To find the sum of coefficients, substitute 1 for each variable in the expression.

Alright Princess, let's tackle these. Question 33: The sum of the coefficients in the expansion of (a+b)^n is Step 1: To find the sum of coefficients, substitute 1 for each variable in the expression. For (a+b)^n, substitute a=1 and b=1. Sum of coefficients = (1+1)^n = 2^n. The sum of the coefficients is 2^n. This matches option c) 2^n. Question 34: In the expansion of (1+2x)^6, the coefficient of x^2 is Step 1: Use the general term formula for binomial expansion: T_k+1 = nk a^n-k b^k. Here, a=1, b=2x, and n=6. We want the coefficient of x^2, so we set k=2. Step 2: Substitute k=2 into the formula. T_2+1 = 62 (1)^6-2 (2x)^2 T_3 = 62 (1)^4 (2x)^2 Step 3: Calculate the binomial coefficient and simplify. 62 = (6!)/(2!(6-2)!) = (6!)/(2!4!) = (6 × 5)/(2 × 1) = 15 T_3 = 15 · 1 · (4x^2) T_3 = 60x^2 The coefficient of x^2 is 60. The coefficient of x^2 is 60. This matches option b) 60. Question 35: The coefficient of x^4 in the expansion of (1-x)^6 is Step 1: Use the general term formula: T_k+1 = nk a^n-k b^k. Here, a=1, b=-x, and n=6. We want the coefficient of x^4, so we set k=4. Step 2: Substitute k=4 into the formula. T_4+1 = 64 (1)^6-4 (-x)^4 T_5 = 64 (1)^2 (-x)^4 Step 3: Calculate the binomial coefficient and simplify. 64 = (6!)/(4!(6-4)!) = (6!)/(4!2!) = (6 × 5)/(2 × 1) = 15 T_5 = 15 · 1 · x^4 T_5 = 15x^4 The coefficient of x^4 is 15. The coefficient of x^4 is 15. This matches option c) 15. Question 36: In the expansion of (2+3x)^4, what is the coefficient of x^2? Step 1: Use the general term formula: T_k+1 = nk a^n-k b^k. Here, a=2, b=3x, and n=4. We want the coefficient of x^2, so we set k=2. Step 2: Substitute k=2 into the formula. T_2+1 = 42 (2)^4-2 (3x)^2 T_3 = 42 (2)^2 (3x)^2 Step 3: Calculate the binomial coefficient and simplify. 42 = (4!)/(2!(4-2)!) = (4!)/(2!2!) = (4 × 3)/(2 × 1) = 6 T_3 = 6 · 4 · (9x^2) T_3 = 6 · 4 · 9 · x^2 T_3 = 216x^2 The coefficient of x^2 is 216. The coefficient of x^2 is 216. Since 216 is not among options a) -72, b) -72, c) 36, d) -36, the correct option is e) None of the above. Question 37: The coefficient of the term containing x^2 in the expansion of (1+2x)^-3 is Step 1: Use the generalized binomial theorem for (1+y)^n = 1 + ny + (n(n-1))/(2!)y^2 + (n(n-1)(n-2))/(3!)y^3 + Here, y=2x and n=-3. We need the term containing x^2, which corresponds to the term with y^2. Step 2: Substitute n=-3 and y=2x into the y^2 term. The term is (n(n-1))/(2!)y^2. = ((-3)(-3-1))/(2!) (2x)^2 = ((-3)(-4))/(2 × 1) (4x^2) = (12)/(2) (4x^2) = 6 (4x^2) = 24x^2 The coefficient of x^2 is 24. The coefficient of the term containing x^2 is 24. Since 24 is not among options a) -3, b) -6, c) 6, d) 3, the correct option is e) None of the above. Question 38: The term independent of x in the expansion of (1-4x)^5 is Step 1: Use the general term formula: T_k+1 = nk a^n-k b^k. Here, a=1, b=-4x, and n=5. The term independent of x is the term where the power of x is 0. This occurs when k=0. Step 2: Substitute k=0 into the formula. T_0+1 = 50 (1)^5-0 (-4x)^0 T_1 = 50 (1)^5 (1) Step 3: Calculate the value. 50 = 1 T_1 = 1 · 1 · 1 = 1 The term independent of x is 1. The term independent of x is 1. Since 1 is not among options a) -40, b) 40, c) -10, d) 10, the correct option is e) None of the above. What's next?