1. **Problem:** Identify the null set among the given sets. - (a) $C=\{\phi\}$: This set contains the empty set as an element, so it is not empty. - (b) $B=\{x : x+3=3\}$: Solve $x+3=3 \Rightarrow x=0$, so $B=\{0\}$, not empty. - (c) $D=\{0\}$: Contains zero, not empty. - (d) $A=\{x : x>4 \text{ and } x<2\}$: No $x$ satisfies both conditions simultaneously, so $A=\phi$ (empty set). **Answer:** (d) $A$ is the null set. 2. **Problem:** Find $A \cap (A \cup B)$. Recall the distributive property of sets: $A \cap (A \cup B) = A$. **Answer:** (c) $A$. 3. **Problem:** Number of relations from $A$ to $A$ where $A=\{3,6,8\}$. - Number of elements in $A$ is 3. - Number of ordered pairs in $A \times A$ is $3 \times 3 = 9$. - Each relation is a subset of $A \times A$, so total relations = $2^{9}$. **Answer:** (b) $2^9$. 4. **Problem:** Which relation is not a function? - A function assigns exactly one output to each input. - From the description, relation D maps some elements to multiple outputs (e.g., 1,2,3,4 all map to 4 and 8), violating the definition of a function. **Answer:** (d) D. 5. **Problem:** Compare $ cos 44^\circ$ and $ sin 44^\circ$. - Since $\cos 44^\circ = \sin (90^\circ - 44^\circ) = \sin 46^\circ$. - $\sin 46^\circ > \sin 44^\circ$, so $\cos 44^\circ > \sin 44^\circ$. **Answer:** (b) $\cos 44^\circ > \sin 44^\circ$. 6. **Problem:** Convert $40^\circ 20'$ to radians. - $40^\circ 20' = 40 + \frac{20}{60} = 40 + \frac{1}{3} = \frac{121}{3}^\circ$. - Convert degrees to radians: $\theta = \frac{121}{3} \times \frac{\pi}{180} = \frac{121\pi}{540}$. **Answer:** (d) $\frac{121\pi}{540}$. 7. **Problem:** Calculate $i^{29} + \frac{1}{i^{29}}$. - $i^{4} = 1$, so $i^{29} = i^{(4 \times 7) + 1} = i^{1} = i$. - $\frac{1}{i^{29}} = \frac{1}{i} = -i$. - Sum: $i + (-i) = 0$. **Answer:** (b) 0. 8. **Problem:** Find conjugate of $Z = 4i - 3$. - Conjugate of $a + bi$ is $a - bi$. - Here, $Z = -3 + 4i$, so conjugate is $-3 - 4i$. **Answer:** (c) $-3 - 4i$. 9. **Problem:** Identify linear inequality in one variable. - (a) $x^2 + 1 < 3$ is quadratic. - (b) $2x + 3 < 7$ is linear. - (c) $x + y > 2$ has two variables. - (d) $xy < 5$ is not linear. **Answer:** (b) $2x + 3 < 7$. 10. **Problem:** Number of ways student will not pass in all five subjects. - Passing each subject is independent. - Total ways to fail in at least one subject = total subsets excluding passing all. - Number of ways to fail in at least one = $2^5 - 1 = 31$. **Answer:** (a) 31. 11. **Problem:** If $\binom{24}{10} = \binom{24}{r}$, find $r$. - Property: $\binom{n}{r} = \binom{n}{n-r}$. - So, $r = 24 - 10 = 14$. **Answer:** (d) 14. 12. **Problem:** Sum of binomial coefficients $\sum_{k=0}^n \binom{n}{k}$. - Known identity: $\sum_{k=0}^n \binom{n}{k} = 2^n$. **Answer:** (b) $2^n$. 13. **Problem:** Number of terms in expansion of $(x + y)^{2024}$. - Number of terms = $n + 1 = 2024 + 1 = 2025$. **Answer:** (a) 2025. 14. **Problem:** Find term number in G.P. $2,8,32,...$ equal to 131072. - General term: $T_n = ar^{n-1}$, $a=2$, $r=4$. - Solve $2 \times 4^{n-1} = 131072$. - $4^{n-1} = \frac{131072}{2} = 65536 = 4^8$. - So, $n-1=8 \Rightarrow n=9$. **Answer:** (b) 9. 15. **Problem:** For G.P. terms $T_4 = p$, $T_7 = q$, $T_{10} = r$, find relation. - $T_n = ar^{n-1}$. - $p = ar^3$, $q = ar^6$, $r = ar^9$. - Compute $q^2 = (ar^6)^2 = a^2 r^{12}$. - Compute $pr = (ar^3)(ar^9) = a^2 r^{12}$. - So, $q^2 = pr$. **Answer:** (c) $q^2 = pr$. 16. **Problem:** Equation of line $L$ perpendicular to $2x + 3y = 6$ passing through $(1,1)$. - Slope of given line: $-\frac{2}{3}$. - Slope of $L$: $m = \frac{3}{2}$ (negative reciprocal). - Equation: $y - 1 = \frac{3}{2}(x - 1)$. - Simplify: $2y - 2 = 3x - 3 \Rightarrow 3x - 2y = 1$. **Answer:** (d) $3x - 2y = 1$. 17. **Problem:** Find $b$ if lines $x - by = 5$ and $\sqrt{3}x + y = 7$ are perpendicular. - Slope of first line: $\frac{1}{b}$. - Slope of second line: $-\sqrt{3}$. - For perpendicular lines: $m_1 \times m_2 = -1$. - So, $\frac{1}{b} \times (-\sqrt{3}) = -1 \Rightarrow -\frac{\sqrt{3}}{b} = -1 \Rightarrow b = \sqrt{3}$. **Answer:** (a) $\sqrt{3}$. 18. **Problem:** Find center of circle $2x^2 + 2y^2 - 8x + 4y - 12 = 0$. - Divide by 2: $x^2 + y^2 - 4x + 2y - 6 = 0$. - Complete the square: - $x^2 - 4x + y^2 + 2y = 6$. - $x^2 - 4x + 4 + y^2 + 2y + 1 = 6 + 4 + 1$. - $(x - 2)^2 + (y + 1)^2 = 11$. - Center: $(2, -1)$. **Answer:** (d) $(2, -1)$. 19. **Assertion-Reason:** - Assertion: $\tan(-\frac{11\pi}{25}) = 1$. - Reason: $\sin(3\pi + \theta) = \frac{\sin \theta}{4}$. - $\sin(3\pi + \theta) = -\sin \theta$, so Reason is false. - Assertion is false as $\tan(-\frac{11\pi}{25}) \neq 1$. **Answer:** (d) Assertion (A) is false but Reason (R) is true. 20. **Assertion-Reason:** - Sum of first 4 terms $S_4 = 15$, sum of first 8 terms $S_8 = 255$. - Using formula $S_n = a \frac{r^n - 1}{r - 1}$. - Solving gives $r=2$. - Reason formula is correct. **Answer:** (a) Both Assertion and Reason are true and Reason is correct explanation. 21. **Problem:** Find $A - B$ and $B - A$ for $A=\{1,2,3,4,5,6\}$, $B=\{2,4,6,8\}$. - $A - B = \{1,3,5\}$ (elements in $A$ not in $B$). - $B - A = \{8\}$ (elements in $B$ not in $A$). 22. **Problem:** Find $\left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + 1} \right|$ where $z_1 = 2 - i$, $z_2 = 1 + i$. - Compute numerator: $z_1 + z_2 + 1 = (2 - i) + (1 + i) + 1 = 4$. - Compute denominator: $z_1 - z_2 + 1 = (2 - i) - (1 + i) + 1 = 2 - i - 1 - i + 1 = 2 - 2i$. - Magnitude numerator: $|4| = 4$. - Magnitude denominator: $|2 - 2i| = \sqrt{2^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}$. - Result: $\frac{4}{2\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$. 23. **Problem:** Prove $\sum_{r=0}^n 3^r \binom{n}{r} = 4^n$. - Use binomial theorem: $(1 + 3)^n = \sum_{r=0}^n \binom{n}{r} 1^{n-r} 3^r = 4^n$. **Answer:** Proven.