CBSE Class 12 Mathematics Set 1- (65/5/1) Question Paper 2026 with Solution PDF

If matrix \( A = \begin{bmatrix} -p & q \\ r & p \end{bmatrix} \) is such that \( A^2 = I \), then :

• (A) \( 1 + p^2 + qr = 0 \)
  
• (B) \( 1 - p^2 - qr = 0 \)
  
• (C) \( 1 - p^2 + qr = 0 \)
  
• (D) \( 1 + p^2 - qr = 0 \)

If \( A \) is a square matrix such that \( A^2 = A \), then \( (A - I)^3 - A \) is equal to :

• (A) \( I \)
  
• (B) \( -I \)
  
• (C) \( A \)
  
• (D) \( A^2 \)

For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined ?

• (A) \( \tan^{-1} : \mathbb{R} \to \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)
  
• (B) \( \sec^{-1} : \mathbb{R} - (-1, 1) \to [0, \pi] - \left\{ \frac{\pi}{2} \right\} \)
  
• (C) \( \cot^{-1} : \mathbb{R} \to (0, \pi) \)
  
• (D) \( \text{cosec}^{-1} : \mathbb{R} - (-1, 1) \to \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)

Let \( A = \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} -3 & 0 & 1 \\ 2 & 4 & 0 \end{bmatrix} \). If \( A + B + C = O \), then matrix \( C \) is :

• (A) \( \begin{bmatrix} -3 & -3 & 5\\ 3 & 4 & 2 \end{bmatrix} \)
  
• (B) \( \begin{bmatrix} 3 & 3 & 5 \\ -3 & -4 & -2 \end{bmatrix} \)
  
• (C) \( \begin{bmatrix} 3 & 3 & -5 \\ -3 & -4 & -2 \end{bmatrix} \)
  
• (D) \( \begin{bmatrix} -3 & -3 & -5 \\ 3 & 4 & 2 \end{bmatrix} \)

If \( A \) is a non-singular matrix, then which of the following is not true ?

• (A) \( \text{adj } A \) is singular
  
• (B) \( (\text{adj } A)^{-1} = (\text{adj } A^{-1}) \)
  
• (C) \( |A| \neq 0 \)
  
• (D) \( A^{-1} \) exists

If \[ f(x)= \begin{cases} \dfrac{x^2-4x-5}{x+1}, & x\neq -1 \\ k, & x=-1 \end{cases} \] is continuous at \(x=-1\), then the value of \(k\) is:

• (A) Any real value
  
• (B) \( 6 \)
  
• (C) \( -1 \)
  
• (D) \( -6 \)

If the area of \( \triangle ABC \) with vertices \( A(3, 1) \), \( B(-2, 1) \) and \( C(0, k) \) is 5 sq. units, then values of \( k \) are :

• (A) \( 3, 1 \)
  
• (B) \( -1, 3 \)
  
• (C) \( -1, 2 \)
  
• (D) \( 0, 2 \)

Derivative of \( \cos^{-1} \left( \frac{\sin x + \cos x}{\sqrt{2}} \right) \), \( -\frac{\pi}{4} < x < \frac{\pi}{4} \) with respect to \( x \) is :

• (A) \( -1 \)
  
• (B) \( 1 \)
  
• (C) \( \frac{\pi}{4} \)
  
• (D) \( -\frac{\pi}{4} \)

Absolute minimum value of \( f(x) = (x - 2)^2 + 5 \) in the interval \( [-3, 2] \) is :

• (A) \( -3 \)
  
• (B) \( 2 \)
  
• (C) \( 5 \)
  
• (D) \( 30 \)

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

• (A) \( \log \cos x + C \)
  
• (B) \( \frac{1}{\sqrt{2}} \log |\sec x + \tan x| + C \)
  
• (C) \( \frac{1}{\sqrt{2}} \log |\sec x - \tan x| + C \)
  
• (D) \( \log \sin 2x + C \)

The value of \(\int_{-5}^{-1} \frac{1}{x}\,dx\) is equal to :

• (A) \(-\log 5\)
• (B) \(x^{6}\)
• (C) \(\log(-5)\)
• (D) \(x^{-6}\)

An ant is observed crawling on a sheet of paper along a straight line given by equation \( y = 2x - 4 \). Area of the surface covered by the ant bounded by y-axis, x-axis and \( x = 1 \) is :

• (A) \( 1 \text{ sq. unit} \)
  
• (B) \( 3 \text{ sq. units} \)
  
• (C) \( 2 \text{ sq. units} \)
  
• (D) \( 4 \text{ sq. units} \)

The order and degree of the differential equation \[ 1+\left(\frac{d^3y}{dx^3}\right)^3 = \lambda \frac{d^2y}{dx^2} \] is: 

• (A) Order = \( 3 \), Degree = \( 3 \)
  
• (B) Order = \( 2 \), Degree = \( 2 \)
  
• (C) Order = \( 3 \), Degree = \( 1 \)
  
• (D) Order = \( 2 \), Degree = \( 1 \)

The general solution for the differential equation \( \frac{dy}{dx} = e^{3x-y} \) is :

• (A) \( 3e^y = e^{3x} + C \)
  
• (B) \( \log (3x - y) = C \)
  
• (C) \( e^{3x-y} = C \)
  
• (D) \( -e^y + 3e^{3x} = C \)

The corner points of the feasible region determined by the system of linear constraints are \( (0, 0) \), \( (0, 40) \), \( (20, 40) \), \( (60, 20) \) and \( (60, 0) \). If the objective function of an LPP is \( Z = 4x + 3y \), then the maximum value is :

• (A) \( 200 \)
  
• (B) \( 300 \)
  
• (C) \( 240 \)
  
• (D) \( 120 \)

If position vector \( \vec{p} \) of a point \( (24, n) \) is such that \( |\vec{p}| = 25 \), then the value of \( n \) is :

• (A) \( \pm 49 \)
  
• (B) \( \pm 5 \)
  
• (C) \( \pm 1 \)
  
• (D) \( \pm 7 \)

If vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k} \) and \( \vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \), represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of \( \lambda \) is :

• (A) \( 1 \)
  
• (B) \( \frac{5}{2} \)
  
• (C) \( \frac{2}{5} \)
  
• (D) \( 0 \)

If \[ 3P(A)=P(B)=\frac{3}{5} \] and \[ P(A|B)=\frac{2}{3}, \] then \(P(A\cup B)\) is: 

• (A) \( \frac{3}{5} \)
  
• (B) \( \frac{1}{5} \)
  
• (C) \( \frac{2}{15} \)
  
• (D) \( \frac{2}{5} \)

Assertion (A): A relation \(R\) on the set \(\{1,2,3\}\) defined as \[ R=\{(1,1),(1,2),(2,1),(2,2),(3,3)\} \] is an equivalence relation. 
 
Reason (R): A relation that is reflexive, symmetric and transitive is an equivalence relation.

• (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) : Consider a Linear Programming Problem with minimise \( Z = x + 2y \) subject to constraints \( 2x + y \geq 3, x + 2y \geq 6, x, y \geq 0 \) which gives minimum \( Z \) at infinitely many points. The corner points of feasible region are \( (0, 3) \) and \( (6, 0) \).
Reason (R) : If two corner points produce the same minimum value of the objective function, then every point on the line segment joining the points will give the same minimum value.

• (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.

Evaluate \(\sin \left[\tan^{-1} \tan\left(\frac{3\pi}{4}\right)\right]\).

Differentiate \( x^x \) with respect to \( x \log x \).

If \( y = P \cos ux + Q \sin ux \), show that \( \frac{d^2y}{dx^2} + u^2y = 0 \).

Determine the values of \(x\) for which \[ f(x)=\frac{x-3}{x+1}, \quad x\neq -1 \] is an increasing function. 

Three honey bees were found flying along the vectors \( \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \), \( \vec{b} = 4\hat{j} - 2\hat{k} \) and \( \vec{c} = 3\hat{i} + 2\hat{k} \) respectively. Find the value of \( \lambda \) such that the path for \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \).

If \( A, B \) and \( C \) be three non-collinear points such that \( \vec{AB} = \hat{i} + 2\hat{j} - \hat{k} \) and \( \vec{AC} = 2\hat{i} - 3\hat{j} \), then find the area of \( \triangle ABC \).

Find the angle between the following pair of lines : \( \frac{x-2}{3} = \frac{y+5}{2} = \frac{1-z}{-6} \) and \( \frac{x-7}{1} = \frac{y}{2} = \frac{6-z}{-2} \).

A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.

Find : \( \int \frac{x - \sin x}{1 - \cos x} \, dx \).

Evaluate : \( \int_0^2 \frac{1}{\sqrt{x^2 + 2x + 3}} \, dx \).

Solve the differential equation \[ (x+2y^3)\frac{dy}{dx}=y. \]

Solve the following Linear Programming Problem graphically :
Maximize \( Z = \frac{2x}{5} + \frac{3y}{10} \)
subject to constraints
\( 2x + y \leq 1000 \)
\( x + y \leq 800 \)
\( x, y \geq 0 \).

Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are \( 55\hat{i} - 2\hat{j} \), \( 5\hat{i} + 8\hat{j} \) and \( a\hat{i} - 52\hat{j} \) respectively, find the value of ‘a’.

If \( \vec{a} \), \( \vec{b} \) and \( \vec{c} \) are unit vectors, then prove that \( |\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 \leq 9 \).

A die is rolled. Consider events : \( A = \{1, 2, 5\}, B = \{3, 5\}, C = \{2, 3, 4, 5\} \) and hence find : (i) \( P(A|C) \) and \( P(C|A) \) (ii) \( P(A \cap B | C) \) and \( P(A \cup B | C) \)

A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find \( P(A \text{ and } B) \) and find whether the events A and B are independent events or not.

A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost Rs.60, 2 apples, 4 oranges and 6 bananas cost Rs.90, whereas 6 apples, 2 oranges and 3 bananas cost Rs.70. Using matrix method, find the cost of one fruit of each kind.

If \[ y\sqrt{x^2+1}=\log\left(\sqrt{x^2+1}-x\right), \] show that \[ (x^2+1)\frac{dy}{dx}+xy+1=0. \] 

Find the differential of \( x^{\cot x} + \frac{2x^2-3}{2x^2-x+2} \) with respect to \( x \).

Sketch the curve \( \{(x, y) : 100x^2 + 25y^2 = 2500\} \) and find the area of the region enclosed by it, using integration.

Find the foot of the perpendicular from the point \( (0, 2, 3) \) on the line \( \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} \) and hence find the length of the perpendicular.

Find the value of \( p \) if the shortest distance between the lines \( \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( \vec{r} = (p\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k}) \) is \( \frac{3}{\sqrt{2}} \) units.

A school wants the students of class XII to do a project on ‘Sustainability’ keeping the world environment in mind. They select the student participants on the basis of an essay writing competition.
7 students out of 80 are selected for the project and are categorized into two sets such that :
Girl students belong to Set A = \( A = \{G_1, G_2, G_3, G_4\} \) and
Boy students belong to Set B = \( B = \{B_1, B_2, B_3\} \) .

There are three types of vaccines \( A_1, A_2, A_3 \), available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine \( A_1 \), 35% of the population was given Vaccine \( A_2 \) and 40% of the population was given Vaccine \( A_3 \). The survey also stated that the probabilities that Vaccines \( A_1, A_2 \) and \( A_3 \) would protect against the infection were 60%, 55% and 50% respectively. Based on the above information, find the probability that:

A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the surface area of the tumblers produced. If for a tumbler, \( V \) is its volume, \( h \) the height and \( r \) the radius of the circular base, then: