PSEB CLASS 11 MATHEMATICS ANSWER KEY 2025

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Section-A — MCQs (1 mark each)

1. Let $A=\{1,\{3\},4,5\}$. Which of the following statements is true?
   A. $5\notin A$
   B. $\{3\}\in A$
   C. $\{1,4\}\not\subset A$
   D. $\{3\}\subset A$

2. If $P=\{0,2,5\}$ and $Q=\{2,0,6\}$ then $Q-P$ is:
   A. $\{0,2\}$
   B. $\{5\}$
   C. $\{0,2,5,6\}$
   D. $\{6\}$

3. Interval form of $\{t:t\in\mathbb{R},\ t\le -2\}$:
   A. $(0,-2]$
   B. $(-\infty,-2)$
   C. $(-\infty,-2]$
   D. $[-2,\infty)$

4. $\{-1\}\times\{1,2\}=$
   A. $\{1,2\}\times\{-1\}$
   B. $\{(-1,1),(-1,2)\}$
   C. $\{(-1,-1),(-1,-2)\}$
   D. $\{-1,-2\}$

5. Let $R=\{(x^2+1,x): x\ \text{is an odd prime number less than 7}\}$. Then Domain of $R$ is:
   A. $\{3,5,7\}$
   B. $\{3,5\}$
   C. $\{10,26\}$
   D. $\{10,26,50\}$

6. Domain of the function $f(x)=\sqrt{x}$ is:
   A. $(0,\infty)$
   B. $[0,\infty)$
   C. $(-\infty,0]$
   D. $(-\infty,0)$

7. $\sin 120^\circ$ is equal to:
   A. $1/2$
   B. $1/\sqrt2$
   C. $\sqrt3/2$
   D. $-\sqrt3/2$

8. $\sin 13A\cos5A-\cos13A\sin5A=$
   A. $\sin 8A$
   B. $\sin 18A$
   C. $\cos 8A$
   D. $\cos 18A$

9. If $\sin x=0.04$ and $\cos x=-0.04$, then $x$ lies in the quadrant:
   A. I
   B. II
   C. III
   D. IV

10. $x+iy=3-4i$ then $x+y=$
    A. $-1$
    B. $-7$
    C. $1$
    D. $7$

11. Imaginary part of $(1+i)^2$ is equal to:
    A. $0$
    B. $2$
    C. $2i$
    D. $-2$

12. Value of $i^{-101}$ is equal to:
    A. $1$
    B. $-1$
    C. $i$
    D. $-i$

13. If $S$ is solution set of $x-1>3x-7$ when $x$ is a natural number, then
    A. $3\in S$
    B. $4\in S$
    C. $-3\in S$
    D. $2\in S$

14. Solution set of $4<2x\le6$ is:
    A. $(4,6]$
    B. $(2,3)$
    C. $(2,3]$
    D. $[2,3]$

15. The graph of a number line (filled dot at 3 and arrow to the left) represents which inequality? 

    
    
    
    A. $x<3$
    B. $x\le3$
    C. $x>3$
    D. $x\ge3$

16. Number of triangles that can be drawn through 10 points on a circle is:
    A. $45$
    B. $90$
    C. $120$
    D. $240$

17. Total number of ways in which “EAGLE” can be arranged are:
    A. $60$
    B. $120$
    C. $240$
    D. $480$

18. How many 4-digit numbers can be formed from the digits $\{2,3,4,5\}$ which are divisible by 5, if no digit is repeated?
    A. $4$
    B. $6$
    C. $12$
    D. $24$

19. Number of terms in the expansion of $(1+x^2)^{50}$ is:
    A. $3$
    B. $51$
    C. $60$
    D. $100$

20. If $(1+x)^5=1+5x+10x^2+ax^3+5x^4+x^5$, then $a=$
    A. $5$
    B. $6$
    C. $10$
    D. $15$

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Answer Key — Section-A (MCQs)

1. B

2. D

3. C

4. B

5. C

6. B

7. C

8. A

9. B

10. A

11. B

12. D

13. D

14. C

15. B

16. C

17. A

18. B

19. B

20. C

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Section-B (2 Marks Each)

Q.2. Let

$$S=\{x : x \text{ is a vowel in the English alphabet which precedes k}\}, \quad T=\{y: y \text{ is a letter in the word “EDUCATION”}\}.$$

Find
(i) $S-T$
(ii) $S \cap T$.

Solution:

• Vowels before ‘k’: $A, E, I$. So $S=\{A,E,I\}$.

• Letters of EDUCATION: $T=\{E,D,U,C,A,T,I,O,N\}$.

(i) $S-T=\emptyset$ (since all $A,E,I$ are in T).
(ii) $S \cap T=\{A,E,I\}$.

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Q.3. Draw the graph of $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=|x|$. Also write domain and range.

Solution:

• Graph: V-shaped, vertex at (0,0).

• Domain: $(-\infty,\infty)$.

• Range: $[0,\infty)$.

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Q.4. A circular disc rotates 600 times in 10 minutes. Calculate the angle in radians turned in 20 seconds.

Solution:

• 10 minutes = 600 seconds.

• Rotations per second = $600/600=1$.

• In 20 seconds → 20 rotations.

• Angle = $20 \times 2\pi = 40\pi$ radians.

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Q.5. Solve: $5-3(x+1)<2(4-x)$.

Solution:
$5-3x-3<8-2x \implies 2-3x<8-2x \implies -x<6 \implies x>-6.$
Answer: $x \in (-6,\infty)$.

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Q.6. Find $\Re\left(\frac{1}{3+4i}\right)$.

Solution:

$$\frac{1}{3+4i}\cdot\frac{3-4i}{3-4i}=\frac{3-4i}{25}.$$

Real part = $\frac{3}{25}$.

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Q.7. A committee of 6 members is to be formed from 7 boys and 9 girls. Find the number of ways the committee can have at least 5 girls.

Solution:

• Case 1: 5 girls + 1 boy = $\binom{9}{5}\binom{7}{1}=126 \times 7=882$.

• Case 2: 6 girls = $\binom{9}{6}=84$.
  Total = $882+84=966$.

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Q.8. Expand $\Big(\frac{1}{x}-\frac{x}{2}\Big)^4$.

Solution:
Let $a=\frac{1}{x},\; b=-\frac{x}{2}$.
$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$.

• $a^4=\frac{1}{x^4}$.

• $4a^3b=4\cdot\frac{1}{x^3}\cdot\left(-\frac{x}{2}\right)=-\frac{2}{x^2}$.

• $6a^2b^2=6\cdot\frac{1}{x^2}\cdot\frac{x^2}{4}=\frac{3}{2}$.

• $4ab^3=4\cdot\frac{1}{x}\cdot\left(-\frac{x^3}{8}\right)=-\frac{x^2}{2}$.

• $b^4=\frac{x^4}{16}$.

Final expansion:

$$\frac{1}{x^4}-\frac{2}{x^2}+\frac{3}{2}-\frac{x^2}{2}+\frac{x^4}{16}.$$

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Section-C (4 marks each)

Q.9. If $A=\{t: t \in \mathbb{Z}, t^{2} \leq 3\},\; B=\{y: y \text{ divides } 6\}$ and $C=\{x: x \in \mathbb{R}, x^{2}-x=0\}$, then find
(i) $(A \cup B) \cap C$
(ii) $C \times (A - B)$

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Q.10. If cosec $x = -\dfrac{25}{7}$ where $x$ is in 3rd quadrant, then find $\sin \left(\dfrac{3\pi}{2}+x\right)$.

Or

If $\tan x = -\dfrac{12}{5}$ where $x$ is in 4th quadrant, then find cosec $\left(\dfrac{3\pi}{2}-x\right)$.

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Q.11. Prove that

$$-4 \cos 300^{\circ} + \sqrt{3} \, \cosec 1500^{\circ} - 2 \sqrt{3} \, \sin (-780^{\circ}) = 3$$

Or

(a) Prove that

$$\dfrac{\cos 10x - \cos 4x}{\sin 4x - \sin 10x} = \tan 7x$$

(b) Find $\cos 105^{\circ}$.

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Q.12. Find the value of $\tan \dfrac{\pi}{8}$.

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Q.13. Find the real numbers $x$ and $y$ if $(x - i y)(3 + 5i)$ is the conjugate of $-6 - 24i$.

Or

If $z_{1}=3-i,\; z_{2}=3+i$, find

$$\left| \dfrac{z_{1}+z_{2}+1}{z_{1}-z_{2}+1} \right|$$

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Q.14. Solve $5(-x+4) - 3(2x+3) \geq 0,\; 2x+19 \leq 6x+47$ and represent the solution graphically on number line.

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Q.15. In how many ways can the letters of the word PERMUTATIONS be arranged if the
(i) words start with R and end with S,
(ii) there are always 6 letters between P and S?

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Section-D (6 marks each)

Q.16. Find $\sin \dfrac{x}{2}, \cos \dfrac{x}{2}$ and $\tan \dfrac{x}{2}$ when $\sec x = -\dfrac{25}{7},\; x$ lies in 2nd quadrant.

Or

Find $\sin \dfrac{x}{2}, \cos \dfrac{x}{2}$ and $\tan \dfrac{x}{2}$ when $\cosec x = \dfrac{25}{7},\; x$ lies in 2nd quadrant.

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Q.17. How many numbers greater than 10,000,000 can be formed using the digits 1, 0, 3, 2, 4, 4, 1, 4?

Or

(a) Determine the number of 4-card combinations out of a deck of 52 cards if each selection of 4 cards has exactly three kings.

(b) In how many ways can 6 girls and 4 boys be seated in a row so that no two boys are together?

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Q.18. Prove that

$$\cos^{2}x + \cos^{2}\left(x+\dfrac{\pi}{3}\right) + \cos^{2}\left(x-\dfrac{\pi}{3}\right) = \dfrac{3}{2}$$

Or

(a) Find the value of $\cos \dfrac{17\pi}{3}$.

(b) Prove that

$$\cos\left(\dfrac{5\pi}{4}-x\right) - \cos\left(\dfrac{5\pi}{4}+x\right) = -\sqrt{2}\sin x$$

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