CBSE Class 12 Mathematics Compartment Set 3 - (65/7/3) Question Paper 2026 PDF

1.

Find the domain of \[ q(x)=\cos^{-1}(4x^2-3). \] Hence, find the value of \(x\) for which \[ q(x)=0. \] Also, write the range of \[ 3q(x)-\pi. \] 

2.

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

3.

In a school, the probability of holding a debate competition is \( \frac{1}{3} \) and that of a quiz competition is \( \frac{2}{3} \). In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.

4.

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

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

5.

Check whether the function \[ f(x)= \begin{cases} \dfrac{|x-3|}{2(x-3)}, & x<3,\\[6pt] \dfrac{x-6}{6}, & x\geq 3 \end{cases} \] is continuous at \(x=3\) or not. 

6.

The least value of \[ f(x)=e^{-x} \] in the interval \[ [0,3] \] is:

• \( e^{-3} \)
  
• \( -1 \)
  
• \( 1 \)
  
• \( -e^3 \)

7.

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

• local maximum value is 2
  
• local minimum value is \( -2 \)
  
• local maximum value is \( -2 \)
  
• local minimum value \( < \) local maximum value

8.

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:

9.

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

10.

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

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

11.

Find: \[ \int \frac{x^2}{(x^2+9)(x^2+16)}\,dx \]

12.

A function \[ f:\mathbb{R}-\left\{\frac{3}{5}\right\} \to \mathbb{R}-\left\{\frac{3}{5}\right\} \] is defined as \[ f(x)=\frac{3x+2}{5x-3}. \] Show that \(f\) is one-one and onto.

13.

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

• \([-1, 1]\)
  
• \([4, 6]\)
  
• \([-7, -3]\)
  
• \([2, 3]\)

14.

Evaluate: \[ \cos\left[\sin^{-1}(-1)-\tan^{-1}(-\sqrt{3})\right]. \]

15.

Assertion (A) : In an experiment of throwing an unbiased die, the probability of getting a prime number given that number appearing on the die being odd is \( \frac{2}{3} \).
Reason (R) : For any two events \( A \) and \( B \), \( P(A|B) = \frac{P(A \cup B)}{P(B)} \).

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