CBSE Class 12 Mathematics Set 3- (65/3/3) Question Paper 2026 with Solution PDF

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

• (A) \( |\vec{a}| = \sqrt{2} \)
  
• (B) \( |\vec{b}| = \sqrt{2} \)
  
• (C) \( |\vec{b}| = 10\sqrt{2} \)
  
• (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 :

• (A) \( \cos \theta = l_1l_2 + m_1m_2 + n_1n_2 \)
  
• (B) \( \sin \theta = l_1l_2 + m_1m_2 + n_1n_2 \)
  
• (C) \( \tan \theta = \frac{l_1}{l_2} + \frac{m_1}{m_2} + \frac{n_1}{n_2} \)
  
• (D) \( \cos \theta = |l_1l_2 + m_1m_2 + n_1n_2| \)

Direction ratios of lines \(l_1\) and \(l_2\) respectively are \(\langle 1, 0, 0\rangle\) and \(\langle 0, -1, 0\rangle\). The direction ratios of the line perpendicular to both \(l_1\) and \(l_2\) are :

• (A) \(\langle 1, 1, 0\rangle\)
• (B) \(\langle 0, 0, -1\rangle\)
• (C) \(\langle 1, 1, 1\rangle\)
• (D) \(\langle 1, 0, -1\rangle\)

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

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

The domain of \(\cos^{-1}(4x+1)\) is :

• (A) \([-1, 1]\)
• (B) \([-3, 5]\)
• (C) \([-4, 4]\)
• (D) \(\left[-\frac{1}{2}, 0\right]\)

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

• (A) \(-1\)
  
• (B) \(1\)
  
• (C) \(\frac{5}{3}\)
  
• (D) \(7\)

If \(A\) and \(B\) are skew-symmetric matrices of same order, then \(AB' + BA'\) is a/an :

• (A) symmetric matrix
  
• (B) skew-symmetric matrix
  
• (C) null matrix
  
• (D) identity matrix

If a matrix X is such that \(\begin{bmatrix}2 & 1\end{bmatrix}X=\begin{bmatrix}3 & 4 & 5\end{bmatrix}\), then the order of matrix X is :

• (A) \(3\times 1\)
• (B) \(2\times 3\)
• (C) \(1\times 2\)
• (D) \(1\times 3\)

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

• (A) \(7\)
  
• (B) \(5\)
  
• (C) \(-7\)
  
• (D) \(-1\)

If \(A(\operatorname{adj}A)=\begin{bmatrix}3 & 0 & 0\\0 & 3 & 0\\0 & 0 & 3\end{bmatrix}\), then the value of \(|2A|\) is :

• (A) 6
• (B) 54
• (C) 12
• (D) 24

The value of \(k\) for which the function \[ f(x)= \begin{cases} x^2\sin\left(\frac{1}{x}\right), & x\neq 0,\\ k(x+1), & x=0 \end{cases} \] is a continuous function, is: 

• (A) \(\frac{1}{4}\)
  
• (B) \(2\)
  
• (C) \(\frac{1}{2}\)
  
• (D) \(0\)

If \(2\cos^{-1}x=y\), then \(\dfrac{dy}{dx}\) is :

• (A) \(-2\sin^{-1}x\)
• (B) \(-\dfrac{1}{2}\sin\dfrac{y}{2}\)
• (C) \(\dfrac{-1}{\sqrt{1-2x^{2}}}\)
• (D) \(-2\operatorname{cosec}\dfrac{y}{2}\)

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

• (A) \(400\pi\text{ cm}^3/\text{cm}\)
  
• (B) \(100\pi\text{ cm}^3/\text{cm}\)
  
• (C) \(50\pi\text{ cm}^3/\text{cm}\)
  
• (D) \(25\pi\text{ cm}^3/\text{cm}\)

\(\displaystyle\int \frac{\cos x}{\sqrt{\sin^2 x + 1}}\,dx\) is equal to :

• (A) \(\tan^{-1}(\sin x) + C\)
• (B) \(\sin^{-1}(\sin x) + C\)
• (C) \(\log\left|\sin x + \sqrt{1+\sin^2 x}\right| + C\)
• (D) \(\log\left|\sin x - \sqrt{\sin^2 x+1}\right| + C\)

\( \int_{-1}^{1} (1 - |x|) \, dx \) is equal to :

• (A) \( 2 \int_{0}^{1} (1 + x) \, dx \)
  
• (B) \( 2 \int_{-1}^{0} (1 + x) \, dx \)
  
• (C) \( 0 \)
  
• (D) \( 2 \int_{-1}^{0} (1 - x) \, dx \)

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

• (A) \( \int_{1}^{3} \sqrt{9 - y^2} \, dy \)
  
• (B) \( 2 \int_{1}^{3} \sqrt{9 - y^2} \, dy \)
  
• (C) \( \int_{0}^{3} \sqrt{9 - x^2} \, dx \)
  
• (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) \( F(x, y) = \sqrt{x^2 + y^2} - y \)

• (A) (i) and (ii)
  
• (B) (i), (ii) and (iii)
  
• (C) (ii), (iii) and (iv)
  
• (D) (ii) and (iii)

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

• (A) \( \vec{a} \cdot \vec{b} \le |\vec{a}| |\vec{b}| \)
  
• (B) \( |\vec{a} + \vec{b}| \ge |\vec{a}| + |\vec{b}| \)
  
• (C) \( |\vec{a} - \vec{b}| = |\vec{a}| - |\vec{b}| \)
  
• (D) \( |\vec{a} \times \vec{b}| \ge |\vec{a}| |\vec{b}| \)

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 \(\frac{dy}{dx} = e^{x+y}\).

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

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

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

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 \(\vec{r} = \mu(\hat{i} - \hat{j} + \hat{k})\).

If \( x = e^{\sin^{-1} t}, y = e^{\cos^{-1} t} \), find \( \frac{dy}{dx} \) at \( t = \frac{1}{\sqrt{2}} \).

Find the sub-interval of \((0, \infty)\) in which \(f(x) = x \log x\) is increasing.

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

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 \(\vec{b}\).

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.

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

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

Evaluate :

\[ \int_0^{\frac{\pi}{2}} \frac{\sin^2 x}{1 + \sin 2x}\,dx \]

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 \( (y^2 - x^2)dx = 2xy dy \)

Find the general solution of the differential equation \((x^2 - y^2)\,dx + 2xy\,dy = 0\).

Solve the differential equation \(\sin x \cos y\,dx + \cos x \sin y\,dy = 0\), given that \(y = \frac{\pi}{4}\) when \(x = 0\).

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

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

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

Show that a function \( f: R_{+} \to A \subset N \), defined as \( f(x) = 4x^2 + 12x + 15 \) is one-one. Find set \( A \) so that \( f \) is onto where \( R_{+} = [0, \infty) \). Also, find if there exists \( a \in R_{+} \) such that \( f(a) = 7 \). Justify.

Solve the following Linear Programming Problem graphically :

Maximize \( Z = 8x + 8y \)

subject to the constraints

\[ \begin{aligned} x - 20 &\leq 0 \\ 2x + 3y &\leq 120 \\ 2x + y &\leq 60 \\ x \geq 0,\ y &\geq 0 \end{aligned} \]

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 \(P = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}\), \(Q = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix}\) and \(R = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix}\), find a matrix \(S\) such that \(PQ - RS\) is a null matrix.

There is a triangular park \(ABC\) in the society. The park is divided into two sections as shown in the figure.

Two vertical light poles of height \(22 \text{ m}\) and \(16 \text{ m}\) stand on the opposite sides of a \(20 \text{ m}\) wide road. Two ladders of length \(l_1\) and \(l_2\) are placed from a common point \(R\) on the road at a distance of \(x \text{ m}\) from the smaller pole.

Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length \( l_1 \) and \( l_2 \) are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :

A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII.