CBSE Class 12 Maths 2025 Question Paper 65/6/1 with Solutions

Let both $A{B}^{\prime }$ and $B^{\prime }A$ be defined for matrices A and B. If order of A is $n\times m$, then the order of B is :

1. (A)$n\times n$
2. (B)$n\times m$
3. (C)$m\times m$
4. (D)$m\times n$

If $A=\left[\begin{array}{ccc}-1& 0& 0\\ 0& 3& 0\\ 0& 0& 5\end{array}\right]$, then A is a/an :

1. (A)scalar matrix
2. (B)identity matrix
3. (C)symmetric matrix
4. (D)skew-symmetric matrix

The following graph is a combination of :

1. (A)$y={\sin }^{-1}x$ and $y={\cos }^{-1}x$
2. (B)$y={\cos }^{-1}x$ and $y=\cos x$
3. (C)$y={\sin }^{-1}x$ and $y=\sin x$
4. (D)$y={\cos }^{-1}x$ and $y=\sin x$

Sum of two skew-symmetric matrices of same order is always a/an :

1. (A)skew-symmetric matrix
2. (B)symmetric matrix
3. (C)null matrix
4. (D)identity matrix

$\left[{\sec }^{-1}(-\sqrt{2})-{\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)\right]$ is equal to :

1. (A)$\frac{11\pi }{12}$
2. (B)$\frac{5\pi }{12}$
3. (C)$-\frac{5\pi }{12}$
4. (D)$\frac{7\pi }{12}$

If $f(x)=\{\begin{array}{ll}\frac{\log (1+ax)+\log (1-bx)}{x},& \text{for }x\ne 0\\ k,& \text{for }x=0\end{array}$ is continuous at $x=0$, then the value of k is :

1. (A)a
2. (B)$a+b$
3. (C)$a-b$
4. (D)b

If ${\tan }^{-1}(x^2-y^2)=a$, where ‘a’ is a constant, then $\frac{dy}{dx}$ is :

1. (A)$\frac{x}{y}$
2. (B)$-\frac{x}{y}$
3. (C)$\frac{a}{x}$
4. (D)$\frac{a}{y}$

If $y=a\cos (\log x)+b\sin (\log x)$, then $x^2{y}_2+x{y}_1$ is :

1. (A)$\cot (\log x)$
2. (B)y
3. (C)$-y$
4. (D)$\tan (\log x)$

Let $f(x)=|x|$, $x\in R$. Then, which of the following statements is incorrect ?

1. (A)f has a minimum value at $x=0$.
2. (B)f has no maximum value in R.
3. (C)f is continuous at $x=0$.
4. (D)f is differentiable at $x=0$.

Let $f^{\prime }(x)=3(x^2+2x)-\frac{4}{x^3}+5$, $f(1)=0$. Then, f(x) is :

1. (A)$x^3+3x^2+\frac{2}{x^2}+5x+11$
2. (B)$x^3+3x^2+\frac{2}{x^2}+5x-11$
3. (C)$x^3+3x^2-\frac{2}{x^2}+5x-11$
4. (D)$x^3-3x^2-\frac{2}{x^2}+5x-11$

$\int \frac{x+5}{(x+6{)}^2}{e}^{x}dx$ is equal to :

1. (A)$\log (x+6)+C$
2. (B)$e^x+C$
3. (C)$\frac{e^x}{x+6}+C$
4. (D)$\frac{-1}{(x+6{)}^2}+C$

The order and degree of the following differential equation are, respectively :
$-\frac{d^{4}y}{d{x}^4}+2e^{dy/dx}+y^2=0$

1. (A)$-4,1$
2. (B)4, not defined
3. (C)1, 1
4. (D)4, 1

The solution for the differential equation $\log \left(\frac{dy}{dx}\right)=3x+4y$ is :

1. (A)$3e^{4y}+4e^{-3x}+C=0$
2. (B)$e^{3x+4y}+C=0$
3. (C)$3e^{-3y}+4e^{4x}+12C=0$
4. (D)$3e^{-4y}+4e^{3x}+12C=0$

For a Linear Programming Problem (LPP), the given objective function is $Z=x+2y$. The feasible region PQRS determined by the set of constraints is shown as a shaded region in the graph.
(Note : The figure is not to scale)
$P\equiv \left(\frac{3}{13},\frac{24}{13}\right)$, $Q\equiv \left(\frac{3}{2},\frac{15}{4}\right)$, $R\equiv \left(\frac{7}{2},\frac{3}{4}\right)$, $S\equiv \left(\frac{18}{7},\frac{2}{7}\right)$
Which of the following statements is correct ?

1. (A)Z is minimum at $S\left(\frac{18}{7},\frac{2}{7}\right)$
2. (B)Z is maximum at $R\left(\frac{7}{2},\frac{3}{4}\right)$
3. (C)(Value of Z at P) $>$ (Value of Z at Q)
4. (D)(Value of Z at Q) $<$ (Value of Z at R)

In a Linear Programming Problem (LPP), the objective function $Z=2x+5y$ is to be maximised under the following constraints :
$x+y\le 4$, $3x+3y\ge 18$, $x,y\ge 0$
Study the graph and select the correct option.
(Note : The figure is not to scale)
The solution of the given LPP :

1. (A)lies in the shaded unbounded region.
2. (B)lies in $△$ AOB.
3. (C)does not exist.
4. (D)lies in the combined region of $△$ AOB and unbounded shaded region.

Let $|\vec{a}|=5$ and $-2\le \lambda \le 1$. Then, the range of $|\lambda \vec{a}|$ is :

1. (A)$[5,10]$
2. (B)$[-2,5]$
3. (C)$[-2,1]$
4. (D)$[-10,5]$

The area of the region bounded by the curve $y^2=x$ between $x=0$ and $x=1$ is :

1. (A)$\frac{3}{2}$ sq units
2. (B)$\frac{2}{3}$ sq units
3. (C)3 sq units
4. (D)$\frac{4}{3}$ sq units

A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is :

1. (A)$\frac{124}{125}$
2. (B)$\frac{1}{125}$
3. (C)$\frac{61}{125}$
4. (D)$\frac{64}{125}$

Assertion (A) : If $|\vec{a}\times \vec{b}{|}^2+|\vec{a}\cdot \vec{b}{|}^2=256$ and $|\vec{b}|=8$, then $|\vec{a}|=2$.
Reason (R) : ${\sin }^2\theta +{\cos }^2\theta =1$ and $|\vec{a}\times \vec{b}|=|\vec{a}||\vec{b}|\sin \theta$ and $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

Assertion (A) : Let $f(x)=e^x$ and $g(x)=\log x$. Then $(f+g)x=e^x+\log x$ where domain of $(f+g)$ is R.
Reason (R) : $\text{Dom}(f+g)=\text{Dom}(f)\cap \text{Dom}(g)$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

Find the domain of $f(x)={\sin }^{-1}(-x^2)$.

Differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$ for $x>0$.

If $(x{)}^y=(y{)}^x$, then find $\frac{dy}{dx}$.

Determine the values of x for which $f(x)=\frac{x-4}{x+1}$, $x\ne -1$ is an increasing or a decreasing function.

If $\vec{a}$ and $\vec{b}$ are position vectors of point A and point B respectively, find the position vector of point C on BA produced such that BC = 3BA.

Vector $\vec{r}$ is inclined at equal angles to the three axes x, y and z. If magnitude of $\vec{r}$ is $5\sqrt{3}$ units, then find $\vec{r}$.

Determine if the lines $\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (3\hat{i}-\hat{j})$ and $\vec{r}=(4\hat{i}-\hat{k})+\mu (2\hat{i}+3\hat{k})$ intersect with each other.

Let $A=\left[\begin{array}{c}1\\ 4\\ -2\end{array}\right]$ and $C=\left[\begin{array}{ccc}3& 4& 2\\ 12& 16& 8\\ -6& -8& -4\end{array}\right]$ be two matrices. Then, find the matrix B if AB = C.

Differentiate $y={\sin }^{-1}(3x-4x^3)$ w.r.t. x, if $x\in \left[-\frac{1}{2},\frac{1}{2}\right]$.

Differentiate $y={\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ with respect to x, when $x\in (0,1)$.

A student wants to pair up natural numbers in such a way that they satisfy the equation $2x+y=41$, $x,y\in N$. Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric and transitive. Hence, state whether it is an equivalence relation or not.

Show that the function $f:N\to N$, where N is a set of natural numbers, given by $f(n)=\{\begin{array}{ll}n-1,& \text{if }n\text{ is even}\\ n+1,& \text{if }n\text{ is odd}\end{array}$ is a bijection.

Consider the Linear Programming Problem, where the objective function $Z=(x+4y)$ needs to be minimized subject to constraints
$2x+y\ge 1000$
$x+2y\ge 800$
$x,y\ge 0$.
Draw a neat graph of the feasible region and find the minimum value of Z.

Find the distance of the point P(2, 4, $-1$) from the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$.

Let the position vectors of the points A, B and C be $3\hat{i}-\hat{j}-2\hat{k}$, $\hat{i}+2\hat{j}-\hat{k}$ and $\hat{i}+5\hat{j}+3\hat{k}$ respectively. Find the vector and cartesian equations of the line passing through A and parallel to line BC.

A person is Head of two independent selection committees I and II. If the probability of making a wrong selection in committee I is 0.03 and that in committee II is 0.01, then find the probability that the person makes the correct decision of selection :
(i) in both committees
(ii) in only one committee

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

Evaluate :
${\int }_0^{\pi /2}\frac{x}{\sin x+\cos x}dx$

Draw a rough sketch for the curve $y=2+|x+1|$. Using integration, find the area of the region bounded by the curve $y=2+|x+1|$, $x=-4$, $x=3$ and $y=0$.

Solve the differential equation : $x^{2}ydx-(x^3+y^3)dy=0$.

Solve the differential equation $(1+x^2)\frac{dy}{dx}+2xy-4x^2=0$ subject to initial condition $y(0)=0$.

Let the polished side of the mirror be along the line $\frac{x}{1}=\frac{1-y}{-2}=\frac{2z-4}{6}$. A point P(1, 6, 3), some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point P and its image.

Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads and 3 erasers.
Based upon the above information, answer the following questions :
(i) Form the equations required to solve the problem of finding the price of each item, and express it in the matrix form AX = B. (1)
(ii) Find |A| and confirm if it is possible to find $A^{-1}$. (1)
(iii) (a) Find $A^{-1}$, if possible, and write the formula to find X. (2)
OR
(iii) (b) Find $A^2-8I$, where I is an identity matrix. (2)