1. Problem: Find the specified sets given \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}, A = \{1, 2, 3, 4\}, B = \{2, 4, 6, 8\}, C = \{3, 4, 5, 6\}. (i) Find \( A' \) (complement of A in U). Step 1: \( A' = U - A = \{5, 6, 7, 8, 9\} \) (ii) Find \( A \cap C \). Step 1: \( A = \{1,2,3,4\}, C = \{3,4,5,6\} \) Step 2: Intersection \( A \cap C = \{3,4\} \) (iii) Find \( A - B' \) (elements in A but not in complement of B). Step 1: \( B' = U - B = \{1,3,5,7,9\} \) Step 2: \( A - B' = A \cap B = \{2,4\} \) (iv) Find \( (A \cup B)' \). Step 1: \( A \cup B = \{1,2,3,4,6,8\} \) Step 2: Complement \( (A \cup B)' = U - (A \cup B) = \{5,7,9\} \) (v) Find \( (A')' \) double complement. Step 1: \( (A')' = A \) so \( \{1,2,3,4\} \) (vi) Find \( (B - C)' \). Step 1: \( B - C = B \cap C' \), where \( C' = U - C = \{1,2,7,8,9\} \) Step 2: \( B - C = \{2,4,6,8\} \cap \{1,2,7,8,9\} = \{2,8\} \) Step 3: Complement \( (B - C)' = U - \{2,8\} = \{1,3,4,5,6,7,9\} \) 2. Problem: For \( A = \{1,2,3\}, B = \{3,4\}, C = \{4,5,6\} \), find (i) \( A \times (B \cap C) \) Step 1: \( B \cap C = \{4\} \) Step 2: \( A \times \{4\} = \{(1,4), (2,4), (3,4)\} \) (ii) \( (A \times B) \cap (A \times C) \) Step 1: \( A \times B = \{(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)\} \) Step 2: \( A \times C = \{(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)\} \) Step 3: Intersection is \( \{(1,4),(2,4),(3,4)\} \) (iii) \( A \times (B \cup C) \) Step 1: \( B \cup C = \{3,4,5,6\} \) Step 2: \( A \times (B \cup C) \) includes all ordered pairs with elements of A and the union, e.g., \( (1,3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6) \) (iv) \( (A \times B) \cup (A \times C) \) Step 1: Union of sets in (ii) and (iii) is same as in (iii). 3. Problem: Define Greatest Integer Function. Step 1: It maps a real number \( x \) to the greatest integer less than or equal to \( x \). Step 2: Its domain is \( (-\infty, \infty) \) and range is all integers \( \mathbb{Z} \). 4. Problem: Define Modulus Function. Step 1: Defined as \( f(x) = |x| \), absolute value of \( x \). Step 2: Its domain is \( (-\infty, \infty) \) and range is \( [0, \infty) \). 5. Problem: Define Identity Function. Step 1: Given by \( f(x) = x \). Step 2: Domain and range are both \( (-\infty, \infty) \). 6. Problem: Prove geometrically \( \cos(x + y) = \cos x \cos y - \sin x \sin y \). Step 1: Use unit circle and coordinate definitions of sine and cosine. Step 2: Use rotation formulas or vector dot product of rotated vectors to prove. 7. Problem: Formulas for trigonometric multiples: \( \sin 2x = 2 \sin x \cos x \), \( \cos 2x = \cos^2 x - \sin^2 x \), \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \), \( \sin 3x = 3 \sin x - 4 \sin^3 x \), \( \cos 3x = 4 \cos^3 x - 3 \cos x \). 8. Problem: Prove \( \frac{\sin 5x - 2 \sin 3x + \sin x}{\cos 5x - \cos x} = \tan x \). Use sine and cosine expansion formulas and simplify numerator and denominator. 9. Problem: Prove \( \tan 4x = \frac{4 \tan x (1 - \tan^2 x)}{1 - 6 \tan^2 x + \tan^4 x} \). Use multiple angle formulas for tangent and algebraic manipulation. 10. Problem: If \( x + iy = \frac{a + ib}{a - ib} \), prove \( x^2 + y^2 = 1 \). Multiply numerator and denominator with conjugate and simplify. 11. Problem: Derive line with intercepts \( a \) and \( b \): Equation is \( \frac{x}{a} + \frac{y}{b} = 1 \). 12. Problem: Derive formula for angle between lines with slopes \( m_1, m_2 \): Angle \( \theta = \arctan\left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \). 13. Problem: Distance of point \( P(x_1,y_1) \) from line \( Ax + By + C = 0 \): Distance = \( \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \). 14. Problem: Prove Binomial theorem for positive integer \( n \): \( (a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k \). Proof by induction. 15. Problem: Sum of sequence \( 7, 77, 777, 7777, \ldots \) to \( n \) terms. Using expression \( 7 \times (10^0 + 10^1 + ... + 10^{n-1}) - \) appropriate correction. 16,17. Problems: Prove standard equations of ellipse and hyperbola from definitions. 18. Problem: Show points \( P(-2,3,5), Q(1,2,3), R(7,0,-1) \) are collinear. Check if vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{QR} \) are scalar multiples. 19. Problem: Check if points \( A(3,6,9), B(10,20,30), C(25,-41,5) \) form right angled triangle. Use distance formula and Pythagoras theorem. 20. Problem: Prove geometrically \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \) using Sandwich Theorem. 21. Problem: Derivative of \( \sin x \) from first principle. Use definition of derivative and limit of trigonometric functions. 22. Problem: Derivative of \( \tan x \) from first principle similarly. 23. Problem: Mean deviation about mean for data \( 6,7,10,12,13,4,8,12 \). Calculate mean, find absolute deviations, average them. 24. Problem: Probability of drawing colored discs from bag. Calculate probabilities by counting favorable outcomes over total discs. (i) Red: \( \frac{4}{9} \) (ii) Yellow: \( \frac{2}{9} \) (iii) Blue: \( \frac{3}{9} = \frac{1}{3} \) (iv) Neither blue: \( 1 - \frac{3}{9} = \frac{6}{9} = \frac{2}{3} \) 25. Problem: Calculating probabilities for Kings in 7 cards from 52. (i) All 4 Kings in 7 cards: \( \frac{{4 \choose 4} {48 \choose 3}}{{52 \choose 7}} \) (ii) Exactly 3 Kings: \( \frac{{4 \choose 3} {48 \choose 4}}{{52 \choose 7}} \) (iii) At least 3 Kings = sum of probabilities for 3 and 4 Kings. Final answers depend on numerical values computed from combinations.