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The most experienced teachers advice their students to follow these ten "Model Papers in mathematics, which are made for 10+2 science students."
The salient features of this Mock tests are : An exhaustive coverage of all type questions on each topic. Before going to appear the test, the students must read the instructions carefully. The students may appear the tests as if they are appearing a rehearsal test. After finishing the test the students must submit the test and will be able to see his/her result.

Instructions to follow these Model Papers :

1.) There are three columns in the top row. If clicked/touched on the "Back to Website", many informations will be shown related to the website. Middle column shows the current time. If clicked/touched on the "Start To Answer", a countdown timer would be noticed which means that the test has started. Examinee will have to finished his/her test within the given period of time.

2.) There are three columns in the 2nd row. Subject and class are mentioned in the 1st column, "Model Paper number" out of ten Model Papers in the 2nd column, and for which year students are appearing in the 3rd column.

3.) There are two columns in the 3rd row. Time is mentioned in the 1st column and "Full Marks" in the 2nd column.

4.) There are many columns in the 4th row. Clicked/touched on the "Previous Set" of the 1st column, then if current set mentioned in next column is not similar to 1, the previous set of current set will be shown. Clicked/touched on the "Next Set" in the 3rd column, then if current set mentioned in before column is not equal to 10, the next set of current set will be shown. "Pages=>" in the 4th column and following that numbers 1, 2, 3... means the number of pages in the Model Papers. Each page contains 5 questions. Clicked/touched on the "numbers followed by Pages=>" will be shown the corresponding page. There are two numbers in the last columns one followed by "Not Answered" and other followed by Answered, the 1st one means that the number of questions is not answered and the 2nd one means that the number of questions is answered.

5.) The next rows before last two rows are easily understandable to the examinees.

6.) There are two numbers in the last row one followed by "Set-" and other followed by "Next Page-", the 1st one means that the Model Paper number and the 2nd one means the page number. Clicked/touched on the "last but one row", the questions of the next page of the current page i.e. numbering the next number as shown in the current page as a last number.

7.) Clicked/touched on the "last row where there are two words "Test Submit", the Test willbe submitted and the result will be shown. If examinee is not submitting his/her test, test will be automatically submitted after time period is over.

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Before going to start the test read carefully the instruction. To start the test touch/click "Start To Answer".
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1

Mathematics - XII

The Model Paper - 1

Year : 2026 - 27

Time - 2 Hours 40 minutes
Full Marks - 73
Previous Set
Cur Set 1
Next Set
Pages=>
1
2
3
4
5
6
Not Answered 28 Answered 0

In questions 1 to 28, 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 17 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 3 questions of 5 marks each.

The Model Paper - 1

Close

×

1 ) If matrix A = [1 2 3], find $AA^ \prime$
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 ) 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*} f(x) = \left\{ \begin{array}{ll} 2x - 1 \qquad x < 0 \\ 2x + 1 \quad x \geq 0 \end{array} \right. \end{equation*}
7 ) Let f(x) = $\sqrt{ x^2 \, - \, 3x \, -4 \, }$. Then domain of f(x) is
8 ) Evaluate $\,\, \lim_{x \to - \infty} \frac{2x^3 + 3x}{x^3 + 2x^2}$
9 ) Find the derivative of $f(x) = \sqrt{x^3 - 2x^2 + 1}.$
10 ) The derivative of $tan^{-1}x$ with respect to $cot^{-1} x$
11 ) The value of sin[$2 tan^{-1} \frac{5}{8}$]
12 ) Evaluate $\int xin^3x \, cos^3x \, dx$ .
13 ) Evaluate $\int_{-\pi}^{\pi} \, sin^{61}x + x^{123} \, dx$ .
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 $\frac{dy}{dx} = 1-x+y-xy$.
16 ) If $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, then $\frac{1}{2}|\hat{a} - \hat{b}|$ = ?
17 ) The vertices of a $\triangle ABC$ are A(-1, 3, 2), B(2, 3, 5) and C(3, 5, -2). Then $\angle B = ?$
18 ) Evaluate $\int_0^{\frac{\pi}{4}} \, log(1+tan \, x) \, dx$ .
19 ) Evaluate $\int \, \sqrt{\frac{1+x}{1-x}} \, dx$ .
20 ) Evaluate $\int \, \sqrt{\frac{1-x}{1+x}}$ .
21 ) The intervals in which the function $f(x)=x^3-12 x^2+36x +17$ is increasing
22 ) Find the equation of tangent at $t=\frac{\pi}{4}$ for the curve x = sin 3t, y = cos 2t.
23 ) The volume of the greatest cylinder which can be inscribed in cone of height h and semi vertical angle $\theta$ is
24 ) 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.
25 ) 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
26 ) Let f : A $\rightarrow$ B and f{$(x, \, x^2)$ : x $\in$ A} the f : A $\rightarrow$ A is
27 ) 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.
28 ) Find the area bounded by curves$x^2=4y$ and the line x = 4y - 2

Close

1 )
If matrix A = [1 2 3], find $AA^ \prime$
2
   
   
   
   
NA
2 )
If A and B are invertible matrices of the same order, then $(AB)^{-1} = ?$
2
   
   
   
   
NA
3 )
Find the value of \begin{vmatrix} a & ab^2 & ac^2 \\ a^2b & 0 & bc^2 \\ a^2c & cb^2 & 0 \\ \end{vmatrix}
2
   
   
   
   
NA
4 )
For square matrices A and B of the same order, we have adj(AB) = ?
2
   
   
   
   
NA
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}
2
   
   
   
   
NA
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*}
2
   
   
   
   
NA
7 )
Let f(x) = $\sqrt{ x^2 \, - \, 3x \, -4 \, }$. Then domain of f(x) is
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to - \infty} \frac{2x^3 + 3x}{x^3 + 2x^2}$
2
   
   
   
   
NA
9 )
Find the derivative of $f(x) = \sqrt{x^3 - 2x^2 + 1}.$
2
   
   
   
   
NA
10 )
The derivative of $tan^{-1}x$ with respect to $cot^{-1} x$
2
   
   
   
   
NA
11 )
The value of sin[$2 tan^{-1} \frac{5}{8}$]
2
   
   
   
   
NA
12 )
Evaluate $\int xin^3x \, cos^3x \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_{-\pi}^{\pi} \, sin^{61}x + x^{123} \, dx$ .
2
   
   
   
   
NA
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
2
   
   
   
   
NA
15 )
The solution of differential equation $\frac{dy}{dx} = 1-x+y-xy$.
2
   
   
   
   
NA
16 )
If $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, then $\frac{1}{2}|\hat{a} - \hat{b}|$ = ?
2
   
   
   
   
NA
17 )
The vertices of a $\triangle ABC$ are A(-1, 3, 2), B(2, 3, 5) and C(3, 5, -2). Then $\angle B = ?$
2
   
   
   
   
NA
18 )
Evaluate $\int_0^{\frac{\pi}{4}} \, log(1+tan \, x) \, dx$ .
3
   
   
   
   
NA
19 )
Evaluate $\int \, \sqrt{\frac{1+x}{1-x}} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \sqrt{\frac{1-x}{1+x}}$ .
3
   
   
   
   
NA
21 )
The intervals in which the function $f(x)=x^3-12 x^2+36x +17$ is increasing
3
   
   
   
   
NA
22 )
Find the equation of tangent at $t=\frac{\pi}{4}$ for the curve x = sin 3t, y = cos 2t.
3
   
   
   
   
NA
23 )
The volume of the greatest cylinder which can be inscribed in cone of height h and semi vertical angle $\theta$ is
3
   
   
   
   
NA
24 )
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.
3
   
   
   
   
NA
25 )
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
3
   
   
   
   
NA
26 )
Let f : A $\rightarrow$ B and f{$(x, \, x^2)$ : x $\in$ A} the f : A $\rightarrow$ A is
5
   
   
   
   
NA
27 )
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.
5
   
   
   
   
NA
28 )
Find the area bounded by curves$x^2=4y$ and the line x = 4y - 2
5
   
   
   
   
NA
Set-1, Next Page-
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Show Answers
1 ) [14] 2 ) $B^{-1}A^{-1}$ 3 ) $2a^3b^3c^3$ 4 ) adj(B) adj(A) 5 ) 0 6 ) f(x) is discontinuous at x = 0 7 ) $-1 \, \le \, x \, \le \, 4 \,$ 8 ) 2 9 ) $\frac{3x^2 - 4x}{2 \sqrt{x^3 - 2x^2 + 1}}$ 10 ) -1 11 ) $\frac{80}{89}$ 12 ) $\frac{1}{4}sin^4x - \frac{1}{6}sin^6x + c$ 13 ) 0 14 ) 0.5 m/s 15 ) $log(1+y)=x-\frac{x^2}{2} +c$ 16 ) $sin \, \frac{\theta}{2}$ 17 ) $cos^{-1}\frac{1}{\sqrt{2}}$ 18 ) $\frac{\pi}{8} \, log \, 2 $ 19 ) $sin^{-1}x-\sqrt{1-x^2} +c$ 20 ) $sin^{-1}x + \sqrt{1-x^2}+c$ 21 ) $- \infty \lt x \lt 2 $ and $6 \lt x \lt \infty $ 22 ) $$4x-3\sqrt{2}y-2\sqrt{2}=0$$ 23 ) $\frac{4}{27}\pi h^3tan^2 \theta$ 24 ) $\frac{36}{61}$ 25 ) None of these 26 ) neither one-one nor onto. 27 ) (2, 4, -3) 28 ) $\frac{9}{8}$ sq. units
2

Mathematics - XII

The Model Paper - 2

Year : 2026 - 27

Time - 2 marks 40 minutes
Full Marks - 73
Previous Set
Cur Set 2
Next Set
Pages=>
1
2
3
4
5
6
Not Answered 28 Answered 0

In questions 1 to 28, 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 17 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 3 questions of 5 marks each.

The Model Paper - 2

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 A is an invertible square matrix then $A^{-1} = ?$
3 ) Find the value of \begin{vmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \\ \end{vmatrix}
4 ) A is 3-rowed square matrix and |A| = 4, then adj(adj A) = ?
5 ) Evaluate \begin{vmatrix} x+1 & x \\ x & x - 1 \\ \end{vmatrix}
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 ) \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 $\,\, \lim_{x \to -2} \frac{x^2 - 4}{x + 2}$
9 ) Calculate the derivative of $g(x) = e^x sin(x).$
10 ) Find the derivative of $x = 3t^2 + 2t$ and $y = t^3 - 5t.$
11 ) The value of tan{$cos^{-1}\frac{4}{5} + tan^{-1}\frac{2}{3}$}
12 ) Evaluate $\int \frac{\sqrt{tan \, x} }{sin \, x \, \, cos \, x} \, dx$ .
13 ) Evaluate $\int_0^{\pi} \, |cos \, x| \, dx$ .
14 ) The radius of a spherical soap bubble is increasing at the rate of 0.05cm/sec. Find the rate of increase in its volume when its radius is 4 cm.
15 ) The solution of differential equation $\frac{dy}{dx} = \frac{-2xy}{1+x^2}$ is.
16 ) If $|\vec{a}|=\sqrt{26}$, $|\vec{b}|=7$ and $|\vec{a} \, \times \, \vec{b}| =35 $, then $\vec{a} \,. \, \vec{b}$ is
17 ) The lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-5}$ are perpendicular to each other, then k = ?
18 ) Evaluate $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \, |sin \, x| \, dx$ .
19 ) Evaluate $\int \, \frac{x^2+1}{x^4+1} \, dx$ .
20 ) Evaluate $\int \, cos^4x \, dx$ .
21 ) The function $f(x)=cos^{-1}(sin \, x +cos \, x)$ strictly decreasing function in the interval -
22 ) If the curves $x=y^2$ and $xy=k$ cut at right angles, then
23 ) 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.
24 ) 8 coins are tossed at a time, the probability of getting at least 6 heads up, is
25 ) 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
26 ) Let $f : R \rightarrow (-1, \, 1)$ ,is given by $f(x)=\frac{10^x-10^{-x}}{10^x+10^{-x}}$ is invertible. Find $f^{-1}(x)$.
27 ) Find the image of the point having position vector $\vec{r}=\hat{i}+3\hat{j}+4\hat{k}$ $\vec{r}.(2\hat{i}-\hat{j}+\hat{k})+3=0$
28 ) 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 & 3 \\ 4 & -1 \end{bmatrix} \end{equation*} and $f(x)=x^2 - 3x +5$, find f(A).
2
   
   
   
   
NA
2 )
If A is an invertible square matrix then $A^{-1} = ?$
2
   
   
   
   
NA
3 )
Find the value of \begin{vmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \\ \end{vmatrix}
2
   
   
   
   
NA
4 )
A is 3-rowed square matrix and |A| = 4, then adj(adj A) = ?
2
   
   
   
   
NA
5 )
Evaluate \begin{vmatrix} x+1 & x \\ x & x - 1 \\ \end{vmatrix}
2
   
   
   
   
NA
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*}
2
   
   
   
   
NA
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
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to -2} \frac{x^2 - 4}{x + 2}$
2
   
   
   
   
NA
9 )
Calculate the derivative of $g(x) = e^x sin(x).$
2
   
   
   
   
NA
10 )
Find the derivative of $x = 3t^2 + 2t$ and $y = t^3 - 5t.$
2
   
   
   
   
NA
11 )
The value of tan{$cos^{-1}\frac{4}{5} + tan^{-1}\frac{2}{3}$}
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{\sqrt{tan \, x} }{sin \, x \, \, cos \, x} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_0^{\pi} \, |cos \, x| \, dx$ .
2
   
   
   
   
NA
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.
2
   
   
   
   
NA
15 )
The solution of differential equation $\frac{dy}{dx} = \frac{-2xy}{1+x^2}$ is.
2
   
   
   
   
NA
16 )
If $|\vec{a}|=\sqrt{26}$, $|\vec{b}|=7$ and $|\vec{a} \, \times \, \vec{b}| =35 $, then $\vec{a} \,. \, \vec{b}$ is
2
   
   
   
   
NA
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 = ?
2
   
   
   
   
NA
18 )
Evaluate $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \, |sin \, x| \, dx$ .
3
   
   
   
   
NA
19 )
Evaluate $\int \, \frac{x^2+1}{x^4+1} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, cos^4x \, dx$ .
3
   
   
   
   
NA
21 )
The function $f(x)=cos^{-1}(sin \, x +cos \, x)$ strictly decreasing function in the interval -
3
   
   
   
   
NA
22 )
If the curves $x=y^2$ and $xy=k$ cut at right angles, then
3
   
   
   
   
NA
23 )
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.
3
   
   
   
   
NA
24 )
8 coins are tossed at a time, the probability of getting at least 6 heads up, is
3
   
   
   
   
NA
25 )
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
3
   
   
   
   
NA
26 )
Let $f : R \rightarrow (-1, \, 1)$ ,is given by $f(x)=\frac{10^x-10^{-x}}{10^x+10^{-x}}$ is invertible. Find $f^{-1}(x)$.
5
   
   
   
   
NA
27 )
Find the image of the point having position vector $\vec{r}=\hat{i}+3\hat{j}+4\hat{k}$ $\vec{r}.(2\hat{i}-\hat{j}+\hat{k})+3=0$
5
   
   
   
   
NA
28 )
Find the area cut off from the parabola $4y=3x^2$ by the straight line 3x - 2y +12 = 0.
5
   
   
   
   
NA
Set-2, Next Page-
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Show Answers
1 ) \begin{bmatrix} 15 & -6 \\ -8 & 21 \end{bmatrix} 2 ) $\frac{1}{|A|}$ 3 ) 0 4 ) 4A 5 ) -1 6 ) g(x) is discontinuous function 7 ) $\frac{1}{2}$ 8 ) -4 9 ) $e^x sin(x) + e^x cos(x)$ 10 ) $\frac{3t^2 - 5}{6t + 2}$ 11 ) $\frac{17}{6}$ 12 ) $2 \sqrt{tan \, x} +c$ 13 ) 2 14 ) $3.2 \pi cm^3/sec$ 15 ) $y(x^2 + 1)=c$ 16 ) 7 17 ) $-\frac{10}{7}$ 18 ) $2-\sqrt{2}$ 19 ) $\frac{1}{\sqrt{2}}tan^{-1}[\frac{1}{\sqrt{2}}(x-\frac{1}{x})]+c$ 20 ) $\frac{3x}{8}+\frac{sin \, 4x}{32}+\frac{sin \, 2x}{4}+c$ 21 ) $[0, \frac{\pi}{4}]$ 22 ) $8k^2=1$ 23 ) $75\sqrt{3}cm^2$. 24 ) $\frac{37}{256}$ 25 ) 120 26 ) $\frac{1}{2} log_{10} \frac{1+x}{1-x}$ 27 ) (-3, 5, 2) 28 ) 27 sq. units
3

Mathematics - XII

The Model Paper - 3

Year : 2026 - 27

Time - 2 marks 40 minutes
Full Marks - 73
Previous Set
Cur Set 3
Next Set
Pages=>
1
2
3
4
5
6
Not Answered 28 Answered 0

In questions 1 to 28, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 3

Instructions

Section-A contains 17 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 3 questions of 5 marks each.

The Model Paper - 3

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 ) Find the co-factor of \begin{vmatrix} 2 & 6 \\ -1 & 4 \\ \end{vmatrix}
4 ) If A and B are square matrices of the same order, then $(A - B)^2$ = ?
5 ) If matrix A=[-5], what is the value of det A ?
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 ) 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 ) Calculate the derivative of $f(x) = 5^x + 3^x.$
10 ) The derivative of $sin^2x$ with respect to cos x
11 ) The value of cot[$ tan^{-1} x + cot^{-1} x $]
12 ) Evaluate $\int \sqrt{e^x - 1} \, dx$ .
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 $\frac{dy}{dx} = - \sqrt{ \frac{1-y^2}{1-x^2}}$ is.
16 ) $\vec{a}, \, \vec{b} \, and \, \vec{c}$ are units vectors such that $\vec{a} + \vec{b} + \vec{c}=0$ then $\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}$ = ?
17 ) The line $\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+5}{4}$ meets the plane at the point
18 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{cos^3 x}{sin^3 x +cos^3 x} \, dx$ .
19 ) Evaluate $\int \, \frac{sin \, 2x}{a^2 \, cos^2x + b^2 \, sin^2x} \, dx$ .
20 ) Evaluate $\int \,[\frac{1}{log \, x}-\frac{1}{(log \, x)^2}] \, dx$ .
21 ) The intervals in which the function $f(x)=x^3-12 x^2+36x +17$ is decreasing
22 ) 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 $- \frac{2}{3}$
23 ) 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.
24 ) 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.
25 ) 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
26 ) 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 $f \, o \, f \, = \, I_R$.
27 ) Find the equation of the plane passing through the point (0, 7, -7) and containing the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$
28 ) Find the area bounded by curve $y^2=2y-x$ and the y-axis.

Close

1 )
Find a 2 x 2 matrix $\, A=[a_{ij} \,] $, $\, a_{ij} \, =\, 2i + 3j - 6$.
2
   
   
   
   
NA
2 )
if A is a 2-rowed square matrix and |A| = 6, then A . Adj A = ?
2
   
   
   
   
NA
3 )
Find the co-factor of \begin{vmatrix} 2 & 6 \\ -1 & 4 \\ \end{vmatrix}
2
   
   
   
   
NA
4 )
If A and B are square matrices of the same order, then $(A - B)^2$ = ?
2
   
   
   
   
NA
5 )
If matrix A=[-5], what is the value of det A ?
2
   
   
   
   
NA
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.
2
   
   
   
   
NA
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.
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to 0} \frac{x^2 - 3x}{x}$
2
   
   
   
   
NA
9 )
Calculate the derivative of $f(x) = 5^x + 3^x.$
2
   
   
   
   
NA
10 )
The derivative of $sin^2x$ with respect to cos x
2
   
   
   
   
NA
11 )
The value of cot[$ tan^{-1} x + cot^{-1} x $]
2
   
   
   
   
NA
12 )
Evaluate $\int \sqrt{e^x - 1} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_{-a}^a \, x|x| \, dx$ .
2
   
   
   
   
NA
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
2
   
   
   
   
NA
15 )
The solution of differential equation $\frac{dy}{dx} = - \sqrt{ \frac{1-y^2}{1-x^2}}$ is.
2
   
   
   
   
NA
16 )
$\vec{a}, \, \vec{b} \, and \, \vec{c}$ are units vectors such that $\vec{a} + \vec{b} + \vec{c}=0$ then $\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}$ = ?
2
   
   
   
   
NA
17 )
The line $\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+5}{4}$ meets the plane at the point
2
   
   
   
   
NA
18 )
Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{cos^3 x}{sin^3 x +cos^3 x} \, dx$ .
3
   
   
   
   
NA
19 )
Evaluate $\int \, \frac{sin \, 2x}{a^2 \, cos^2x + b^2 \, sin^2x} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \,[\frac{1}{log \, x}-\frac{1}{(log \, x)^2}] \, dx$ .
3
   
   
   
   
NA
21 )
The intervals in which the function $f(x)=x^3-12 x^2+36x +17$ is decreasing
3
   
   
   
   
NA
22 )
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 $- \frac{2}{3}$
3
   
   
   
   
NA
23 )
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.
3
   
   
   
   
NA
24 )
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.
3
   
   
   
   
NA
25 )
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
3
   
   
   
   
NA
26 )
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 $f \, o \, f \, = \, I_R$.
5
   
   
   
   
NA
27 )
Find the equation of the plane passing through the point (0, 7, -7) and containing the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$
5
   
   
   
   
NA
28 )
Find the area bounded by curve $y^2=2y-x$ and the y-axis.
5
   
   
   
   
NA
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1 ) \begin{bmatrix} -1 & 2 \\ 1 & 4 \\ \end{bmatrix} 2 ) \begin{bmatrix} 6 & 0 \\ 0 & 6 \\ \end{bmatrix} 3 ) -6 4 ) $A^2 - AB - BA + B^2$ 5 ) -5 6 ) 0 7 ) -1 8 ) -3 9 ) $ln(5) 5^x + ln(3) 3^x$ 10 ) -2cos x 11 ) 0 12 ) $2 \sqrt{e^x - 1} - 2 tan^{-1}\sqrt{e^x - 1} + c$ 13 ) 0 14 ) 15 m/s 15 ) $sin^{-1}y+sin^{-1}x=c$ 16 ) $\frac{-3}{2}$ 17 ) (3, -1, -1) 18 ) $\frac{\pi}{4}$ 19 ) $\frac{log|a^2 \, cos^2x + b^2 sin^2x|}{b^2 - a^2} +c$ 20 ) $\frac{x}{log \, x} +c$ 21 ) $2 \lt x \lt 6$ 22 ) (2a, a) 23 ) 1024 cu. cm. 24 ) $\frac{1}{5}$ 25 ) none of these 26 ) either a=1 and b=0 or a=-1 and $b \, \in \, R$ 27 ) x + y + z = 0 28 ) $\frac{4}{3}$ sq. units
4

Mathematics - XII

The Model Paper - 4

Year : 2026 - 27

Time - 2 marks 40 minutes
Full Marks - 73
Previous Set
Cur Set 4
Next Set
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1
2
3
4
5
6
Not Answered 28 Answered 0

In questions 1 to 28, 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 17 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 3 questions of 5 marks each.

The Model Paper - 4

Close

×

1 ) If A be a 2x2 matrix and is defined by A = $\, [a_{ij}] \, $ where $\, [a_{ij}] \, = \, \frac{(i+2j)^2}{2}$, then find the value of $a_{21} $
2 ) If A is an invertible matrix and \begin{equation*} A^{-1} = \begin{bmatrix} 3 & 4 \\ 5 & 6 \\ \end{bmatrix} \end{equation*}, then A = ?
3 ) The area of $\triangle ABC$ having vertices A(2, -6), B(5, 4) and C(k, 4) is 35 sq units. Find the value of k.
4 ) If \begin{equation*} A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \end{equation*} , then adj.A = ?
5 ) Find the value of \begin{equation} \begin{vmatrix} -1 & 3 & 4 \\ 1 & 9 & 12 \\ 9 & 9 & 12 \\ \end{vmatrix} \end{equation}
6 ) Find the value of p for which \begin{equation*} f(x)=\begin{cases} 5x - 4 \quad &\text{if} \, 0 < x \le 1 \\ 4x^2 +3px \quad &\text{if} \, 1 < x < 2 \\ \end{cases} \end{equation*}, is continuous at x = 1.
7 ) \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 $\,\, \lim_{x \to 1} \frac{x^2 - 1}{x - 1}$
9 ) Find the derivative of $f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 1.$
10 ) The derivative of $sec^2x$ with respect to $tan^2x$.
11 ) The value of $ \, cos^{-1}( cos \, 1540^o)$
12 ) Evaluate $\int \frac{x^4 + 1}{x^2 + 1} \, dx$ .
13 ) Evaluate $\int_1^2 \, |x^2 -3x + 2| \, dx$ .
14 ) Find the length of the edge of the cube such that the rate of increase in its volume is the same as the rate of increase in its surface area.
15 ) The solution of differential equation $\frac{dy}{dx} = \frac{1-cos \, x}{1+cos \, x}$.
16 ) What is the projection of $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ on $\vec{b}=\hat{i}- 2\hat{j}+\hat{k}$ ?
17 ) The 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 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{sin^3 x}{sin^3 x +cos^3 x} \, dx$ .
19 ) Evaluate $\int \, tan^{-1} \sqrt{\frac{1-sin \, x}{1+sin \, x}} \, dx$ .
20 ) Evaluate $\int \, \frac{x^2+4}{x^4+16} \, dx$
21 ) The intervals in which the function $f(x)=log(1+ x) - \frac{x}{1+x}$ is increasing or decreasing
22 ) Find the angle of intersection of the curves $y^2=2a$ and $x^2+y^2=8$
23 ) 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
24 ) 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.
25 ) The maximum value of P=5x+3y subject to constraints $5x+2y \le 10$, $x \ge 0$ and $y \ge 0$ is
26 ) Let $f : [-1, \infty) \rightarrow [-1, \infty)$ ,is given by $f(x)=(x+1)^2-1$, where $x\ge -1$. Find S={$x : f(x)=f^{-1}(x)$}.
27 ) Find the image of the point (1, 6, 3) in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$.
28 ) Find the area bounded by curves $y^2 = 2x - x^2 $ and the x-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} $
2
   
   
   
   
NA
2 )
If A is an invertible matrix and \begin{equation*} A^{-1} = \begin{bmatrix} 3 & 4 \\ 5 & 6 \\ \end{bmatrix} \end{equation*}, then A = ?
2
   
   
   
   
NA
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.
2
   
   
   
   
NA
4 )
If \begin{equation*} A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \end{equation*} , then adj.A = ?
2
   
   
   
   
NA
5 )
Find the value of \begin{equation} \begin{vmatrix} -1 & 3 & 4 \\ 1 & 9 & 12 \\ 9 & 9 & 12 \\ \end{vmatrix} \end{equation}
2
   
   
   
   
NA
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.
2
   
   
   
   
NA
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
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to 1} \frac{x^2 - 1}{x - 1}$
2
   
   
   
   
NA
9 )
Find the derivative of $f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 1.$
2
   
   
   
   
NA
10 )
The derivative of $sec^2x$ with respect to $tan^2x$.
2
   
   
   
   
NA
11 )
The value of $ \, cos^{-1}( cos \, 1540^o)$
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{x^4 + 1}{x^2 + 1} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_1^2 \, |x^2 -3x + 2| \, dx$ .
2
   
   
   
   
NA
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.
2
   
   
   
   
NA
15 )
The solution of differential equation $\frac{dy}{dx} = \frac{1-cos \, x}{1+cos \, x}$.
2
   
   
   
   
NA
16 )
What is the projection of $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ on $\vec{b}=\hat{i}- 2\hat{j}+\hat{k}$ ?
2
   
   
   
   
NA
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 = ?
2
   
   
   
   
NA
18 )
Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{sin^3 x}{sin^3 x +cos^3 x} \, dx$ .
3
   
   
   
   
NA
19 )
Evaluate $\int \, tan^{-1} \sqrt{\frac{1-sin \, x}{1+sin \, x}} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{x^2+4}{x^4+16} \, dx$
3
   
   
   
   
NA
21 )
The intervals in which the function $f(x)=log(1+ x) - \frac{x}{1+x}$ is increasing or decreasing
3
   
   
   
   
NA
22 )
Find the angle of intersection of the curves $y^2=2a$ and $x^2+y^2=8$
3
   
   
   
   
NA
23 )
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
3
   
   
   
   
NA
24 )
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.
3
   
   
   
   
NA
25 )
The maximum value of P=5x+3y subject to constraints $5x+2y \le 10$, $x \ge 0$ and $y \ge 0$ is
3
   
   
   
   
NA
26 )
Let $f : [-1, \infty) \rightarrow [-1, \infty)$ ,is given by $f(x)=(x+1)^2-1$, where $x\ge -1$. Find S={$x : f(x)=f^{-1}(x)$}.
5
   
   
   
   
NA
27 )
Find the image of the point (1, 6, 3) in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$.
5
   
   
   
   
NA
28 )
Find the area bounded by curves $y^2 = 2x - x^2 $ and the x-axis.
5
   
   
   
   
NA
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1 ) 8 2 ) \begin{bmatrix} -3 & 2 \\ \frac{5}{2} & - \frac{3}{2} \\ \end{bmatrix} 3 ) 12, -2 4 ) \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix} 5 ) 0 6 ) -1 7 ) $\frac{3}{4}$ 8 ) 2 9 ) $12x^3 - 6x^2 + 10x - 7$ 10 ) 1 11 ) $100^o$ 12 ) $\frac{x^2}{3} - x - 2 tan^{-1}x + c$ 13 ) $\frac{1}{6}$ 14 ) 4 cm 15 ) $y=2tan \frac{x}{2}-x+c$ 16 ) $\frac{5}{\sqrt{6}}$ 17 ) $-\frac{10}{7}$ 18 ) $\frac{\pi}{4}$ 19 ) $\frac{\pi x}{4} - \frac{x^2}{4} + c$ 20 ) $\frac{1}{2\sqrt{2}} tan^{-1}\frac{x^2-4}{2\sqrt{2} \, x}+c$ 21 ) increasing in $0 \lt x \lt \infty$ ; decreasing in $ - \infty \lt x \lt 0$ 22 ) $tan^{-1}3$ 23 ) $\frac{\pi}{3}$ 24 ) $\frac{5}{9}$ 25 ) 15 26 ) {0, -1} 27 ) (1, 0, 7) 28 ) $\frac{4}{3}$ sq. units$
5

Mathematics - XII

The Model Paper - 5

Year : 2026 - 27

Time - 2 marks 40 minutes
Full Marks - 73
Previous Set
Cur Set 5
Next Set
Pages=>
1
2
3
4
5
6
Not Answered 28 Answered 0

In questions 1 to 28, 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 17 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 3 questions of 5 marks each.

The Model Paper - 5

Close

×

1 ) If \begin{equation*} A = \begin{bmatrix} 2 & 0 \\ 3 & -5 \end{bmatrix} \end{equation*} and $A^2 - 3A - 10I_2 = 0 $, find $A^{-1}$.
2 ) \begin{equation*} A = \begin{bmatrix} ab & b^2 \\ -a^2 & ab \\ \end{bmatrix} \end{equation*}, is
3 ) Find the cofactor of 4 in the determinant \begin{vmatrix} 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \\ \end{vmatrix}
4 ) If A and B are symmetric square matrices of the same order, then (AB - BA) is always
5 ) Find the value of \begin{equation} \begin{vmatrix} 0 & b & -c \\ -b & 0 & a \\ c & -a & 0 \\ \end{vmatrix} \end{equation}
6 ) If \begin{equation*} g(x)=\begin{cases} \frac{x}{sin 3x} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, then the value of k.
7 ) The set of points where the f(x) is defined by f(x) = |x - 3| cos x is differentiable, is
8 ) Evaluate $\,\, \lim_{x \to 0} \frac{1 - cos x}{x^2}$
9 ) Find the derivative of the function $y = (2x - 1)^3 (x + 2)^2.$
10 ) Determine the derivative of $x = e^t$ and $y = ln t.$
11 ) if $tan^{-1}x = \frac{\pi}{4} - tan^{-1} \frac{1}{3}$, then x = ?
12 ) Evaluate $\int \frac{log \, tan \, x }{sin \, x \, cos \, x} \, dx$ .
13 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{sin \, x}{sin \, x + cos \, x} \, dx$ .
14 ) Find the intervals on which the following function $f(x) = (x-1) (x-3)^2$, is decreasing.
15 ) The solution of differential equation $log(\frac{dy}{dx} )= ax+by$.
16 ) If $\hat{a}, \, \hat{b}, \, \hat{c} $ are mutually perpendicular unit vectors, then $ |\vec{a} + \vec{b} + \vec{c} | = ? $
17 ) The angle between any two diagonals of cube is
18 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x tan \, x}{sec \, x +cos \, x} \, dx$ .
19 ) Evaluate $\int \, \frac{x^5}{\sqrt{1+x^3}} \, dx$ .
20 ) Evaluate $\int \, \frac{2}{(1-x)(1+x^2)} \, dx$ .
21 ) The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is decreasing -
22 ) At what points on the curve $x^2+y^2 -2x-4y+1=0$, is the tangent parallel to y-axis ?
23 ) 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 ?
24 ) 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.
25 ) 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
26 ) Let $f : [1, \infty) \rightarrow [1, \infty)$ ,is given by $f(x)=2^{x(x-1)}$ is invertible. Find $f^{-1}(x)$.
27 ) 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.
28 ) Find the area bounded by curves {$(x, y) \, : \, x^2 + y^2 \le 1 \le x+y $}.

Close

1 )
If \begin{equation*} A = \begin{bmatrix} 2 & 0 \\ 3 & -5 \end{bmatrix} \end{equation*} and $A^2 - 3A - 10I_2 = 0 $, find $A^{-1}$.
2
   
   
   
   
NA
2 )
\begin{equation*} A = \begin{bmatrix} ab & b^2 \\ -a^2 & ab \\ \end{bmatrix} \end{equation*}, is
2
   
   
   
   
NA
3 )
Find the cofactor of 4 in the determinant \begin{vmatrix} 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \\ \end{vmatrix}
2
   
   
   
   
NA
4 )
If A and B are symmetric square matrices of the same order, then (AB - BA) is always
2
   
   
   
   
NA
5 )
Find the value of \begin{equation} \begin{vmatrix} 0 & b & -c \\ -b & 0 & a \\ c & -a & 0 \\ \end{vmatrix} \end{equation}
2
   
   
   
   
NA
6 )
If \begin{equation*} g(x)=\begin{cases} \frac{x}{sin 3x} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, then the value of k.
2
   
   
   
   
NA
7 )
The set of points where the f(x) is defined by f(x) = |x - 3| cos x is differentiable, is
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to 0} \frac{1 - cos x}{x^2}$
2
   
   
   
   
NA
9 )
Find the derivative of the function $y = (2x - 1)^3 (x + 2)^2.$
2
   
   
   
   
NA
10 )
Determine the derivative of $x = e^t$ and $y = ln t.$
2
   
   
   
   
NA
11 )
if $tan^{-1}x = \frac{\pi}{4} - tan^{-1} \frac{1}{3}$, then x = ?
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{log \, tan \, x }{sin \, x \, cos \, x} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{sin \, x}{sin \, x + cos \, x} \, dx$ .
2
   
   
   
   
NA
14 )
Find the intervals on which the following function $f(x) = (x-1) (x-3)^2$, is decreasing.
2
   
   
   
   
NA
15 )
The solution of differential equation $log(\frac{dy}{dx} )= ax+by$.
2
   
   
   
   
NA
16 )
If $\hat{a}, \, \hat{b}, \, \hat{c} $ are mutually perpendicular unit vectors, then $ |\vec{a} + \vec{b} + \vec{c} | = ? $
2
   
   
   
   
NA
17 )
The angle between any two diagonals of cube is
2
   
   
   
   
NA
18 )
Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x tan \, x}{sec \, x +cos \, x} \, dx$ .
3
   
   
   
   
NA
19 )
Evaluate $\int \, \frac{x^5}{\sqrt{1+x^3}} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{2}{(1-x)(1+x^2)} \, dx$ .
3
   
   
   
   
NA
21 )
The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is decreasing -
3
   
   
   
   
NA
22 )
At what points on the curve $x^2+y^2 -2x-4y+1=0$, is the tangent parallel to y-axis ?
3
   
   
   
   
NA
23 )
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 ?
3
   
   
   
   
NA
24 )
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.
3
   
   
   
   
NA
25 )
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
3
   
   
   
   
NA
26 )
Let $f : [1, \infty) \rightarrow [1, \infty)$ ,is given by $f(x)=2^{x(x-1)}$ is invertible. Find $f^{-1}(x)$.
5
   
   
   
   
NA
27 )
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.
5
   
   
   
   
NA
28 )
Find the area bounded by curves {$(x, y) \, : \, x^2 + y^2 \le 1 \le x+y $}.
5
   
   
   
   
NA
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1 ) \begin{bmatrix} \frac{1}{2} & 0 \\ \frac{3}{10} & - \frac{1}{5} \end{bmatrix} 2 ) nilpotent 3 ) 3 4 ) a skew symmetric square matrix 5 ) 0 6 ) $\frac{1}{3}$ 7 ) R - {3} 8 ) $\frac{1}{2}$ 9 ) $3(2x - 1)^2 (x + 2)^2 + 2(2x - 1)^3 2(x + 2)$ 10 ) $\frac{1}{te^t}$ 11 ) $\frac{1}{2}$ 12 ) $\frac{1}{2} (log \, tan \, x)^2 + c$ 13 ) $\frac{\pi}{4}$ 14 ) $]\frac{5}{3}, 3[$ 15 ) $\frac{-e^{-by}}{b}=\frac{e^{ax}}{a} + c$ 16 ) $\sqrt{3}$ 17 ) $cos^{-1}\frac{1}{3}$ 18 ) $\frac{\pi ^2}{4}$ 19 ) $\frac{2}{9}(1+x^3)^\frac{3}{2}-\frac{2}{3}(1+x^3)^\frac{1}{2}+c$ 20 ) $\frac{1}{2} \frac{1+x^2}{(1-x)^2} +tan^{-1}x +c$ 21 ) $- \infty \lt x \lt 1$ 22 ) (3, 2), (-1, 2) 23 ) $\frac{25\pi}{\pi +4}$ and $\frac{100}{\pi +4}$ 24 ) 0.22 25 ) x=2, y=6, z=36 26 ) $f^{-1}(x)= \frac{1+\sqrt{1+4log_2 x}}{2}$. 27 ) (-1, -7, -4) 28 ) $\frac{\pi -2}{4} sq.units$
6

Mathematics - XII

The Model Paper - 6

Year : 2026 - 27

Time - 2 marks 40 minutes
Full Marks - 73
Previous Set
Cur Set 6
Next Set
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1
2
3
4
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6
Not Answered 28 Answered 0

In questions 1 to 28, 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 17 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 3 questions of 5 marks each.

The Model Paper - 6

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 two non-zero square matrices of the same order such that AB = 0, then
3 ) Find the value of \begin{vmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \\ \end{vmatrix}
4 ) Solve for x and y : \begin{equation*} 2 \begin{bmatrix} 3 & 4 \\ 5 & x \\ \end{bmatrix} + \begin{bmatrix} 1 & y \\ 0 & 1 \\ \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 10 & 5 \\ \end{bmatrix} \end{equation*}
5 ) Find the value of \begin{equation} \begin{vmatrix} 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 ) Let $\, f(x) \, = \, [tan^2 x] \, $, [.] denotes the greatest function. Then
8 ) Evaluate $ \, \lim_{x \to \infty} \frac{2x^3 + 3x}{x^3 + 2x^2}$
9 ) Find the derivative of g(x) = ln(cos x).
10 ) Calculate the derivative of $cos \, x = \, \frac{1}{\sqrt{1+t^2}} $ and $sin \, y = \, \frac{t}{\sqrt{1+t^2}} $
11 ) If $tan^{-1}3x + tan^{-1}2x = \frac{\pi}{4} $ , then x = ?
12 ) Evaluate $\int sin^3(2x+1) \, dx$ .
13 ) Evaluate $\int_{\pi}^{2\pi} \, |sin \, x| \, dx$ .
14 ) A spherical balloon of radius 5 cm has been compressed to a spherical balloon of radius 4.9 cm. Find the approximate decrease in its volume.
15 ) The solution of differential equation $x \frac{dy}{dx} = cot \, y$.
16 ) If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$, then
17 ) If the planes 2x - 4y +3z = 5 and $x + 2y + \lambda z = 12$ are perpendicular to each other, then $\lambda= ?$
18 ) Evaluate $\int_0^{\frac{\pi}{2}} \, log(sin \, x) \, dx$ .
19 ) Evaluate $\int \, \frac{cos(x+a)}{sin(x+b)} \, dx$ .
20 ) Evaluate $\int \, e^x \frac{sin \, 4x - 4}{1- cos \, 4x} \, dx$ .
21 ) The intervals in which the function $f(x)=sin \, x + cos \, x$ is increasing or decreasing
22 ) Find the point on the curve $y=x^3-11x+5$ at which the equation of the tangent is y=x-11
23 ) 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.
24 ) 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 ?
25 ) 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
26 ) Let f : R $\rightarrow$ R, such that $f(x)=(3-x^3)^{\frac{1}{3}}$. Find f o f.
27 ) 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 $\vec{r}.2\hat{i}-2\hat{j}+\hat{k}=5$
28 ) Find the area bounded by curves y = sin x and y = cos x for $0 \le x\le \frac{\pi}{2}$

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}
2
   
   
   
   
NA
2 )
If A and B are two non-zero square matrices of the same order such that AB = 0, then
2
   
   
   
   
NA
3 )
Find the value of \begin{vmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \\ \end{vmatrix}
2
   
   
   
   
NA
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*}
2
   
   
   
   
NA
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}
2
   
   
   
   
NA
6 )
Find f(0), so that $\, f(x) = \frac{x}{1 - \sqrt{1-x}} \,is continuous at x=0 $
2
   
   
   
   
NA
7 )
Let $\, f(x) \, = \, [tan^2 x] \, $, [.] denotes the greatest function. Then
2
   
   
   
   
NA
8 )
Evaluate $ \, \lim_{x \to \infty} \frac{2x^3 + 3x}{x^3 + 2x^2}$
2
   
   
   
   
NA
9 )
Find the derivative of g(x) = ln(cos x).
2
   
   
   
   
NA
10 )
Calculate the derivative of $cos \, x = \, \frac{1}{\sqrt{1+t^2}} $ and $sin \, y = \, \frac{t}{\sqrt{1+t^2}} $
2
   
   
   
   
NA
11 )
If $tan^{-1}3x + tan^{-1}2x = \frac{\pi}{4} $ , then x = ?
2
   
   
   
   
NA
12 )
Evaluate $\int sin^3(2x+1) \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_{\pi}^{2\pi} \, |sin \, x| \, dx$ .
2
   
   
   
   
NA
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.
2
   
   
   
   
NA
15 )
The solution of differential equation $x \frac{dy}{dx} = cot \, y$.
2
   
   
   
   
NA
16 )
If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$, then
2
   
   
   
   
NA
17 )
If the planes 2x - 4y +3z = 5 and $x + 2y + \lambda z = 12$ are perpendicular to each other, then $\lambda= ?$
2
   
   
   
   
NA
18 )
Evaluate $\int_0^{\frac{\pi}{2}} \, log(sin \, x) \, dx$ .
3
   
   
   
   
NA
19 )
Evaluate $\int \, \frac{cos(x+a)}{sin(x+b)} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, e^x \frac{sin \, 4x - 4}{1- cos \, 4x} \, dx$ .
3
   
   
   
   
NA
21 )
The intervals in which the function $f(x)=sin \, x + cos \, x$ is increasing or decreasing
3
   
   
   
   
NA
22 )
Find the point on the curve $y=x^3-11x+5$ at which the equation of the tangent is y=x-11
3
   
   
   
   
NA
23 )
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.
3
   
   
   
   
NA
24 )
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 ?
3
   
   
   
   
NA
25 )
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
3
   
   
   
   
NA
26 )
Let f : R $\rightarrow$ R, such that $f(x)=(3-x^3)^{\frac{1}{3}}$. Find f o f.
5
   
   
   
   
NA
27 )
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 $\vec{r}.2\hat{i}-2\hat{j}+\hat{k}=5$
5
   
   
   
   
NA
28 )
Find the area bounded by curves y = sin x and y = cos x for $0 \le x\le \frac{\pi}{2}$
5
   
   
   
   
NA
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1 ) \begin{bmatrix} 2 & -1 \\ 2 & 4 \\ 3 & 6 \end{bmatrix} 2 ) |A|=0 and |B|=0 3 ) 1 4 ) x=2, y=-8 5 ) 2 6 ) 3 7 ) none of these 8 ) 2 9 ) -tan x 10 ) 1 11 ) $\frac{1}{6} $ or -1 12 ) $-\frac{1}{2}cos(2x+1) + \frac{1}{6}cos^3(2x+1) + c$ 13 ) 2 14 ) $10 \pi cu cm$ 15 ) x cos y = c 16 ) $\vec{a} \perp \vec{b}$ 17 ) 2 18 ) $- \frac{\pi}{2} \, log \, 2 $ 19 ) cos(a-b) log|sin(x+b)| - xsin(a-b) +c 20 ) $e^x cot \, 2x+c$ 21 ) increasing in $0 \lt x \lt \frac{3\pi}{4} $ and $\frac{3\pi}{4} \lt x \lt 2\pi $ ; decreasing in $\f 22 ) (2, -9) 23 ) $\frac{16}{6-\sqrt{3}}$m 24 ) $\frac{2}{5}$ 25 ) 440 26 ) x 27 ) 13 28 ) $2(\sqrt{2} -1)$ sq. units
7

Mathematics - XII

The Model Paper - 7

Year : 2026 - 27

Time - 2 marks 40 minutes
Full Marks - 73
Previous Set
Cur Set 7
Next Set
Pages=>
1
2
3
4
5
6
Not Answered 28 Answered 0

In questions 1 to 28, 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 17 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 3 questions of 5 marks each.

The Model Paper - 7

Close

×

1 ) If \begin{equation*} A = \begin{bmatrix} sin \alpha & cos \alpha \\ -cos \alpha & sin \alpha \\ \end{bmatrix} \end{equation*}, then $A^TA \, = \, ?$
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 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 ) \begin{equation} \begin{vmatrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \\ \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 ) The function f(x) = x - [x], where [.] denotes the greatest integer function is
8 ) Evaluate $\,\, \lim_{x \to 3} \frac{x - 3}{|x - 3|}$
9 ) Determine the derivative of $h(x) = ln(2x + 1) - e^x.$
10 ) Derivative of $x^2$ w.r.t. $x^3$
11 ) $sin^{-1}x + sin^{-1}y = \frac{2 \pi}{3} $, then $cos^{-1}x + cos^{-1}y = ? $
12 ) Evaluate $\int \frac{sin x}{sin(x-\alpha)} \, dx$ .
13 ) Evaluate $\int_{-2}^1 \, \frac{|x|}{x} \, dx$ .
14 ) The least value of k such that $x^2+kx+1$ is increasing on ]1, 2[
15 ) The solution of differential equation $\frac{dy}{dx} + \frac{y}{x}=x^2$.
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 ) 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 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x}{sin \, x +cos \, x} \, dx$ .
19 ) Evaluate $\int \, (1-x)\sqrt{x} \, dx$ .
20 ) Evaluate $\int \,x. log|x+1| \, dx$ .
21 ) The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is increasing
22 ) The tangents to the curve $y=2x^3-4$ at the points x=2 and x=-2 are
23 ) 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.
24 ) 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.
25 ) 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
26 ) Find the domain and range of $f(x)=\frac{1}{\sqrt{x+2}}$
27 ) Find the distance of the point (2, 3, 4) from the plane 3x + 2y + 2z + 5= 0, measured parallel to the line $\frac{x+3}{3}=\frac{y-2}{6}=\frac{z}{2}$
28 ) 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} sin \alpha & cos \alpha \\ -cos \alpha & sin \alpha \\ \end{bmatrix} \end{equation*}, then $A^TA \, = \, ?$
2
   
   
   
   
NA
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
2
   
   
   
   
NA
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}
2
   
   
   
   
NA
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*}
2
   
   
   
   
NA
5 )
\begin{equation} \begin{vmatrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \\ \end{vmatrix} \end{equation}
2
   
   
   
   
NA
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
2
   
   
   
   
NA
7 )
The function f(x) = x - [x], where [.] denotes the greatest integer function is
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to 3} \frac{x - 3}{|x - 3|}$
2
   
   
   
   
NA
9 )
Determine the derivative of $h(x) = ln(2x + 1) - e^x.$
2
   
   
   
   
NA
10 )
Derivative of $x^2$ w.r.t. $x^3$
2
   
   
   
   
NA
11 )
$sin^{-1}x + sin^{-1}y = \frac{2 \pi}{3} $, then $cos^{-1}x + cos^{-1}y = ? $
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{sin x}{sin(x-\alpha)} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_{-2}^1 \, \frac{|x|}{x} \, dx$ .
2
   
   
   
   
NA
14 )
The least value of k such that $x^2+kx+1$ is increasing on ]1, 2[
2
   
   
   
   
NA
15 )
The solution of differential equation $\frac{dy}{dx} + \frac{y}{x}=x^2$.
2
   
   
   
   
NA
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
2
   
   
   
   
NA
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
2
   
   
   
   
NA
18 )
Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x}{sin \, x +cos \, x} \, dx$ .
3
   
   
   
   
NA
19 )
Evaluate $\int \, (1-x)\sqrt{x} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \,x. log|x+1| \, dx$ .
3
   
   
   
   
NA
21 )
The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is increasing
3
   
   
   
   
NA
22 )
The tangents to the curve $y=2x^3-4$ at the points x=2 and x=-2 are
3
   
   
   
   
NA
23 )
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.
3
   
   
   
   
NA
24 )
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.
3
   
   
   
   
NA
25 )
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
3
   
   
   
   
NA
26 )
Find the domain and range of $f(x)=\frac{1}{\sqrt{x+2}}$
5
   
   
   
   
NA
27 )
Find the distance of the point (2, 3, 4) from the plane 3x + 2y + 2z + 5= 0, measured parallel to the line $\frac{x+3}{3}=\frac{y-2}{6}=\frac{z}{2}$
5
   
   
   
   
NA
28 )
Find the area bounded by curve $y=x^2+2 $, and the lines y=x, x=0 and x=3 .
5
   
   
   
   
NA
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1 ) \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} 2 ) 1 3 ) $\pm 2 \sqrt{3}$ 4 ) x=-7, y=-3 or x=3, y=-3 5 ) 0 6 ) none of these 7 ) continuous at non-integer points only 8 ) The limit does not exist. 9 ) $\frac{2}{2x + 1} - e^x$ 10 ) $\frac{2}{3x}$ 11 ) $\frac{\pi}{3}$ 12 ) $x cos \alpha +(sin \alpha) log|sin(x-\alpha)| + c$ 13 ) -1 14 ) -2 15 ) $4xy = x^4 + c$ 16 ) $\frac{\pi}{2}$ 17 ) 20x + 23y + 26z = 69 18 ) $\frac{\pi}{4\sqrt{2}} log|\frac{\sqrt{2}+1}{\sqrt{2}-1}|$ 19 ) $\frac{2}{15}x \sqrt{x} (5-3x) + c$ 20 ) $\frac{1}{2}(x^2-1)log|x+1| - \frac{1}{4}x^2 +\frac{1}{2}x +c$ 21 ) $1 \le x \lt \infty $ 22 ) parallel 23 ) $\frac{\pi}{2}$, Area=$\frac{1}{2}ab$ 24 ) $\frac{5}{9}$ 25 ) 31 26 ) $Domain=(2, \, \infty )$ and $Range= x \in R^+$ 27 ) 7 units 28 ) $\frac{21}{2}$ sq. units
8

Mathematics - XII

The Model Paper - 8

Year : 2026 - 27

Time - 2 marks 40 minutes
Full Marks - 73
Previous Set
Cur Set 8
Next Set
Pages=>
1
2
3
4
5
6
Not Answered 28 Answered 0

In questions 1 to 28, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 8

Instructions

Section-A contains 17 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 3 questions of 5 marks each.

The Model Paper - 8

Close

×

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 \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 \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} cos \, 50^o & sin \, 10^o \\ sin \, 50^o & cos \, 10^o \\ \end{vmatrix} \end{equation}
6 ) The value of k, \begin{equation*} f(x) = \lvert x \rvert = \left\{ \begin{array}{ll} kx^2 & \quad & \text{if} \, x \leq 2 \\ 3 & \quad & \text{if} \, x \ge 2 \end{array} \right. \end{equation*}
7 ) Let $\, f(x) \, = \, [tan^2 x] \, $, [.] denotes the greatest function. Then
8 ) Evaluate $\,\, \lim_{x \to 2} \frac{x^2 - 4}{x - 2}$
9 ) Determine the derivative of $h(x) = e^x sin(2x).$
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 cos[$2 tan^{-1} \frac{1}{2}$]
12 ) Evaluate $\int sin \, 3x \, sin \, 2x \, dx$ .
13 ) Evaluate $\int_{-a}^a \, f(x) \, dx$ .
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 $\frac{dy}{dx} = 2^{x+y}$.
16 ) If $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, then $\frac{1}{2}|\hat{a} - \hat{b}|$ = ?
17 ) The ratio in which the plane 3x + 4y -5z = 1 divides the line joining the points (-2, 4 -6) and (3, -5, 8) is
18 ) Evaluate $\int_0^a \, \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} \, dx$ .
19 ) Evaluate $\int \, (2x+4) \sqrt{x^2+4x+3} \, dx$ .
20 ) Evaluate $\int \, \frac{3x+1}{(x+2)(x-2)^2} \, dx$ .
21 ) The intervals in which the function $f(x)=( x+2)e^{-x}$ is increasing or decreasing
22 ) 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
23 ) The maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3}$ is
24 ) 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.
25 ) 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
26 ) Let f : R $\rightarrow$ R, such that $f(x)=\frac{2x-7}{4}$ be an invertible function. Find $f^{-1}$
27 ) Find the point of intersection of the lines $\frac{x-1}{2}=\frac{y-1}{2}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-8}{5}=\frac{z}{1}$
28 ) 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 3 matrix $\, B=[a_{ij} ] \,$, where $\, b_{ij} \, =\, \frac{1}{2}(i - j)^2$.
2
   
   
   
   
NA
2 )
\begin{equation*} A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \\ \end{bmatrix} \end{equation*}, is
2
   
   
   
   
NA
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.
2
   
   
   
   
NA
4 )
If \begin{equation*} A= \begin{bmatrix} 1 & k & 3 \\ 3 & k & -2 \\ 2 & 3 & -4 \\ \end{bmatrix} \end{equation*} is singular, then k = ?
2
   
   
   
   
NA
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}
2
   
   
   
   
NA
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*}
2
   
   
   
   
NA
7 )
Let $\, f(x) \, = \, [tan^2 x] \, $, [.] denotes the greatest function. Then
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to 2} \frac{x^2 - 4}{x - 2}$
2
   
   
   
   
NA
9 )
Determine the derivative of $h(x) = e^x sin(2x).$
2
   
   
   
   
NA
10 )
In a curve $y=t^2+2t$ and $x=t^3$, find the slope of $\frac{dy}{dx}$ at x=5.
2
   
   
   
   
NA
11 )
The value of cos[$2 tan^{-1} \frac{1}{2}$]
2
   
   
   
   
NA
12 )
Evaluate $\int sin \, 3x \, sin \, 2x \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_{-a}^a \, f(x) \, dx$ .
2
   
   
   
   
NA
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.
2
   
   
   
   
NA
15 )
The solution of differential equation $\frac{dy}{dx} = 2^{x+y}$.
2
   
   
   
   
NA
16 )
If $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, then $\frac{1}{2}|\hat{a} - \hat{b}|$ = ?
2
   
   
   
   
NA
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
2
   
   
   
   
NA
18 )
Evaluate $\int_0^a \, \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} \, dx$ .
3
   
   
   
   
NA
19 )
Evaluate $\int \, (2x+4) \sqrt{x^2+4x+3} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{3x+1}{(x+2)(x-2)^2} \, dx$ .
3
   
   
   
   
NA
21 )
The intervals in which the function $f(x)=( x+2)e^{-x}$ is increasing or decreasing
3
   
   
   
   
NA
22 )
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
3
   
   
   
   
NA
23 )
The maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3}$ is
3
   
   
   
   
NA
24 )
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.
3
   
   
   
   
NA
25 )
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
3
   
   
   
   
NA
26 )
Let f : R $\rightarrow$ R, such that $f(x)=\frac{2x-7}{4}$ be an invertible function. Find $f^{-1}$
5
   
   
   
   
NA
27 )
Find the point of intersection of the lines $\frac{x-1}{2}=\frac{y-1}{2}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-8}{5}=\frac{z}{1}$
5
   
   
   
   
NA
28 )
Find the area bounded by curves {$(x, y) \, : \, x^2 + y^2 \le 2ax , \, y^2 \gt ax, a \gt 0, \, x \gt 0, \, y \gt 0, \,$}.
5
   
   
   
   
NA
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1 ) \begin{bmatrix} 0 & \frac{1}{2} & 2 \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \end{bmatrix} 2 ) idempotent 3 ) 0 4 ) $\frac{33}{2}$ 5 ) $\frac{1}{2}$ 6 ) 1 7 ) none of these 8 ) 4 9 ) $e^x (sin(2x) + 2cos(2x))$ 10 ) $\frac{4}{25}$ 11 ) $\frac{3}{5}$ 12 ) $\frac{1}{2} sin \, x - \frac{1}{10} sin \, 5x +c $ 13 ) $\int_{0}^a \, [f(x)+f(-x)] \, dx$ 14 ) 6% 15 ) $2^x+2^{-y}=c$ 16 ) $sin \, \frac{\theta}{2}$ 17 ) 3 : 4 18 ) $\frac{\pi}{4}$ 19 ) $\frac{2}{3}(x^2+4x+3)^\frac{3}{2}$ 20 ) $\frac{5}{16} log|\frac{x-2}{x+2}| - \frac{7}{4(x-2)} +c$ 21 ) inc in $(- \infty, -1) $; dec in $(-1, \infty) $ 22 ) a=-2, b=-5 23 ) $500 \pi cm^3$ 24 ) $\frac{2}{3}$ 25 ) 14 26 ) $f^{-1}(x)=\frac{4x+7}{2}$ 27 ) (-1, -1, -1) 28 ) $\frac{a^2}{12}(3\pi-8) sq. \, units$
9

Mathematics - XII

The Model Paper - 9

Year : 2026 - 27

Time - 2 marks 40 minutes
Full Marks - 73
Previous Set
Cur Set 9
Next Set
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1
2
3
4
5
6
Not Answered 28 Answered 0

In questions 1 to 28, 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 17 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 3 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 $A^2$
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 ) 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 function $f(x) = \frac{4-x^2}{4x-x^3}$
7 ) The function $\, f(x) = e^{-|x|} \,$ is
8 ) Evaluate $\,\, \lim_{x \to -\frac{\pi}{2}} tan x$
9 ) Calculate the derivative of $g(x) = \frac{ln(x^2 + 1)}{2x + 1} .$
10 ) If $x=t^2$ and $y=t^3$, then $\frac{d^2y}{dx^2}=?$
11 ) The value of $cot^{-1}9 + cosec^{-1} \frac{\sqrt{41}}{4}$
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 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 ) Two adjacent sides of a triangle are represented by the vectors $\vec{a}=3\hat{i}+4\hat{j}$ and $\vec{b}=-5\hat{i}+7\hat{j}$ . The area of the triangle is
17 ) The foot of the perpendicular from the point A(1, 3, 4) on the plane 2x - y + z + 3 = 0 is
18 ) Evaluate $\int_0^{\pi} \, \frac{x}{a^2 \, cos^2 x +b^2 \, sin^2 x} \, dx$ .
19 ) Evaluate $\int \, \frac{sin \, 2x}{(a+b \, cos \, x)^2} \, dx$ .
20 ) Evaluate $\int \, \frac{4x+3}{\sqrt{2x^2 + 2x - 3}} \, dx$ .
21 ) The function $f(x)=-\frac{x}{2}+sin\, x$ is strictly increasing in
22 ) At what points on the curve $x^2+y^2 -2x-3=0$, is the tangent parallel to x-axis ?
23 ) The semi-vertical angle of a cone of maximum volume and given slant height is
24 ) 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
25 ) The constraints $-x+y \le 1$, $-x+3y \le 9$ and $x \ge 0$, $y \ge 0$ defines on
26 ) Find the value of parameter $\alpha$ for which function $f(x)=1+\alpha x$, $\alpha \ne 0$ is the inverse of itself.
27 ) 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 $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$
28 ) 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 $A^2$
2
   
   
   
   
NA
2 )
If \begin{equation*} A = \begin{bmatrix} cos \theta & -sin \theta \\ -sin \theta & cos \theta \\ \end{bmatrix} \end{equation*}, then $\, A^{-1} \,$ is
2
   
   
   
   
NA
3 )
Let \begin{equation*} A= \begin{bmatrix} -1 & 2 \\ 1 & 4 \\ \end{bmatrix} \end{equation*}, find A(adj A)
2
   
   
   
   
NA
4 )
Matrices A and B are inverses of each other only when
2
   
   
   
   
NA
5 )
Find the value of \begin{equation} \begin{vmatrix} 1 \, + \, 2i & 1 \, - \, i \\ 1 \, + \, i & 1 \, - \, 2i \\ \end{vmatrix} \end{equation} where $\, i^2 = -1$
2
   
   
   
   
NA
6 )
The function $f(x) = \frac{4-x^2}{4x-x^3}$
2
   
   
   
   
NA
7 )
The function $\, f(x) = e^{-|x|} \,$ is
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to -\frac{\pi}{2}} tan x$
2
   
   
   
   
NA
9 )
Calculate the derivative of $g(x) = \frac{ln(x^2 + 1)}{2x + 1} .$
2
   
   
   
   
NA
10 )
If $x=t^2$ and $y=t^3$, then $\frac{d^2y}{dx^2}=?$
2
   
   
   
   
NA
11 )
The value of $cot^{-1}9 + cosec^{-1} \frac{\sqrt{41}}{4}$
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{sin^3x+cos^3x}{sin^2x \, cos^2x} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_0^{2\pi} \, |sin \, x| \, dx$ .
2
   
   
   
   
NA
14 )
Find the intervals on which the following function f(x) = sin x - cos x, $0 \lt x \lt 2\pi$ is increasing.
2
   
   
   
   
NA
15 )
The solution of differential equation $x \sqrt{1+y^2} dx +y \sqrt{1+x^2} dy = 0 $ is.
2
   
   
   
   
NA
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
2
   
   
   
   
NA
17 )
The foot of the perpendicular from the point A(1, 3, 4) on the plane 2x - y + z + 3 = 0 is
2
   
   
   
   
NA
18 )
Evaluate $\int_0^{\pi} \, \frac{x}{a^2 \, cos^2 x +b^2 \, sin^2 x} \, dx$ .
3
   
   
   
   
NA
19 )
Evaluate $\int \, \frac{sin \, 2x}{(a+b \, cos \, x)^2} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{4x+3}{\sqrt{2x^2 + 2x - 3}} \, dx$ .
3
   
   
   
   
NA
21 )
The function $f(x)=-\frac{x}{2}+sin\, x$ is strictly increasing in
3
   
   
   
   
NA
22 )
At what points on the curve $x^2+y^2 -2x-3=0$, is the tangent parallel to x-axis ?
3
   
   
   
   
NA
23 )
The semi-vertical angle of a cone of maximum volume and given slant height is
3
   
   
   
   
NA
24 )
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
3
   
   
   
   
NA
25 )
The constraints $-x+y \le 1$, $-x+3y \le 9$ and $x \ge 0$, $y \ge 0$ defines on
3
   
   
   
   
NA
26 )
Find the value of parameter $\alpha$ for which function $f(x)=1+\alpha x$, $\alpha \ne 0$ is the inverse of itself.
5
   
   
   
   
NA
27 )
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 $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$
5
   
   
   
   
NA
28 )
Find the area bounded by curves {$(x, y) \, : \, x^2 \le y \le |x| $}.
5
   
   
   
   
NA
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1 ) \begin{bmatrix} cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix} 2 ) adj A 3 ) \begin{bmatrix} 53 & 0 \\ 0 & 53 \\ \end{bmatrix} 4 ) AB = BA = I 5 ) 3 6 ) discontinuous at only one point 7 ) continuous everywhere but not differentiable at x = 0 8 ) $-\infty$ 9 ) $\frac{2x}{x^2 + 1} - \frac{2 ln(x^2 + 1) }{(2x + 1)^2} $ 10 ) $\frac{3}{4t}$ 11 ) $\frac{\pi}{4}$ 12 ) sec x - cosec x + c 13 ) 4 14 ) $] 0, \frac{3\pi}{4}[$ 15 ) $ \sqrt{1+y^2} +\sqrt{1+x^2} = c $ 16 ) $\frac{41}{2}$ sq units 17 ) (-1, 4, 3) 18 ) $\frac{\pi^2}{2ab}$ 19 ) $-\frac{2}{b^2}[log|a+b \, cos \, x|+\frac{a}{a+b \, cos \, x}]+c$ 20 ) $2\sqrt{2x^2 + 2x - 3} + \frac{1}{\sqrt{2}} log|x + \frac{1}{2} +\sqrt{x^2 + x - \frac{3}{2}} | +c$ 21 ) $-\frac{\pi}{3} \le x \le \frac{\pi}{3} $ 22 ) (1, 2), (1, -2) 23 ) $tan^{-1} \sqrt{2}$ 24 ) $\frac{37}{56}$ 25 ) both bounded and unbounded space 26 ) $\alpha=-1$ 27 ) $3\sqrt{30}$ units, $\frac{x-3}{2}=\frac{y-8}{5}=\frac{z-3}{-1}$ 28 ) $\frac{1}{3}$ sq. units
10

Mathematics - XII

The Model Paper - 10

Year : 2026 - 27

Time - 2 marks 40 minutes
Full Marks - 73
Previous Set
Cur Set 10
Next Set
Pages=>
1
2
3
4
5
6
Not Answered 28 Answered 0

In questions 1 to 28, 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 - 10

Instructions

Section-A contains 17 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 3 questions of 5 marks each.

The Model Paper - 10

Close

×

1 ) The product of the matrices $[1 \,\,\, 2 \,\,\, 3 ]^T$ and $[2 \,\,\, 3 \,\,\, 4 ] $ is
2 ) If Ais an invertible square matrix and k is an non-negative real number, then $(k .A)^{-1} \, = \, ?$
3 ) The area of triangle whose vertices are A(-2, -3), B(3, 2) and C(-1, 8).
4 ) If A is 3-rowed square matrix and |A| = 5, then |adj A| = ?
5 ) Find the value of \begin{equation} \begin{vmatrix} a \, + \, ib & c \, + \, id \\ -c \, + \, id & a \, - \, ib \\ \end{vmatrix} \end{equation} where $\, i^2 = -1$
6 ) If \begin{equation*} g(x)=\begin{cases} \frac{1 - cos x}{x^2} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, find k.
7 ) If f(x) = |sin x|, then
8 ) Evaluate $\,\, \lim_{x \to 0} \frac{sec x - 1}{x}$
9 ) Find the derivative of $g(x) = \frac{x^2 + 1}{x^3 + 2x}$
10 ) The derivative of x = a(t - sin t) and y = a(1 - cos t)
11 ) The value of sin[$2 sin^{-1} \frac{4}{5}$]
12 ) Evaluate $\int \frac{ax + b}{cx + d} \, dx$ .
13 ) Evaluate $\int_0^1 \, \frac{xe^x}{(1+x)^2} \, dx$ .
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 $\frac{dy}{dx} = e^{x+y} + x^2.e^y$.
16 ) If $|\vec{a}|=2$, $|\vec{b}|=7$ and $\vec{a} \, \times \, \vec{b} = 3\hat{i}+2\hat{j}+6\hat{k}$, then angle between $\vec{a} \, and \, \vec{b}$ is
17 ) The angle between the lines $\frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2}$ and $\frac{x+3}{3}=\frac{y-2}{5}=\frac{z+5}{4}$ is
18 ) Evaluate $\int_0^1 \, cot^{-1}(1-x+x^2) \, dx$ .
19 ) Evaluate $\int \, \frac{sin \, x + cos \, x}{\sqrt{sin \, 2x}} \, dx$ .
20 ) Evaluate $\int \, \frac{sin^{-1}}{x^2} \, dx$ .
21 ) The least possible value of k for which the function $f(x)= x^2+kx+1$ may be increasing on [1, 2].
22 ) 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).
23 ) 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
24 ) 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.
25 ) 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
26 ) If $f(x)=\frac{3x-2}{2x-3}$, then f(f(x)) = ? . where x is areal number and $x \ne \frac{3}{2}$
27 ) Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane $\vec{r}.(-2\hat{j}+4\hat{k})+5=0$.
28 ) 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
2
   
   
   
   
NA
2 )
If Ais an invertible square matrix and k is an non-negative real number, then $(k .A)^{-1} \, = \, ?$
2
   
   
   
   
NA
3 )
The area of triangle whose vertices are A(-2, -3), B(3, 2) and C(-1, 8).
2
   
   
   
   
NA
4 )
If A is 3-rowed square matrix and |A| = 5, then |adj A| = ?
2
   
   
   
   
NA
5 )
Find the value of \begin{equation} \begin{vmatrix} a \, + \, ib & c \, + \, id \\ -c \, + \, id & a \, - \, ib \\ \end{vmatrix} \end{equation} where $\, i^2 = -1$
2
   
   
   
   
NA
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.
2
   
   
   
   
NA
7 )
If f(x) = |sin x|, then
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to 0} \frac{sec x - 1}{x}$
2
   
   
   
   
NA
9 )
Find the derivative of $g(x) = \frac{x^2 + 1}{x^3 + 2x}$
2
   
   
   
   
NA
10 )
The derivative of x = a(t - sin t) and y = a(1 - cos t)
2
   
   
   
   
NA
11 )
The value of sin[$2 sin^{-1} \frac{4}{5}$]
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{ax + b}{cx + d} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_0^1 \, \frac{xe^x}{(1+x)^2} \, dx$ .
2
   
   
   
   
NA
14 )
If $y=x^4 + 10$ and if x changes from 2 to 1.99, what is the approximate change in y.
2
   
   
   
   
NA
15 )
The solution of differential equation $\frac{dy}{dx} = e^{x+y} + x^2.e^y$.
2
   
   
   
   
NA
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
2
   
   
   
   
NA
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
2
   
   
   
   
NA
18 )
Evaluate $\int_0^1 \, cot^{-1}(1-x+x^2) \, dx$ .
3
   
   
   
   
NA
19 )
Evaluate $\int \, \frac{sin \, x + cos \, x}{\sqrt{sin \, 2x}} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{sin^{-1}}{x^2} \, dx$ .
3
   
   
   
   
NA
21 )
The least possible value of k for which the function $f(x)= x^2+kx+1$ may be increasing on [1, 2].
3
   
   
   
   
NA
22 )
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).
3
   
   
   
   
NA
23 )
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
3
   
   
   
   
NA
24 )
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.
3
   
   
   
   
NA
25 )
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
3
   
   
   
   
NA
26 )
If $f(x)=\frac{3x-2}{2x-3}$, then f(f(x)) = ? . where x is areal number and $x \ne \frac{3}{2}$
5
   
   
   
   
NA
27 )
Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane $\vec{r}.(-2\hat{j}+4\hat{k})+5=0$.
5
   
   
   
   
NA
28 )
Find the area bounded by curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x}{a}+\frac{y}{b}=1$.
5
   
   
   
   
NA
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1 ) \begin{bmatrix} 2 & 3 & 6 \\ 4 & 6 & 8 \\ 6 & 9 & 12 \\ \end{bmatrix} 2 ) $\frac{1}{k} .A^{-1}$ 3 ) 15 sq units 4 ) 25 5 ) $a^2+b^2+c^2+d^2$ 6 ) $\frac{1}{2}$ 7 ) f(x) is not differentiable at $ \, x = n \pi, \, x \in Z $ 8 ) 0 9 ) $(2x (x^3 + 2x) - (x^2 + 1) \frac{3x^2 + 2}{(x^3 + 2x)^2} $ 10 ) $cot \frac{t}{2}$ 11 ) $\frac{24}{25}$ 12 ) $\frac{ax}{c} + \frac{bc - ad}{c^2} log|cx + d|$ + C 13 ) $\frac{e}{2} -1$ 14 ) -0.32 15 ) $e^x+e^{-y} + \frac{x^3}{3}=c$ 16 ) $\frac{\pi}{6}$ 17 ) $cos^{-1}\frac{8\sqrt{3}}{15}$ 18 ) $\frac{\pi}{2} - log \, 2$ 19 ) $tan^{-1}\frac{tan \, x -1}{\sqrt{2 \, tan \, x}}+c$ 20 ) $-\frac{sin^{-1}x}{x}+log| \frac{1}{x} - \frac{\sqrt{1-x^2}}{x}|$ 21 ) -2 22 ) $( \frac{7}{4}, \, \frac{1}{4})$ 23 ) $\frac{c^3}{6\sqrt{3}}$ 24 ) $\frac{25}{52}$ 25 ) (2, 0) 26 ) x 27 ) $\frac{13}{12} \sqrt{6}$, $(-\frac{1}{12}, \frac{25}{12},- \frac{2}{12})$ 28 ) $(\frac{\pi ab}{4} -\frac{ab}{2})$ sq. units