CBSE Class 12 Mathematics Set 1- (65/2/1) Question Paper 2026 with Solution PDF

A relation R on set A = {1, 2, 3} defined as R = {(1, 1), (2, 2), (1, 2)} is

• (A) Reflexive only
• (B) Reflexive and Transitive
• (C) Symmetric and Transitive
• (D) Transitive only

If \(A\) and \(B\) are square matrices of the same order, then which of the following statements is/are always true?
(i) \((A + B)(A - B) = A^2 - B^2\)
(ii) \(AB = BA\)
(iii) \((A + B)^2 = A^2 + AB + BA + B^2\)
(iv) \(AB = 0 \implies A = 0\) or \(B = 0\)

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

If \( A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix} \) is a symmetric matrix, then the value of \( 3a + b + c \) is

• (A) \( 2 \)
  
• (B) \( 6 \)
  
• (C) \( 4 \)
  
• (D) \( 0 \)

If \( A = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix} \) and \( A + A^{\prime} = I \), then the value of \( x \in \left[ 0, \dfrac{\pi}{2} \right] \) is

• (A) \( 0 \)
• (B) \( \dfrac{\pi}{4} \)
• (C) \( \dfrac{\pi}{3} \)
• (D) \( \dfrac{\pi}{2} \)

For a square matrix \(A\), \[ (3A)^{-1}= \]

• (A) \( 3A^{-1} \)
  
• (B) \( 9A^{-1} \)
  
• (C) \( \frac{1}{3} A^{-1} \)
  
• (D) \( \frac{1}{9} A^{-1} \)

If \( \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86 \), then the sum of all possible values of \( a \) is

• (A) \( 4 \)
  
• (B) \( 5 \)
  
• (C) \( -4 \)
  
• (D) \( 9 \)

If \(e^{-x} + e^{-y} = 2\), then \(\frac{dy}{dx}\) is

• (A) \(e^{x-y}\)
• (B) \(e^{y-x}\)
• (C) \(-e^{x-y}\)
• (D) \(-e^{y-x}\)

For \[ f(x)=x+\frac{1}{x}, \quad x\neq 0. \]

• (A) local maximum value is 2
  
• (B) local minimum value is \( -2 \)
  
• (C) local maximum value is \( -2 \)
  
• (D) local minimum value \( < \) local maximum value

If \(\displaystyle\int_{0}^{2a}\frac{1}{1+4x^{2}}\,dx = \frac{\pi}{6}\), then the value of a is

• (A) \(\frac{\sqrt{3}}{4}\)
• (B) \(\frac{\sqrt{3}}{2}\)
• (C) \(\sqrt{3}\)
• (D) \(2\sqrt{3}\)

Which of the following expressions will give the area of region bounded by the curve \( y = x^2 \) and line \( y = 16 \)?

• (A) \( \int_0^4 x^2 dx \)
  
• (B) \( 2 \int_0^4 x^2 dx \)
  
• (C) \( \int_0^{16} \sqrt{y} dy \)
  
• (D) \( 2 \int_0^{16} \sqrt{y} dy \)

The general solution of the differential equation \(\frac{dy}{dx} = \frac{\sqrt{y}}{\sqrt{x}}\) is

• (A) \(\log\sqrt{y} = \log\sqrt{x} + C\)
• (B) \(\sqrt{y} + \sqrt{x} = C\)
• (C) \(\sqrt{y} - \sqrt{x} = C\)
• (D) \(\log\sqrt{y} + \log\sqrt{x} = C\)

The integrating factor of the differential equation \( 2x \frac{dy}{dx} - y = 3 \) is

• (A) \( \sqrt{x} \)
  
• (B) \( \frac{1}{\sqrt{x}} \)
  
• (C) \( e^x \)
  
• (D) \( e^{-x} \)

If \( |\vec{a}| = 5 \) and \( -2 \leq \lambda \leq 1 \), then the sum of greatest and the smallest value of \( |\lambda \vec{a}| \) is

• (A) \( -5 \)
  
• (B) \( 5 \)
  
• (C) \( 10 \)
  
• (D) \( 15 \)

Vector of magnitude 3 making equal angles with \( x \) and \( y \) axes and perpendicular to \( z \) axis is

• (A) \( \hat{i} + 2\sqrt{2}\hat{j} \)
  
• (B) \( 3\hat{k} \)
  
• (C) \( \frac{3\sqrt{2}}{2}\hat{i} + \frac{3\sqrt{2}}{2}\hat{j} \)
  
• (D) \( \sqrt{3}\hat{i} + \sqrt{3}\hat{j} + \sqrt{3}\hat{k} \)

Direction cosines of the line given by equations : \(\frac{2x-1}{4} = \frac{1-y}{3} = \frac{-z}{6}\) are

• (A) \(2, -3, -6\)
• (B) \(\frac{2}{7}, \frac{-3}{7}, \frac{-6}{7}\)
• (C) \(\frac{2}{7}, \frac{-3}{7}, \frac{6}{7}\)
• (D) \(\frac{4}{\sqrt{61}}, \frac{-3}{\sqrt{61}}, \frac{-6}{\sqrt{61}}\)

In a linear programming problem, the linear function which has to be maximized or minimized is called

• (A) a feasible function
  
• (B) an objective function
  
• (C) an optimal function
  
• (D) a constraint

For the feasible region shown below, the non-trivial constraints of the linear programming problem are

• (A) \( x + y \leq 5, x + 3y \leq 9 \)
  
• (B) \( x + y \leq 5, x + 3y \geq 9 \)
  
• (C) \( x + y \geq 5, x + 3y \leq 9 \)
  
• (D) \( x + y \geq 5, 3x + y \leq 9 \)

For two events \( A \) and \( B \) such that \( P(A) \neq 0 \) and \( P(B) \neq 1 \), \( P(A' / B') = \)

• (A) \( 1 - P(A / B) \)
  
• (B) \( 1 - P(A' / B) \)
  
• (C) \( \frac{1 - P(A \cap B)}{P(B')} \)
  
• (D) \( \frac{1 - P(A \cup B)}{P(B')} \)

For two vectors \(\vec{a}\) and \(\vec{b}\):  

Assertion (A): \[ |\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 \] Reason (R): \[ |\vec{a}\times\vec{b}| = (\vec{a}\cdot\vec{b})\tan\theta, \quad \theta\neq\frac{\pi}{2}. \]

• (A) Both Assertion (A) and Reason (R) are true and the 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): A line can have direction cosines \(\langle 1, 1, 1 \rangle\).
Reason (R): \(\cos\theta = 1\) is possible for \(\theta = 0\).

• (A) Both Assertion (A) and Reason (R) are true and the 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.

Check whether \[ f:\mathbb{R}-\{3\}\rightarrow\mathbb{R} \] defined as \[ f(x)=\frac{x-2}{x-3} \] is onto or not. 

Check whether \( f : Z \times Z \rightarrow Z \times Z \) (where \( Z \) is the set of integers) defined as \( f(x, y) = (2y, 3x) \) is injective or not.

If \(x = a\sin^{3}t,\ y = b\cos^{3}t\), then find \(\frac{dy}{dx}\) at \(t = \frac{\pi}{4}\).

Find the absolute maximum value of \( f(x) = \cos x + \sin^2 x, x \in [0, \pi] \).

If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.

If \( \vec{AB} = \hat{j} + \hat{k} \) and \( \vec{AC} = 3\hat{i} - \hat{j} + 4\hat{k} \) represent the two vectors along the sides \( AB \) and \( AC \) of \( \Delta ABC \), prove that the median \( \vec{AD} = \frac{\vec{AB} + \vec{AC}}{2} \), where \( D \) is midpoint of \( BC \). Hence, find the length of median \( AD \).

Find the co-ordinates of the point on the line \(\vec{r} = -\hat{j} + 3\hat{k} + \lambda(2\hat{i} - 2\hat{j} + \hat{k})\) such that the sum of co-ordinates is 3.

Find : \(\displaystyle\int \dfrac{x+2}{\sqrt{9x-x^{2}}}\,dx\)

Evaluate : \( \int_{\pi/12}^{5\pi/12} \frac{dx}{1 + \sqrt{\cot x}} \)

Evaluate : \( \int_{-\pi/6}^{\pi/2} (\sin |x| + \cos |x|) dx \)

If \(\dfrac{d}{dx}(F(x))=\dfrac{1}{e^x+1}\), then find \(F(x)\) given that \(F(0)=\log\left(\dfrac{1}{2}\right)\).

Solve the following differential equation : \( x \frac{dy}{dx} = y - x \sin^2 \left( \frac{y}{x} \right) \), given that y(1) = \( \frac{\pi}{6} \).

Find the general solution of the differential equation : \( y \log y \frac{dx}{dy} + x = \frac{2}{y} \).

Solve the following linear programming problem graphically :

Maximize \(Z = 10500x + 9000y\)

Subject to constraints

\(x + y \le 50\)

\(2x + y \le 80\)

\(x, y \ge 0\)

The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that:
(i) target is hit
(ii) atleast one shot misses the target.

Mother, Father and Son line up at random for a family picture. Let events E : Son on one end and F : Father in the middle. Find P(E/F).

If \( P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \) and \( Q = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix} \), find \( (QP) \) and hence solve the following system of equations using matrices : \( x - y = 3, 2x + 3y + 4z = 17, y + 2z = 7 \)

Obtain the value of \( \Delta = \begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{vmatrix} \) in terms of \( x, y \) and \( z \). Further, if \( \Delta = 0 \) and \( x, y, z \) are non-zero real numbers, prove that \( x^{-1} + y^{-1} + z^{-1} = -1 \).

Find the sub intervals in which \(f(x) = \cot^{-1}(\sin x + \cos x),\ x \in (0, \pi)\) is increasing and decreasing.

Find the domain of \(g(x) = \cos^{-1}(x^{2} - 1)\). Hence, find the value of \(x\) for which \(g(x) = \dfrac{\pi}{3}\).

Also, write the range of \(\cos^{-1}x\) other than its principal branch.

A line passing through the points \( A(1, 2, 3) \) and \( B(5, 8, 11) \) intersects the line \( \vec{r} = 4\hat{i} + \hat{j} + \lambda(5\hat{i} + 2\hat{j} + \hat{k}) \). Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.

Smoking increases the risk of lung problems.

Smoking increases the risk of lung problems. A study revealed that \(170\) in \(1000\) males who smoke develop lung complications, while \(120\) out of \(1000\) females who smoke develop lung related problems. In a colony, \(50\) people were found to be smokers of which \(30\) are males. A person is selected at random from these \(50\) people and tested for lung related problems. Based on the given information, answer the following question:

A racing track is build around an elliptical ground whose equation is given by \( 9x^2 + 16y^2 = 144 \).The width of the track is 3 m as shown below :

A racing track is build around an elliptical ground whose equation is given by \(9x^2 + 16y^2 = 144\). The width of the track is \(3\text{ m}\). Based on the given information, answer the following question:

Sports car racing is a form of motorsport which uses sports car prototypes.The competition is held on special tracks designed in various shapes. 

The equation of a sports car racing track is given as: \[ f(x)= \begin{cases} x^4-4x^2+4, & 0\leq x<3,\\ x^2+40, & x\geq 3 \end{cases} \] Based on this information:

Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes. The piecewise function describing the boundary of one such track is given as follows: \[ f(x) = \begin{cases} x^2 + 4, & 0 \le x \\ 4x^2 - 4x + 40, & x \ge 3 \end{cases} \] Based on the given information, answer the following question: