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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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1

Mathematics - XII

The Model Paper - 1

Year : 2025 - 26


Time - 3 Hours 0 minutes
Full Marks - 80

Previous Set
Cur Set 1
Next Set
Pages=>
1
2
3
4
5
6
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

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 A is an invertible matrix and \begin{equation*} A^{-1} = \begin{bmatrix} 3 & 4 \\ 5 & 6 \\ \end{bmatrix} \end{equation*}, then A = ?
3 ) Find the co-factor of \begin{vmatrix} 2 & 6 \\ -1 & 4 \\ \end{vmatrix}
4 ) Matrices A and B are inverses of each other only when
5 ) \begin{equation} \begin{vmatrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \\ \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 ) \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 ) Determine the derivative of h(x) = sin(2x) + cos(3x).
10 ) In a curve $y=t^2+2t$ and $x=t^3$, find the slope of $\frac{dy}{dx}$ at x=5.
11 ) The value of tan{$cos^{-1}\frac{4}{5} + tan^{-1}\frac{2}{3}$}
12 ) Evaluate $\int sin \, 3x \, sin \, 2x \, dx$ .
13 ) Evaluate $\int_{-a}^a \, f(x) \, dx$ .
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 $\frac{dy}{dx} = 2^{x+y}$.
16 ) If $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, then $\frac{1}{2}|\hat{a} - \hat{b}|$ = ?
17 ) A plane through line of intersection of the planes x + y + z = 6 and 2x + 3y + 4z + 5 = 0 and passing through the point P(1, 1, 1) is
18 ) A coin is tossed 5 times. What is the probability that tail appears an odd number of times ?
19 ) Evaluate $\int_0^{\frac{\pi}{4}} \, log(1+tan \, x) \, dx$ .
20 ) Evaluate $\int \, \frac{x^5}{\sqrt{1+x^3}} \, dx$ .
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 ) 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 ) A square piece of tin of side 24 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut so that the volume of the box is maximum ? Also find the maximum volume.
25 ) A purse contains 4 copper coins, 3 silver coins, the second purse contains 6 copper coins, 2 silver coins. A coin is taken out from any purse, the probability that it is copper coin is
26 ) The 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 image of the point (1, 6, 3) in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$.
30 ) Find the area bounded by curve $y=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 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 )
Find the co-factor of \begin{vmatrix} 2 & 6 \\ -1 & 4 \\ \end{vmatrix}
2
   
   
   
   
NA
4 )
Matrices A and B are inverses of each other only when
2
   
   
   
   
NA
5 )
\begin{equation} \begin{vmatrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \\ \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 )
\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 $\,\, \lim_{x \to 3} \frac{x - 3}{|x - 3|}$
2
   
   
   
   
NA
9 )
Determine the derivative of h(x) = sin(2x) + cos(3x).
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 tan{$cos^{-1}\frac{4}{5} + tan^{-1}\frac{2}{3}$}
2
   
   
   
   
NA
12 )
Evaluate $\int sin \, 3x \, sin \, 2x \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_{-a}^a \, f(x) \, dx$ .
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 $\frac{dy}{dx} = 2^{x+y}$.
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 )
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 )
A coin is tossed 5 times. What is the probability that tail appears an odd number of times ?
2
   
   
   
   
NA
19 )
Evaluate $\int_0^{\frac{\pi}{4}} \, log(1+tan \, x) \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{x^5}{\sqrt{1+x^3}} \, dx$ .
3
   
   
   
   
NA
21 )
Evaluate $\int \, \frac{2}{(1-x)(1+x^2)} \, dx$ .
3
   
   
   
   
NA
22 )
The least possible value of k for which the function $f(x)= x^2+kx+1$ may be increasing on [1, 2].
3
   
   
   
   
NA
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
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
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
3
   
   
   
   
NA
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
3
   
   
   
   
NA
27 )
Find the domain and range of $f(x)=\frac{1}{\sqrt{x+2}}$
5
   
   
   
   
NA
28 )
Solve the following system of equations using matrix method : x + y + z = 1 x - 2y + 3z = 2 5x - 3y + z = 3
5
   
   
   
   
NA
29 )
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
30 )
Find the area bounded by curve $y=x^2+2 $, and the lines y=x, x=0 and x=3 .
5
   
   
   
   
NA
Set-1, Next Page-

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1 ) \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} 2 ) \begin{bmatrix} -3 & 2 \\ \frac{5}{2} & - \frac{3}{2} \\ \end{bmatrix} 3 ) -6 4 ) AB = BA = I 5 ) 0 6 ) f(x) is discontinuous at x = 0 7 ) $\frac{1}{2}$ 8 ) The limit does not exist. 9 ) 2cos(2x) - 3sin(3x) 10 ) $\frac{4}{25}$ 11 ) $\frac{17}{6}$ 12 ) $\frac{1}{2} sin \, x - \frac{1}{10} sin \, 5x +c $ 13 ) $\int_{0}^a \, [f(x)+f(-x)] \, dx$ 14 ) 15 m/s 15 ) $2^x+2^{-y}=c$ 16 ) $sin \, \frac{\theta}{2}$ 17 ) 20x + 23y + 26z = 69 18 ) $\frac{1}{2}$ 19 ) $\frac{\pi}{8} \, log \, 2 $ 20 ) $\frac{2}{9}(1+x^3)^\frac{3}{2}-\frac{2}{3}(1+x^3)^\frac{1}{2}+c$ 21 ) $\frac{1}{2} \frac{1+x^2}{(1-x)^2} +tan^{-1}x +c$ 22 ) -2 23 ) a=-2, b=-5 24 ) 1024 cu. cm. 25 ) $\frac{37}{56}$ 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 ) (1, 0, 7) 30 ) $\frac{21}{2}$ sq. units
2

Mathematics - XII

The Model Paper - 2

Year : 2025 - 26


Time - 3 marks 0 minutes
Full Marks - 80

Previous Set
Cur Set 2
Next Set
Pages=>
1
2
3
4
5
6
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 3 matrix $\, B=[a_{ij} ] \,$, where $\, b_{ij} \, =\, \frac{1}{2}(i - j)^2$.
2 ) \begin{equation*} A = \begin{bmatrix} ab & b^2 \\ -a^2 & ab \\ \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 ) If A and B are symmetric square matrices of the same order, then (AB - BA) is always
5 ) Evaluate \begin{vmatrix} x+1 & x \\ x & x - 1 \\ \end{vmatrix}
6 ) Find f(0), so that $\, f(x) = \frac{x}{1 - \sqrt{1-x}} \,is continuous at x=0 $
7 ) \begin{equation*} Let f(x) = \lvert x \rvert = \left\{ \begin{array}{ll} kx^2 & \quad x \le 2 \\ 3 & \quad x \gt 2 \end{array} \right. \end{equation*}, if f(x) is continuous then the value of k is
8 ) Evaluate $\,\, \lim_{x \to 0} \frac{sec x - 1}{x}$
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 ) The value of sin[$2 tan^{-1} \frac{5}{8}$]
12 ) Evaluate $\int \sqrt{e^x - 1} \, dx$ .
13 ) Evaluate $\int_1^2 \, |x^2 -3x + 2| \, dx$ .
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 $\hat{a}, \, \hat{b}, \, \hat{c} $ are mutually perpendicular unit vectors, then $ |\vec{a} + \vec{b} + \vec{c} | = ? $
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 ) 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 $\int_0^2 \, x\sqrt{2-x} \, dx$ .
20 ) Evaluate $\int \, \frac{sin \, 2x}{a^2 \, cos^2x + b^2 \, sin^2x} \, dx$ .
21 ) Evaluate $\int \, \frac{x^2+4}{x^4+16} \, dx$
22 ) The intervals in which the function $f(x)=sin \, x + cos \, x$ 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 ) 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 ) Bag A contains 2 white and 3 red balls, and bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from bag B.
26 ) The maximum value of P=5x+3y subject to constraints $5x+2y \le 10$, $x \ge 0$ and $y \ge 0$ 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 + 1y + z = 4 -x + y + z = 0 x - 3y + z = 1
29 ) Find the point of intersection of the lines $\frac{x-1}{2}=\frac{y-1}{2}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-8}{5}=\frac{z}{1}$
30 ) Find the area bounded by curves {$(x, y) \, : \, x^2 \le y \le |x| $}.

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} ab & b^2 \\ -a^2 & ab \\ \end{bmatrix} \end{equation*}, 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 )
If A and B are symmetric square matrices of the same order, then (AB - BA) is always
2
   
   
   
   
NA
5 )
Evaluate \begin{vmatrix} x+1 & x \\ x & x - 1 \\ \end{vmatrix}
2
   
   
   
   
NA
6 )
Find f(0), so that $\, f(x) = \frac{x}{1 - \sqrt{1-x}} \,is continuous at x=0 $
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 $\,\, \lim_{x \to 0} \frac{sec x - 1}{x}$
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 )
The value of sin[$2 tan^{-1} \frac{5}{8}$]
2
   
   
   
   
NA
12 )
Evaluate $\int \sqrt{e^x - 1} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_1^2 \, |x^2 -3x + 2| \, dx$ .
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 $\hat{a}, \, \hat{b}, \, \hat{c} $ are mutually perpendicular unit vectors, then $ |\vec{a} + \vec{b} + \vec{c} | = ? $
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 )
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 ?
2
   
   
   
   
NA
19 )
Evaluate $\int_0^2 \, x\sqrt{2-x} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{sin \, 2x}{a^2 \, cos^2x + b^2 \, sin^2x} \, dx$ .
3
   
   
   
   
NA
21 )
Evaluate $\int \, \frac{x^2+4}{x^4+16} \, dx$
3
   
   
   
   
NA
22 )
The intervals in which the function $f(x)=sin \, x + cos \, x$ is increasing or decreasing
3
   
   
   
   
NA
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).
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
26 )
The maximum value of P=5x+3y subject to constraints $5x+2y \le 10$, $x \ge 0$ and $y \ge 0$ is
3
   
   
   
   
NA
27 )
Find the value of parameter $\alpha$ for which function $f(x)=1+\alpha x$, $\alpha \ne 0$ is the inverse of itself.
5
   
   
   
   
NA
28 )
Solve the following system of equations using matrix method : x + 1y + z = 4 -x + y + z = 0 x - 3y + z = 1
5
   
   
   
   
NA
29 )
Find the point of intersection of the lines $\frac{x-1}{2}=\frac{y-1}{2}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-8}{5}=\frac{z}{1}$
5
   
   
   
   
NA
30 )
Find the area bounded by curves {$(x, y) \, : \, x^2 \le y \le |x| $}.
5
   
   
   
   
NA
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1 ) \begin{bmatrix} 0 & \frac{1}{2} & 2 \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \end{bmatrix} 2 ) nilpotent 3 ) $\pm 2 \sqrt{3}$ 4 ) a skew symmetric square matrix 5 ) -1 6 ) 3 7 ) $\frac{3}{4}$ 8 ) 0 9 ) -tan x 10 ) 1 11 ) $\frac{80}{89}$ 12 ) $2 \sqrt{e^x - 1} - 2 tan^{-1}\sqrt{e^x - 1} + c$ 13 ) $\frac{1}{6}$ 14 ) -0.32 15 ) $e^x+e^{-y} + \frac{x^3}{3}=c$ 16 ) $\sqrt{3}$ 17 ) 3 : 4 18 ) $\frac{5}{9}$ 19 ) $\frac{16\sqrt{2}}{15}$ 20 ) $\frac{log|a^2 \, cos^2x + b^2 sin^2x|}{b^2 - a^2} +c$ 21 ) $\frac{1}{2\sqrt{2}} tan^{-1}\frac{x^2-4}{2\sqrt{2} \, x}+c$ 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 ) $( \frac{7}{4}, \, \frac{1}{4})$ 24 ) $\frac{\pi}{2}$, Area=$\frac{1}{2}ab$ 25 ) $\frac{25}{52}$ 26 ) 15 27 ) $\alpha=-1$ 28 ) x = 3, y = 0, z = 2 29 ) (-1, -1, -1) 30 ) $\frac{1}{3}$ sq. units
3

Mathematics - XII

The Model Paper - 3

Year : 2025 - 26


Time - 3 marks 0 minutes
Full Marks - 80

Previous Set
Cur Set 3
Next Set
Pages=>
1
2
3
4
5
6
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 - 3

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 - 3

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 Ais an invertible square matrix and k is an non-negative real number, then $(k .A)^{-1} \, = \, ?$
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 square matrices of the same order, then (A + B) (A - B) = ?
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 ) If \begin{equation*} g(x)=\begin{cases} \frac{sin^{-1}}{x} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, find k.
7 ) Let $\, f(x) \, = \, [tan^2 x] \, $, [.] denotes the greatest function. Then
8 ) Evaluate $\,\, \lim_{x \to \infty} \frac{3x^2 + 2x + 1}{x^2 - 1}$
9 ) Determine the derivative of $h(x) = e^x sin(2x).$
10 ) If $x=t^2$ and $y=t^3$, then $\frac{d^2y}{dx^2}=?$
11 ) $tan^{-1}(1+x) + tan^{-1}(1 - x) = \frac{\pi}{2}$
12 ) Evaluate $\int \frac{\sqrt{tan \, x} }{sin \, x \, \, cos \, x} \, dx$ .
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 $x \frac{dy}{dx} = cot \, y$.
16 ) If $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, then $\frac{1}{2}|\hat{a} - \hat{b}|$ = ?
17 ) The vertices of a $\triangle ABC$ are A(-1, 3, 2), B(2, 3, 5) and C(3, 5, -2). Then $\angle B = ?$
18 ) If A and B are events such that $P(A \cup B)=\frac{5}{6}$, $P(A \cap B)=\frac{1}{3}$ and $P(\bar{B} )=\frac{1}{2}$, then the events A and B are
19 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x tan \, x}{sec \, x +cos \, x} \, dx$ .
20 ) Evaluate $\int \, (2x+4) \sqrt{x^2+4x+3} \, dx$ .
21 ) Evaluate $\int \,[\frac{1}{log \, x}-\frac{1}{(log \, x)^2}] \, dx$ .
22 ) The function $f(x)=cos^{-1}(sin \, x +cos \, x)$ strictly decreasing function in the interval -
23 ) Find the angle of intersection of the curves $y^2=2a$ and $x^2+y^2=8$
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 ) A bag contains 1 white and 6 red balls. Another bag contains 4 white and 3 red balls. One of the two bag is selected at random and a ball is drawn from it, which is found to be white. Find the probability that the ball drawn is from the bag A.
26 ) The constraints $-x+y \le 1$, $-x+3y \le 9$ and $x \ge 0$, $y \ge 0$ defines on
27 ) Let $f : [1, \infty) \rightarrow [1, \infty)$ ,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 : 2x + 8y + 5z = 5 x + y + z = -2 x + 2y - z = 2
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 $y^2 = 2x - x^2 $ and the x-axis.

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 Ais an invertible square matrix and k is an non-negative real number, then $(k .A)^{-1} \, = \, ?$
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 square matrices of the same order, then (A + B) (A - B) = ?
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 )
If \begin{equation*} g(x)=\begin{cases} \frac{sin^{-1}}{x} \quad &\text{if} \, x \ne 0 \\ k \quad &\text{if} \, x = 0 \\ \end{cases} \end{equation*}, is continuous at x = 0, find k.
2
   
   
   
   
NA
7 )
Let $\, f(x) \, = \, [tan^2 x] \, $, [.] denotes the greatest function. Then
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to \infty} \frac{3x^2 + 2x + 1}{x^2 - 1}$
2
   
   
   
   
NA
9 )
Determine the derivative of $h(x) = e^x sin(2x).$
2
   
   
   
   
NA
10 )
If $x=t^2$ and $y=t^3$, then $\frac{d^2y}{dx^2}=?$
2
   
   
   
   
NA
11 )
$tan^{-1}(1+x) + tan^{-1}(1 - x) = \frac{\pi}{2}$
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{\sqrt{tan \, x} }{sin \, x \, \, cos \, x} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_{\pi}^{2\pi} \, |sin \, x| \, dx$ .
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 $x \frac{dy}{dx} = cot \, y$.
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 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 )
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
2
   
   
   
   
NA
19 )
Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x tan \, x}{sec \, x +cos \, x} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, (2x+4) \sqrt{x^2+4x+3} \, dx$ .
3
   
   
   
   
NA
21 )
Evaluate $\int \,[\frac{1}{log \, x}-\frac{1}{(log \, x)^2}] \, dx$ .
3
   
   
   
   
NA
22 )
The function $f(x)=cos^{-1}(sin \, x +cos \, x)$ strictly decreasing function in the interval -
3
   
   
   
   
NA
23 )
Find the angle of intersection of the curves $y^2=2a$ and $x^2+y^2=8$
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
26 )
The constraints $-x+y \le 1$, $-x+3y \le 9$ and $x \ge 0$, $y \ge 0$ defines on
3
   
   
   
   
NA
27 )
Let $f : [1, \infty) \rightarrow [1, \infty)$ ,is given by $f(x)=2^{x(x-1)}$ is invertible. Find $f^{-1}(x)$.
5
   
   
   
   
NA
28 )
Solve the following system of equations using matrix method : 2x + 8y + 5z = 5 x + y + z = -2 x + 2y - z = 2
5
   
   
   
   
NA
29 )
Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane $\vec{r}.(-2\hat{j}+4\hat{k})+5=0$.
5
   
   
   
   
NA
30 )
Find the area bounded by curves $y^2 = 2x - x^2 $ and the x-axis.
5
   
   
   
   
NA
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1 ) \begin{bmatrix} 2 & -1 \\ 2 & 4 \\ 3 & 6 \end{bmatrix} 2 ) $\frac{1}{k} .A^{-1}$ 3 ) 3 4 ) $A^2 - AB +BA + B^2 $ 5 ) $\frac{1}{2}$ 6 ) 1 7 ) none of these 8 ) 3 9 ) $e^x (sin(2x) + 2cos(2x))$ 10 ) $\frac{3}{4t}$ 11 ) 0 12 ) $2 \sqrt{tan \, x} +c$ 13 ) 2 14 ) -2 15 ) x cos y = c 16 ) $sin \, \frac{\theta}{2}$ 17 ) $cos^{-1}\frac{1}{\sqrt{2}}$ 18 ) independent 19 ) $\frac{\pi ^2}{4}$ 20 ) $\frac{2}{3}(x^2+4x+3)^\frac{3}{2}$ 21 ) $\frac{x}{log \, x} +c$ 22 ) $[0, \frac{\pi}{4}]$ 23 ) $tan^{-1}3$ 24 ) $75\sqrt{3}cm^2$. 25 ) $\frac{1}{5}$ 26 ) both bounded and unbounded space 27 ) $f^{-1}(x)= \frac{1+\sqrt{1+4log_2 x}}{2}$. 28 ) x = -3, y = 2, z = -1 29 ) $\frac{13}{12} \sqrt{6}$, $(-\frac{1}{12}, \frac{25}{12},- \frac{2}{12})$ 30 ) $\frac{4}{3}$ sq. units$
4

Mathematics - XII

The Model Paper - 4

Year : 2025 - 26


Time - 3 marks 0 minutes
Full Marks - 80

Previous Set
Cur Set 4
Next Set
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Not Answered 30 Answered 0

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

This Model Paper contains 3 Sections.

All Questions of Model Paper - 4

Instructions

Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 4

Close

×

1 ) If \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 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 ) 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 ) Find the value of p for which \begin{equation*} f(x)=\begin{cases} 5x - 4 \quad &\text{if} \, 0 < x \le 1 \\ 4x^2 +3px \quad &\text{if} \, 1 < x < 2 \\ \end{cases} \end{equation*}, is continuous at x = 1.
7 ) Let f(x) = $\sqrt{ x^2 \, - \, 3x \, -4 \, }$. Then domain of f(x) is
8 ) Evaluate $\, \, \lim_{x \to 1}\frac{x^2 - 1}{x - 1}$.
9 ) Find the derivative of $f(x) = \frac{x^4}{1 + x^2} $
10 ) Calculate the derivative of x = sin t and y = cos t.
11 ) If $tan^{-1}3x + tan^{-1}2x = \frac{\pi}{4} $ , then x = ?
12 ) Evaluate $\int \frac{sin^3x+cos^3x}{sin^2x \, cos^2x} \, dx$ .
13 ) Evaluate $\int_0^1 \, \frac{xe^x}{(1+x)^2} \, dx$ .
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 $\frac{dy}{dx} = 1-x+y-xy$.
16 ) If $|\vec{a}|=\sqrt{26}$, $|\vec{b}|=7$ and $|\vec{a} \, \times \, \vec{b}| =35 $, then $\vec{a} \,. \, \vec{b}$ 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 ) 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 $\int_0^1 \, \frac{log(1+x)}{(1+x)^2} \, dx$ .
20 ) Evaluate $\int \, \frac{sin \, 2x}{(a+b \, cos \, x)^2} \, dx$ .
21 ) Evaluate $\int \,x. log|x+1| \, dx$ .
22 ) The intervals in which the function $f(x)=log(1+ x) - \frac{x}{1+x}$ is increasing or decreasing
23 ) If the curves $x=y^2$ and $xy=k$ cut at right angles, then
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 ) Three urns A, B and C contains 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls respectively. An urn is choose at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from the urn A.
26 ) The maximum value of P=8x+3y subject to constraints $x+y \le 3$, $4x+y \le 6$ and $x \ge 0$, $y \ge 0$ is
27 ) 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 : 2x + 3y + 3z = 5 x - 2y + z = -4 3x - y - 2z = 3
29 ) Find the image of the point having position vector $\vec{r}=\hat{i}+3\hat{j}+4\hat{k}$ $\vec{r}.(2\hat{i}-\hat{j}+\hat{k})+3=0$
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} 2 & 0 \\ 3 & -5 \end{bmatrix} \end{equation*} and $A^2 - 3A - 10I_2 = 0 $, find $A^{-1}$.
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 \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 )
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 )
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 )
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 )
Let f(x) = $\sqrt{ x^2 \, - \, 3x \, -4 \, }$. Then domain of f(x) is
2
   
   
   
   
NA
8 )
Evaluate $\, \, \lim_{x \to 1}\frac{x^2 - 1}{x - 1}$.
2
   
   
   
   
NA
9 )
Find the derivative of $f(x) = \frac{x^4}{1 + x^2} $
2
   
   
   
   
NA
10 )
Calculate the derivative of x = sin t and y = cos t.
2
   
   
   
   
NA
11 )
If $tan^{-1}3x + tan^{-1}2x = \frac{\pi}{4} $ , then x = ?
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{sin^3x+cos^3x}{sin^2x \, cos^2x} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_0^1 \, \frac{xe^x}{(1+x)^2} \, dx$ .
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 $\frac{dy}{dx} = 1-x+y-xy$.
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 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 )
If A and B are events such that P(A)=0.3, P(B)=0.2 and $P(A \cap B)=0.1$, then $P(\bar{A} \cap B)= ?$
2
   
   
   
   
NA
19 )
Evaluate $\int_0^1 \, \frac{log(1+x)}{(1+x)^2} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{sin \, 2x}{(a+b \, cos \, x)^2} \, dx$ .
3
   
   
   
   
NA
21 )
Evaluate $\int \,x. log|x+1| \, dx$ .
3
   
   
   
   
NA
22 )
The intervals in which the function $f(x)=log(1+ x) - \frac{x}{1+x}$ is increasing or decreasing
3
   
   
   
   
NA
23 )
If the curves $x=y^2$ and $xy=k$ cut at right angles, then
3
   
   
   
   
NA
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
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
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
3
   
   
   
   
NA
27 )
Let f : R $\rightarrow$ R, such that $f(x)=\frac{2x-7}{4}$ be an invertible function. Find $f^{-1}$
5
   
   
   
   
NA
28 )
Solve the following system of equations using matrix method : 2x + 3y + 3z = 5 x - 2y + z = -4 3x - y - 2z = 3
5
   
   
   
   
NA
29 )
Find the image of the point having position vector $\vec{r}=\hat{i}+3\hat{j}+4\hat{k}$ $\vec{r}.(2\hat{i}-\hat{j}+\hat{k})+3=0$
5
   
   
   
   
NA
30 )
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} \frac{1}{2} & 0 \\ \frac{3}{10} & - \frac{1}{5} \end{bmatrix} 2 ) 1 3 ) 0 4 ) x=-7, y=-3 or x=3, y=-3 5 ) $a^2+b^2+c^2+d^2$ 6 ) -1 7 ) $-1 \, \le \, x \, \le \, 4 \,$ 8 ) 2 9 ) $\frac{4x^3 + 2x}{(1 + x^2)^2}$ 10 ) -tan t 11 ) $\frac{1}{6} $ or -1 12 ) sec x - cosec x + c 13 ) $\frac{e}{2} -1$ 14 ) $10 \pi cu cm$ 15 ) $log(1+y)=x-\frac{x^2}{2} +c$ 16 ) 7 17 ) $cos^{-1}\frac{8\sqrt{3}}{15}$ 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 ) $\frac{1}{2}(x^2-1)log|x+1| - \frac{1}{4}x^2 +\frac{1}{2}x +c$ 22 ) increasing in $0 \lt x \lt \infty$ ; decreasing in $ - \infty \lt x \lt 0$ 23 ) $8k^2=1$ 24 ) $\frac{\pi}{3}$ 25 ) $\frac{36}{61}$ 26 ) 14 27 ) $f^{-1}(x)=\frac{4x+7}{2}$ 28 ) x = 1, y = 2, z = -1 29 ) (-3, 5, 2) 30 ) $2(\sqrt{2} -1)$ sq. units
5

Mathematics - XII

The Model Paper - 5

Year : 2025 - 26


Time - 3 marks 0 minutes
Full Marks - 80

Previous Set
Cur Set 5
Next Set
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1
2
3
4
5
6
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 - 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

Close

×

1 ) If matrix A = [1 2 3], find $AA^ \prime$
2 ) If \begin{equation*} A = \begin{bmatrix} cos \theta & -sin \theta \\ -sin \theta & cos \theta \\ \end{bmatrix} \end{equation*}, then $\, A^{-1} \,$ is
3 ) The area of triangle whose vertices are A(-2, -3), B(3, 2) and C(-1, 8).
4 ) For square matrices A and B of the same order, we have adj(AB) = ?
5 ) Find the value of \begin{equation} \begin{vmatrix} -1 & 3 & 4 \\ 1 & 9 & 12 \\ 9 & 9 & 12 \\ \end{vmatrix} \end{equation}
6 ) \begin{equation*} f(x) = \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 ) If \begin{equation*} f(x)=\begin{cases} -log_ex \quad &\text{if} \, 0 < x < 1 \\ k log_ex \quad &\text{if} \, x \ge 1 \\ \end{cases} \end{equation*}, find the value of k, so that f(x) is differentiable at x = 1.
8 ) Evaluate $\,\, \lim_{x \to -2} \frac{x^2 - 4}{x + 2}$
9 ) Calculate the derivative of $g(x) = \frac{ln(x^2 + 1)}{2x + 1} .$
10 ) Derivative of $x^2$ w.r.t. $x^3$
11 ) if $tan^{-1}x = \frac{\pi}{4} - tan^{-1} \frac{1}{3}$, then x = ?
12 ) Evaluate $\int xin^3x \, cos^3x \, dx$ .
13 ) Evaluate $\int_0^1 \, |2x - 1| \, dx$ .
14 ) Find the intervals on which the following function $f(x) = (x-1) (x-3)^2$, is decreasing.
15 ) The solution of differential equation $\frac{dy}{dx} = \frac{1-cos \, x}{1+cos \, x}$.
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 ) 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 $\int_0^{\frac{\pi}{2}} \, \frac{sin^3 x}{sin^3 x +cos^3 x} \, dx$ .
20 ) Evaluate $\int \, (1-x)\sqrt{x} \, dx$ .
21 ) Evaluate $\int \, \sqrt{\frac{1-x}{1+x}}$ .
22 ) The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is decreasing -
23 ) Find the equation of tangent at $t=\frac{\pi}{4}$ for the curve x = sin 3t, y = cos 2t.
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 ) 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 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 - 3y + 5z = 11 3x + 2y - 4z = -5 x + y - 2z = -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 {$(x, y) \, : \, x^2 + y^2 \le 2ax , \, y^2 \gt ax, a \gt 0, \, x \gt 0, \, y \gt 0, \,$}.

Close


1 )
If matrix A = [1 2 3], find $AA^ \prime$
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 )
The area of triangle whose vertices are A(-2, -3), B(3, 2) and C(-1, 8).
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} -1 & 3 & 4 \\ 1 & 9 & 12 \\ 9 & 9 & 12 \\ \end{vmatrix} \end{equation}
2
   
   
   
   
NA
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
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 $\,\, \lim_{x \to -2} \frac{x^2 - 4}{x + 2}$
2
   
   
   
   
NA
9 )
Calculate the derivative of $g(x) = \frac{ln(x^2 + 1)}{2x + 1} .$
2
   
   
   
   
NA
10 )
Derivative of $x^2$ w.r.t. $x^3$
2
   
   
   
   
NA
11 )
if $tan^{-1}x = \frac{\pi}{4} - tan^{-1} \frac{1}{3}$, then x = ?
2
   
   
   
   
NA
12 )
Evaluate $\int xin^3x \, cos^3x \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_0^1 \, |2x - 1| \, dx$ .
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 $\frac{dy}{dx} = \frac{1-cos \, x}{1+cos \, x}$.
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 )
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 ?
2
   
   
   
   
NA
19 )
Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{sin^3 x}{sin^3 x +cos^3 x} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, (1-x)\sqrt{x} \, dx$ .
3
   
   
   
   
NA
21 )
Evaluate $\int \, \sqrt{\frac{1-x}{1+x}}$ .
3
   
   
   
   
NA
22 )
The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is decreasing -
3
   
   
   
   
NA
23 )
Find the equation of tangent at $t=\frac{\pi}{4}$ for the curve x = sin 3t, y = cos 2t.
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
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
3
   
   
   
   
NA
27 )
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
28 )
Solve the following system of equations using matrix method : 2x - 3y + 5z = 11 3x + 2y - 4z = -5 x + y - 2z = -3
5
   
   
   
   
NA
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 $\vec{r}.2\hat{i}-2\hat{j}+\hat{k}=5$
5
   
   
   
   
NA
30 )
Find the area bounded by curves {$(x, y) \, : \, x^2 + y^2 \le 2ax , \, y^2 \gt ax, a \gt 0, \, x \gt 0, \, y \gt 0, \,$}.
5
   
   
   
   
NA
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1 ) [14] 2 ) adj A 3 ) 15 sq units 4 ) adj(B) adj(A) 5 ) 0 6 ) 1 7 ) -1 8 ) -4 9 ) $\frac{2x}{x^2 + 1} - \frac{2 ln(x^2 + 1) }{(2x + 1)^2} $ 10 ) $\frac{2}{3x}$ 11 ) $\frac{1}{2}$ 12 ) $\frac{1}{4}sin^4x - \frac{1}{6}sin^6x + c$ 13 ) $\frac{1}{2}$ 14 ) $]\frac{5}{3}, 3[$ 15 ) $y=2tan \frac{x}{2}-x+c$ 16 ) $\frac{5}{\sqrt{6}}$ 17 ) $-\frac{10}{7}$ 18 ) 0.72 19 ) $\frac{\pi}{4}$ 20 ) $\frac{2}{15}x \sqrt{x} (5-3x) + c$ 21 ) $sin^{-1}x + \sqrt{1-x^2}+c$ 22 ) $- \infty \lt x \lt 1$ 23 ) $$4x-3\sqrt{2}y-2\sqrt{2}=0$$ 24 ) $\frac{16}{6-\sqrt{3}}$m 25 ) $\frac{5}{9}$ 26 ) (2, 0) 27 ) x 28 ) x = 1, y = 2, z = 3 29 ) 13 30 ) $\frac{a^2}{12}(3\pi-8) sq. \, units$
6

Mathematics - XII

The Model Paper - 6

Year : 2025 - 26


Time - 3 marks 0 minutes
Full Marks - 80

Previous Set
Cur Set 6
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6
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 ) Find a 2 x 2 matrix $\, A=[a_{ij} \,] $, $\, a_{ij} \, =\, 2i + 3j - 6$.
2 ) \begin{equation*} A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \\ \end{bmatrix} \end{equation*}, is
3 ) Let \begin{equation*} A= \begin{bmatrix} -1 & 2 \\ 1 & 4 \\ \end{bmatrix} \end{equation*}, find A(adj A)
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 ) 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 f(x) = |sin x|, then
8 ) Evaluate $ \, \lim_{x \to \infty} \frac{2x^3 + 3x}{x^3 + 2x^2}$
9 ) Calculate the derivative of $g(x) = e^x sin(x).$
10 ) The derivative of $sin^2x$ with respect to cos x
11 ) The value of $cot^{-1}9 + cosec^{-1} \frac{\sqrt{41}}{4}$
12 ) Evaluate $\int \frac{sin x}{sin(x-\alpha)} \, dx$ .
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 $log(\frac{dy}{dx} )= ax+by$.
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 ) 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 $\int_0^1 \, cot^{-1}(1-x+x^2) \, dx$ .
20 ) Evaluate $\int \, tan^{-1} \sqrt{\frac{1-sin \, x}{1+sin \, x}} \, dx$ .
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 ) At what points on the curve $x^2+y^2 -2x-4y+1=0$, is the tangent parallel to y-axis ?
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 ) 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 ) By graphical method the solution of linear programming problem : maximum value of z=3x+5y subject to constraints $3x+2y \le 18$, $x \le 4$, $y \le 6$ and $x \ge 0$, $y \ge 0$ is
27 ) Let $f : [-1, \infty) \rightarrow [-1, \infty)$ ,is given by $f(x)=(x+1)^2-1$, where $x\ge -1$. Find S={$x : f(x)=f^{-1}(x)$}.
28 ) Solve the following system of equations using matrix method : 3x - 4y + 2z = -1 2x + 3y + 5z = 7 x + z = 2
29 ) Find the distance of the point (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$x^2=4y$ and the line x = 4y - 2

Close


1 )
Find a 2 x 2 matrix $\, A=[a_{ij} \,] $, $\, a_{ij} \, =\, 2i + 3j - 6$.
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 )
Let \begin{equation*} A= \begin{bmatrix} -1 & 2 \\ 1 & 4 \\ \end{bmatrix} \end{equation*}, find A(adj A)
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 )
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 f(x) = |sin x|, then
2
   
   
   
   
NA
8 )
Evaluate $ \, \lim_{x \to \infty} \frac{2x^3 + 3x}{x^3 + 2x^2}$
2
   
   
   
   
NA
9 )
Calculate the derivative of $g(x) = e^x sin(x).$
2
   
   
   
   
NA
10 )
The derivative of $sin^2x$ with respect to cos x
2
   
   
   
   
NA
11 )
The value of $cot^{-1}9 + cosec^{-1} \frac{\sqrt{41}}{4}$
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{sin x}{sin(x-\alpha)} \, dx$ .
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 $log(\frac{dy}{dx} )= ax+by$.
2
   
   
   
   
NA
16 )
If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$, then
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 )
If A and B are events such that P(A)=0.4, P(B)=0.8 and $P(B/A)=0.6$, then $P(A/B)= ?$
2
   
   
   
   
NA
19 )
Evaluate $\int_0^1 \, cot^{-1}(1-x+x^2) \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, tan^{-1} \sqrt{\frac{1-sin \, x}{1+sin \, x}} \, dx$ .
3
   
   
   
   
NA
21 )
Evaluate $\int \, \frac{4x+3}{\sqrt{2x^2 + 2x - 3}} \, dx$ .
3
   
   
   
   
NA
22 )
The intervals in which the function $f(x)=x^3-12 x^2+36x +17$ is increasing
3
   
   
   
   
NA
23 )
At what points on the curve $x^2+y^2 -2x-4y+1=0$, is the tangent parallel to y-axis ?
3
   
   
   
   
NA
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 ?
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
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
3
   
   
   
   
NA
27 )
Let $f : [-1, \infty) \rightarrow [-1, \infty)$ ,is given by $f(x)=(x+1)^2-1$, where $x\ge -1$. Find S={$x : f(x)=f^{-1}(x)$}.
5
   
   
   
   
NA
28 )
Solve the following system of equations using matrix method : 3x - 4y + 2z = -1 2x + 3y + 5z = 7 x + z = 2
5
   
   
   
   
NA
29 )
Find the distance of the point (2, 3, 4) from the plane 3x + 2y + 2z + 5= 0, measured parallel to the line $\frac{x+3}{3}=\frac{y-2}{6}=\frac{z}{2}$
5
   
   
   
   
NA
30 )
Find the area bounded by curves$x^2=4y$ and the line x = 4y - 2
5
   
   
   
   
NA
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1 ) \begin{bmatrix} -1 & 2 \\ 1 & 4 \\ \end{bmatrix} 2 ) idempotent 3 ) \begin{bmatrix} 53 & 0 \\ 0 & 53 \\ \end{bmatrix} 4 ) x=2, y=-8 5 ) -5 6 ) 0 7 ) f(x) is not differentiable at $ \, x = n \pi, \, x \in Z $ 8 ) 2 9 ) $e^x sin(x) + e^x cos(x)$ 10 ) -2cos x 11 ) $\frac{\pi}{4}$ 12 ) $x cos \alpha +(sin \alpha) log|sin(x-\alpha)| + c$ 13 ) 2 14 ) $3.2 \pi cm^3/sec$ 15 ) $\frac{-e^{-by}}{b}=\frac{e^{ax}}{a} + c$ 16 ) $\vec{a} \perp \vec{b}$ 17 ) $-\frac{10}{7}$ 18 ) 0.3 19 ) $\frac{\pi}{2} - log \, 2$ 20 ) $\frac{\pi x}{4} - \frac{x^2}{4} + c$ 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 ) (3, 2), (-1, 2) 24 ) $\frac{25\pi}{\pi +4}$ and $\frac{100}{\pi +4}$ 25 ) $\frac{2}{3}$ 26 ) x=2, y=6, z=36 27 ) {0, -1} 28 ) x = 2, y = 3, z = -1 29 ) 7 units 30 ) $\frac{9}{8}$ sq. units
7

Mathematics - XII

The Model Paper - 7

Year : 2025 - 26


Time - 3 marks 0 minutes
Full Marks - 80

Previous Set
Cur Set 7
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3
4
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6
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 - 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 and B are invertible matrices of the same order, then $(AB)^{-1} = ?$
3 ) If the points A(a, 0), B(0, b) and C(1, 1) are collinear, then the value of $\frac{1}{a}+\frac{1}{b}$ is
4 ) If A and B are square matrices of the same order, then $(A - B)^2$ = ?
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 $\,\, \lim_{x \to - \infty} \frac{2x^3 + 3x}{x^3 + 2x^2}$
9 ) Determine the derivative of $h(x) = ln(2x + 1) - e^x.$
10 ) Find the derivative of $x = 3t^2 + 2t$ and $y = t^3 - 5t.$
11 ) The value of cot[$ tan^{-1} x + cot^{-1} x $]
12 ) Evaluate $\int \frac{x^4 + 1}{x^2 + 1} \, dx$ .
13 ) Evaluate $\int_{-2}^1 \, \frac{|x|}{x} \, dx$ .
14 ) Find the length of the edge of the cube such that the rate of increase in its volume is the same as the rate of increase in its surface area.
15 ) The solution of differential equation $\frac{dy}{dx} = - \sqrt{ \frac{1-y^2}{1-x^2}}$ 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 angle between any two diagonals of cube is
18 ) 8 coins are tossed simultaneously. The probability of getting 6 heads is
19 ) Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x}{sin \, x +cos \, x} \, dx$ .
20 ) Evaluate $\int \, \frac{x^2+1}{x^4+1} \, dx$ .
21 ) Evaluate $\int \, cos^4x \, dx$ .
22 ) The intervals in which the function $f(x)=( x+2)e^{-x}$ is increasing or decreasing
23 ) At what points on the curve $x^2+y^2 -2x-3=0$, is the tangent parallel to x-axis ?
24 ) The semi-vertical angle of a cone of maximum volume and given slant height 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=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 - 3y + 5z = 16 3x +2 y - 4z = -4 x + y - 2z = -3
29 ) Find the co-ordinates of the point where the line $\frac{x-2}{3}=\frac{y+3}{4}=\frac{z-2}{6}$ meets the plane x - y + z = 2.
30 ) Find the area cut off from the parabola $4y=3x^2$ by the straight line 3x - 2y +12 = 0.

Close


1 )
The product of the matrices $[1 \,\,\, 2 \,\,\, 3 ]^T$ and $[2 \,\,\, 3 \,\,\, 4 ] $ is
2
   
   
   
   
NA
2 )
If A and B are invertible matrices of the same order, then $(AB)^{-1} = ?$
2
   
   
   
   
NA
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
2
   
   
   
   
NA
4 )
If A and B are square matrices of the same order, then $(A - B)^2$ = ?
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{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 )
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 )
Determine the derivative of $h(x) = ln(2x + 1) - e^x.$
2
   
   
   
   
NA
10 )
Find the derivative of $x = 3t^2 + 2t$ and $y = t^3 - 5t.$
2
   
   
   
   
NA
11 )
The value of cot[$ tan^{-1} x + cot^{-1} x $]
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{x^4 + 1}{x^2 + 1} \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_{-2}^1 \, \frac{|x|}{x} \, dx$ .
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 $\frac{dy}{dx} = - \sqrt{ \frac{1-y^2}{1-x^2}}$ 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 angle between any two diagonals of cube is
2
   
   
   
   
NA
18 )
8 coins are tossed simultaneously. The probability of getting 6 heads is
2
   
   
   
   
NA
19 )
Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{x}{sin \, x +cos \, x} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{x^2+1}{x^4+1} \, dx$ .
3
   
   
   
   
NA
21 )
Evaluate $\int \, cos^4x \, dx$ .
3
   
   
   
   
NA
22 )
The intervals in which the function $f(x)=( x+2)e^{-x}$ is increasing or decreasing
3
   
   
   
   
NA
23 )
At what points on the curve $x^2+y^2 -2x-3=0$, is the tangent parallel to x-axis ?
3
   
   
   
   
NA
24 )
The semi-vertical angle of a cone of maximum volume and given slant height is
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
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
3
   
   
   
   
NA
27 )
Let f : A $\rightarrow$ B and f{$(x, \, x^2)$ : x $\in$ A} the f : A $\rightarrow$ A is
5
   
   
   
   
NA
28 )
Solve the following system of equations using matrix method : 2x - 3y + 5z = 16 3x +2 y - 4z = -4 x + y - 2z = -3
5
   
   
   
   
NA
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.
5
   
   
   
   
NA
30 )
Find the area cut off from the parabola $4y=3x^2$ by the straight line 3x - 2y +12 = 0.
5
   
   
   
   
NA
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1 ) \begin{bmatrix} 2 & 3 & 6 \\ 4 & 6 & 8 \\ 6 & 9 & 12 \\ \end{bmatrix} 2 ) $B^{-1}A^{-1}$ 3 ) 1 4 ) $A^2 - AB - BA + B^2$ 5 ) 0 6 ) $\frac{1}{2}$ 7 ) none of these 8 ) 2 9 ) $\frac{2}{2x + 1} - e^x$ 10 ) $\frac{3t^2 - 5}{6t + 2}$ 11 ) 0 12 ) $\frac{x^2}{3} - x - 2 tan^{-1}x + c$ 13 ) -1 14 ) 4 cm 15 ) $sin^{-1}y+sin^{-1}x=c$ 16 ) $\frac{-3}{2}$ 17 ) $cos^{-1}\frac{1}{3}$ 18 ) $\frac{37}{256}$ 19 ) $\frac{\pi}{4\sqrt{2}} log|\frac{\sqrt{2}+1}{\sqrt{2}-1}|$ 20 ) $\frac{1}{\sqrt{2}}tan^{-1}[\frac{1}{\sqrt{2}}(x-\frac{1}{x})]+c$ 21 ) $\frac{3x}{8}+\frac{sin \, 4x}{32}+\frac{sin \, 2x}{4}+c$ 22 ) inc in $(- \infty, -1) $; dec in $(-1, \infty) $ 23 ) (1, 2), (1, -2) 24 ) $tan^{-1} \sqrt{2}$ 25 ) $\frac{5}{9}$ 26 ) 120 27 ) neither one-one nor onto. 28 ) x = 2, y = 1, z = 3 29 ) (-1, -7, -4) 30 ) 27 sq. units
8

Mathematics - XII

The Model Paper - 8

Year : 2025 - 26


Time - 3 marks 0 minutes
Full Marks - 80

Previous Set
Cur Set 8
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In questions 1 to 30, out of the four options, only one answer is correct. Choose the correct answer.

This Model Paper contains 3 Sections.

All Questions of Model Paper - 8

Instructions

Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 8

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1 ) If \begin{equation*} A = \begin{bmatrix} sin \alpha & cos \alpha \\ -cos \alpha & sin \alpha \\ \end{bmatrix} \end{equation*}, then $A^2$
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 ) 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} x + 1 & x-1 \\ x^2 + x +1 & x^2 - x +1 \\ \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 $\, f(x) = e^{-|x|} \,$ is
8 ) Evaluate $\,\, \lim_{x \to 0} \frac{sin x}{x}$
9 ) Find the derivative of $f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 1.$
10 ) Determine the derivative of $x = e^t$ and $y = ln t.$
11 ) The value of cos[$2 tan^{-1} \frac{1}{2}$]
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 ) A man 1.8 meters tall walks directly away from a lamp post, whose height is 9 meters, at the rate 2 m/s. The rate at which his shadow lengthens is
15 ) The solution of differential equation $x \sqrt{1+y^2} dx +y \sqrt{1+x^2} dy = 0 $ is.
16 ) 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 ) If the planes 2x - 4y +3z = 5 and $x + 2y + \lambda z = 12$ are perpendicular to each other, then $\lambda= ?$
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 $\int_0^{\pi} \, \frac{x}{a^2 \, cos^2 x +b^2 \, sin^2 x} \, dx$ .
20 ) Evaluate $\int \, \sqrt{\frac{1+x}{1-x}} \, dx$ .
21 ) Evaluate $\int \, \frac{3x+1}{(x+2)(x-2)^2} \, dx$ .
22 ) The intervals in which the function $f(x)=x^3-12 x^2+36x +17$ is decreasing
23 ) The tangents to the curve $y=2x^3-4$ at the points x=2 and x=-2 are
24 ) The maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3}$ 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 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 $f : R \rightarrow (-1, \, 1)$ ,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 : x - y + z = 1 2x + y - z = 2 x - 2y - z = 4
29 ) Find the point of intersection of the lines $\frac{x+4}{3}=\frac{y+6}{5}=\frac{z-1}{-2}$ and 3x - 2y + z + 5 = 0 = 2x + 3y + 4z - 4.
30 ) Find the area bounded by curves {$(x, y) \, : \, x^2 + y^2 \le 1 \le x+y $}.

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 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 )
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} x + 1 & x-1 \\ x^2 + x +1 & x^2 - x +1 \\ \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 $\, f(x) = e^{-|x|} \,$ is
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to 0} \frac{sin x}{x}$
2
   
   
   
   
NA
9 )
Find the derivative of $f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 1.$
2
   
   
   
   
NA
10 )
Determine the derivative of $x = e^t$ and $y = ln t.$
2
   
   
   
   
NA
11 )
The value of cos[$2 tan^{-1} \frac{1}{2}$]
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 )
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 $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 )
If the planes 2x - 4y +3z = 5 and $x + 2y + \lambda z = 12$ are perpendicular to each other, then $\lambda= ?$
2
   
   
   
   
NA
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 ?
2
   
   
   
   
NA
19 )
Evaluate $\int_0^{\pi} \, \frac{x}{a^2 \, cos^2 x +b^2 \, sin^2 x} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \sqrt{\frac{1+x}{1-x}} \, dx$ .
3
   
   
   
   
NA
21 )
Evaluate $\int \, \frac{3x+1}{(x+2)(x-2)^2} \, dx$ .
3
   
   
   
   
NA
22 )
The intervals in which the function $f(x)=x^3-12 x^2+36x +17$ is decreasing
3
   
   
   
   
NA
23 )
The tangents to the curve $y=2x^3-4$ at the points x=2 and x=-2 are
3
   
   
   
   
NA
24 )
The maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3}$ is
3
   
   
   
   
NA
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 ?
3
   
   
   
   
NA
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
3
   
   
   
   
NA
27 )
Let $f : R \rightarrow (-1, \, 1)$ ,is given by $f(x)=\frac{10^x-10^{-x}}{10^x+10^{-x}}$ is invertible. Find $f^{-1}(x)$.
5
   
   
   
   
NA
28 )
Solve the following system of equations using matrix method : x - y + z = 1 2x + y - z = 2 x - 2y - z = 4
5
   
   
   
   
NA
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.
5
   
   
   
   
NA
30 )
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} cos 2 \alpha & sin 2 \alpha \\ -sin 2 \alpha & cos 2 \alpha \\ \end{bmatrix} 2 ) |A|=0 and |B|=0 3 ) 1 4 ) \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix} 5 ) 2 6 ) none of these 7 ) continuous everywhere but not differentiable at x = 0 8 ) 1 9 ) $12x^3 - 6x^2 + 10x - 7$ 10 ) $\frac{1}{te^t}$ 11 ) $\frac{3}{5}$ 12 ) $\frac{1}{2} (log \, tan \, x)^2 + c$ 13 ) $\frac{\pi}{4}$ 14 ) 0.5 m/s 15 ) $ \sqrt{1+y^2} +\sqrt{1+x^2} = c $ 16 ) $\frac{41}{2}$ sq units 17 ) 2 18 ) $\frac{1}{3}$ 19 ) $\frac{\pi^2}{2ab}$ 20 ) $sin^{-1}x-\sqrt{1-x^2} +c$ 21 ) $\frac{5}{16} log|\frac{x-2}{x+2}| - \frac{7}{4(x-2)} +c$ 22 ) $2 \lt x \lt 6$ 23 ) parallel 24 ) $500 \pi cm^3$ 25 ) $\frac{2}{5}$ 26 ) 31 27 ) $\frac{1}{2} log_{10} \frac{1+x}{1-x}$ 28 ) x = 1, y = -1, z = -1 29 ) (2, 4, -3) 30 ) $\frac{\pi -2}{4} sq.units$
9

Mathematics - XII

The Model Paper - 9

Year : 2025 - 26


Time - 3 marks 0 minutes
Full Marks - 80

Previous Set
Cur Set 9
Next Set
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Not Answered 30 Answered 0

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

This Model Paper contains 3 Sections.

All Questions of Model Paper - 9

Instructions

Section-A contains 18 questions of 2 marks each.

Section-B contains 8 questions of 3 marks each.

Section-C contains 4 questions of 5 marks each.

The Model Paper - 9

Close

×

1 ) If A be a 2x2 matrix and is defined by A = $\, [a_{ij}] \, $ where $\, [a_{ij}] \, = \, \frac{(i+2j)^2}{2}$, then find the value of $a_{21} $
2 ) if A is a 2-rowed square matrix and |A| = 6, then A . Adj A = ?
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 ) 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 ) \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) = x - [x], where [.] denotes the greatest integer function is
8 ) Evaluate $\,\, \lim_{x \to 1} \frac{x^2 - 1}{x - 1}$
9 ) Calculate the derivative of $f(x) = \sqrt{x^2 + 4x} - x^2.$
10 ) The derivative of $tan^{-1}x$ with respect to $cot^{-1} x$
11 ) $sin^{-1}x + sin^{-1}y = \frac{2 \pi}{3} $, then $cos^{-1}x + cos^{-1}y = ? $
12 ) Evaluate $\int \frac{ax + b}{cx + d} \, dx$ .
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 $\frac{dy}{dx} = \frac{-2xy}{1+x^2}$ 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 foot of the perpendicular from the point A(1, 3, 4) on the plane 2x - y + z + 3 = 0 is
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 $\int_0^{\frac{\pi}{2}} \, log(sin \, x) \, dx$ .
20 ) Evaluate $\int \, \frac{sin \, x + cos \, x}{\sqrt{sin \, 2x}} \, dx$ .
21 ) Evaluate $\int \, \frac{sin^{-1}}{x^2} \, dx$ .
22 ) The function $f(x)=-\frac{x}{2}+sin\, x$ is strictly increasing in
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 ) An open box with a square base is to be made out of a given cardboard of area $c^2$ square units. The maximum volume of the box is
25 ) A 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=6x+11y subject to constraints $2x+y \le 104$, $x+2y \le 76$ 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 : x - y = 3 2x + 3y + 4z = 17 y + 2z = 7
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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x}{a}+\frac{y}{b}=1$.

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 a 2-rowed square matrix and |A| = 6, then A . Adj A = ?
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 )
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 )
\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 )
The function f(x) = x - [x], where [.] denotes the greatest integer function is
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to 1} \frac{x^2 - 1}{x - 1}$
2
   
   
   
   
NA
9 )
Calculate the derivative of $f(x) = \sqrt{x^2 + 4x} - x^2.$
2
   
   
   
   
NA
10 )
The derivative of $tan^{-1}x$ with respect to $cot^{-1} x$
2
   
   
   
   
NA
11 )
$sin^{-1}x + sin^{-1}y = \frac{2 \pi}{3} $, then $cos^{-1}x + cos^{-1}y = ? $
2
   
   
   
   
NA
12 )
Evaluate $\int \frac{ax + b}{cx + d} \, dx$ .
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 $\frac{dy}{dx} = \frac{-2xy}{1+x^2}$ is.
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 foot of the perpendicular from the point A(1, 3, 4) on the plane 2x - y + z + 3 = 0 is
2
   
   
   
   
NA
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 ?
2
   
   
   
   
NA
19 )
Evaluate $\int_0^{\frac{\pi}{2}} \, log(sin \, x) \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{sin \, x + cos \, x}{\sqrt{sin \, 2x}} \, dx$ .
3
   
   
   
   
NA
21 )
Evaluate $\int \, \frac{sin^{-1}}{x^2} \, dx$ .
3
   
   
   
   
NA
22 )
The function $f(x)=-\frac{x}{2}+sin\, x$ is strictly increasing in
3
   
   
   
   
NA
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 $- \frac{2}{3}$
3
   
   
   
   
NA
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
3
   
   
   
   
NA
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.
3
   
   
   
   
NA
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
3
   
   
   
   
NA
27 )
Let f : R $\rightarrow$ R, such that $f(x)=(3-x^3)^{\frac{1}{3}}$. Find f o f.
5
   
   
   
   
NA
28 )
Solve the following system of equations using matrix method : x - y = 3 2x + 3y + 4z = 17 y + 2z = 7
5
   
   
   
   
NA
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 $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$
5
   
   
   
   
NA
30 )
Find the area bounded by curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x}{a}+\frac{y}{b}=1$.
5
   
   
   
   
NA
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1 ) 8 2 ) \begin{bmatrix} 6 & 0 \\ 0 & 6 \\ \end{bmatrix} 3 ) 0 4 ) 4A 5 ) 3 6 ) g(x) is discontinuous function 7 ) continuous at non-integer points only 8 ) 2 9 ) $\frac{1}{2} (x^2 + 4x)^{-\frac{1}{2}} (2x + 4) - 2x$ 10 ) -1 11 ) $\frac{\pi}{3}$ 12 ) $\frac{ax}{c} + \frac{bc - ad}{c^2} log|cx + d|$ + C 13 ) 4 14 ) $] 0, \frac{3\pi}{4}[$ 15 ) $y(x^2 + 1)=c$ 16 ) $\frac{\pi}{6}$ 17 ) (-1, 4, 3) 18 ) $\frac{3}{5}$ 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 ) $-\frac{\pi}{3} \le x \le \frac{\pi}{3} $ 23 ) (2a, a) 24 ) $\frac{c^3}{6\sqrt{3}}$ 25 ) 0.22 26 ) 440 27 ) x 28 ) x = 2, y = -1, z = 4 29 ) $3\sqrt{30}$ units, $\frac{x-3}{2}=\frac{y-8}{5}=\frac{z-3}{-1}$ 30 ) $(\frac{\pi ab}{4} -\frac{ab}{2})$ sq. units
10

Mathematics - XII

The Model Paper - 10

Year : 2025 - 26


Time - 3 marks 0 minutes
Full Marks - 80

Previous Set
Cur Set 10
Next Set
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1
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3
4
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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 - 10

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 - 10

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 ) 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 \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 \, 70^o & sin \, 20^o \\ sin \, 70^o & cos \, 20^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 ) The set of points where the f(x) is defined by f(x) = |x - 3| cos x is differentiable, is
8 ) Evaluate $\,\, \lim_{x \to 2} \frac{x^2 - 4}{x - 2}$
9 ) Find the derivative of $f(x) = \sqrt{x^3 - 2x^2 + 1}.$
10 ) The derivative of $sec^2x$ with respect to $tan^2x$.
11 ) The value of sin[$2 sin^{-1} \frac{4}{5}$]
12 ) Evaluate $\int sin^3(2x+1) \, dx$ .
13 ) Evaluate $\int_{-a}^a \, x|x| \, dx$ .
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 $\frac{dy}{dx} + \frac{y}{x}=x^2$.
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 line $\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+5}{4}$ meets the plane at the point
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 $\int_0^{\frac{\pi}{2}} \, \frac{cos^3 x}{sin^3 x +cos^3 x} \, dx$ .
20 ) Evaluate $\int \, \frac{cos(x+a)}{sin(x+b)} \, dx$ .
21 ) Evaluate $\int \, e^x \frac{sin \, 4x - 4}{1- cos \, 4x} \, dx$ .
22 ) The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ 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 ) The volume of the greatest cylinder which can be inscribed in cone of height h and semi vertical angle $\theta$ is
25 ) 8 coins are tossed at a time, the probability of getting at least 6 heads up, is
26 ) The minimum value of P=x+3y subject to constraints $2x+y \le 20$, $x+2y \le 20$ and $x \ge 0$, $y \ge 0$ is
27 ) Let f : R $\rightarrow$ R be a function given by f(x)=ax+b for all $x \, \in \, R$. Find the constants a and b such that $f \, o \, f \, = \, I_R$.
28 ) Solve the following system of equations using matrix method : 2x + y - z = 1 x - y + z = 2 3x + y - 2z = -1
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.

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1 )
if \begin{equation*} A = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix} \end{equation*} and $f(x)=x^2 - 3x +5$, find f(A).
2
   
   
   
   
NA
2 )
If A is an invertible square matrix then $A^{-1} = ?$
2
   
   
   
   
NA
3 )
A and B are two determinant of the same order, such that |A|=20 and |A|=-20, then find the value of |AB|.
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 \, 70^o & sin \, 20^o \\ sin \, 70^o & cos \, 20^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 )
The set of points where the f(x) is defined by f(x) = |x - 3| cos x is differentiable, is
2
   
   
   
   
NA
8 )
Evaluate $\,\, \lim_{x \to 2} \frac{x^2 - 4}{x - 2}$
2
   
   
   
   
NA
9 )
Find the derivative of $f(x) = \sqrt{x^3 - 2x^2 + 1}.$
2
   
   
   
   
NA
10 )
The derivative of $sec^2x$ with respect to $tan^2x$.
2
   
   
   
   
NA
11 )
The value of sin[$2 sin^{-1} \frac{4}{5}$]
2
   
   
   
   
NA
12 )
Evaluate $\int sin^3(2x+1) \, dx$ .
2
   
   
   
   
NA
13 )
Evaluate $\int_{-a}^a \, x|x| \, dx$ .
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 $\frac{dy}{dx} + \frac{y}{x}=x^2$.
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 )
The line $\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+5}{4}$ meets the plane at the point
2
   
   
   
   
NA
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 ?
2
   
   
   
   
NA
19 )
Evaluate $\int_0^{\frac{\pi}{2}} \, \frac{cos^3 x}{sin^3 x +cos^3 x} \, dx$ .
3
   
   
   
   
NA
20 )
Evaluate $\int \, \frac{cos(x+a)}{sin(x+b)} \, dx$ .
3
   
   
   
   
NA
21 )
Evaluate $\int \, e^x \frac{sin \, 4x - 4}{1- cos \, 4x} \, dx$ .
3
   
   
   
   
NA
22 )
The intervals in which the function $f(x)=(x+1)^3(x-3)^3$ is increasing
3
   
   
   
   
NA
23 )
Find the point on the curve $y=x^3-11x+5$ at which the equation of the tangent is y=x-11
3
   
   
   
   
NA
24 )
The volume of the greatest cylinder which can be inscribed in cone of height h and semi vertical angle $\theta$ is
3
   
   
   
   
NA
25 )
8 coins are tossed at a time, the probability of getting at least 6 heads up, is
3
   
   
   
   
NA
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
3
   
   
   
   
NA
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 $f \, o \, f \, = \, I_R$.
5
   
   
   
   
NA
28 )
Solve the following system of equations using matrix method : 2x + y - z = 1 x - y + z = 2 3x + y - 2z = -1
5
   
   
   
   
NA
29 )
Find the equation of the plane passing through the point (0, 7, -7) and containing the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$
5
   
   
   
   
NA
30 )
Find the area bounded by curve $y^2=2y-x$ and the y-axis.
5
   
   
   
   
NA
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1 ) \begin{bmatrix} 15 & -6 \\ -8 & 21 \end{bmatrix} 2 ) $\frac{1}{|A|}$ 3 ) -400 4 ) $\frac{33}{2}$ 5 ) 0 6 ) 1 7 ) R - {3} 8 ) 4 9 ) $\frac{3x^2 - 4x}{2 \sqrt{x^3 - 2x^2 + 1}}$ 10 ) 1 11 ) $\frac{24}{25}$ 12 ) $-\frac{1}{2}cos(2x+1) + \frac{1}{6}cos^3(2x+1) + c$ 13 ) 0 14 ) 6% 15 ) $4xy = x^4 + c$ 16 ) $\frac{\pi}{2}$ 17 ) (3, -1, -1) 18 ) $\frac{5}{8}$ 19 ) $\frac{\pi}{4}$ 20 ) cos(a-b) log|sin(x+b)| - xsin(a-b) +c 21 ) $e^x cot \, 2x+c$ 22 ) $1 \le x \lt \infty $ 23 ) (2, -9) 24 ) $\frac{4}{27}\pi h^3tan^2 \theta$ 25 ) $\frac{37}{256}$ 26 ) none of these 27 ) either a=1 and b=0 or a=-1 and $b \, \in \, R$ 28 ) x = 1, y = 2, z = 3 29 ) x + y + z = 0 30 ) $\frac{4}{3}$ sq. units