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Mathematics - XII

The Model Paper - 1

Year : 2024 - 25

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 ) 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 \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 ) Find the co-factor of \begin{vmatrix} 2 & 6 \\ -1 & 4 \\ \end{vmatrix}

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

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, \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 $\,\, \lim_{x \to -2} \frac{x^2 - 4}{x + 2}$

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

10 ) Derivative of $x^2$ w.r.t. $x^3$

11 ) If $tan^{-1}3x + tan^{-1}2x = \frac{\pi}{4} $ , then x = ?

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

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

14 ) The least value of k such that $x^2+kx+1$ is increasing on ]1, 2[

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

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

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

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

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

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

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 ) 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 : 2x + 8y + 5z = 5 x + y + z = -2 x + 2y - z = 2

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 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 \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 )

Find the co-factor of \begin{vmatrix} 2 & 6 \\ -1 & 4 \\ \end{vmatrix}

1) -4

2) -2

3) -1

4) -6

4 )

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

1) 4A

2) 16A

3) 64A

4) 128A

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, \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 $\,\, \lim_{x \to -2} \frac{x^2 - 4}{x + 2}$

1) 0

2) 1

3) -1

4) -4

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 )

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 )

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

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

1) -2

2) -1

3) 2

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 $\frac{dy}{dx} = e^{x+y} + x^2.e^y$.

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 )

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 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 )

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

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 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 )

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 )

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 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 )

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 : 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 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 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 ) 1 3 ) -6 4 ) 4A 5 ) 3 6 ) 1 7 ) none of these 8 ) -4 9 ) $e^x (sin(2x) + 2cos(2x))$ 10 ) $\frac{2}{3x}$ 11 ) $\frac{1}{6} $ or -1 12 ) $\frac{x^2}{3} - x - 2 tan^{-1}x + c$ 13 ) 2 14 ) -2 15 ) $e^x+e^{-y} + \frac{x^3}{3}=c$ 16 ) $sin \, \frac{\theta}{2}$ 17 ) (3, -1, -1) 18 ) 0.72 19 ) $\frac{\pi}{4\sqrt{2}} log|\frac{\sqrt{2}+1}{\sqrt{2}-1}|$ 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 ) $-\frac{\pi}{3} \le x \le \frac{\pi}{3} $ 23 ) a=-2, b=-5 24 ) $\frac{4}{27}\pi h^3tan^2 \theta$ 25 ) $\frac{2}{5}$ 26 ) (2, 0) 27 ) x 28 ) x = -3, y = 2, z = -1 29 ) (-1, -1, -1) 30 ) 27 sq. units

Mathematics - XII

The Model Paper - 2

Year : 2024 - 25

Time - 3 marks 0 minutes

Full Marks - 80

Previous Set

Cur Set 2

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

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

9 ) Find the derivative of $f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 1.$

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

11 ) The value of sin[$2 tan^{-1} \frac{5}{8}$]

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

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

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 $log(\frac{dy}{dx} )= ax+by$.

16 ) If $|\vec{a}|=\sqrt{26}$, $|\vec{b}|=7$ and $|\vec{a} \, \times \, \vec{b}| =35 $, then $\vec{a} \,. \, \vec{b}$ 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 ) A coin is tossed 5 times. What is the probability that tail appears an odd number of times ?

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

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

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

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

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

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 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 ) 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 $f : R \rightarrow (-1, \, 1)$ ,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 {$(x, y) \, : \, x^2 + y^2 \le 2ax , \, y^2 \gt ax, a \gt 0, \, x \gt 0, \, y \gt 0, \,$}.

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 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 )

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 )

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

1) -5

2) 5

3) 0

4) none of these

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 )

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

1) 1

2) 4

3) 0

4) -3

9 )

Find the derivative of $f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 1.$

1) $12x^3 - 6x^2 + 10x - 7$

2) $12x^3 - 6x^2 - 10x - 7$

3) $12x^3 + 6x^2 + 10x - 7$

4) $12x^3 - 6x^2 + 10x + 7$

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

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 )

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 $log(\frac{dy}{dx} )= ax+by$.

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

1) 4

2) 5

3) 8

4) 7

17 )

The 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 )

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

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

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

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

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

20 )

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

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 $\int \, \frac{4x+3}{\sqrt{2x^2 + 2x - 3}} \, dx$ .

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 $f(x)=x^3-12 x^2+36x +17$ 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 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 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 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 )

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 $f : R \rightarrow (-1, \, 1)$ ,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 {$(x, y) \, : \, x^2 + y^2 \le 2ax , \, y^2 \gt ax, a \gt 0, \, x \gt 0, \, y \gt 0, \,$}.

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 & 2 \\ 1 & 4 \\ \end{bmatrix} 2 ) \begin{bmatrix} 6 & 0 \\ 0 & 6 \\ \end{bmatrix} 3 ) 12, -2 4 ) x=2, y=-8 5 ) -5 6 ) -1 7 ) -1 8 ) -3 9 ) $12x^3 - 6x^2 + 10x - 7$ 10 ) -tan t 11 ) $\frac{80}{89}$ 12 ) sec x - cosec x + c 13 ) 4 14 ) 4 cm 15 ) $\frac{-e^{-by}}{b}=\frac{e^{ax}}{a} + c$ 16 ) 7 17 ) $cos^{-1}\frac{1}{\sqrt{2}}$ 18 ) $\frac{1}{2}$ 19 ) $\frac{\pi^2}{2ab}$ 20 ) $\frac{2}{9}(1+x^3)^\frac{3}{2}-\frac{2}{3}(1+x^3)^\frac{1}{2}+c$ 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 ) $- \infty \lt x \lt 2 $ and $6 \lt x \lt \infty $ 23 ) (2, -9) 24 ) $\frac{25\pi}{\pi +4}$ and $\frac{100}{\pi +4}$ 25 ) $\frac{1}{5}$ 26 ) x=2, y=6, z=36 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 ) $\frac{a^2}{12}(3\pi-8) sq. \, units$

Mathematics - XII

The Model Paper - 3

Year : 2024 - 25

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.

The Model Paper - 3

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

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

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

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 ) 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} 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 $\,\, \lim_{x \to 3} \frac{x - 3}{|x - 3|}$

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

10 ) The derivative of $sec^2x$ with respect to $tan^2x$.

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

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

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

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 $\frac{dy}{dx} = 2^{x+y}$.

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

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

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

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

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

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

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

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

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 + 1y + z = 4 -x + y + z = 0 x - 3y + 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.

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

Close

1 )

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

1) [12]

2) [14]

3) [24]

4) [0]

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 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 )

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 )

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 )

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} 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 $\,\, \lim_{x \to 3} \frac{x - 3}{|x - 3|}$

1) The limit does not exist.

2) 1

3) -1

4) none of this

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 )

The derivative of $sec^2x$ with respect to $tan^2x$.

1) 0

2) 1

3) 2

4) -2

11 )

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

1) -1

2) 0

3) 2

4) 1

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

1) 2

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

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 $\frac{dy}{dx} = 2^{x+y}$.

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 )

$\vec{a}, \, \vec{b} \, and \, \vec{c}$ 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 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 )

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

1) $2-\sqrt{2}$

2) $3-\sqrt{2}$

3) $2-\sqrt{3}$

4) $3-\sqrt{3}$

20 )

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

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

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 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 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 )

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 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 $f : [1, \infty) \rightarrow [1, \infty)$ ,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 + 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 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 ) [14] 2 ) \begin{bmatrix} -3 & 2 \\ \frac{5}{2} & - \frac{3}{2} \\ \end{bmatrix} 3 ) 1 4 ) $A^2 - AB +BA + B^2 $ 5 ) 2 6 ) 3 7 ) $\frac{1}{2}$ 8 ) The limit does not exist. 9 ) $3(2x - 1)^2 (x + 2)^2 + 2(2x - 1)^3 2(x + 2)$ 10 ) 1 11 ) 0 12 ) $\frac{1}{2} sin \, x - \frac{1}{10} sin \, 5x +c $ 13 ) $\frac{1}{2}$ 14 ) $10 \pi cu cm$ 15 ) $2^x+2^{-y}=c$ 16 ) $\frac{-3}{2}$ 17 ) (-1, 4, 3) 18 ) $\frac{37}{256}$ 19 ) $2-\sqrt{2}$ 20 ) $\frac{log|a^2 \, cos^2x + b^2 sin^2x|}{b^2 - a^2} +c$ 21 ) $e^x cot \, 2x+c$ 22 ) $2 \lt x \lt 6$ 23 ) $$4x-3\sqrt{2}y-2\sqrt{2}=0$$ 24 ) $\frac{\pi}{3}$ 25 ) $\frac{2}{3}$ 26 ) both bounded and unbounded space 27 ) $f^{-1}(x)= \frac{1+\sqrt{1+4log_2 x}}{2}$. 28 ) x = 3, y = 0, z = 2 29 ) (2, 4, -3) 30 ) $\frac{4}{3}$ sq. units$

Mathematics - XII

The Model Paper - 4

Year : 2024 - 25

Time - 3 marks 0 minutes

Full Marks - 80

Previous Set

Cur Set 4

Next Set

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

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

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 ) 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 ) \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 $\,\, \lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

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

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 $\int \frac{ax + b}{cx + d} \, dx$ .

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

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

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

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 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 ) 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 $\int_0^a \, \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} \, dx$ .

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

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

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

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

24 ) The semi-vertical angle of a cone of maximum volume and given slant height is

25 ) 8 coins are tossed at a time, the probability of getting at least 6 heads up, is

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 ) 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 : x - y = 3 2x + 3y + 4z = 17 y + 2z = 7

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 curves$x^2=4y$ and the line x = 4y - 2

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 )

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 \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 )

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 \, 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 )

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 )

\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 $\,\, \lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

1) 1

2) 4

3) 0

4) 3

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 )

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 $\int \frac{ax + b}{cx + d} \, dx$ .

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

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 )

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 $\frac{dy}{dx} = \frac{-2xy}{1+x^2}$ 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 $|\vec{a}|=2$, $|\vec{b}|=7$ 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 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 )

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 $\int_0^a \, \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} \, dx$ .

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

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

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

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

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

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 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 )

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 )

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 )

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=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 )

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 : 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 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 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} 2 & -1 \\ 2 & 4 \\ 3 & 6 \end{bmatrix} 2 ) $B^{-1}A^{-1}$ 3 ) $2a^3b^3c^3$ 4 ) 25 5 ) $\frac{1}{2}$ 6 ) none of these 7 ) $\frac{3}{4}$ 8 ) 4 9 ) $e^x sin(x) + e^x cos(x)$ 10 ) 1 11 ) $\frac{3}{5}$ 12 ) $\frac{ax}{c} + \frac{bc - ad}{c^2} log|cx + d|$ + C 13 ) $\int_{0}^a \, [f(x)+f(-x)] \, dx$ 14 ) $]\frac{5}{3}, 3[$ 15 ) $y(x^2 + 1)=c$ 16 ) $\frac{\pi}{6}$ 17 ) $-\frac{10}{7}$ 18 ) 0.3 19 ) $\frac{\pi}{4}$ 20 ) $\frac{2}{15}x \sqrt{x} (5-3x) + c$ 21 ) $\frac{1}{2}(x^2-1)log|x+1| - \frac{1}{4}x^2 +\frac{1}{2}x +c$ 22 ) $- \infty \lt x \lt 1$ 23 ) $8k^2=1$ 24 ) $tan^{-1} \sqrt{2}$ 25 ) $\frac{37}{256}$ 26 ) 440 27 ) $\alpha=-1$ 28 ) x = 2, y = -1, z = 4 29 ) (1, 0, 7) 30 ) $\frac{9}{8}$ sq. units

Mathematics - XII

The Model Paper - 5

Year : 2024 - 25

Time - 3 marks 0 minutes

Full Marks - 80

Previous Set

Cur Set 5

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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

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 - 5

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×

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 two non-zero square matrices of the same order such that AB = 0, then

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 \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 ) If f(x) = x sin $\frac{1}{x}$ , $x \ne 0 $ then the value of f(0) so that the function is continuous at x = 0.

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

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

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

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 $\frac{dy}{dx} = 1-x+y-xy$.

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 angle between any two diagonals of cube 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 $\int_0^1 \, \frac{log(1+x)}{(1+x)^2} \, dx$ .

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

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

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

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

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 ) 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 maximum value of P=5x+3y subject to constraints $5x+2y \le 10$, $x \ge 0$ and $y \ge 0$ 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 : 2x - 3y + 5z = 16 3x +2 y - 4z = -4 x + y - 2z = -3

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 curve $y^2=2y-x$ and the y-axis.

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 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 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 \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 )

If f(x) = x sin $\frac{1}{x}$ , $x \ne 0 $ then the value of f(0) so that the function is continuous at x = 0.

1) -1

2) 1

3) 0

4) indeterminate

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 )

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

1) tan x

2) -tan x

3) cot x

4) -cot x

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

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

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

2) -a

3) 1

4) 0

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 $\frac{dy}{dx} = 1-x+y-xy$.

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 )

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 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 )

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

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 $\int \, \sqrt{\frac{1+x}{1-x}} \, dx$ .

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

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 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 )

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 )

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 )

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 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 )

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 : 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 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 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 ) 8 2 ) |A|=0 and |B|=0 3 ) 1 4 ) $\frac{33}{2}$ 5 ) 0 6 ) 0 7 ) R - {3} 8 ) 0 9 ) -tan x 10 ) $\frac{3t^2 - 5}{6t + 2}$ 11 ) $\frac{24}{25}$ 12 ) $\frac{1}{4}sin^4x - \frac{1}{6}sin^6x + c$ 13 ) 0 14 ) 15 m/s 15 ) $log(1+y)=x-\frac{x^2}{2} +c$ 16 ) $\frac{5}{\sqrt{6}}$ 17 ) $cos^{-1}\frac{1}{3}$ 18 ) $\frac{5}{8}$ 19 ) $\frac{\pi}{8} log \, 2$ 20 ) $sin^{-1}x-\sqrt{1-x^2} +c$ 21 ) $\frac{1}{2} \frac{1+x^2}{(1-x)^2} +tan^{-1}x +c$ 22 ) -2 23 ) (3, 2), (-1, 2) 24 ) $\frac{\pi}{2}$, Area=$\frac{1}{2}ab$ 25 ) 0.22 26 ) 15 27 ) {0, -1} 28 ) x = 2, y = 1, z = 3 29 ) x + y + z = 0 30 ) $\frac{4}{3}$ sq. units

Mathematics - XII

The Model Paper - 6

Year : 2024 - 25

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

3 ) Let \begin{equation*} A= \begin{bmatrix} -1 & 2 \\ 1 & 4 \\ \end{bmatrix} \end{equation*}, find A(adj A)

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

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

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 f(x) = |sin x|, then

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

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

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

13 ) Evaluate $\int_{-\pi}^{\pi} \, sin^{61}x + x^{123} \, dx$ .

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 $\frac{dy}{dx} + \frac{y}{x}=x^2$.

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 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 $\int_0^{\frac{\pi}{4}} \, log(1+tan \, x) \, dx$ .

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

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

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

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

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 ) 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=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, 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 : 3x - 4y + 2z = -1 2x + 3y + 5z = 7 x + z = 2

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

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} 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 )

Let \begin{equation*} A= \begin{bmatrix} -1 & 2 \\ 1 & 4 \\ \end{bmatrix} \end{equation*}, find A(adj A)

1) \begin{bmatrix} 53 & 0 \\ 0 & 1 \\ \end{bmatrix}

2) \begin{bmatrix} 0 & 0 \\ 0 & 53 \\ \end{bmatrix}

3) \begin{bmatrix} 53 & 0 \\ 0 & 53 \\ \end{bmatrix}

4) \begin{bmatrix} 53 & 0 \\ 0 & 0 \\ \end{bmatrix}

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 )

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

1) 2

2) 1

3) 0

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

1) 0

2) 1

3) -1

4) none of this

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 )

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

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 $\int_{-\pi}^{\pi} \, sin^{61}x + x^{123} \, dx$ .

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

2) -2

3) $\pi$

4) 0

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 $\frac{dy}{dx} + \frac{y}{x}=x^2$.

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}=\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 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 $\int_0^{\frac{\pi}{4}} \, log(1+tan \, x) \, dx$ .

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

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

3) $\frac{\pi}{5} \, log \, 2 $

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

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

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 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 )

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 )

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 )

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=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, 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 : 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 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 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} 15 & -6 \\ -8 & 21 \end{bmatrix} 2 ) adj A 3 ) \begin{bmatrix} 53 & 0 \\ 0 & 53 \\ \end{bmatrix} 4 ) AB = BA = I 5 ) -1 6 ) f(x) is discontinuous at x = 0 7 ) f(x) is not differentiable at $ \, x = n \pi, \, x \in Z $ 8 ) 1 9 ) 2cos(2x) - 3sin(3x) 10 ) $\frac{1}{te^t}$ 11 ) $\frac{\pi}{4}$ 12 ) $\frac{1}{2} (log \, tan \, x)^2 + c$ 13 ) 0 14 ) $3.2 \pi cm^3/sec$ 15 ) $4xy = x^4 + c$ 16 ) $\frac{\pi}{2}$ 17 ) 3 : 4 18 ) 0.1 19 ) $\frac{\pi}{8} \, log \, 2 $ 20 ) $-\frac{2}{b^2}[log|a+b \, cos \, x|+\frac{a}{a+b \, cos \, x}]+c$ 21 ) $sin^{-1}x + \sqrt{1-x^2}+c$ 22 ) $[0, \frac{\pi}{4}]$ 23 ) parallel 24 ) $\frac{c^3}{6\sqrt{3}}$ 25 ) $\frac{5}{9}$ 26 ) none of these 27 ) $f^{-1}(x)=\frac{4x+7}{2}$ 28 ) x = 2, y = 3, z = -1 29 ) (-1, -7, -4) 30 ) $\frac{21}{2}$ sq. units

Mathematics - XII

The Model Paper - 7

Year : 2024 - 25

Time - 3 marks 0 minutes

Full Marks - 80

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Cur Set 7

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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

Close

×

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

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

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

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 $\,\, \lim_{x \to 0} \frac{1 - cos x}{x^2}$

9 ) Determine the derivative of $h(x) = ln(2x + 1) - e^x.$

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

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

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

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

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 $\frac{dy}{dx} = - \sqrt{ \frac{1-y^2}{1-x^2}}$ 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 ) 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 ) 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 $\int_0^{\frac{\pi}{2}} \, \frac{cos^3 x}{sin^3 x +cos^3 x} \, dx$ .

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

21 ) Evaluate $\int \,[\frac{1}{log \, x}-\frac{1}{(log \, x)^2}] \, dx$ .

22 ) The intervals in which the function $f(x)=sin \, x + cos \, x$ 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 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 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 ) 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 : x - y + z = 1 2x + y - z = 2 x - 2y - z = 4

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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ 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 an invertible square matrix then $A^{-1} = ?$

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

2) |A|

3) 1

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 )

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 )

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 $\,\, \lim_{x \to 0} \frac{1 - cos x}{x^2}$

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

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

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

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

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 )

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

1) -1

2) 0

3) 1

4) non of these

11 )

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

1) 0

2) 3

3) 2

4) -1

12 )

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

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 )

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 $\frac{dy}{dx} = - \sqrt{ \frac{1-y^2}{1-x^2}}$ 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 )

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 )

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

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

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

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

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

20 )

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

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 $\int \,[\frac{1}{log \, x}-\frac{1}{(log \, x)^2}] \, dx$ .

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 $f(x)=sin \, x + cos \, x$ 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 )

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 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 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 )

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 : 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 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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ 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 ) $\frac{1}{|A|}$ 3 ) 9 4 ) adj(B) adj(A) 5 ) 0 6 ) g(x) is discontinuous function 7 ) continuous everywhere but not differentiable at x = 0 8 ) $\frac{1}{2}$ 9 ) $\frac{2}{2x + 1} - e^x$ 10 ) -1 11 ) 0 12 ) $2 \sqrt{tan \, x} +c$ 13 ) -1 14 ) -0.32 15 ) $sin^{-1}y+sin^{-1}x=c$ 16 ) $\frac{41}{2}$ sq units 17 ) 20x + 23y + 26z = 69 18 ) $\frac{1}{3}$ 19 ) $\frac{\pi}{4}$ 20 ) $\frac{1}{\sqrt{2}}tan^{-1}[\frac{1}{\sqrt{2}}(x-\frac{1}{x})]+c$ 21 ) $\frac{x}{log \, x} +c$ 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 ) (1, 2), (1, -2) 24 ) $500 \pi cm^3$ 25 ) $\frac{5}{9}$ 26 ) 14 27 ) either a=1 and b=0 or a=-1 and $b \, \in \, R$ 28 ) x = 1, y = -1, z = -1 29 ) 7 units 30 ) $(\frac{\pi ab}{4} -\frac{ab}{2})$ sq. units

Mathematics - XII

The Model Paper - 8

Year : 2024 - 25

Time - 3 marks 0 minutes

Full Marks - 80

Previous Set

Cur Set 8

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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 - 8

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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 - 8

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

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

3 ) Find the value of \begin{vmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \\ \end{vmatrix} where $\omega$ is the cubic root of unity.

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} 0 & b & -c \\ -b & 0 & a \\ c & -a & 0 \\ \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 ) 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 ) Calculate the derivative of $f(x) = \sqrt{x^2 + 4x} - x^2.$

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

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

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

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

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

20 ) Evaluate $\int \, tan^{-1} \sqrt{\frac{1-sin \, x}{1+sin \, x}} \, dx$ .

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

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

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 ) 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 ) 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 : 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 + y - z = 1 x - y + z = 2 3x + y - 2z = -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 y = sin x and y = cos x for $0 \le x\le \frac{\pi}{2}$

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 )

Find the value of \begin{vmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \\ \end{vmatrix} where $\omega$ is the cubic root of unity.

1) 1

2) 0

3) -1

4) none of these

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} 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{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 )

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

1) 3

2) 4

3) 0

4) -3

9 )

Calculate the derivative of $f(x) = \sqrt{x^2 + 4x} - x^2.$

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

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

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

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

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 )

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

1) $\frac{19}{6}$

2) $\frac{17}{6}$

3) $\frac{13}{6}$

4) $\frac{23}{6}$

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 )

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 $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 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 )

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

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 $\int \, tan^{-1} \sqrt{\frac{1-sin \, x}{1+sin \, x}} \, dx$ .

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

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 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 )

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 )

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 )

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 : 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 + 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 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 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 ) \begin{bmatrix} cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix} 2 ) $\frac{1}{k} .A^{-1}$ 3 ) 0 4 ) $A^2 - AB - BA + B^2$ 5 ) 0 6 ) $\frac{1}{2}$ 7 ) none of these 8 ) 3 9 ) $\frac{1}{2} (x^2 + 4x)^{-\frac{1}{2}} (2x + 4) - 2x$ 10 ) $\frac{4}{25}$ 11 ) $\frac{17}{6}$ 12 ) $2 \sqrt{e^x - 1} - 2 tan^{-1}\sqrt{e^x - 1} + c$ 13 ) $\frac{1}{6}$ 14 ) $] 0, \frac{3\pi}{4}[$ 15 ) $ \sqrt{1+y^2} +\sqrt{1+x^2} = c $ 16 ) $\sqrt{3}$ 17 ) $cos^{-1}\frac{8\sqrt{3}}{15}$ 18 ) independent 19 ) $\frac{16\sqrt{2}}{15}$ 20 ) $\frac{\pi x}{4} - \frac{x^2}{4} + c$ 21 ) $\frac{5}{16} log|\frac{x-2}{x+2}| - \frac{7}{4(x-2)} +c$ 22 ) $1 \le x \lt \infty $ 23 ) (2a, a) 24 ) $\frac{16}{6-\sqrt{3}}$m 25 ) $\frac{37}{56}$ 26 ) 120 27 ) neither one-one nor onto. 28 ) x = 1, y = 2, z = 3 29 ) (-3, 5, 2) 30 ) $2(\sqrt{2} -1)$ sq. units

Mathematics - XII

The Model Paper - 9

Year : 2024 - 25

Time - 3 marks 0 minutes

Full Marks - 80

Previous Set

Cur Set 9

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

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

5 ) \begin{equation} \begin{vmatrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \\ \end{vmatrix} \end{equation}

6 ) The function $f(x) = \frac{4-x^2}{4x-x^3}$

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

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

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

10 ) The derivative of x = a(t - sin t) and y = a(1 - cos t)

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

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

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

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 $x \frac{dy}{dx} = cot \, y$.

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

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

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 $\int_0^1 \, cot^{-1}(1-x+x^2) \, dx$ .

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

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

22 ) The intervals in which the function $f(x)=( x+2)e^{-x}$ is increasing or decreasing

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 ) 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 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 ) 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 x - 2y + 3z = 2 5x - 3y + z = 3

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 \le y \le |x| $}.

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 )

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 )

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 )

\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 )

The function $f(x) = \frac{4-x^2}{4x-x^3}$

1) discontinuous at exactly three point

2) discontinuous at exactly two point

3) continuous at only one point

4) discontinuous at only one point

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

1) 2

2) -2

3) 1

4) 0

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 )

The derivative of x = a(t - sin t) and y = a(1 - cos t)

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

2) $- cot \frac{t}{2}$

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

4) $- tan \frac{t}{2}$

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

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

1) 2

2) 4

3) 1

4) none of these

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 $x \frac{dy}{dx} = cot \, y$.

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 )

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 )

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 $\int_0^1 \, cot^{-1}(1-x+x^2) \, dx$ .

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

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 $f(x)=( x+2)e^{-x}$ 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 )

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 )

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 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 )

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 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 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 \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} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} 2 ) nilpotent 3 ) -400 4 ) a skew symmetric square matrix 5 ) 0 6 ) discontinuous at only one point 7 ) $-1 \, \le \, x \, \le \, 4 \,$ 8 ) 2 9 ) $(6x + 2) e^{3x^2 + 2x}$ 10 ) $cot \frac{t}{2}$ 11 ) $\frac{1}{2}$ 12 ) $x cos \alpha +(sin \alpha) log|sin(x-\alpha)| + c$ 13 ) 2 14 ) 0.5 m/s 15 ) x cos y = c 16 ) $sin \, \frac{\theta}{2}$ 17 ) 2 18 ) $\frac{5}{9}$ 19 ) $\frac{\pi}{2} - log \, 2$ 20 ) $tan^{-1}\frac{tan \, x -1}{\sqrt{2 \, tan \, x}}+c$ 21 ) $-\frac{sin^{-1}x}{x}+log| \frac{1}{x} - \frac{\sqrt{1-x^2}}{x}|$ 22 ) inc in $(- \infty, -1) $; dec in $(-1, \infty) $ 23 ) $( \frac{7}{4}, \, \frac{1}{4})$ 24 ) $75\sqrt{3}cm^2$. 25 ) $\frac{25}{52}$ 26 ) None of these 27 ) $Domain=(2, \, \infty )$ and $Range= x \in R^+$ 28 ) $x = \frac{1}{2}$ , y = 0, $z = \frac{1}{2}$ 29 ) 13 30 ) $\frac{1}{3}$ sq. units

Mathematics - XII

The Model Paper - 10

Year : 2024 - 25

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.

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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 - 10

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1 ) Find a 2 x 3 matrix $\, B=[a_{ij} ] \,$, where $\, b_{ij} \, =\, \frac{1}{2}(i - j)^2$.

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 x, \begin{equation} \begin{vmatrix} 2 & x \\ x & 1 \\ \end{vmatrix} = \begin{vmatrix} 2 & 3 \\ 4 & 1 \\ \end{vmatrix} \end{equation}

4 ) Solve for x and y : \begin{equation*} \begin{bmatrix} x^2 \\ y^2 \\ \end{bmatrix} + 2 \begin{bmatrix} 2x \\ 3y \\ \end{bmatrix} = 3 \begin{bmatrix} 7 \\ -3 \\ \end{bmatrix} \end{equation*}

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 ) \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 ) The function f(x) = x - [x], where [.] denotes the greatest integer function is

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

9 ) Find the derivative of $f(x) = \frac{x^4}{1 + x^2} $

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

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

12 ) Evaluate $\int sin^3(2x+1) \, dx$ .

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

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 $\frac{dy}{dx} = \frac{1-cos \, x}{1+cos \, x}$.

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

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

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

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

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

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

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 ) 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 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 : 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 + 3z = 5 x - 2y + z = -4 3x - y - 2z = 3

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 curves {$(x, y) \, : \, x^2 + y^2 \le 1 \le x+y $}.

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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 )

\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 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 )

Solve for x and y : \begin{equation*} \begin{bmatrix} x^2 \\ y^2 \\ \end{bmatrix} + 2 \begin{bmatrix} 2x \\ 3y \\ \end{bmatrix} = 3 \begin{bmatrix} 7 \\ -3 \\ \end{bmatrix} \end{equation*}

1) x=-7, y=-3 or x=3, y=-3

2) x=7, y=3 or x=3, y=-3

3) x=7, y=-3 or x=3, y=-3

4) x=-7, y=3 or x=3, y=-3

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 )

\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 )

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

1) 0

2) $\infty$

3) $-\infty$

4) 1

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 )

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 )

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

1) $105^o$

2) $60^o$

3) $100^o$

4) $80^o$

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

1) $\pi$

2) 0

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

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

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 $\frac{dy}{dx} = \frac{1-cos \, x}{1+cos \, x}$.

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 )

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 )

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 )

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

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

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

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

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

20 )

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

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

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 $f(x)=log(1+ x) - \frac{x}{1+x}$ 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 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 )

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 )

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 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 : 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 + 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 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 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 ) \begin{bmatrix} 0 & \frac{1}{2} & 2 \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \end{bmatrix} 2 ) idempotent 3 ) $\pm 2 \sqrt{3}$ 4 ) x=-7, y=-3 or x=3, y=-3 5 ) $a^2+b^2+c^2+d^2$ 6 ) 1 7 ) continuous at non-integer points only 8 ) $-\infty$ 9 ) $\frac{4x^3 + 2x}{(1 + x^2)^2}$ 10 ) $\frac{3}{4t}$ 11 ) $100^o$ 12 ) $-\frac{1}{2}cos(2x+1) + \frac{1}{6}cos^3(2x+1) + c$ 13 ) $\frac{\pi}{4}$ 14 ) 6% 15 ) $y=2tan \frac{x}{2}-x+c$ 16 ) $\vec{a} \perp \vec{b}$ 17 ) $-\frac{10}{7}$ 18 ) $\frac{3}{5}$ 19 ) $\frac{\pi}{4}$ 20 ) cos(a-b) log|sin(x+b)| - xsin(a-b) +c 21 ) $\frac{3x}{8}+\frac{sin \, 4x}{32}+\frac{sin \, 2x}{4}+c$ 22 ) increasing in $0 \lt x \lt \infty$ ; decreasing in $ - \infty \lt x \lt 0$ 23 ) $tan^{-1}3$ 24 ) 1024 cu. cm. 25 ) $\frac{36}{61}$ 26 ) 31 27 ) x 28 ) x = 1, y = 2, z = -1 29 ) $3\sqrt{30}$ units, $\frac{x-3}{2}=\frac{y-8}{5}=\frac{z-3}{-1}$ 30 ) $\frac{\pi -2}{4} sq.units$

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