CBSE Class 12 Maths 2026 Question Paper 65/3/1 with Solutions

The domain of $f(x)={\cos }^{-1}(2x-5)$ is :

1. (A)$[-1,1]$
2. (B)$[4,6]$
3. (C)$[-7,-3]$
4. (D)$[2,3]$

If $A^2=4A+3I$ and $A^{-1}=xA+yI$, then the value of $(x+y)$ is :

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

If A and B are skew-symmetric matrices of same order, then $A{B}^{\prime }+B{A}^{\prime }$ is a/an :

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

If $\left[\begin{array}{c}4\\ 1\\ 3\end{array}\right]A=\left[\begin{array}{ccc}-4& 8& 4\\ -1& 2& 1\\ -3& 6& 3\end{array}\right]$, then order of A must be :

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

If a square matrix A is such that $A^2=A$ and $(I-A{)}^3=xA+I$, then value of x must be :

1. (A)$7$
2. (B)$5$
3. (C)$-7$
4. (D)$-1$

If $B(\text{adj }B)=\left[\begin{array}{ccc}\frac{1}{3}& 0& 0\\ 0& \frac{1}{3}& 0\\ 0& 0& \frac{1}{3}\end{array}\right]$, then the value of $\det (B^{-1})$ =

1. (A)$\frac{1}{3}$
2. (B)$\frac{1}{9}$
3. (C)$3$
4. (D)$9$

The value of k for which the function $f(x)=\{\begin{array}{ll}{x}^2\sin \frac{1}{x},& x\ne 0\\ k(x+1),& x=0\end{array}$ is a continuous function, is :

1. (A)$\frac{1}{4}$
2. (B)$2$
3. (C)$\frac{1}{2}$
4. (D)$0$

If ${\sin }^{-1}x=y$, then $\frac{dy}{dx}$ is :

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

The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is :

1. (A)$400\pi$ cm${}^3$/cm
2. (B)$100\pi$ cm${}^3$/cm
3. (C)$50\pi$ cm${}^3$/cm
4. (D)$25\pi$ cm${}^3$/cm

$\int \frac{dx}{2^x+2^{-x}}$ is equal to :

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

${\int }_{-1}^1(1-|x|)dx$ is equal to :

1. (A)$2{\int }_0^1(1+x)dx$
2. (B)$2{\int }_{-1}^0(1+x)dx$
3. (C)$0$
4. (D)$2{\int }_{-1}^0(1-x)dx$

The area of the shaded region of the circle given below is equal to :

1. (A)${\int }_1^3\sqrt{9-y^2}dy$
2. (B)$2{\int }_1^3\sqrt{9-y^2}dy$
3. (C)${\int }_0^3\sqrt{9-x^2}dx$
4. (D)$2{\int }_0^3\sqrt{9-x^2}dx$

$\frac{dy}{dx}=F(x,y)$ will be a homogeneous differential equation for which of the following functions ?
(i) $F(x,y)=3x+2y$
(ii) $F(x,y)=\sin \frac{y}{x}+\log y-\log x$
(iii) $F(x,y)=e^{y/x}+1$
(iv) $F(x,y)=\sqrt{x^2+y^2}-y$

1. (A)(i) and (ii)
2. (B)(i), (ii) and (iii)
3. (C)(ii), (iii) and (iv)
4. (D)(ii) and (iii)

For any two vectors $\vec{a}$ and $\vec{b}$, which of the following statements is always true ?

1. (A)$\vec{a}\cdot \vec{b}\le |\vec{a}||\vec{b}|$
2. (B)$|\vec{a}+\vec{b}|\ge |\vec{a}|+|\vec{b}|$
3. (C)$|\vec{a}-\vec{b}|=|\vec{a}|-|\vec{b}|$
4. (D)$|\vec{a}\times \vec{b}|\ge |\vec{a}||\vec{b}|$

If $(\vec{a}+\vec{b})\cdot (\vec{a}-\vec{b})=198$ and $|\vec{a}|=10|\vec{b}|$, then :

1. (A)$|\vec{a}|=\sqrt{2}$
2. (B)$|\vec{b}|=\sqrt{2}$
3. (C)$|\vec{b}|=10\sqrt{2}$
4. (D)$|\vec{a}|=\frac{10}{\sqrt{2}}$

If $l_1,m_1,n_1$ and $l_2,m_2,n_2$ are direction cosines of lines $L_1$ and $L_2$ respectively and $\theta$ is the acute angle between them, then :

1. (A)$\cos \theta =l_1{l}_2+m_1{m}_2+n_1{n}_2$
2. (B)$\sin \theta =l_1{l}_2+m_1{m}_2+n_1{n}_2$
3. (C)$\tan \theta =\frac{l_1}{l_2}+\frac{m_1}{m_2}+\frac{n_1}{n_2}$
4. (D)$\cos \theta =|l_1{l}_2+m_1{m}_2+n_1{n}_2|$

Direction ratios of lines $l_1$ and $l_2$ are $⟨12,-3,9⟩$ and $⟨4,q,-p⟩$ respectively. The values of p and q for which $l_1$ and $l_2$ are parallel are respectively :

1. (A)$-1,3$
2. (B)$3,1$
3. (C)$-3,-1$
4. (D)$-1,-3$

If E and F are two independent events such that $P(E)=\frac{3}{10}$, $P(E\cup F)=\frac{1}{2}$, then $P(E|F)-P(F|E)$ is equal to :

1. (A)$\frac{2}{7}$
2. (B)$\frac{3}{35}$
3. (C)$\frac{1}{70}$
4. (D)$\frac{1}{7}$

Assertion (A) : One of the particular solutions of the differential equation $\frac{dy}{dx}=e^{x+y}$ can be $e^x+e^{-y}=-2$.
Reason (R) : $e^x+e^{-y}=C$ is the general solution of the differential equation $\frac{dy}{dx}=e^{x+y}$.

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) : The vectors $\vec{a}$ and $(-2\vec{a})$, where $\vec{a}\ne \vec{0}$ are collinear vectors.
Reason (R) : $\vec{a}\cdot (-2\vec{a})=0$.

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.

If $x=t+\frac{1}{t}$ and $y=t-\frac{1}{t}$, find $\frac{dy}{dx}$ at $t=2$.

Find the sub-interval(s) of $\left(0,\frac{\pi }{2}\right)$ in which $f(x)=\tan x-4x$ is increasing.

Find the value of $\sin \left[{\cot }^{-1}\sqrt{2}(\cos ({\tan }^{-1}1))\right]$.

A relation R on $A=\{1,2,3\}$ is defined as $R=\{(1,1)(3,3),(1,2)\}$. Is R a symmetric relation ? Justify. Write the smallest relation set $R_1$ such that $R\cup R_1$ becomes an equivalence relation on the set $\{1,2,3\}$.

If for two unit vectors $\vec{a}$ and $\vec{b}$, $|\vec{a}+2\vec{b}|=|2\vec{a}-\vec{b}|$, then find the angle between $\vec{a}$ and $\vec{b}$.

If the lines $\frac{x-3}{1}=\frac{1-y}{1}=\frac{z+2}{p}$ and $\frac{2-x}{3}=\frac{y+1}{5}=\frac{z+56}{2p}$ are perpendicular to each other, then find the value(s) of p.

Find the vector equation of a line passing through the origin and perpendicular to both the lines $\vec{r}=2\hat{i}-\hat{j}+2\hat{k}+\lambda (3\hat{i}+4\hat{j}+2\hat{k})$ and $\vec{r}=\mu (\hat{i}-\hat{j}+\hat{k})$.

Find :
$\int \frac{dx}{x^{1/2}+x^{1/3}}$

Find :
$\int {\tan }^{-1}\left(\frac{1-x}{1+x}\right)dx$

Evaluate :
${\int }_0^{\pi }\frac{{\sin }^{2026}x}{{\sin }^{2026}x+{\cos }^{2026}x}dx$

Find :
$\int \frac{\cos x}{(2+\sin x)(4+\sin x)}dx$

Find :
$\int \frac{x+3}{x^2+4x+5}dx$

Find the general solution of the differential equation $2x^2\frac{dy}{dx}=y^2+2xy$.

Find a particular solution of the differential equation $(x+1)\frac{dy}{dx}=2e^{-y}-1$, given that $y=0$ when $x=0$.

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

A survey was conducted on the patients who have undergone knee replacement surgeries.
It was found that, Robotic Knee replacement surgeries have 90% success rate.
On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that :
(i) exactly one surgery is successful,
(ii) at most two surgeries are successful.

Show that $f:R\to R$ defined as $f(x)=\frac{x}{\sqrt{1+x^2}}$ is one-one but not onto.

Solve the following Linear Programming Problem graphically :
Maximise $Z=600x+400y$
subject to the constraints
$x+2y\le 12$
$4x+5y\ge 20$
$2x+y\le 12$
$x,y\ge 0$

On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners.
A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.

Given that $P=\left[\begin{array}{cc}2& -1\\ 3& 4\end{array}\right]$, $Q=\left[\begin{array}{cc}5& 2\\ 7& 4\end{array}\right]$ and $R=\left[\begin{array}{cc}2& 5\\ 3& 8\end{array}\right]$, find a matrix S such that $PQ-RS$ is a null matrix.

Represent the equations of lines $l_1$ and $l_2$ in vector form and check whether they are intersecting or not.
$l_1:\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$
$l_2:\frac{x+1}{-1}=\frac{2-y}{-2}=\frac{z-5}{5}$

Opposite sides of a square are along the lines :
$\vec{r}=\hat{i}+2\hat{j}-4\hat{k}+\lambda (2\hat{i}+3\hat{j}+6\hat{k})$
$\vec{r}=3\hat{i}+3\hat{j}-5\hat{k}+\mu (2\hat{i}+3\hat{j}+6\hat{k})$
Find the area of the square if direction ratios of other pair of opposite sides of the square are given by $⟨-3,6,p⟩$. Also, find the value of p.