is

27 ) 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 $f \, o \, f \, = \, I_R$.

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

29 ) Find the length and the equations of the line of shortest distance between the lines $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$

30 ) Find the area bounded by curve $y=x^2+2 $, and the lines y=x, x=0 and x=3 .

Close

1 )

Find a 2 x 2 matrix $\, A=[a_{ij} \,] $, $\, a_{ij} \, =\, 2i + 3j - 6$.

1) \begin{bmatrix} -1 & 2 \\ 1 & 4 \\ \end{bmatrix}

2) \begin{bmatrix} -1 & 3 \\ 1 & 4 \\ \end{bmatrix}

3) \begin{bmatrix} -1 & 2 \\ 2 & 4 \\ \end{bmatrix}

4) \begin{bmatrix} -1 & -2 \\ 1 & 4 \\ \end{bmatrix}

2 )

If A is an invertible square matrix then $A^{-1} = ?$

1) $\frac{1}{|A|}$

2) |A|

3) 1

4) none of these

3 )

Find the value of \begin{vmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \\ \end{vmatrix}

1) -1

2) 1

3) 0

4) none of these

4 )

If A is 3-rowed square matrix and |A| = 5, then |adj A| = ?

1) 5

2) 25

3) 125

4) 75

5 )

Find the value of \begin{equation} \begin{vmatrix} cos \, 70^o & sin \, 20^o \\ sin \, 70^o & cos \, 20^o \\ \end{vmatrix} \end{equation}

1) 1

2) 0

3) $cos \, 50^o$

4) $sin \, 50^o$

6 )

\begin{equation*} g(x)=\begin{cases} x \quad &\text{if} \, x \in \mathbb{Q} \\ -x \quad &\text{if} \, x \notin \mathbb{Q} \\ \end{cases} \end{equation*}

1) g(x) is continuous function at x=1

2) g(x) is continuous function

3) g(x) is discontinuous function

4) none of these

7 )

The function $\, f(x) = e^{-|x|} \,$ is

1) continuous everywhere and differentiable at x = 0

2) not continuous at x = 0

3) continuous differentiable everywhere

4) continuous everywhere but not differentiable at x = 0

8 )

Evaluate $\,\, \lim_{x \to 1} \frac{x^2 - 1}{x - 1}$

1) 1

2) 4

3) 2

4) 3

9 )

Determine the derivative of h(x) = sin(2x) + cos(3x).

1) 2cos(2x) + 3sin(3x)

2) 2cos(2x) - 3sin(3x)

3) 3cos(2x) - 2sin(3x)

4) 3cos(2x) + 2sin(3x)

10 )

In a curve $y=t^2+2t$ and $x=t^3$, find the slope of $\frac{dy}{dx}$ at x=5.

1) $\frac{6}{25}$

2) $\frac{3}{25}$

3) $\frac{4}{25}$

4) $\frac{4}{35}$

11 )

if $tan^{-1}x = \frac{\pi}{4} - tan^{-1} \frac{1}{3}$, then x = ?

1) 2

2) $\frac{1}{2}$

3) $\frac{1}{3}$

4) 1

12 )

Evaluate $\int \sqrt{e^x - 1} \, dx$ .

1) $2 \sqrt{e^x - 1} + c$

2) $2 \sqrt{e^x - 1} - 2 tan^{-1}\sqrt{e^x - 1} + c$

3) $2 \sqrt{e^x - 1} + 2 tan^{-1}\sqrt{e^x - 1} + c$

4) $2 \sqrt{e^x - 1} - tan^{-1}\sqrt{e^x - 1} + c$

13 )

Evaluate $\int_1^2 \, |x^2 -3x + 2| \, dx$ .

1) $- \frac{1}{6}$

2) $\frac{1}{6}$

3) $\frac{2}{3}$

4) $\frac{2}{5}$

14 )

A circular metal plate expands upon heating so that its radius increases by 3%. Find the approximate increase in the area of the plate, it being given that the radius of the plate was 10 cm before heating.

1) 9%

2) 4%

3) 6%

4) none of these

15 )

The solution of differential equation $\frac{dy}{dx} = 2^{x+y}$.

1) $2^x+2^{-y}=c$

2) $2^x+2^{y}=c$

3) $2^x-2^{-y}=c$

4) $2^{-x}+2^y=c$

16 )

If $|\vec{a}|=\sqrt{26}$, $|\vec{b}|=7$ and $|\vec{a} \, \times \, \vec{b}| =35 $, then $\vec{a} \,. \, \vec{b}$ is

1) 4

2) 5

3) 8

4) 7

17 )

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 = ?

1) $\frac{10}{7}$

2) $-\frac{10}{7}$

3) $-\frac{5}{7}$

4) $\frac{5}{7}$

18 )

8 coins are tossed simultaneously. The probability of getting 6 heads is

1) $\frac{249}{256}$

2) $\frac{37}{256}$

3) $\frac{57}{64}$

4) $\frac{7}{64}$

19 )

Evaluate $\int_0^2 \, x\sqrt{2-x} \, dx$ .

1) $\frac{16\sqrt{2}}{15}$

2) $\frac{14\sqrt{2}}{15}$

3) $\frac{16\sqrt{3}}{15}$

4) $\frac{12\sqrt{2}}{15}$

20 )

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

1) $\frac{4}{15}x \sqrt{x} (5-3x) + c$

2) $\frac{2}{15}x \sqrt{x} (5+3x) + c$

3) $\frac{2}{15}x \sqrt{x} (5-3x) + c$

4) none of these

21 )

Evaluate $\int \, \frac{3x+1}{(x+2)(x-2)^2} \, dx$ .

1) $\frac{5}{16} log|\frac{x-2}{x+2}| + \frac{7}{4(x-2)} +c$

2) $\frac{5}{18} log|\frac{x-2}{x+2}| - \frac{7}{4(x-2)} +c$

3) $\frac{5}{16} log|\frac{x-2}{x+2}| - \frac{7}{4(x-2)} +c$

4) $\frac{5}{16} log|\frac{x-2}{x+2}| - \frac{5}{4(x-2)} +c$

22 )

The function $f(x)=-\frac{x}{2}+sin\, x$ is strictly increasing in

1) $-\frac{\pi}{3} \le x \lt \frac{\pi}{3} $

2) $-\frac{\pi}{3} \le x \le \frac{\pi}{3} $

3) $-\frac{\pi}{3} \le x \lt \frac{\pi}{3} $

4) $-\frac{\pi}{3} \lt x \lt \frac{\pi}{3} $

23 )

Find the angle of intersection of the curves $y^2=2a$ and $x^2+y^2=8$

1) $tan^{-1}2$

2) $tan^{-1}3$

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

4) $tan^{-1}(-2)$

24 )

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, the area of the triangle is maximum when angle between is

1) $\frac{\pi}{3}$

2) $\frac{\pi}{4}$

3) $\frac{2\pi}{3}$

4) $\frac{\pi}{6}$

25 )

Suppose 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

1) $\frac{5}{6}$

2) $\frac{3}{5}$

3) $\frac{2}{3}$

4) $\frac{3}{4}$

26 )

The minimum value of P=x+3y subject to constraints $2x+y \le 20$, $x+2y \le 20$ and $x \ge 0$, $y \ge 0$ is

1) 10

2) 20

3) 25

4) none of these

27 )

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 $f \, o \, f \, = \, I_R$.

1) either a=1 and b=0 or a=1 and $b \, \in \, R$

2) either a=0 and b=1 or a=-1 and $b \, \in \, R$

3) either a=1 and b=1 or a=-1 and $b \, \in \, R$

4) either a=1 and b=0 or a=-1 and $b \, \in \, R$

28 )

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

1) x = -3, y = 2, z = -1

2) x = 3, y = 2, z = -1

3) x = -3, y = 2, z = 1

4) x = -3, y = 2, z = -2

29 )

Find the length and the equations of the line of shortest distance between the lines $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$

1) $3\sqrt{30}$ units, $\frac{x-3}{2}=\frac{y-8}{5}=\frac{z-3}{-1}$

2) $3\sqrt{30}$ units, $\frac{x-3}{-2}=\frac{y-8}{5}=\frac{z-3}{-1}$

3) $3\sqrt{30}$ units, $\frac{x-3}{3}=\frac{y-8}{5}=\frac{z-3}{-1}$

4) $3\sqrt{30}$ units, $\frac{x-3}{2}=\frac{y-8}{5}=\frac{z-5}{-1}$

30 )

Find the area bounded by curve $y=x^2+2 $, and the lines y=x, x=0 and x=3 .

1) $\frac{18}{5}$ sq. units

2) $\frac{21}{5}$ sq. units

3) $\frac{21}{4}$ sq. units

4) $\frac{21}{2}$ sq. units

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1 ) \begin{bmatrix} -1 & 2 \\ 1 & 4 \\ \end{bmatrix} 2 ) $\frac{1}{|A|}$ 3 ) 0 4 ) 25 5 ) 0 6 ) g(x) is discontinuous function 7 ) continuous everywhere but not differentiable at x = 0 8 ) 2 9 ) 2cos(2x) - 3sin(3x) 10 ) $\frac{4}{25}$ 11 ) $\frac{1}{2}$ 12 ) $2 \sqrt{e^x - 1} - 2 tan^{-1}\sqrt{e^x - 1} + c$ 13 ) $\frac{1}{6}$ 14 ) 6% 15 ) $2^x+2^{-y}=c$ 16 ) 7 17 ) $-\frac{10}{7}$ 18 ) $\frac{37}{256}$ 19 ) $\frac{16\sqrt{2}}{15}$ 20 ) $\frac{2}{15}x \sqrt{x} (5-3x) + c$ 21 ) $\frac{5}{16} log|\frac{x-2}{x+2}| - \frac{7}{4(x-2)} +c$ 22 ) $-\frac{\pi}{3} \le x \le \frac{\pi}{3} $ 23 ) $tan^{-1}3$ 24 ) $\frac{\pi}{3}$ 25 ) $\frac{2}{3}$ 26 ) none of these 27 ) either a=1 and b=0 or a=-1 and $b \, \in \, R$ 28 ) x = -3, y = 2, z = -1 29 ) $3\sqrt{30}$ units, $\frac{x-3}{2}=\frac{y-8}{5}=\frac{z-3}{-1}$ 30 ) $\frac{21}{2}$ sq. units

Mathematics - XII

The Model Paper - 3

Year : 2025 - 26

Time - 3 marks 0 minutes

Full Marks - 80

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Cur Set 3

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In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 3

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Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 3

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1 ) if \begin{equation*} A = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix} \end{equation*} and $f(x)=x^2 - 3x +5$, find f(A).

2 ) If \begin{equation*} A = \begin{bmatrix} 1 & \lambda & 2 \\ 1 & 2 & 5 \\ 2 & 1 & 1 \\ \end{bmatrix} \end{equation*}, is not invertible then the value of $\, \lambda \,$ is

3 ) A and B are two determinant of the same order, such that |A|=20 and |A|=-20, then find the value of |AB|.

4 ) For square matrices A and B of the same order, we have adj(AB) = ?

5 ) If matrix A=[-5], what is the value of det A ?

6 ) \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

7 ) \begin{equation*} Let f(x) = \lvert x \rvert = \left\{ \begin{array}{ll} ax^2 + 1 & \quad x > 1 \\ x + \frac{1}{2} & \quad x \le 1 \end{array} \right. \end{equation*}, then f(x) is derivable at x = 1 if

8 ) Evaluate $\,\, \lim_{x \to 0} \frac{x^2 - 3x}{x}$

9 ) Find the derivative of the function $y = (2x - 1)^3 (x + 2)^2.$

10 ) Calculate the derivative of x = sin t and y = cos t.

11 ) The value of cot[$ tan^{-1} x + cot^{-1} x $]

12 ) Evaluate $\int xin^3x \, cos^3x \, dx$ .

13 ) Evaluate $\int_0^1 \, |2x - 1| \, dx$ .

14 ) A man 1.8 meters tall walks directly away from a lamp post, whose height is 9 meters, at the rate 2 m/s. The rate at which his shadow lengthens is

15 ) The solution of differential equation $x \sqrt{1+y^2} dx +y \sqrt{1+x^2} dy = 0 $ is.

16 ) If $\hat{a}, \, \hat{b}, \, \hat{c} $ are mutually perpendicular unit vectors, then $ |\vec{a} + \vec{b} + \vec{c} | = ? $

17 ) The angle between any two diagonals of cube is

18 ) An unbiased die is tossed twice. What is the probability of getting a 4, 5 or 6 on the first toss and a 1, 2, 3 or 4 on the second toss ?

19 ) Evaluate $\int_0^{\frac{\pi}{2}} \, log(sin \, x) \, dx$ .

20 ) Evaluate $\int \, \frac{cos(x+a)}{sin(x+b)} \, dx$ .

21 ) Evaluate $\int \,x. log|x+1| \, dx$ .

22 ) The least possible value of k for which the function $f(x)= x^2+kx+1$ may be increasing on [1, 2].

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

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 ) 8 coins are tossed at a time, the probability of getting at least 6 heads up, is

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 domain and range of $f(x)=\frac{1}{\sqrt{x+2}}$

28 ) Solve the following system of equations using matrix method : x - y + z = 1 2x + y - z = 2 x - 2y - z = 4

29 ) Find the point of intersection of the lines $\frac{x+4}{3}=\frac{y+6}{5}=\frac{z-1}{-2}$ and 3x - 2y + z + 5 = 0 = 2x + 3y + 4z - 4.

30 ) Find the area bounded by curves $y^2 = 2x - x^2 $ and the x-axis.

Close

1 )

if \begin{equation*} A = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix} \end{equation*} and $f(x)=x^2 - 3x +5$, find f(A).

1) \begin{bmatrix} 15 & -6 \\ -6 & 21 \end{bmatrix}

2) \begin{bmatrix} 15 & -6 \\ -8 & 21 \end{bmatrix}

3) \begin{bmatrix} 15 & -5 \\ -8 & 21 \end{bmatrix}

4) \begin{bmatrix} 14 & -6 \\ -8 & 21 \end{bmatrix}

2 )

If \begin{equation*} A = \begin{bmatrix} 1 & \lambda & 2 \\ 1 & 2 & 5 \\ 2 & 1 & 1 \\ \end{bmatrix} \end{equation*}, is not invertible then the value of $\, \lambda \,$ is

1) 0

2) -1

3) 2

4) 1

3 )

A and B are two determinant of the same order, such that |A|=20 and |A|=-20, then find the value of |AB|.

1) -400

2) 400

3) 200

4) 500

4 )

For square matrices A and B of the same order, we have adj(AB) = ?

1) adj(B) adj(A)

2) adj(A) adj(B)

3) |AB|

4) none of these

5 )

If matrix A=[-5], what is the value of det A ?

1) -5

2) 5

3) 0

4) none of these

6 )

\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

1) 0

2) 1

3) -1

4) none of these

7 )

\begin{equation*} Let f(x) = \lvert x \rvert = \left\{ \begin{array}{ll} ax^2 + 1 & \quad x > 1 \\ x + \frac{1}{2} & \quad x \le 1 \end{array} \right. \end{equation*}, then f(x) is derivable at x = 1 if

1) 0

2) $\frac{1}{2}$

3) 1

4) 2

8 )

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

1) 1

2) 4

3) 0

4) -3

9 )

Find the derivative of the function $y = (2x - 1)^3 (x + 2)^2.$

1) $3(2x - 1)^2 (x + 3)^2 + 2(2x - 1)^3 2(x + 2)$

2) $3(2x - 1)^2 (x + 2)^2 + 2(2x - 1)^3 3(x + 2)$

3) $3(2x - 1)^2 (x + 2)^2 - 2(2x - 1)^3 2(x + 2)$

4) $3(2x - 1)^2 (x + 2)^2 + 2(2x - 1)^3 2(x + 2)$

10 )

Calculate the derivative of x = sin t and y = cos t.

1) tan t

2) -tan t

3) -cot t

4) cot t

11 )

The value of cot[$ tan^{-1} x + cot^{-1} x $]

1) -1

2) 0

3) 2

4) 1

12 )

Evaluate $\int xin^3x \, cos^3x \, dx$ .

1) $\frac{1}{4}sin^4x + \frac{1}{6}sin^6x + c$

2) $\frac{1}{4}sin^4x - \frac{1}{6}cos^6x + c$

3) $\frac{1}{4}sin^4x - \frac{1}{6}sin^6x + c$

4) none of these

13 )

Evaluate $\int_0^1 \, |2x - 1| \, dx$ .

1) 2

2) $\frac{1}{2}$

3) 1

4) 0

14 )

A man 1.8 meters tall walks directly away from a lamp post, whose height is 9 meters, at the rate 2 m/s. The rate at which his shadow lengthens is

1) 0.5 m/s

2) 0.2 m/s

3) 0.6 m/s

4) 0.4 m/s

15 )

The solution of differential equation $x \sqrt{1+y^2} dx +y \sqrt{1+x^2} dy = 0 $ is.

1) $ \sqrt{1+y^2} - \sqrt{1+x^2} = c $

2) $ sin^x +sin^y = c $

3) $ tan^x +tan^y = c $

4) $ \sqrt{1+y^2} +\sqrt{1+x^2} = c $

16 )

If $\hat{a}, \, \hat{b}, \, \hat{c} $ are mutually perpendicular unit vectors, then $ |\vec{a} + \vec{b} + \vec{c} | = ? $

1) $\sqrt{3}$

2) $\sqrt{2}$

3) 1

4) none of these

17 )

The angle between any two diagonals of cube is

1) $cos^{-1}\frac{2}{3}$

2) $cos^{-1}\frac{1}{3}$

3) $cos^{-1}\frac{3}{4}$

4) none of these

18 )

An unbiased die is tossed twice. What is the probability of getting a 4, 5 or 6 on the first toss and a 1, 2, 3 or 4 on the second toss ?

1) $\frac{3}{5}$

2) $\frac{2}{5}$

3) $\frac{2}{3}$

4) $\frac{1}{3}$

19 )

Evaluate $\int_0^{\frac{\pi}{2}} \, log(sin \, x) \, dx$ .

1) $\frac{\pi}{8} \, log \, 2 $

2) $- \frac{\pi}{2} \, log \, 2 $

3) $- \frac{\pi}{3} \, log \, 2 $

4) $- \frac{\pi}{4} \, log \, 2 $

20 )

Evaluate $\int \, \frac{cos(x+a)}{sin(x+b)} \, dx$ .

1) cos(a-b) log|cos(x+b)| - x sin(a-b) +c

2) cos(a-b) log|sin(x+b)| - x sin(a-b) +c

3) cos(a-b) log|sin(x+b)| - x cos(a-b) +c

4) none of these

21 )

Evaluate $\int \,x. log|x+1| \, dx$ .

1) $\frac{1}{3}(x^2-1)log|x+1| - \frac{1}{4}x^2 +\frac{1}{2}x +c$

2) $\frac{1}{2}(x^2-1)log|x+1| - \frac{1}{4}x^2 -\frac{1}{2}x +c$

3) $\frac{1}{2}(x^2-1)log|x+1| - \frac{1}{4}x^2 +\frac{1}{2}x +c$

4) none of these

22 )

The least possible value of k for which the function $f(x)= x^2+kx+1$ may be increasing on [1, 2].

1) -1

2) -2

3) 1

4) 2

23 )

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

1) (2, -7)

2) (2, -9)

3) (2, -5)

4) (2, 9)

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.

1) $\frac{15}{5-\sqrt{3}}$m

2) $\frac{16}{6-\sqrt{3}}$m

3) $\frac{14}{4-\sqrt{3}}$m

4) $\frac{15}{6-\sqrt{3}}$m

25 )

8 coins are tossed at a time, the probability of getting at least 6 heads up, is

1) $\frac{57}{64}$

2) $\frac{37}{256}$

3) $\frac{7}{64}$

4) none of these

26 )

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

1) 10

2) 6

3) 15

4) 25

27 )

Find the domain and range of $f(x)=\frac{1}{\sqrt{x+2}}$

1) $Domain=(1, \, \infty )$ and $Range= x \in R^+$

2) $Domain=(2, \, \infty )$ and $Range= x \in R$

3) $Domain=(2, \, \infty )$ and $Range= x \in R^+$

4) $Domain=(2, \, \infty )$ and $Range= x \in R^-$

28 )

Solve the following system of equations using matrix method : x - y + z = 1 2x + y - z = 2 x - 2y - z = 4

1) x = 2, y = -1, z = -1

2) x = 1, y = -1, z = -1

3) x = -1, y = 1, z = -1

4) x = 1, y = 2, z = -1

29 )

Find the point of intersection of the lines $\frac{x+4}{3}=\frac{y+6}{5}=\frac{z-1}{-2}$ and 3x - 2y + z + 5 = 0 = 2x + 3y + 4z - 4.

1) (2, 4, 3)

2) (2, 4, -3)

3) (2, -4, -3)

4) (-2, 4, -3)

30 )

Find the area bounded by curves $y^2 = 2x - x^2 $ and the x-axis.

1) $\frac{5}{3}$ sq. units$

2) $\frac{4}{5}$ sq. units$

3) $\frac{4}{3}$ sq. units$

4) $\frac{3}{2}$ sq. units$

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1 ) \begin{bmatrix} 15 & -6 \\ -8 & 21 \end{bmatrix} 2 ) 1 3 ) -400 4 ) adj(B) adj(A) 5 ) -5 6 ) 1 7 ) $\frac{1}{2}$ 8 ) -3 9 ) $3(2x - 1)^2 (x + 2)^2 + 2(2x - 1)^3 2(x + 2)$ 10 ) -tan t 11 ) 0 12 ) $\frac{1}{4}sin^4x - \frac{1}{6}sin^6x + c$ 13 ) $\frac{1}{2}$ 14 ) 0.5 m/s 15 ) $ \sqrt{1+y^2} +\sqrt{1+x^2} = c $ 16 ) $\sqrt{3}$ 17 ) $cos^{-1}\frac{1}{3}$ 18 ) $\frac{1}{3}$ 19 ) $- \frac{\pi}{2} \, log \, 2 $ 20 ) cos(a-b) log|sin(x+b)| - xsin(a-b) +c 21 ) $\frac{1}{2}(x^2-1)log|x+1| - \frac{1}{4}x^2 +\frac{1}{2}x +c$ 22 ) -2 23 ) (2, -9) 24 ) $\frac{16}{6-\sqrt{3}}$m 25 ) $\frac{37}{256}$ 26 ) 15 27 ) $Domain=(2, \, \infty )$ and $Range= x \in R^+$ 28 ) x = 1, y = -1, z = -1 29 ) (2, 4, -3) 30 ) $\frac{4}{3}$ sq. units$

Mathematics - XII

The Model Paper - 4

Year : 2025 - 26

Time - 3 marks 0 minutes

Full Marks - 80

Previous Set

Cur Set 4

Next Set

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Not Answered 30 Answered 0

In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 4

Instructions

Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 4

Close

×

1 ) If matrix A = [1 2 3], find $AA^ \prime$

2 ) If \begin{equation*} A = \begin{bmatrix} cos \theta & -sin \theta \\ -sin \theta & cos \theta \\ \end{bmatrix} \end{equation*}, then $\, A^{-1} \,$ is

3 ) The area of triangle whose vertices are A(-2, -3), B(3, 2) and C(-1, 8).

4 ) If \begin{equation*} A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \end{equation*} , then adj.A = ?

5 ) Find the value of \begin{equation} \begin{vmatrix} a \, + \, ib & c \, + \, id \\ -c \, + \, id & a \, - \, ib \\ \end{vmatrix} \end{equation} where $\, i^2 = -1$

6 ) Find f(0), so that $\, f(x) = \frac{x}{1 - \sqrt{1-x}} \,is continuous at x=0 $

7 ) \begin{equation*} Let f(x) = \lvert x \rvert = \left\{ \begin{array}{ll} kx^2 & \quad x \le 2 \\ 3 & \quad x \gt 2 \end{array} \right. \end{equation*}, if f(x) is continuous then the value of k is

8 ) Evaluate $\,\, \lim_{x \to 0} \frac{sin x}{x}$

9 ) Calculate the derivative of $g(x) = e^x sin(x).$

10 ) The derivative of $tan^{-1}x$ with respect to $cot^{-1} x$

11 ) The value of $ \, cos^{-1}( cos \, 1540^o)$

12 ) Evaluate $\int sin \, 3x \, sin \, 2x \, dx$ .

13 ) Evaluate $\int_0^{\pi} \, |cos \, x| \, dx$ .

14 ) A ladder 5 meter long is leaning against a wall,the foot of the ladder being 3 meter away from the wall. If the lower end of the ladder is pulled away form the wall at the rate of 0.2 m/s. The rate of decrease of the upper end along the wall is

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

16 ) If $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, then $\frac{1}{2}|\hat{a} - \hat{b}|$ = ?

17 ) The ratio in which the plane 3x + 4y -5z = 1 divides the line joining the points (-2, 4 -6) and (3, -5, 8) is

18 ) If A and B are events such that P(A)=0.3, P(B)=0.2 and $P(A \cap B)=0.1$, then $P(\bar{A} \cap B)= ?$

19 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x tan \, x}{sec \, x +cos \, x} \, dx$ .

20 ) Evaluate $\int \, \frac{sin \, 2x}{a^2 \, cos^2x + b^2 \, sin^2x} \, dx$ .

21 ) Evaluate $\int \, \frac{sin^{-1}}{x^2} \, dx$ .

22 ) The intervals in which the function $f(x)=sin \, x + cos \, x$ is increasing or decreasing

23 ) The tangents to the curve $y=2x^3-4$ at the points x=2 and x=-2 are

24 ) A wire of length 25m is to be cut into two pieces. One of the wires is to be made into square and the other into circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum ?

25 ) 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 ?

26 ) By graphical method the solution of linear programming problem : maximum value of z=3x+5y subject to constraints $3x+2y \le 18$, $x \le 4$, $y \le 6$ and $x \ge 0$, $y \ge 0$ is

27 ) Let $f : [-1, \infty) \rightarrow [-1, \infty)$ ,is given by $f(x)=(x+1)^2-1$, where $x\ge -1$. Find S={$x : f(x)=f^{-1}(x)$}.

28 ) Solve the following system of equations using matrix method : 3x - 4y + 2z = -1 2x + 3y + 5z = 7 x + z = 2

29 ) Find the distance of the point with position vector $-\hat{i}-5\hat{j}-10\hat{k}$ from the point of intersection of the line $\vec{r}=2\hat{i}-\hat{j}-2\hat{k}+\lambda (3\hat{i}+4\hat{j}+12\hat{k})$ with the plane $\vec{r}.2\hat{i}-2\hat{j}+\hat{k}=5$

30 ) Find the area bounded by curves {$(x, y) \, : \, x^2 + y^2 \le 1 \le x+y $}.

Close

1 )

If matrix A = [1 2 3], find $AA^ \prime$

1) [12]

2) [14]

3) [24]

4) [0]

2 )

If \begin{equation*} A = \begin{bmatrix} cos \theta & -sin \theta \\ -sin \theta & cos \theta \\ \end{bmatrix} \end{equation*}, then $\, A^{-1} \,$ is

1) adj A

2) A

3) -adj A

4) -A

3 )

The area of triangle whose vertices are A(-2, -3), B(3, 2) and C(-1, 8).

1) 10 sq units

2) 15 sq units

3) 25 sq units

4) 20 sq units

4 )

If \begin{equation*} A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \end{equation*} , then adj.A = ?

1) \begin{bmatrix} d & -b \\ -c & -a \\ \end{bmatrix}

2) \begin{bmatrix} d & b \\ -c & a \\ \end{bmatrix}

3) \begin{bmatrix} d & -b \\ c & a \\ \end{bmatrix}

4) \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}

5 )

Find the value of \begin{equation} \begin{vmatrix} a \, + \, ib & c \, + \, id \\ -c \, + \, id & a \, - \, ib \\ \end{vmatrix} \end{equation} where $\, i^2 = -1$

1) $a^2+b^2+c^2-d^2$

2) $a^2+b^2-c^2+d^2$

3) $a^2+b^2+c^2+d^2$

4) $a^2-b^2+c^2+d^2$

6 )

Find f(0), so that $\, f(x) = \frac{x}{1 - \sqrt{1-x}} \,is continuous at x=0 $

1) 2

2) 3

3) 1

4) 0

7 )

\begin{equation*} Let f(x) = \lvert x \rvert = \left\{ \begin{array}{ll} kx^2 & \quad x \le 2 \\ 3 & \quad x \gt 2 \end{array} \right. \end{equation*}, if f(x) is continuous then the value of k is

1) $\frac{3}{5}$

2) $\frac{3}{4}$

3) $\frac{2}{5}$

4) $\frac{5}{4}$

8 )

Evaluate $\,\, \lim_{x \to 0} \frac{sin x}{x}$

1) 0

2) 1

3) -1

4) none of this

9 )

Calculate the derivative of $g(x) = e^x sin(x).$

1) $e^x sin(x) + e^x sin(x)$

2) $e^x sin(x) + e^x cos(x)$

3) $e^x sin(-x) + e^x cos(x)$

4) $e^x sin(x) + e^x cos(-x)$

10 )

The derivative of $tan^{-1}x$ with respect to $cot^{-1} x$

1) -1

2) 0

3) 1

4) non of these

11 )

The value of $ \, cos^{-1}( cos \, 1540^o)$

1) $105^o$

2) $60^o$

3) $100^o$

4) $80^o$

12 )

Evaluate $\int sin \, 3x \, sin \, 2x \, dx$ .

1) $\frac{1}{2} sin \, x - \frac{1}{10} sin \, 5x +c $

2) $\frac{1}{2} sin \, x + \frac{1}{10} sin \, 5x +c $

3) $- \frac{1}{2} sin \, x - \frac{1}{10} sin \, 5x +c $

4) $- \frac{1}{2} sin \, x + \frac{1}{10} sin \, 5x +c $

13 )

Evaluate $\int_0^{\pi} \, |cos \, x| \, dx$ .

1) 2

2) 4

3) 1

4) none of these

14 )

A ladder 5 meter long is leaning against a wall,the foot of the ladder being 3 meter away from the wall. If the lower end of the ladder is pulled away form the wall at the rate of 0.2 m/s. The rate of decrease of the upper end along the wall is

1) 5 m/s

2) 15 m/s

3) 12 m/s

4) 8 m/s

15 )

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

1) x tan y = c

2) x cos y = c

3) x sec y = c

4) x cot y = c

16 )

If $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, then $\frac{1}{2}|\hat{a} - \hat{b}|$ = ?

1) $sin \, \frac{\theta}{2}$

2) $cos \, \frac{\theta}{2}$

3) $tan \, \frac{\theta}{2}$

4) none of these

17 )

The ratio in which the plane 3x + 4y -5z = 1 divides the line joining the points (-2, 4 -6) and (3, -5, 8) is

1) 3 : 4

2) 3 : 5

3) 2 : 5

4) 1 : 4

18 )

If A and B are events such that P(A)=0.3, P(B)=0.2 and $P(A \cap B)=0.1$, then $P(\bar{A} \cap B)= ?$

1) 0.2

2) 0.4

3) 0.1

4) 0.5

19 )

Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x tan \, x}{sec \, x +cos \, x} \, dx$ .

1) $\frac{\pi ^2}{2}$

2) $\frac{\pi ^2}{4}$

3) $\frac{\pi ^2}{3}$

4) $\frac{2\pi ^2}{3}$

20 )

Evaluate $\int \, \frac{sin \, 2x}{a^2 \, cos^2x + b^2 \, sin^2x} \, dx$ .

1) $\frac{log|a^2 \, cos^2x + b^2 sin^2x|}{b^2 - a^2} +c$

2) $- \frac{log|a^2 \, cos^2x + b^2 sin^2x|}{b^2 - a^2} +c$

3) $\frac{log|a^2 \, cos^2x + b^2 sin^2x|}{a^2 - b^2} +c$

4) none of these

21 )

Evaluate $\int \, \frac{sin^{-1}}{x^2} \, dx$ .

1) $\frac{sin^{-1}x}{x}+log| \frac{1}{x} - \frac{\sqrt{1-x^2}}{x}|$

2) $-\frac{sin^{-1}x}{x}+log| \frac{1}{x} + \frac{\sqrt{1-x^2}}{x}|$

3) $-\frac{sin^{-1}x}{x}+log| \frac{1}{x} - \frac{\sqrt{1-x^2}}{x}|$

4) none of these

22 )

The intervals in which the function $f(x)=sin \, x + cos \, x$ is increasing or decreasing

1) increasing in $0 \le x \le \frac{3\pi}{4} $ and $\frac{3\pi}{4} \lt x \lt 2\pi $ ; decreasing in $\f

2) increasing in $0 \lt x \lt \frac{3\pi}{4} $ and $\frac{3\pi}{4} \lt x \lt 2\pi $ ; decreasing in $\f

3) increasing in $0 \lt x \lt \frac{3\pi}{4} $ and $\frac{3\pi}{4} \lt x \lt 2\pi $ ; decreasing in $\f

4) none of these

23 )

The tangents to the curve $y=2x^3-4$ at the points x=2 and x=-2 are

1) perpendicular

2) intersecting

3) parallel

4) none of these

24 )

A wire of length 25m is to be cut into two pieces. One of the wires is to be made into square and the other into circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum ?

1) $\frac{25\pi}{\pi +4}$ and $\frac{100}{\pi +4}$

2) $\frac{20\pi}{\pi +4}$ and $\frac{80}{\pi +4}$

3) $\frac{30\pi}{\pi +4}$ and $\frac{120}{\pi +4}$

4) none of these

25 )

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 ?

1) $\frac{3}{5}$

2) $\frac{2}{3}$

3) $\frac{2}{5}$

4) $\frac{4}{5}$

26 )

By graphical method the solution of linear programming problem : maximum value of z=3x+5y subject to constraints $3x+2y \le 18$, $x \le 4$, $y \le 6$ and $x \ge 0$, $y \ge 0$ is

1) x=2, y=0, z=36

2) x=4, y=6, z=27

3) x=2, y=6, z=36

4) x=4, y=6, z=48

27 )

Let $f : [-1, \infty) \rightarrow [-1, \infty)$ ,is given by $f(x)=(x+1)^2-1$, where $x\ge -1$. Find S={$x : f(x)=f^{-1}(x)$}.

1) {1, 2}

2) {-1, 0}

3) {0, 1}

4) {0, -1}

28 )

Solve the following system of equations using matrix method : 3x - 4y + 2z = -1 2x + 3y + 5z = 7 x + z = 2

1) x = 2, y = 3, z = -1

2) x = 2, y = 3, z = 1

3) x = -2, y = 3, z = -1

4) x = 2, y = -3, z = -1

29 )

Find the distance of the point with position vector $-\hat{i}-5\hat{j}-10\hat{k}$ from the point of intersection of the line $\vec{r}=2\hat{i}-\hat{j}-2\hat{k}+\lambda (3\hat{i}+4\hat{j}+12\hat{k})$ with the plane $\vec{r}.2\hat{i}-2\hat{j}+\hat{k}=5$

1) 10

2) 15

3) 13

4) 9

30 )

Find the area bounded by curves {$(x, y) \, : \, x^2 + y^2 \le 1 \le x+y $}.

1) $\frac{\pi -2}{4} sq.units$

2) $\frac{\pi +2}{4} sq.units$

3) $\frac{\pi -2}{5} sq.units$

4) $\frac{\pi -3}{4} sq.units$

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1 ) [14] 2 ) adj A 3 ) 15 sq units 4 ) \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix} 5 ) $a^2+b^2+c^2+d^2$ 6 ) 3 7 ) $\frac{3}{4}$ 8 ) 1 9 ) $e^x sin(x) + e^x cos(x)$ 10 ) -1 11 ) $100^o$ 12 ) $\frac{1}{2} sin \, x - \frac{1}{10} sin \, 5x +c $ 13 ) 2 14 ) 15 m/s 15 ) x cos y = c 16 ) $sin \, \frac{\theta}{2}$ 17 ) 3 : 4 18 ) 0.1 19 ) $\frac{\pi ^2}{4}$ 20 ) $\frac{log|a^2 \, cos^2x + b^2 sin^2x|}{b^2 - a^2} +c$ 21 ) $-\frac{sin^{-1}x}{x}+log| \frac{1}{x} - \frac{\sqrt{1-x^2}}{x}|$ 22 ) increasing in $0 \lt x \lt \frac{3\pi}{4} $ and $\frac{3\pi}{4} \lt x \lt 2\pi $ ; decreasing in $\f 23 ) parallel 24 ) $\frac{25\pi}{\pi +4}$ and $\frac{100}{\pi +4}$ 25 ) $\frac{2}{5}$ 26 ) x=2, y=6, z=36 27 ) {0, -1} 28 ) x = 2, y = 3, z = -1 29 ) 13 30 ) $\frac{\pi -2}{4} sq.units$

Mathematics - XII

The Model Paper - 5

Year : 2025 - 26

Time - 3 marks 0 minutes

Full Marks - 80

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In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 5

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Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 5

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1 ) If \begin{equation*} A = \begin{bmatrix} sin \alpha & cos \alpha \\ -cos \alpha & sin \alpha \\ \end{bmatrix} \end{equation*}, then $A^TA \, = \, ?$

2 ) \begin{equation*} A = \begin{bmatrix} ab & b^2 \\ -a^2 & ab \\ \end{bmatrix} \end{equation*}, is

3 ) Find the value of \begin{vmatrix} a & ab^2 & ac^2 \\ a^2b & 0 & bc^2 \\ a^2c & cb^2 & 0 \\ \end{vmatrix}

4 ) Solve for x and y : \begin{equation*} 2 \begin{bmatrix} 3 & 4 \\ 5 & x \\ \end{bmatrix} + \begin{bmatrix} 1 & y \\ 0 & 1 \\ \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 10 & 5 \\ \end{bmatrix} \end{equation*}

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

6 ) Find the value of p for which \begin{equation*} f(x)=\begin{cases} 5x - 4 \quad &\text{if} \, 0 < x \le 1 \\ 4x^2 +3px \quad &\text{if} \, 1 < x < 2 \\ \end{cases} \end{equation*}, is continuous at x = 1.

7 ) Let f(x) = $\sqrt{ x^2 \, - \, 3x \, -4 \, }$. Then domain of f(x) is

8 ) Evaluate $\,\, \lim_{x \to - \infty} \frac{2x^3 + 3x}{x^3 + 2x^2}$

9 ) Find the derivative of g(x) = ln(cos x).

10 ) The derivative of $sin^2x$ with respect to cos x

11 ) $sin^{-1}x + sin^{-1}y = \frac{2 \pi}{3} $, then $cos^{-1}x + cos^{-1}y = ? $

12 ) Evaluate $\int \frac{\sqrt{tan \, x} }{sin \, x \, \, cos \, x} \, dx$ .

13 ) Evaluate $\int_{-2}^1 \, \frac{|x|}{x} \, dx$ .

14 ) Find the length of the edge of the cube such that the rate of increase in its volume is the same as the rate of increase in its surface area.

15 ) The solution of differential equation $\frac{dy}{dx} = \frac{1-cos \, x}{1+cos \, x}$.

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

17 ) The foot of the perpendicular from the point A(1, 3, 4) on the plane 2x - y + z + 3 = 0 is

18 ) If A and B are events such that P(A)=0.4, P(B)=0.8 and $P(B/A)=0.6$, then $P(A/B)= ?$

19 ) Evaluate $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \, |sin \, x| \, dx$ .

20 ) Evaluate $\int \, (2x+4) \sqrt{x^2+4x+3} \, dx$ .

21 ) Evaluate $\int \, \frac{x^2+4}{x^4+16} \, dx$

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

23 ) Find the point on the parabola $y=(x-3)^2$, where the tangent is parallel to the chord joining the points (3, 0) and (4, 1).

24 ) If the length of three sides of a trapezium other than base are equal to 10 cm, then find the area of trapezium when it is maximum.

25 ) 12 cards numbered 1 to 12, are placed in a box, mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the card is more than 3, find the probability that it is an even number.

26 ) The minimum value of P=5x+7y subject to constraints $3x+2y \le 12$, $2x+3y \le 13$ and $x \ge 0$, $y \ge 0$ is

27 ) Let f : A $\rightarrow$ B and f{$(x, \, x^2)$ : x $\in$ A} the f : A $\rightarrow$ A is

28 ) Solve the following system of equations using matrix method : 2x + 3y + 3z = 5 x - 2y + z = -4 3x - y - 2z = 3

29 ) Find the point of intersection of the lines $\frac{x-1}{2}=\frac{y-1}{2}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-8}{5}=\frac{z}{1}$

30 ) Find the area bounded by curves {$(x, y) \, : \, x^2 + y^2 \le 2ax , \, y^2 \gt ax, a \gt 0, \, x \gt 0, \, y \gt 0, \,$}.

Close

1 )

If \begin{equation*} A = \begin{bmatrix} sin \alpha & cos \alpha \\ -cos \alpha & sin \alpha \\ \end{bmatrix} \end{equation*}, then $A^TA \, = \, ?$

1) \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}

2) \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}

3) \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}

4) \begin{bmatrix} -1 & 0 \\ 0 & 1 \\ \end{bmatrix}

2 )

\begin{equation*} A = \begin{bmatrix} ab & b^2 \\ -a^2 & ab \\ \end{bmatrix} \end{equation*}, is

1) idempotent

2) nilpotent

3) orthogonal

4) none of these

3 )

Find the value of \begin{vmatrix} a & ab^2 & ac^2 \\ a^2b & 0 & bc^2 \\ a^2c & cb^2 & 0 \\ \end{vmatrix}

1) $a^3b^3c^3$

2) $3a^3b^3c^3$

3) $4a^3b^3c^3$

4) $2a^3b^3c^3$

4 )

Solve for x and y : \begin{equation*} 2 \begin{bmatrix} 3 & 4 \\ 5 & x \\ \end{bmatrix} + \begin{bmatrix} 1 & y \\ 0 & 1 \\ \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 10 & 5 \\ \end{bmatrix} \end{equation*}

1) x=-2, y=-8

2) x=2, y=-8

3) x=-2, y=8

4) x=2, y=8

5 )

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

1) 1

2) $\frac{2}{3}$

3) $\frac{1}{2}$

4) $\frac{1}{3}$

6 )

Find the value of p for which \begin{equation*} f(x)=\begin{cases} 5x - 4 \quad &\text{if} \, 0 < x \le 1 \\ 4x^2 +3px \quad &\text{if} \, 1 < x < 2 \\ \end{cases} \end{equation*}, is continuous at x = 1.

1) 0

2) -1

3) 1

4) none of these

7 )

Let f(x) = $\sqrt{ x^2 \, - \, 3x \, -4 \, }$. Then domain of f(x) is

1) $-1 \, \le \, x \, \le \, 4 \,$

2) $1 \, \le \, x \, \le \, 4 \,$

3) $-4 \, \le \, x \, \le \, 1 \,$

4) none of these

8 )

Evaluate $\,\, \lim_{x \to - \infty} \frac{2x^3 + 3x}{x^3 + 2x^2}$

1) 1

2) 2

3) 3

4) -3

9 )

Find the derivative of g(x) = ln(cos x).

1) tan x

2) -tan x

3) cot x

4) -cot x

10 )

The derivative of $sin^2x$ with respect to cos x

1) 2sin x

2) 2cos x

3) -2sin x

4) -2cos x

11 )

$sin^{-1}x + sin^{-1}y = \frac{2 \pi}{3} $, then $cos^{-1}x + cos^{-1}y = ? $

1) $\frac{2 \pi}{3}$

2) $\pi$

3) $\frac{\pi}{4}$

4) $\frac{\pi}{3}$

12 )

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

1) $ \sqrt{tan \, x} +c$

2) $2 \sqrt{sec \, x} +c$

3) $2 \sqrt{tan \, x} +c$

4) $2 \sqrt{cot \, x} +c$

13 )

Evaluate $\int_{-2}^1 \, \frac{|x|}{x} \, dx$ .

1) 3

2) 2.5

3) -1

4) none of these

14 )

Find the length of the edge of the cube such that the rate of increase in its volume is the same as the rate of increase in its surface area.

1) 4.5 cm

2) 2 cm

3) 4 cm

4) none of these

15 )

The solution of differential equation $\frac{dy}{dx} = \frac{1-cos \, x}{1+cos \, x}$.

1) $y=2tan \frac{x}{2}+x+c$

2) $y=2tan \frac{x}{2}-x+c$

3) $y=2cos \frac{x}{2}-x+c$

4) $y=2tan \, x-x+c$

16 )

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

1) $\frac{5}{\sqrt{7}}$

2) $\frac{5}{\sqrt{3}}$

3) $\frac{3}{\sqrt{6}}$

4) $\frac{5}{\sqrt{6}}$

17 )

The foot of the perpendicular from the point A(1, 3, 4) on the plane 2x - y + z + 3 = 0 is

1) (-1, 4, 3)

2) (-1, -4, 3)

3) (1, 4, -3)

4) none of these

18 )

If A and B are events such that P(A)=0.4, P(B)=0.8 and $P(B/A)=0.6$, then $P(A/B)= ?$

1) 0.6

2) 0.2

3) 0.4

4) 0.3

19 )

Evaluate $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \, |sin \, x| \, dx$ .

1) $2-\sqrt{2}$

2) $3-\sqrt{2}$

3) $2-\sqrt{3}$

4) $3-\sqrt{3}$

20 )

Evaluate $\int \, (2x+4) \sqrt{x^2+4x+3} \, dx$ .

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

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

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

4) $\frac{2}{3}(x^2+4x+3)^\frac{5}{2}$

21 )

Evaluate $\int \, \frac{x^2+4}{x^4+16} \, dx$

1) $\frac{1}{2\sqrt{2}} tan^{-1}\frac{x^2+4}{2\sqrt{2} \, x} +c$

2) $\frac{1}{2\sqrt{2}} tan^{-1}\frac{x^2-4}{2\sqrt{2} \, x}+c$

3) $-\frac{1}{2\sqrt{2}} tan^{-1}\frac{x^2-4}{2\sqrt{2} \, x}+c$

4) none of these

22 )

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

1) $- \infty \lt x \lt 2 $ and $5 \lt x \lt \infty $

2) $- \infty \lt x \lt -2 $ and $6 \lt x \lt \infty $

3) $- \infty \lt x \lt 2 $ and $6 \lt x \lt \infty $

4) $- \infty \lt x \lt 3 $ and $6 \lt x \lt \infty $

23 )

Find the point on the parabola $y=(x-3)^2$, where the tangent is parallel to the chord joining the points (3, 0) and (4, 1).

1) $( \frac{3}{4}, \, \frac{1}{4})$

2) $( \frac{7}{4}, \, \frac{3}{4})$

3) $( \frac{5}{4}, \, \frac{1}{4})$

4) $( \frac{7}{4}, \, \frac{1}{4})$

24 )

If the length of three sides of a trapezium other than base are equal to 10 cm, then find the area of trapezium when it is maximum.

1) $75\sqrt{3}cm^2$.

2) $75\sqrt{2}cm^2$.

3) $80\sqrt{3}cm^2$.

4) $65\sqrt{3}cm^2$.

25 )

12 cards numbered 1 to 12, are placed in a box, mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the card is more than 3, find the probability that it is an even number.

1) $\frac{3}{4}$

2) $\frac{2}{3}$

3) $\frac{4}{9}$

4) $\frac{5}{9}$

26 )

The minimum value of P=5x+7y subject to constraints $3x+2y \le 12$, $2x+3y \le 13$ and $x \ge 0$, $y \ge 0$ is

1) 20

2) 31

3) 15

4) none of these

27 )

Let f : A $\rightarrow$ B and f{$(x, \, x^2)$ : x $\in$ A} the f : A $\rightarrow$ A is

1) one-one but not onto.

2) one-one and onto.

3) neither one-one nor onto.

4) none of these

28 )

Solve the following system of equations using matrix method : 2x + 3y + 3z = 5 x - 2y + z = -4 3x - y - 2z = 3

1) x = 1, y = 2, z = -2

2) x = 1, y = 2, z = 1

3) x = -1, y = 2, z = -1

4) x = 1, y = 2, z = -1

29 )

Find the point of intersection of the lines $\frac{x-1}{2}=\frac{y-1}{2}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-8}{5}=\frac{z}{1}$

1) (1, -1, -1)

2) (-1, -1, -1)

3) (-1, 1, -1)

4) (-1, -1, 1)

30 )

Find the area bounded by curves {$(x, y) \, : \, x^2 + y^2 \le 2ax , \, y^2 \gt ax, a \gt 0, \, x \gt 0, \, y \gt 0, \,$}.

1) $\frac{a^2}{12}(3\pi-6) sq. \, units$

2) $\frac{a^2}{12}(3\pi-8) sq. \, units$

3) $\frac{a^2}{10}(3\pi+8) sq. \, units$

4) $\frac{a^2}{12}(3\pi-8) sq. \, units$

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1 ) \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} 2 ) nilpotent 3 ) $2a^3b^3c^3$ 4 ) x=2, y=-8 5 ) $\frac{1}{2}$ 6 ) -1 7 ) $-1 \, \le \, x \, \le \, 4 \,$ 8 ) 2 9 ) -tan x 10 ) -2cos x 11 ) $\frac{\pi}{3}$ 12 ) $2 \sqrt{tan \, x} +c$ 13 ) -1 14 ) 4 cm 15 ) $y=2tan \frac{x}{2}-x+c$ 16 ) $\frac{5}{\sqrt{6}}$ 17 ) (-1, 4, 3) 18 ) 0.3 19 ) $2-\sqrt{2}$ 20 ) $\frac{2}{3}(x^2+4x+3)^\frac{3}{2}$ 21 ) $\frac{1}{2\sqrt{2}} tan^{-1}\frac{x^2-4}{2\sqrt{2} \, x}+c$ 22 ) $- \infty \lt x \lt 2 $ and $6 \lt x \lt \infty $ 23 ) $( \frac{7}{4}, \, \frac{1}{4})$ 24 ) $75\sqrt{3}cm^2$. 25 ) $\frac{5}{9}$ 26 ) 31 27 ) neither one-one nor onto. 28 ) x = 1, y = 2, z = -1 29 ) (-1, -1, -1) 30 ) $\frac{a^2}{12}(3\pi-8) sq. \, units$

Mathematics - XII

The Model Paper - 6

Year : 2025 - 26

Time - 3 marks 0 minutes

Full Marks - 80

Previous Set

Cur Set 6

Next Set

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Not Answered 30 Answered 0

In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 6

Instructions

Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 6

Close

×

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

2 ) If Ais an invertible square matrix and k is an non-negative real number, then $(k .A)^{-1} \, = \, ?$

3 ) If the points A(a, 0), B(0, b) and C(1, 1) are collinear, then the value of $\frac{1}{a}+\frac{1}{b}$ is

4 ) If A and B are square matrices of the same order, then (A + B) (A - B) = ?

5 ) Evaluate \begin{vmatrix} x+1 & x \\ x & x - 1 \\ \end{vmatrix}

6 ) If \begin{equation*} g(x)=\begin{cases} \frac{x}{sin 3x} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, then the value of k.

7 ) Let $\, f(x) \, = \, [tan^2 x] \, $, [.] denotes the greatest function. Then

8 ) Evaluate $\,\, \lim_{x \to -\frac{\pi}{2}} tan x$

9 ) Determine the derivative of $h(x) = e^{3x^2 + 2x}.$

10 ) Find the derivative of $x = 3t^2 + 2t$ and $y = t^3 - 5t.$

11 ) The value of sin[$2 sin^{-1} \frac{4}{5}$]

12 ) Evaluate $\int \frac{sin^3x+cos^3x}{sin^2x \, cos^2x} \, dx$ .

13 ) Evaluate $\int_0^{2\pi} \, |sin \, x| \, dx$ .

14 ) The radius of a spherical soap bubble is increasing at the rate of 0.05cm/sec. Find the rate of increase in its volume when its radius is 4 cm.

15 ) The solution of differential equation $\frac{dy}{dx} = e^{x+y} + x^2.e^y$.

16 ) $\vec{a}, \, \vec{b} \, and \, \vec{c}$ are units vectors such that $\vec{a} + \vec{b} + \vec{c}=0$ then $\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}$ = ?

17 ) The line $\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+5}{4}$ meets the plane at the point

18 ) If A and B are events such that $P(A \cup B)=\frac{5}{6}$, $P(A \cap B)=\frac{1}{3}$ and $P(\bar{B} )=\frac{1}{2}$, then the events A and B are

19 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{cos^3 x}{sin^3 x +cos^3 x} \, dx$ .

20 ) Evaluate $\int \, \frac{sin \, 2x}{(a+b \, cos \, x)^2} \, dx$ .

21 ) Evaluate $\int \, cos^4x \, dx$ .

22 ) The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is decreasing -

23 ) At what points on the curve $x^2+y^2 -2x-4y+1=0$, is the tangent parallel to y-axis ?

24 ) A square piece of tin of side 24 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut so that the volume of the box is maximum ? Also find the maximum volume.

25 ) A purse contains 4 copper coins, 3 silver coins, the second purse contains 6 copper coins, 2 silver coins. A coin is taken out from any purse, the probability that it is copper coin is

26 ) The maximum value of P=6x+8y subject to constraints $2x+y \le 30$, $x+2y \le 24$ and $x \ge 0$, $y \ge 0$ is

27 ) Let f : R $\rightarrow$ R, such that $f(x)=\frac{2x-7}{4}$ be an invertible function. Find $f^{-1}$

28 ) Solve the following system of equations using matrix method : x + 1y + z = 4 -x + y + z = 0 x - 3y + z = 1

29 ) Find the image of the point having position vector $\vec{r}=\hat{i}+3\hat{j}+4\hat{k}$ $\vec{r}.(2\hat{i}-\hat{j}+\hat{k})+3=0$

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

Close

1 )

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

1) \begin{bmatrix} -cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix}

2) \begin{bmatrix} cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix}

3) \begin{bmatrix} cos 2 \alpha & sin 2 \alpha \\ sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix}

4) \begin{bmatrix} cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & -cos 2 \alpha \\ \end{bmatrix}

2 )

If Ais an invertible square matrix and k is an non-negative real number, then $(k .A)^{-1} \, = \, ?$

1) $k .A^{-1}$

2) $\frac{1}{k} .A^{-1}$

3) $- \frac{1}{k} .A^{-1}$

4) $-k .A^{-1}$

3 )

If the points A(a, 0), B(0, b) and C(1, 1) are collinear, then the value of $\frac{1}{a}+\frac{1}{b}$ is

1) 0

2) 1

3) 2

4) -1

4 )

If A and B are square matrices of the same order, then (A + B) (A - B) = ?

1) $A^2 - AB +BA + B^2 $

2) $A^2 - B^2 $

3) $A^2 + B^2 $

4) $A^2 - AB - BA + B^2 $

5 )

Evaluate \begin{vmatrix} x+1 & x \\ x & x - 1 \\ \end{vmatrix}

1) 2

2) 1

3) 0

4) -1

6 )

If \begin{equation*} g(x)=\begin{cases} \frac{x}{sin 3x} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, then the value of k.

1) $\frac{1}{4}$

2) $\frac{5}{3}$

3) $\frac{1}{3}$

4) $\frac{2}{3}$

7 )

Let $\, f(x) \, = \, [tan^2 x] \, $, [.] denotes the greatest function. Then

1) $f^{\prime}(1)$

2) $f^{\prime}(0)$

3) f(x) is not differentiable at x=0

4) none of these

8 )

Evaluate $\,\, \lim_{x \to -\frac{\pi}{2}} tan x$

1) 0

2) $\infty$

3) $-\infty$

4) 1

9 )

Determine the derivative of $h(x) = e^{3x^2 + 2x}.$

1) $(6x + 2) e^{3x^2 + 2}$

2) $(6x - 2) e^{3x^2 + 2x}$

3) $(6x + 2) e^{3x^2 + 2x}$

4) $(6x + 2) e^{3x^2 - 2x}$

10 )

Find the derivative of $x = 3t^2 + 2t$ and $y = t^3 - 5t.$

1) $\frac{3t^2 - 5}{6t + 2}$

2) $\frac{3t^2 + 5}{6t + 2}$

3) $\frac{3t^2 - 5}{6t - 2}$

4) none of these

11 )

The value of sin[$2 sin^{-1} \frac{4}{5}$]

1) $\frac{13}{25}$

2) $\frac{16}{25}$

3) $\frac{24}{25}$

4) none of these

12 )

Evaluate $\int \frac{sin^3x+cos^3x}{sin^2x \, cos^2x} \, dx$ .

1) sin x + cos x + c

2) sin x - cos x + c

3) sec x - cosec x + c

4) sec x + cosec x + c

13 )

Evaluate $\int_0^{2\pi} \, |sin \, x| \, dx$ .

1) 2

2) 4

3) -1

4) 3

14 )

The radius of a spherical soap bubble is increasing at the rate of 0.05cm/sec. Find the rate of increase in its volume when its radius is 4 cm.

1) $5 \pi cm^3/sec$

2) $2.2 \pi cm^3/sec$

3) $3 \pi cm^3/sec$

4) $3.2 \pi cm^3/sec$

15 )

The solution of differential equation $\frac{dy}{dx} = e^{x+y} + x^2.e^y$.

1) $e^{x-y} + \frac{x^3}{3}=c$

2) $e^{x+y} + \frac{x^3}{3}=c$

3) $e^x-e^{-y} + \frac{x^3}{3}=c$

4) $e^x+e^{-y} + \frac{x^3}{3}=c$

16 )

$\vec{a}, \, \vec{b} \, and \, \vec{c}$ are units vectors such that $\vec{a} + \vec{b} + \vec{c}=0$ then $\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}$ = ?

1) $\frac{3}{2}$

2) $\frac{-3}{2}$

3) $\frac{-1}{2}$

4) $\frac{1}{2}$

17 )

The line $\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+5}{4}$ meets the plane at the point

1) (-3, 1, 1)

2) (3, -1, 1)

3) (3, 1, -1)

4) (3, -1, -1)

18 )

If A and B are events such that $P(A \cup B)=\frac{5}{6}$, $P(A \cap B)=\frac{1}{3}$ and $P(\bar{B} )=\frac{1}{2}$, then the events A and B are

1) independent

2) dependent

3) mutually exclusive

4) none of these

19 )

Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{cos^3 x}{sin^3 x +cos^3 x} \, dx$ .

1) $\frac{2\pi}{3}$

2) $\frac{\pi}{3}$

3) $\frac{3\pi}{4}$

4) $\frac{\pi}{4}$

20 )

Evaluate $\int \, \frac{sin \, 2x}{(a+b \, cos \, x)^2} \, dx$ .

1) $\frac{2}{b^2}[log|a+b \, cos \, x|+\frac{a}{a+b \, cos \, x}]+c$

2) $-\frac{2}{b^2}[log|a+b \, cos \, x|-\frac{a}{a+b \, cos \, x}]+c$

3) $-\frac{2}{b^2}[log|a+b \, cos \, x|+\frac{a}{a+b \, cos \, x}]+c$

4) none of these

21 )

Evaluate $\int \, cos^4x \, dx$ .

1) $\frac{3x}{8}+\frac{sin \, 4x}{32}+\frac{sin \, 2x}{4}+c$

2) $\frac{3x}{5}+\frac{sin \, 4x}{25}+\frac{sin \, 2x}{4}+c$

3) $\frac{3x}{8}+\frac{sin \, 4x}{15}+\frac{sin \, 2x}{4}+c$

4) none of these

22 )

The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is decreasing -

1) $- \infty \lt x \lt 1$

2) $- \infty \le x \lt 1$

3) $- \infty \lt x \lt 0$

4) none of these

23 )

At what points on the curve $x^2+y^2 -2x-4y+1=0$, is the tangent parallel to y-axis ?

1) (1, 2), (-1, 2)

2) (3, 2), (-1, 2)

3) (-3, 2), (-1, 2)

4) (3, -2), (1, 2)

24 )

A square piece of tin of side 24 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut so that the volume of the box is maximum ? Also find the maximum volume.

1) 1044 cu. cm.

2) 1010 cu. cm.

3) 1020 cu. cm.

4) 1024 cu. cm.

25 )

A purse contains 4 copper coins, 3 silver coins, the second purse contains 6 copper coins, 2 silver coins. A coin is taken out from any purse, the probability that it is copper coin is

1) $\frac{37}{56}$

2) $\frac{4}{7}$

3) $\frac{3}{4}$

4) $\frac{3}{7}$

26 )

The maximum value of P=6x+8y subject to constraints $2x+y \le 30$, $x+2y \le 24$ and $x \ge 0$, $y \ge 0$ is

1) 120

2) 90

3) 86

4) 0

27 )

Let f : R $\rightarrow$ R, such that $f(x)=\frac{2x-7}{4}$ be an invertible function. Find $f^{-1}$

1) $f^{-1}(x)=\frac{4x+7}{2}$

2) $f^{-1}(x)=\frac{2x+7}{2}$

3) $f^{-1}(x)=\frac{4x+7}{5}$

4) $f^{-1}(x)=\frac{2x-7}{5}$

28 )

Solve the following system of equations using matrix method : x + 1y + z = 4 -x + y + z = 0 x - 3y + z = 1

1) x = 3, y = 0, z = 2

2) x = 3, y = 1, z = 2

3) x = -3, y = 0, z = 2

4) x = 3, y = -1, z = 2

29 )

Find the image of the point having position vector $\vec{r}=\hat{i}+3\hat{j}+4\hat{k}$ $\vec{r}.(2\hat{i}-\hat{j}+\hat{k})+3=0$

1) (-3, -5, 2)

2) (3, 5, 2)

3) (-3, 5, 2)

4) (-3, 5, -2)

30 )

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

1) $\frac{9}{8}$ sq. units

2) $\frac{10}{7}$ sq. units

3) $\frac{13}{8}$ sq. units

4) none of these

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1 ) \begin{bmatrix} cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix} 2 ) $\frac{1}{k} .A^{-1}$ 3 ) 1 4 ) $A^2 - AB +BA + B^2 $ 5 ) -1 6 ) $\frac{1}{3}$ 7 ) none of these 8 ) $-\infty$ 9 ) $(6x + 2) e^{3x^2 + 2x}$ 10 ) $\frac{3t^2 - 5}{6t + 2}$ 11 ) $\frac{24}{25}$ 12 ) sec x - cosec x + c 13 ) 4 14 ) $3.2 \pi cm^3/sec$ 15 ) $e^x+e^{-y} + \frac{x^3}{3}=c$ 16 ) $\frac{-3}{2}$ 17 ) (3, -1, -1) 18 ) independent 19 ) $\frac{\pi}{4}$ 20 ) $-\frac{2}{b^2}[log|a+b \, cos \, x|+\frac{a}{a+b \, cos \, x}]+c$ 21 ) $\frac{3x}{8}+\frac{sin \, 4x}{32}+\frac{sin \, 2x}{4}+c$ 22 ) $- \infty \lt x \lt 1$ 23 ) (3, 2), (-1, 2) 24 ) 1024 cu. cm. 25 ) $\frac{37}{56}$ 26 ) 120 27 ) $f^{-1}(x)=\frac{4x+7}{2}$ 28 ) x = 3, y = 0, z = 2 29 ) (-3, 5, 2) 30 ) $\frac{9}{8}$ sq. units

Mathematics - XII

The Model Paper - 7

Year : 2025 - 26

Time - 3 marks 0 minutes

Full Marks - 80

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In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 7

Instructions

Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 7

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×

1 ) Find a 3 x 2 matrix $\, C=[c_{ij} \,] $, where $c_{ij}$ = \begin{cases} i - j \qquad when i < j \\ i + j \qquad i = j \\ i . j \qquad i > j \end{cases}

2 ) \begin{equation*} A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \\ \end{bmatrix} \end{equation*}, is

3 ) Find the value of \begin{vmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \\ \end{vmatrix}

4 ) Matrices A and B are inverses of each other only when

5 ) Find the value of \begin{equation} \begin{vmatrix} x + 1 & x-1 \\ x^2 + x +1 & x^2 - x +1 \\ \end{vmatrix} \end{equation}

6 ) The value of k, \begin{equation*} f(x) = \lvert x \rvert = \left\{ \begin{array}{ll} kx^2 & \quad & \text{if} \, x \leq 2 \\ 3 & \quad & \text{if} \, x \ge 2 \end{array} \right. \end{equation*}

7 ) Let $\, f(x) \, = \, [tan^2 x] \, $, [.] denotes the greatest function. Then

8 ) Evaluate $\,\, \lim_{x \to \infty} \frac{3x^2 + 2x + 1}{x^2 - 1}$

9 ) Determine the derivative of $h(x) = e^x sin(2x).$

10 ) Calculate the derivative of $cos \, x = \, \frac{1}{\sqrt{1+t^2}} $ and $sin \, y = \, \frac{t}{\sqrt{1+t^2}} $

11 ) The value of cos[$2 tan^{-1} \frac{1}{2}$]

12 ) Evaluate $\int \frac{sin x}{sin(x-\alpha)} \, dx$ .

13 ) Evaluate $\int_{-a}^a \, x|x| \, dx$ .

14 ) A spherical balloon of radius 5 cm has been compressed to a spherical balloon of radius 4.9 cm. Find the approximate decrease in its volume.

15 ) The solution of differential equation $\frac{dy}{dx} = - \sqrt{ \frac{1-y^2}{1-x^2}}$ is.

16 ) Two adjacent sides of a triangle are represented by the vectors $\vec{a}=3\hat{i}+4\hat{j}$ and $\vec{b}=-5\hat{i}+7\hat{j}$ . The area of the triangle is

17 ) The vertices of a $\triangle ABC$ are A(-1, 3, 2), B(2, 3, 5) and C(3, 5, -2). Then $\angle B = ?$

18 ) 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 ?

19 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{sin^3 x}{sin^3 x +cos^3 x} \, dx$ .

20 ) Evaluate $\int \, \frac{x^5}{\sqrt{1+x^3}} \, dx$ .

21 ) Evaluate $\int \, \sqrt{\frac{1-x}{1+x}}$ .

22 ) The intervals in which the function $f(x)=log(1+ x) - \frac{x}{1+x}$ is increasing or decreasing

23 ) Find the equation of tangent at $t=\frac{\pi}{4}$ for the curve x = sin 3t, y = cos 2t.

24 ) The volume of the greatest cylinder which can be inscribed in cone of height h and semi vertical angle $\theta$ is

25 ) A box contains 2 gold and 3 silver coins. Another box contains 3 gold and 3 silver coins. One of the two box is selected at random and a coin is drawn from it, which is found to be gold. Find the probability that the ball drawn is from the second box.

26 ) The point at which the maximum value of z=3x+2y subject to constraints $x+y \lt 2$ and $x \ge 0$, $y \ge 0$ is obtained, is

27 ) If $f(x)=\frac{3x-2}{2x-3}$, then f(f(x)) = ? . where x is areal number and $x \ne \frac{3}{2}$

28 ) Solve the following system of equations using matrix method : x - y = 3 2x + 3y + 4z = 17 y + 2z = 7

29 ) Find the equation of the plane passing through the point (0, 7, -7) and containing the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$

30 ) Find the area bounded by curves {$(x, y) \, : \, x^2 \le y \le |x| $}.

Close

1 )

Find a 3 x 2 matrix $\, C=[c_{ij} \,] $, where $c_{ij}$ = \begin{cases} i - j \qquad when i < j \\ i + j \qquad i = j \\ i . j \qquad i > j \end{cases}

1) \begin{bmatrix} 2 & -1 \\ 2 & 4 \\ 3 & 6 \end{bmatrix}

2) \begin{bmatrix} 2 & -1 \\ -2 & 4 \\ 3 & 6 \end{bmatrix}

3) \begin{bmatrix} 2 & -1 \\ 2 & 4 \\ 3 & -6 \end{bmatrix}

4) \begin{bmatrix} 2 & -1 \\ 2 & 4 \\ -3 & 6 \end{bmatrix}

2 )

\begin{equation*} A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \\ \end{bmatrix} \end{equation*}, is

1) nilpotent

2) orthogonal

3) idempotent

4) none of these

3 )

Find the value of \begin{vmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \\ \end{vmatrix}

1) 0

2) 1

3) -1

4) 2

4 )

Matrices A and B are inverses of each other only when

1) AB = O, BA = I

2) AB = BA = O

3) AB = BA

4) AB = BA = I

5 )

Find the value of \begin{equation} \begin{vmatrix} x + 1 & x-1 \\ x^2 + x +1 & x^2 - x +1 \\ \end{vmatrix} \end{equation}

1) 0

2) 1

3) 3

4) 2

6 )

The value of k, \begin{equation*} f(x) = \lvert x \rvert = \left\{ \begin{array}{ll} kx^2 & \quad & \text{if} \, x \leq 2 \\ 3 & \quad & \text{if} \, x \ge 2 \end{array} \right. \end{equation*}

1) 1

2) 1

3) 1

4) 1

7 )

Let $\, f(x) \, = \, [tan^2 x] \, $, [.] denotes the greatest function. Then

1) $ \lim_{x\to 0} f(x)$ does not exist

2) $f^{\prime}(0)=1$

3) f(x) is not differentiable at x=0

4) none of these

8 )

Evaluate $\,\, \lim_{x \to \infty} \frac{3x^2 + 2x + 1}{x^2 - 1}$

1) 3

2) 4

3) 0

4) -3

9 )

Determine the derivative of $h(x) = e^x sin(2x).$

1) $e^x (sin(2x) + 2cos(2x))$

2) $e^x (sin(2x) - 2cos(2x))$

3) $e^x (sin(x) + 2cos(2x))$

4) $e^x (sin(2x) + 2cos(x))$

10 )

Calculate the derivative of $cos \, x = \, \frac{1}{\sqrt{1+t^2}} $ and $sin \, y = \, \frac{t}{\sqrt{1+t^2}} $

1) 0

2) 1

3) -1

4) none of these

11 )

The value of cos[$2 tan^{-1} \frac{1}{2}$]

1) $\frac{3}{5}$

2) $\frac{4}{5}$

3) $\frac{7}{8}$

4) none of these

12 )

Evaluate $\int \frac{sin x}{sin(x-\alpha)} \, dx$ .

1) $x cos \alpha -(sin \alpha) log|sin(x-\alpha)| + c$

2) $x cos \alpha +(sin \alpha) log|sin(x-\alpha)| + c$

3) $x cos \alpha +(sin \alpha) log|sin(x+\alpha)| + c$

4) $-x cos \alpha +(sin \alpha) log|sin(x-\alpha)| + c$

13 )

Evaluate $\int_{-a}^a \, x|x| \, dx$ .

1) $\frac{2a^2}{3}$

2) -a

3) 1

4) 0

14 )

A spherical balloon of radius 5 cm has been compressed to a spherical balloon of radius 4.9 cm. Find the approximate decrease in its volume.

1) $8 \pi cu cm$

2) $10 \pi cu cm$

3) $5 \pi cu cm$

4) none of these

15 )

The solution of differential equation $\frac{dy}{dx} = - \sqrt{ \frac{1-y^2}{1-x^2}}$ is.

1) $sin^{-1}y+sin^{-1}x=c$

2) $y + sin^{-1}y+sin^{-1}x=c$

3) $y + sin^{-1}y-sin^{-1}x=c$

4) $sin^{-1}y+cos^{-1}x=c$

16 )

Two adjacent sides of a triangle are represented by the vectors $\vec{a}=3\hat{i}+4\hat{j}$ and $\vec{b}=-5\hat{i}+7\hat{j}$ . The area of the triangle is

1) $\frac{41}{2}$ sq units

2) $\frac{45}{2}$ sq units

3) $\frac{41}{3}$ sq units

4) none of these

17 )

The vertices of a $\triangle ABC$ are A(-1, 3, 2), B(2, 3, 5) and C(3, 5, -2). Then $\angle B = ?$

1) $cos^{-1}\frac{1}{3}$

2) $cos^{-1}\frac{1}{\sqrt{3}}$

3) $cos^{-1}\frac{1}{\sqrt{2}}$

4) $cos^{-1}\frac{1}{2}$

18 )

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 ?

1) $\frac{5}{9}$

2) $\frac{4}{9}$

3) $\frac{1}{3}$

4) none of these

19 )

Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{sin^3 x}{sin^3 x +cos^3 x} \, dx$ .

1) $\frac{\pi}{2}$

2) $\frac{\pi}{3}$

3) $\frac{\pi}{4}$

4) $\frac{3\pi}{4}$

20 )

Evaluate $\int \, \frac{x^5}{\sqrt{1+x^3}} \, dx$ .

1) $\frac{2}{9}(1+x^3)^\frac{3}{2}+\frac{2}{3}(1+x^3)^\frac{1}{2}+c$

2) $\frac{2}{9}(1+x^3)^\frac{3}{2}-\frac{2}{3}(1+x^3)^\frac{1}{2}+c$

3) $\frac{2}{9}(1+x^3)^\frac{3}{2}-\frac{2}{3}(1+x^3)^\frac{3}{4}+c$

4) none of these

21 )

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

1) $cos^{-1}x + \sqrt{1-x^2}+c$

2) $tan^{-1}x + \sqrt{1-x^2}+c$

3) $sin^{-1}x + \sqrt{1-x^2}+c$

4) $sin^{-1}x - \sqrt{1-x^2}+c$

22 )

The intervals in which the function $f(x)=log(1+ x) - \frac{x}{1+x}$ is increasing or decreasing

1) increasing in $0 \lt x \lt \infty$ ; decreasing in $- \infty \lt x \le 0$

2) increasing in $0 \le x \lt \infty$ ; decreasing in $- \infty \lt x \lt 0$

3) increasing in $1 \lt x \lt \infty$ ; decreasing in $- \infty \lt x \lt 1$

4) increasing in $0 \lt x \lt \infty$ ; decreasing in $- \infty \lt x \lt 0$

23 )

Find the equation of tangent at $t=\frac{\pi}{4}$ for the curve x = sin 3t, y = cos 2t.

1) $$4x-3\sqrt{2}y+2\sqrt{2}=0$$

2) $$4x+3\sqrt{2}y-2\sqrt{2}=0$$

3) $$4x-3\sqrt{2}y-2\sqrt{2}=0$$

4) $$4x-3\sqrt{2}y-2\sqrt{3}=0$$

24 )

The volume of the greatest cylinder which can be inscribed in cone of height h and semi vertical angle $\theta$ is

1) $\frac{4}{27}\pi h^3cot^2 \theta$

2) $\frac{4}{27}\pi h^2tan^2 \theta$

3) $\frac{4}{27}\pi h^3tan^2 \theta$

4) $\frac{4}{25}\pi h^3tan^2 \theta$

25 )

A box contains 2 gold and 3 silver coins. Another box contains 3 gold and 3 silver coins. One of the two box is selected at random and a coin is drawn from it, which is found to be gold. Find the probability that the ball drawn is from the second box.

1) $\frac{5}{9}$

2) $\frac{4}{9}$

3) $\frac{3}{5}$

4) $\frac{2}{5}$

26 )

The point at which the maximum value of z=3x+2y subject to constraints $x+y \lt 2$ and $x \ge 0$, $y \ge 0$ is obtained, is

1) (2, 3)

2) (2, 0)

3) (1.5, 1.5)

4) (0, 2)

27 )

If $f(x)=\frac{3x-2}{2x-3}$, then f(f(x)) = ? . where x is areal number and $x \ne \frac{3}{2}$

1) x

2) -x

3) $x^2$

4) $-x^2$

28 )

Solve the following system of equations using matrix method : x - y = 3 2x + 3y + 4z = 17 y + 2z = 7

1) x = 2, y = 1, z = 4

2) x = 2, y = -1, z = 4

3) x = 2, y = -1, z = -4

4) x = -2, y = -1, z = 4

29 )

Find the equation of the plane passing through the point (0, 7, -7) and containing the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$

1) x + y + z = 0

2) x + y + z +1 = 0

3) x + y + z -1 = 0

4) x - y + z = 0

30 )

Find the area bounded by curves {$(x, y) \, : \, x^2 \le y \le |x| $}.

1) $\frac{2}{3}$ sq. units

2) $\frac{1}{4}$ sq. units

3) $\frac{1}{3}$ sq. units

4) $\frac{3}{4}$ sq. units

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1 ) \begin{bmatrix} 2 & -1 \\ 2 & 4 \\ 3 & 6 \end{bmatrix} 2 ) idempotent 3 ) 1 4 ) AB = BA = I 5 ) 2 6 ) 1 7 ) none of these 8 ) 3 9 ) $e^x (sin(2x) + 2cos(2x))$ 10 ) 1 11 ) $\frac{3}{5}$ 12 ) $x cos \alpha +(sin \alpha) log|sin(x-\alpha)| + c$ 13 ) 0 14 ) $10 \pi cu cm$ 15 ) $sin^{-1}y+sin^{-1}x=c$ 16 ) $\frac{41}{2}$ sq units 17 ) $cos^{-1}\frac{1}{\sqrt{2}}$ 18 ) $\frac{5}{9}$ 19 ) $\frac{\pi}{4}$ 20 ) $\frac{2}{9}(1+x^3)^\frac{3}{2}-\frac{2}{3}(1+x^3)^\frac{1}{2}+c$ 21 ) $sin^{-1}x + \sqrt{1-x^2}+c$ 22 ) increasing in $0 \lt x \lt \infty$ ; decreasing in $ - \infty \lt x \lt 0$ 23 ) $$4x-3\sqrt{2}y-2\sqrt{2}=0$$ 24 ) $\frac{4}{27}\pi h^3tan^2 \theta$ 25 ) $\frac{5}{9}$ 26 ) (2, 0) 27 ) x 28 ) x = 2, y = -1, z = 4 29 ) x + y + z = 0 30 ) $\frac{1}{3}$ sq. units

Mathematics - XII

The Model Paper - 8

Year : 2025 - 26

Time - 3 marks 0 minutes

Full Marks - 80

Previous Set

Cur Set 8

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Not Answered 30 Answered 0

In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 8

Instructions

Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 8

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×

1 ) If \begin{equation*} A = \begin{bmatrix} 2 & 0 \\ 3 & -5 \end{bmatrix} \end{equation*} and $A^2 - 3A - 10I_2 = 0 $, find $A^{-1}$.

2 ) If A is an invertible matrix and \begin{equation*} A^{-1} = \begin{bmatrix} 3 & 4 \\ 5 & 6 \\ \end{bmatrix} \end{equation*}, then A = ?

3 ) Find the co-factor of \begin{vmatrix} 2 & 6 \\ -1 & 4 \\ \end{vmatrix}

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

5 ) Find the value of \begin{equation} \begin{vmatrix} -1 & 3 & 4 \\ 1 & 9 & 12 \\ 9 & 9 & 12 \\ \end{vmatrix} \end{equation}

6 ) \begin{equation*} f(x) = \left\{ \begin{array}{ll} 2x - 1 \qquad x < 0 \\ 2x + 1 \quad x \geq 0 \end{array} \right. \end{equation*}

7 ) If \begin{equation*} f(x)=\begin{cases} -log_ex \quad &\text{if} \, 0 < x < 1 \\ k log_ex \quad &\text{if} \, x \ge 1 \\ \end{cases} \end{equation*}, find the value of k, so that f(x) is differentiable at x = 1.

8 ) Evaluate $\,\, \lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

9 ) Find the derivative of $f(x) = \sqrt{x^3 - 2x^2 + 1}.$

10 ) The derivative of $sec^2x$ with respect to $tan^2x$.

11 ) If $tan^{-1}3x + tan^{-1}2x = \frac{\pi}{4} $ , then x = ?

12 ) Evaluate $\int sin^3(2x+1) \, dx$ .

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

14 ) The least value of k such that $x^2+kx+1$ is increasing on ]1, 2[

15 ) The solution of differential equation $\frac{dy}{dx} + \frac{y}{x}=x^2$.

16 ) If $|\vec{a}|=2$, $|\vec{b}|=7$ and $\vec{a} \, \times \, \vec{b} = 3\hat{i}+2\hat{j}+6\hat{k}$, then angle between $\vec{a} \, and \, \vec{b}$ is

17 ) The angle between the lines $\frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2}$ and $\frac{x+3}{3}=\frac{y-2}{5}=\frac{z+5}{4}$ is

18 ) Two numbers are selected at random from integers 1 through 9. If the sum is even, What is the probability that both numbers are odd ?

19 ) Evaluate $\int_0^{\pi} \, \frac{x}{a^2 \, cos^2 x +b^2 \, sin^2 x} \, dx$ .

20 ) Evaluate $\int \, \sqrt{\frac{1+x}{1-x}} \, dx$ .

21 ) Evaluate $\int \,[\frac{1}{log \, x}-\frac{1}{(log \, x)^2}] \, dx$ .

22 ) The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is increasing

23 ) If the tangent to the curve $y=x^3+ax+b $ at [1,-6] is parallel to the line x - y + 5 = 0, then the values of a and b are

24 ) Two sides of a triangle have lengths a and b and the angle between is $\theta$. What value of $\theta$ will maximize the area of the triangle ? Find the maximum area of the triangle also.

25 ) Three urns A, B and C contains 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls respectively. An urn is choose at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from the urn A.

26 ) The maximum value of P=8x+3y subject to constraints $x+y \le 3$, $4x+y \le 6$ and $x \ge 0$, $y \ge 0$ is

27 ) Find the value of parameter $\alpha$ for which function $f(x)=1+\alpha x$, $\alpha \ne 0$ is the inverse of itself.

28 ) Solve the following system of equations using matrix method : 2x + y - z = 1 x - y + z = 2 3x + y - 2z = -1

29 ) Find the co-ordinates of the point where the line $\frac{x-2}{3}=\frac{y+3}{4}=\frac{z-2}{6}$ meets the plane x - y + z = 2.

30 ) Find the area cut off from the parabola $4y=3x^2$ by the straight line 3x - 2y +12 = 0.

Close

1 )

If \begin{equation*} A = \begin{bmatrix} 2 & 0 \\ 3 & -5 \end{bmatrix} \end{equation*} and $A^2 - 3A - 10I_2 = 0 $, find $A^{-1}$.

1) \begin{bmatrix} \frac{3}{2} & 0 \\ \frac{3}{10} & - \frac{1}{5} \end{bmatrix}

2) \begin{bmatrix} - \frac{1}{2} & 0 \\ \frac{3}{10} & - \frac{1}{5} \end{bmatrix}

3) \begin{bmatrix} \frac{1}{2} & 0 \\ \frac{3}{10} & - \frac{1}{5} \end{bmatrix}

4) \begin{bmatrix} \frac{1}{2} & 0 \\ \frac{3}{10} & - \frac{3}{5} \end{bmatrix}

2 )

If A is an invertible matrix and \begin{equation*} A^{-1} = \begin{bmatrix} 3 & 4 \\ 5 & 6 \\ \end{bmatrix} \end{equation*}, then A = ?

1) \begin{bmatrix} 3 & 2 \\ \frac{5}{2} & - \frac{3}{2} \\ \end{bmatrix}

2) \begin{bmatrix} 3 & -2 \\ \frac{5}{2} & - \frac{3}{2} \\ \end{bmatrix}

3) \begin{bmatrix} -3 & 2 \\ \frac{5}{4} & - \frac{3}{2} \\ \end{bmatrix}

4) \begin{bmatrix} -3 & 2 \\ \frac{5}{2} & - \frac{3}{2} \\ \end{bmatrix}

3 )

Find the co-factor of \begin{vmatrix} 2 & 6 \\ -1 & 4 \\ \end{vmatrix}

1) -4

2) -2

3) -1

4) -6

4 )

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

1) $\frac{16}{3}$

2) $- \frac{33}{2}$

3) $\frac{33}{2}$

4) $\frac{25}{2}$

5 )

Find the value of \begin{equation} \begin{vmatrix} -1 & 3 & 4 \\ 1 & 9 & 12 \\ 9 & 9 & 12 \\ \end{vmatrix} \end{equation}

1) 0

2) 1

3) 2

4) -1

6 )

\begin{equation*} f(x) = \left\{ \begin{array}{ll} 2x - 1 \qquad x < 0 \\ 2x + 1 \quad x \geq 0 \end{array} \right. \end{equation*}

1) f(x) is discontinuous at x = 0

2) f(x) is continuous at x = 0

3) f(x) is continuous at all points except at x = 0

4) none of these

7 )

If \begin{equation*} f(x)=\begin{cases} -log_ex \quad &\text{if} \, 0 < x < 1 \\ k log_ex \quad &\text{if} \, x \ge 1 \\ \end{cases} \end{equation*}, find the value of k, so that f(x) is differentiable at x = 1.

1) -1

2) 2

3) 1

4) 0

8 )

Evaluate $\,\, \lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

1) 1

2) 4

3) 0

4) 3

9 )

Find the derivative of $f(x) = \sqrt{x^3 - 2x^2 + 1}.$

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

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

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

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

10 )

The derivative of $sec^2x$ with respect to $tan^2x$.

1) 0

2) 1

3) 2

4) -2

11 )

If $tan^{-1}3x + tan^{-1}2x = \frac{\pi}{4} $ , then x = ?

1) $\frac{1}{5} $ or -1

2) $\frac{1}{6} $ or -2

3) $\frac{1}{6} $ or -1

4) $\frac{1}{4} $ or -1

12 )

Evaluate $\int sin^3(2x+1) \, dx$ .

1) $-\frac{1}{2}cos(2x+1) + $\frac{1}{6}cos^3(2x+1) + c$

2) $-\frac{1}{2}cos(2x+1) + c$

3) $\frac{1}{6}cos^3(2x+1) + c$

4) $-\frac{1}{2}cos(2x+1) - \frac{1}{6}cos^3(2x+1) + c$

13 )

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

1) $\frac{e}{2} -1$

2) e-1

3) e+1

4) none of these

14 )

The least value of k such that $x^2+kx+1$ is increasing on ]1, 2[

1) -2

2) -1

3) 2

4) 1

15 )

The solution of differential equation $\frac{dy}{dx} + \frac{y}{x}=x^2$.

1) $xy = x^4 + c$

2) $4xy = x^4 + c$

3) $4xy = x^4 + cx$

4) $4xy^2 = x^4 + c$

16 )

If $|\vec{a}|=2$, $|\vec{b}|=7$ and $\vec{a} \, \times \, \vec{b} = 3\hat{i}+2\hat{j}+6\hat{k}$, then angle between $\vec{a} \, and \, \vec{b}$ is

1) $\frac{\pi}{4}$

2) $\frac{3\pi}{4}$

3) $\frac{\pi}{6}$

4) $\frac{\pi}{2}$

17 )

The angle between the lines $\frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2}$ and $\frac{x+3}{3}=\frac{y-2}{5}=\frac{z+5}{4}$ is

1) $cos^{-1}\frac{4}{15}$

2) $cos^{-1}\frac{8\sqrt{2}}{15}$

3) $cos^{-1}\frac{8}{15}$

4) $cos^{-1}\frac{8\sqrt{3}}{15}$

18 )

Two numbers are selected at random from integers 1 through 9. If the sum is even, What is the probability that both numbers are odd ?

1) $\frac{4}{9}$

2) $\frac{5}{8}$

3) $\frac{2}{3}$

4) $\frac{3}{8}$

19 )

Evaluate $\int_0^{\pi} \, \frac{x}{a^2 \, cos^2 x +b^2 \, sin^2 x} \, dx$ .

1) $\frac{2\pi^2}{ab}$

2) $\frac{\pi}{2ab}$

3) $\frac{\pi^2}{ab}$

4) $\frac{\pi^2}{2ab}$

20 )

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

1) $cos^{-1}x+\sqrt{1-x^2} +c$

2) $cos^{-1}x-\sqrt{1-x^2} +c$

3) $sin^{-1}x-\sqrt{1+x^2} +c$

4) $sin^{-1}x-\sqrt{1-x^2} +c$

21 )

Evaluate $\int \,[\frac{1}{log \, x}-\frac{1}{(log \, x)^2}] \, dx$ .

1) $\frac{2x}{log \, x} +c$

2) $\frac{x}{2log \, x} +c$

3) $-\frac{x}{log \, x} +c$

4) $\frac{x}{log \, x} +c$

22 )

The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is increasing

1) $1 \le x \lt \infty $

2) $0 \le x \lt \infty $

3) $0 \lt x \lt \infty $

4) $1 \lt x \lt \infty $

23 )

If the tangent to the curve $y=x^3+ax+b $ at [1,-6] is parallel to the line x - y + 5 = 0, then the values of a and b are

1) a=-2, b=-5

2) a=-2, b=-3

3) a=2, b=-5

4) a=-2, b=5

24 )

Two sides of a triangle have lengths a and b and the angle between is $\theta$. What value of $\theta$ will maximize the area of the triangle ? Find the maximum area of the triangle also.

1) $\frac{\pi}{2}$, Area=$\frac{2}{3}ab$

2) $\frac{\pi}{2}$, Area=$\frac{1}{3}ab$

3) $\frac{\pi}{3}$, Area=$\frac{1}{2}ab$

4) $\frac{\pi}{2}$, Area=$\frac{1}{2}ab$

25 )

Three urns A, B and C contains 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls respectively. An urn is choose at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from the urn A.

1) $\frac{35}{61}$

2) $\frac{36}{61}$

3) $\frac{25}{61}$

4) $\frac{30}{61}$

26 )

The maximum value of P=8x+3y subject to constraints $x+y \le 3$, $4x+y \le 6$ and $x \ge 0$, $y \ge 0$ is

1) 14

2) 5

3) 12

4) 20

27 )

Find the value of parameter $\alpha$ for which function $f(x)=1+\alpha x$, $\alpha \ne 0$ is the inverse of itself.

1) $\alpha=1$

2) $\alpha=-1$

3) $\alpha=2$

4) $\alpha=-3$

28 )

Solve the following system of equations using matrix method : 2x + y - z = 1 x - y + z = 2 3x + y - 2z = -1

1) x = -1, y = 2, z = 3

2) x = 1, y = 2, z = 3

3) x = 1, y = -2, z = 3

4) x = 1, y = 2, z = -3

29 )

Find the co-ordinates of the point where the line $\frac{x-2}{3}=\frac{y+3}{4}=\frac{z-2}{6}$ meets the plane x - y + z = 2.

1) (-1, -7, 4)

2) (1, -7, -4)

3) (-1, 7, -4)

4) (-1, -7, -4)

30 )

Find the area cut off from the parabola $4y=3x^2$ by the straight line 3x - 2y +12 = 0.

1) 25 sq. units

2) 27 sq. units

3) 30 sq. units

4) none of these

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1 ) \begin{bmatrix} \frac{1}{2} & 0 \\ \frac{3}{10} & - \frac{1}{5} \end{bmatrix} 2 ) \begin{bmatrix} -3 & 2 \\ \frac{5}{2} & - \frac{3}{2} \\ \end{bmatrix} 3 ) -6 4 ) $\frac{33}{2}$ 5 ) 0 6 ) f(x) is discontinuous at x = 0 7 ) -1 8 ) 4 9 ) $\frac{3x^2 - 4x}{2 \sqrt{x^3 - 2x^2 + 1}}$ 10 ) 1 11 ) $\frac{1}{6} $ or -1 12 ) $-\frac{1}{2}cos(2x+1) + \frac{1}{6}cos^3(2x+1) + c$ 13 ) $\frac{e}{2} -1$ 14 ) -2 15 ) $4xy = x^4 + c$ 16 ) $\frac{\pi}{6}$ 17 ) $cos^{-1}\frac{8\sqrt{3}}{15}$ 18 ) $\frac{5}{8}$ 19 ) $\frac{\pi^2}{2ab}$ 20 ) $sin^{-1}x-\sqrt{1-x^2} +c$ 21 ) $\frac{x}{log \, x} +c$ 22 ) $1 \le x \lt \infty $ 23 ) a=-2, b=-5 24 ) $\frac{\pi}{2}$, Area=$\frac{1}{2}ab$ 25 ) $\frac{36}{61}$ 26 ) 14 27 ) $\alpha=-1$ 28 ) x = 1, y = 2, z = 3 29 ) (-1, -7, -4) 30 ) 27 sq. units

Mathematics - XII

The Model Paper - 9

Year : 2025 - 26

Time - 3 marks 0 minutes

Full Marks - 80

Previous Set

Cur Set 9

Next Set

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Not Answered 30 Answered 0

In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 9

Instructions

Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 9

Close

×

1 ) If A be a 2x2 matrix and is defined by A = $\, [a_{ij}] \, $ where $\, [a_{ij}] \, = \, \frac{(i+2j)^2}{2}$, then find the value of $a_{21} $

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 A and B are square matrices of the same order, then $(A - B)^2$ = ?

5 ) Find the value of \begin{equation} \begin{vmatrix} 1 \, + \, 2i & 1 \, - \, i \\ 1 \, + \, i & 1 \, - \, 2i \\ \end{vmatrix} \end{equation} where $\, i^2 = -1$

6 ) The value of k, so that \begin{equation*} g(x)=\begin{cases} sin \frac{1}{x} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, continuous at x = 0

7 ) The function f(x) = x - [x], where [.] denotes the greatest integer function is

8 ) Evaluate $\, \, \lim_{x \to 1}\frac{x^2 - 1}{x - 1}$.

9 ) Determine the derivative of $h(x) = ln(2x + 1) - e^x.$

10 ) Determine the derivative of $x = e^t$ and $y = ln t.$

11 ) The value of $cot^{-1}9 + cosec^{-1} \frac{\sqrt{41}}{4}$

12 ) Evaluate $\int \frac{ax + b}{cx + d} \, dx$ .

13 ) Evaluate $\int_{-\pi}^{\pi} \, sin^{61}x + x^{123} \, dx$ .

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

15 ) The solution of differential equation $\frac{dy}{dx} = 1-x+y-xy$.

16 ) If $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, then $\frac{1}{2}|\hat{a} - \hat{b}|$ = ?

17 ) A plane through line of intersection of the planes x + y + z = 6 and 2x + 3y + 4z + 5 = 0 and passing through the point P(1, 1, 1) is

18 ) A coin is tossed 5 times. What is the probability that tail appears an odd number of times ?

19 ) Evaluate $\int_0^1 \, cot^{-1}(1-x+x^2) \, dx$ .

20 ) Evaluate $\int \, \frac{sin \, x + cos \, x}{\sqrt{sin \, 2x}} \, dx$ .

21 ) Evaluate $\int \, \frac{4x+3}{\sqrt{2x^2 + 2x - 3}} \, dx$ .

22 ) The intervals in which the function $f(x)=( x+2)e^{-x}$ is increasing or decreasing

23 ) At what points on the curve $x^2+y^2 -2x-3=0$, is the tangent parallel to x-axis ?

24 ) The maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3}$ is

25 ) A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both spades. Find the probability of the lost card being a spade.

26 ) The minimum value of z=4x+2y subject to constraints $2x+3y \ge 18$, $x+y \le 10$ and $x \ge 0$, $y \ge 0$ is

27 ) Let f : R $\rightarrow$ R, such that $f(x)=(3-x^3)^{\frac{1}{3}}$. Find f o f.

28 ) Solve the following system of equations using matrix method : 2x - 3y + 5z = 16 3x +2 y - 4z = -4 x + y - 2z = -3

29 ) Find the distance of the point (2, 3, 4) from the plane 3x + 2y + 2z + 5= 0, measured parallel to the line $\frac{x+3}{3}=\frac{y-2}{6}=\frac{z}{2}$

30 ) Find the area bounded by curves y = sin x and y = cos x for $0 \le x\le \frac{\pi}{2}$

Close

1 )

If A be a 2x2 matrix and is defined by A = $\, [a_{ij}] \, $ where $\, [a_{ij}] \, = \, \frac{(i+2j)^2}{2}$, then find the value of $a_{21} $

1) 5

2) 12

3) 8

4) 6

2 )

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

1) $A^{-1}B^{-1}$

2) $B^{-1}A^{-1}$

3) $BA^{-1}$

4) $AB^{-1}$

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}

1) $\pm \sqrt{5}$

2) $\pm \sqrt{3}$

3) $\pm 2 \sqrt{3}$

4) $\pm 3 \sqrt{3}$

4 )

If A and B are square matrices of the same order, then $(A - B)^2$ = ?

1) $A^2 - 2AB + B^2$

2) $A^2 - AB - BA + B^2$

3) $A^2 - 2BA + B^2$

4) none of these

5 )

Find the value of \begin{equation} \begin{vmatrix} 1 \, + \, 2i & 1 \, - \, i \\ 1 \, + \, i & 1 \, - \, 2i \\ \end{vmatrix} \end{equation} where $\, i^2 = -1$

1) 3

2) 4

3) 2

4) 1

6 )

The value of k, so that \begin{equation*} g(x)=\begin{cases} sin \frac{1}{x} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, continuous at x = 0

1) 8

2) 1

3) -1

4) none of these

7 )

The function f(x) = x - [x], where [.] denotes the greatest integer function is

1) continuous everywhere

2) continuous at non-integer points only

3) continuous at integer points only

4) differential everywhere

8 )

Evaluate $\, \, \lim_{x \to 1}\frac{x^2 - 1}{x - 1}$.

1) 2

2) -2

3) 1

4) 0

9 )

Determine the derivative of $h(x) = ln(2x + 1) - e^x.$

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

2) $\frac{3}{2x + 1} - e^x$

3) $\frac{2}{2x + 1} - e^x$

4) $\frac{2}{2x - 1} - e^x$

10 )

Determine the derivative of $x = e^t$ and $y = ln t.$

1) $\frac{1}{e^t}$

2) $- \frac{1}{te^t}$

3) $\frac{1}{te^t}$

4) $\frac{2}{te^t}$

11 )

The value of $cot^{-1}9 + cosec^{-1} \frac{\sqrt{41}}{4}$

1) $\frac{\pi}{4}$

2) $\frac{\pi}{6}$

3) $\frac{\pi}{3}$

4) $\frac{3 \pi}{4}$

12 )

Evaluate $\int \frac{ax + b}{cx + d} \, dx$ .

1) $\frac{ax}{c} - \frac{bc + ad}{c^2} log|cx + d|$ + C

2) $\frac{ax}{c} + log|cx + d|$ + C

3) $\frac{ax}{c} + \frac{bc - ad}{c^2} $ + C

4) $\frac{ax}{c} + \frac{bc - ad}{c^2} log|cx + d|$ + C

13 )

Evaluate $\int_{-\pi}^{\pi} \, sin^{61}x + x^{123} \, dx$ .

1) $\frac{\pi}{2}$

2) -2

3) $\pi$

4) 0

14 )

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

1) $]\frac{2}{3}, 3[$

2) $]\frac{5}{3}, 3[$

3) $]\frac{5}{3}, 2[$

4) $]\frac{1}{3}, 3[$

15 )

The solution of differential equation $\frac{dy}{dx} = 1-x+y-xy$.

1) $-e^{1+y}=x-\frac{x^2}{2} +c$

2) $log(1+y)=x+\frac{x^2}{2} +c$

3) $log(1+y)=x-\frac{x^2}{2} +c$

4) $e^{1+y}=x-\frac{x^2}{2} +c$

16 )

If $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, then $\frac{1}{2}|\hat{a} - \hat{b}|$ = ?

1) $sin \, \frac{\theta}{2}$

2) $cos \, \frac{\theta}{2}$

3) $tan \, \frac{\theta}{2}$

4) none of these

17 )

A plane through line of intersection of the planes x + y + z = 6 and 2x + 3y + 4z + 5 = 0 and passing through the point P(1, 1, 1) is

1) 20x + 23y + 26z = 59

2) 20x + 23y + 26z = 69

3) 20x + 23y + 26z = 36

4) 3x + 8y - z = 6

18 )

A coin is tossed 5 times. What is the probability that tail appears an odd number of times ?

1) $\frac{1}{2}$

2) $\frac{1}{3}$

3) $\frac{3}{5}$

4) $\frac{2}{3}$

19 )

Evaluate $\int_0^1 \, cot^{-1}(1-x+x^2) \, dx$ .

1) $\frac{\pi}{4} + log \, 2$

2) $\frac{\pi}{4} - log \, 2$

3) $\frac{\pi}{2} - log \, 2$

4) $\frac{\pi}{2} +log \, 2$

20 )

Evaluate $\int \, \frac{sin \, x + cos \, x}{\sqrt{sin \, 2x}} \, dx$ .

1) $tan^{-1}\frac{tan \, x -1}{\sqrt{2 \, tan \, x}}+c$

2) $-tan^{-1}\frac{tan \, x -1}{\sqrt{2 \, tan \, x}}+c$

3) $cot^{-1}\frac{tan \, x -1}{\sqrt{2 \, tan \, x}}+c$

4) $tan^{-1}\frac{tan \, x -1}{\sqrt{2 \, cot \, x}}+c$

21 )

Evaluate $\int \, \frac{4x+3}{\sqrt{2x^2 + 2x - 3}} \, dx$ .

1) $2\sqrt{2x^2 + 2x - 3} + \frac{1}{\sqrt{2}} log|x + \frac{1}{2} +\sqrt{x^2 + x - \frac{3}{2}} | +c$

2) $2\sqrt{2x^2 + 2x - 3} - \frac{1}{\sqrt{2}} log|x + \frac{1}{2} +\sqrt{x^2 + x - \frac{3}{2}} | +c$

3) $2\sqrt{2x^2 + 2x - 3} + \frac{1}{\sqrt{2}} log|x + \frac{1}{2} +\sqrt{x^2 - x - \frac{3}{2}} | +c$

4) none of these

22 )

The intervals in which the function $f(x)=( x+2)e^{-x}$ is increasing or decreasing

1) inc in $(- \infty, -1) $; dec in $( -1, \infty) $

2) inc in $(- \infty, 0) $; dec in $(0, \infty) $

3) inc in $(- \infty, -1) $; dec in $(1, \infty) $

4) none of these

23 )

At what points on the curve $x^2+y^2 -2x-3=0$, is the tangent parallel to x-axis ?

1) (1, 2), (1, -2)

2) (1, 2), (2, -2)

3) (-1, 2), (1, -2)

4) (-1, 2), (1, 2)

24 )

The maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3}$ is

1) $525 \pi cm^3$

2) $490 \pi cm^3$

3) $500 \pi cm^3$

4) $550 \pi cm^3$

25 )

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both spades. Find the probability of the lost card being a spade.

1) 0.33

2) 0.22

3) 0.32

4) 0.25

26 )

The minimum value of z=4x+2y subject to constraints $2x+3y \ge 18$, $x+y \le 10$ and $x \ge 0$, $y \ge 0$ is

1) 36

2) 40

3) 20

4) None of these

27 )

Let f : R $\rightarrow$ R, such that $f(x)=(3-x^3)^{\frac{1}{3}}$. Find f o f.

1) -x

2) x

3) 2x

4) $x^2$

28 )

Solve the following system of equations using matrix method : 2x - 3y + 5z = 16 3x +2 y - 4z = -4 x + y - 2z = -3

1) x = 1, y = 1, z = 3

2) x = 2, y = 1, z = -3

3) x = 2, y = 1, z = 3

4) x = 2, y = -1, z = 3

29 )

Find the distance of the point (2, 3, 4) from the plane 3x + 2y + 2z + 5= 0, measured parallel to the line $\frac{x+3}{3}=\frac{y-2}{6}=\frac{z}{2}$

1) 5 units

2) 7 units

3) 6 units

4) 10 units

30 )

Find the area bounded by curves y = sin x and y = cos x for $0 \le x\le \frac{\pi}{2}$

1) $3(\sqrt{2} -1)$ sq. units

2) $2(\sqrt{3} -1)$ sq. units

3) $2(\sqrt{2} +1)$ sq. units

4) $2(\sqrt{2} -1)$ sq. units

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1 ) 8 2 ) $B^{-1}A^{-1}$ 3 ) $\pm 2 \sqrt{3}$ 4 ) $A^2 - AB - BA + B^2$ 5 ) 3 6 ) none of these 7 ) continuous at non-integer points only 8 ) 2 9 ) $\frac{2}{2x + 1} - e^x$ 10 ) $\frac{1}{te^t}$ 11 ) $\frac{\pi}{4}$ 12 ) $\frac{ax}{c} + \frac{bc - ad}{c^2} log|cx + d|$ + C 13 ) 0 14 ) $]\frac{5}{3}, 3[$ 15 ) $log(1+y)=x-\frac{x^2}{2} +c$ 16 ) $sin \, \frac{\theta}{2}$ 17 ) 20x + 23y + 26z = 69 18 ) $\frac{1}{2}$ 19 ) $\frac{\pi}{2} - log \, 2$ 20 ) $tan^{-1}\frac{tan \, x -1}{\sqrt{2 \, tan \, x}}+c$ 21 ) $2\sqrt{2x^2 + 2x - 3} + \frac{1}{\sqrt{2}} log|x + \frac{1}{2} +\sqrt{x^2 + x - \frac{3}{2}} | +c$ 22 ) inc in $(- \infty, -1) $; dec in $(-1, \infty) $ 23 ) (1, 2), (1, -2) 24 ) $500 \pi cm^3$ 25 ) 0.22 26 ) None of these 27 ) x 28 ) x = 2, y = 1, z = 3 29 ) 7 units 30 ) $2(\sqrt{2} -1)$ sq. units

Mathematics - XII

The Model Paper - 10

Year : 2025 - 26

Time - 3 marks 0 minutes

Full Marks - 80

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In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

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Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

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1 ) Find a 2 x 3 matrix $\, B=[a_{ij} ] \,$, where $\, b_{ij} \, =\, \frac{1}{2}(i - j)^2$.

2 ) If A and B are two non-zero square matrices of the same order such that AB = 0, then

3 ) If \begin{equation} A = \begin{vmatrix} 11 & 20 \\ 31 & 41 \\ \end{vmatrix} \end{equation} and |3A| = k|A|, then the value of k is

4 ) A is 3-rowed square matrix and |A| = 4, then adj(adj A) = ?

5 ) \begin{equation} \begin{vmatrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \\ \end{vmatrix} \end{equation}

6 ) If \begin{equation*} g(x)=\begin{cases} \frac{1 - cos x}{x^2} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, find k.

7 ) If f(x) = |sin x|, then

8 ) Evaluate $\,\, \lim_{x \to 0} \frac{1 - cos x}{x^2}$

9 ) Find the derivative of $f(x) = \frac{x^4}{1 + x^2} $

10 ) Derivative of $x^2$ w.r.t. $x^3$

11 ) The value of sin[$2 tan^{-1} \frac{5}{8}$]

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

13 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{sin \, x}{sin \, x + cos \, x} \, dx$ .

14 ) Find the intervals on which the following function f(x) = sin x - cos x, $0 \lt x \lt 2\pi$ is increasing.

15 ) The solution of differential equation $log(\frac{dy}{dx} )= ax+by$.

16 ) If $ \vec{a}=\hat{i}+2\hat{j}-3\hat{k} $ and $ \vec{b}=3\hat{i}-\hat{j}+2\hat{k}$ then the angle between $(\vec{a} + \vec{b})$ and $(\vec{a} + \vec{b})$ is

17 ) 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 = ?

18 ) A machine operates only when all its three components function. The probabilities of the failures of the first, second and third components are 0.2, 0.3 and 0.5 respectively. What is the probability that the machine will fail ?

19 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x}{sin \, x +cos \, x} \, dx$ .

20 ) Evaluate $\int \, tan^{-1} \sqrt{\frac{1-sin \, x}{1+sin \, x}} \, dx$ .

21 ) Evaluate $\int \, \frac{2}{(1-x)(1+x^2)} \, dx$ .

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

23 ) Find the point in the first quadrant at which the slope of the normal to the curve $ x^3=8a^2y$, $a \gt 0$ is $- \frac{2}{3}$

24 ) The semi-vertical angle of a cone of maximum volume and given slant height is

25 ) Bag A contains 2 white and 3 red balls, and bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from bag B.

26 ) The maximum value of P=6x+11y subject to constraints $2x+y \le 104$, $x+2y \le 76$ and $x \ge 0$, $y \ge 0$ is

27 ) Let $f : [1, \infty) \rightarrow [1, \infty)$ ,is given by $f(x)=2^{x(x-1)}$ is invertible. Find $f^{-1}(x)$.

28 ) Solve the following system of equations using matrix method : x + y + z = 1 x - 2y + 3z = 2 5x - 3y + z = 3

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

30 ) Find the area bounded by curve $y^2=2y-x$ and the y-axis.

Close

1 )

Find a 2 x 3 matrix $\, B=[a_{ij} ] \,$, where $\, b_{ij} \, =\, \frac{1}{2}(i - j)^2$.

1) \begin{bmatrix} 0 & \frac{1}{2} & 2 \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \end{bmatrix}

2) \begin{bmatrix} 0 & \frac{1}{2} & 3 \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \end{bmatrix}

3) \begin{bmatrix} 0 & \frac{1}{2} & 2 \\ \frac{1}{3} & 0 & \frac{1}{2} \\ \end{bmatrix}

4) \begin{bmatrix} 1 & \frac{1}{2} & 2 \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \end{bmatrix}

2 )

If A and B are two non-zero square matrices of the same order such that AB = 0, then

1) |A|=0 or |B|=0

2) $|A| \ne 0 and |B| \ne 0$

3) |A|=0 and |B|=0

4) none of these

3 )

If \begin{equation} A = \begin{vmatrix} 11 & 20 \\ 31 & 41 \\ \end{vmatrix} \end{equation} and |3A| = k|A|, then the value of k is

1) 12

2) 6

3) 9

4) 8

4 )

A is 3-rowed square matrix and |A| = 4, then adj(adj A) = ?

1) 4A

2) 16A

3) 64A

4) 128A

5 )

\begin{equation} \begin{vmatrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \\ \end{vmatrix} \end{equation}

1) 2

2) -1

3) 0

4) 1

6 )

If \begin{equation*} g(x)=\begin{cases} \frac{1 - cos x}{x^2} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, find k.

1) 1

2) $- \frac{1}{2}$

3) $\frac{1}{3}$

4) $\frac{1}{2}$

7 )

If f(x) = |sin x|, then

1) f(x) is everywhere differentiable

2) f(x) is not differentiable at $ \, x = (2n + 1) \frac{\pi}{2}, \, x \in Z $

3) f(x) is not differentiable at $ \, x = n \pi, \, x \in Z $

4) none of these

8 )

Evaluate $\,\, \lim_{x \to 0} \frac{1 - cos x}{x^2}$

1) $\frac{2}{3}$

2) $\frac{1}{3}$

3) $\frac{3}{2}$

4) $\frac{1}{2}$

9 )

Find the derivative of $f(x) = \frac{x^4}{1 + x^2} $

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

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

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

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

10 )

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

1) $- \frac{2}{3x}$

2) $\frac{2}{x}$

3) $\frac{1}{3x}$

4) $\frac{2}{3x}$

11 )

The value of sin[$2 tan^{-1} \frac{5}{8}$]

1) $\frac{75}{128}$

2) $\frac{40}{89}$

3) $\frac{25}{84}$

4) $\frac{80}{89}$

12 )

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

1) $- \frac{1}{2} (log \, tan \, x)^2$

2) $\frac{1}{2} (log \, cot \, x)^2$

3) $ (log \, tan \, x)^2 + c$

4) $\frac{1}{2} (log \, tan \, x)^2 + c$

13 )

Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{sin \, x}{sin \, x + cos \, x} \, dx$ .

1) $\pi$

2) 0

3) $\frac{\pi}{4}$

4) $\frac{\pi}{3}$

14 )

Find the intervals on which the following function f(x) = sin x - cos x, $0 \lt x \lt 2\pi$ is increasing.

1) $] 0, \frac{3\pi}{2}[$

2) $] 0, \frac{\pi}{4}[$

3) $] \frac{\pi}{4}, \frac{3\pi}{4}[$

4) $] 0, \frac{3\pi}{4}[$

15 )

The solution of differential equation $log(\frac{dy}{dx} )= ax+by$.

1) $\frac{-e^{-by}}{b}=\frac{e^{ax}}{a} + c$

2) $-be^{-by}=ae^{ax} + c$

3) $be^{-by}+ae^{ax} = c$

4) none of these

16 )

If $ \vec{a}=\hat{i}+2\hat{j}-3\hat{k} $ and $ \vec{b}=3\hat{i}-\hat{j}+2\hat{k}$ then the angle between $(\vec{a} + \vec{b})$ and $(\vec{a} + \vec{b})$ is

1) $\frac{\pi}{3}$

2) $\frac{\pi}{2}$

3) $\frac{\pi}{4}$

4) $\frac{\pi}{6}$

17 )

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 = ?

1) $\frac{10}{7}$

2) $-\frac{10}{7}$

3) $-\frac{5}{7}$

4) $\frac{5}{7}$

18 )

A machine operates only when all its three components function. The probabilities of the failures of the first, second and third components are 0.2, 0.3 and 0.5 respectively. What is the probability that the machine will fail ?

1) 0.70

2) 0.72

3) 0.50

4) 0.61

19 )

Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x}{sin \, x +cos \, x} \, dx$ .

1) $\frac{\pi}{2\sqrt{2}} log|\frac{\sqrt{2}+1}{\sqrt{2}-1}|$

2) $\frac{\pi}{4\sqrt{2}} log|\frac{\sqrt{2}+1}{\sqrt{2}-1}|$

3) $\frac{\pi}{4\sqrt{2}} log|\frac{\sqrt{2}-1}{\sqrt{2}+1}|$

4) $\frac{\pi}{3\sqrt{2}} log|\frac{\sqrt{2}+1}{\sqrt{2}-1}|$

20 )

Evaluate $\int \, tan^{-1} \sqrt{\frac{1-sin \, x}{1+sin \, x}} \, dx$ .

1) $\frac{\pi x}{4} - \frac{x^2}{4} + c$

2) $\frac{\pi x^3}{4} - \frac{x^2}{4} + c$

3) $\frac{\pi x}{4} + \frac{x^2}{4} + c$

4) $\frac{\pi x^2}{4} - \frac{x}{4} + c$

21 )

Evaluate $\int \, \frac{2}{(1-x)(1+x^2)} \, dx$ .

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

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

3) $\frac{1}{2} \frac{1-x^2}{(1-x)^2} +tan^{-1}x +c$

4) $\frac{1}{2} \frac{1+x^2}{(1-x)^2} +tan^{-1}x +c$

22 )

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

1) $-2 \lt x \lt 6$

2) $2 \lt x \lt 6$

3) $3 \lt x \lt 6$

4) none of these

23 )

Find the point in the first quadrant at which the slope of the normal to the curve $ x^3=8a^2y$, $a \gt 0$ is $- \frac{2}{3}$

1) (2a, a)

2) (a, a)

3) (2a, 3a)

4) (3a, a)

24 )

The semi-vertical angle of a cone of maximum volume and given slant height is

1) $cot^{-1} \sqrt{2}$

2) $tan^{-1} \sqrt{2}$

3) $tan^{-1} \sqrt{3}$

4) $cot^{-1} \sqrt{3}$

25 )

Bag A contains 2 white and 3 red balls, and bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from bag B.

1) $\frac{25}{52}$

2) $\frac{35}{52}$

3) $\frac{15}{52}$

4) $\frac{5}{13}$

26 )

The maximum value of P=6x+11y subject to constraints $2x+y \le 104$, $x+2y \le 76$ and $x \ge 0$, $y \ge 0$ is

1) 220

2) 440

3) 300

4) 400

27 )

Let $f : [1, \infty) \rightarrow [1, \infty)$ ,is given by $f(x)=2^{x(x-1)}$ is invertible. Find $f^{-1}(x)$.

1) $f^{-1}(x)= \frac{1+\sqrt{1+4log_2 x}}{2}$.

2) $f^{-1}(x)= \frac{1+\sqrt{1-4log_2 x}}{2}$.

3) $f^{-1}(x)= \frac{1-\sqrt{1+4log_2 x}}{2}$.

4) none of these

28 )

Solve the following system of equations using matrix method : x + y + z = 1 x - 2y + 3z = 2 5x - 3y + z = 3

1) $x = \frac{1}{2}$ , y = 0, $z = -\frac{1}{2}$

2) $x = -\frac{1}{2}$ , y = 0, $z = \frac{1}{2}$

3) $x = \frac{1}{2}$ , y = 1, $z = \frac{1}{2}$

4) $x = \frac{1}{2}$ , y = 0, $z = \frac{1}{2}$

29 )

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

1) (1, 0, 7)

2) (1, 1, 7)

3) (-1, 0, 7)

4) (1, 0, 6)

30 )

Find the area bounded by curve $y^2=2y-x$ and the y-axis.

1) $\frac{4}{3}$ sq. units

2) $\frac{5}{3}$ sq. units

3) $\frac{5}{2}$ sq. units

4) none of these

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1 ) \begin{bmatrix} 0 & \frac{1}{2} & 2 \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \end{bmatrix} 2 ) |A|=0 and |B|=0 3 ) 9 4 ) 4A 5 ) 0 6 ) $\frac{1}{2}$ 7 ) f(x) is not differentiable at $ \, x = n \pi, \, x \in Z $ 8 ) $\frac{1}{2}$ 9 ) $\frac{4x^3 + 2x}{(1 + x^2)^2}$ 10 ) $\frac{2}{3x}$ 11 ) $\frac{80}{89}$ 12 ) $\frac{1}{2} (log \, tan \, x)^2 + c$ 13 ) $\frac{\pi}{4}$ 14 ) $] 0, \frac{3\pi}{4}[$ 15 ) $\frac{-e^{-by}}{b}=\frac{e^{ax}}{a} + c$ 16 ) $\frac{\pi}{2}$ 17 ) $-\frac{10}{7}$ 18 ) 0.72 19 ) $\frac{\pi}{4\sqrt{2}} log|\frac{\sqrt{2}+1}{\sqrt{2}-1}|$ 20 ) $\frac{\pi x}{4} - \frac{x^2}{4} + c$ 21 ) $\frac{1}{2} \frac{1+x^2}{(1-x)^2} +tan^{-1}x +c$ 22 ) $2 \lt x \lt 6$ 23 ) (2a, a) 24 ) $tan^{-1} \sqrt{2}$ 25 ) $\frac{25}{52}$ 26 ) 440 27 ) $f^{-1}(x)= \frac{1+\sqrt{1+4log_2 x}}{2}$. 28 ) $x = \frac{1}{2}$ , y = 0, $z = \frac{1}{2}$ 29 ) (1, 0, 7) 30 ) $\frac{4}{3}$ sq. units

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