Answer: Given, Coefficients of (r + 1)th term in (1 + x)n+1 =  n+1Cr Sum of the coefficients of the rth and (r + 1)th terms in (1 + x)n , (1 + x)n = nCr-1 + nCr...

Answer: Coefficient of the r term in the expansion of (1 + x)n - nCr-1 Coefficients of the (2r + 1) and (r + 2) terms in the given expansion are 43C2r+1-1 and 43Cr+2-1 Equaling two coefficients,...

Answer: Coefficient of the r term in the expansion of (1 + x)n is nCr-1 Coefficients of the (2r + 4) and (r – 2) terms in the given expansion are 18C2r+4-1 and 18Cr-2-1 Equaling two coefficients,...

Answers: (ii) If (r + 1)th term in the given expression is independent of x. Tr+1 = nCr xn-r ar For this term to be independent of r, (18-r)/3 – r/3 = 0 (18 – r – r)/3 = 0 18 – 2r = 0 2r = 18 r =...

Answers: (i)  If (r + 1)th term in the given expression is independent of x. Tr+1 = nCr xn-r ar For this term to be independent of x, (8-r)/3 – r/5 = 0 (40 – 5r – 3r)/15 = 0 40 – 5r – 3r = 0 40 – 8r...

Answers: (i) If (r + 1)th term in the given expression is independent of x. Tr+1 = nCr xn-r ar For this term to be independent of x, (10-r)/2 – 2r = 0 10 – 5r = 0 5r = 10 r = 10/5 r = 2 The term is...

Answers: (i) If (r + 1)th term in the given expression is independent of x. Tr+1 = nCr xn-r ar = 25Cr (2x2)25-r (-3/x3)r = (-1)r 25Cr × 225-r × 3r x50-2r-3r For this term to be independent of x, 50...

Answers: (i)  If (r + 1)th term in the given expression is independent of x. Tr+1 = nCr xn-r ar For this term to be independent of x, 18 – 3r = 0 3r = 18 r = 18/3 r = 6 The term is 7th term. T7 =...

Answers: (i) Given, (p/x + x/p)9  [n = 9] Middle terms - ((n+1)/2) = ((9+1)/2) = 10/2 = 5 and ((n+1)/2 + 1) = ((9+1)/2 + 1) = (10/2 + 1) = (5 + 1) = 6 The terms are 5th and 6th. T5 = T4+1 T6 = T5+1...

Answers: (i) Given, (3 – x3/6)7  [n = 7] Middle terms are ((n+1)/2) = ((7+1)/2) = 8/2 = 4 and ((n+1)/2 + 1) = ((7+1)/2 + 1) = (8/2 + 1) = (4 + 1) = 5 The terms are 4th and 5th. T4 = T3+1...

Answers: (v) Given, (x – 1/x)2n+1  [n = (2n + 1)] Middle terms - ((n+1)/2) = ((2n+1+1)/2) = (2n+2)/2 = (n + 1) and ((n+1)/2 + 1) = ((2n+1+1)/2 + 1) = ((2n+2)/2 + 1) = (n + 1 + 1) = (n + 2) The terms...

Answers: (iii) Given, (1 + 3x + 3x2 + x3)2n = (1 + x)6n  [n is an even number] Middle term - (n/2 + 1) = (6n/2 + 1) = (3n + 1)th term. T2n = T3n+1 = 6nC3n x3n = (6n)!/(3n!)2 x3n Middle term is...

Answers: (i) Given, (x – 1/x)10  [n = 10] Middle term - (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6. The term is 6th term T6 = T5+1 Middle term is -252. (ii)  Given, (1 – 2x + x2)n = (1 – x)2n  [n is an...

Answer: This expression contains 26 terms. 11th term from the end is the (26 − 11 + 1) th term from the beginning. T16 = T15+1 = 25C15 (2x)25-15 (-1/x2)15 T16 = 25C15 (210) (x)10 (-1/x30) T16 =...

Answer: Consider, 7th term as T7 T7 = T6+1 T7 = 10C6 (3x2)10-6 (-1/x3)6 T7 = 10C6 (3)4 (x)8 (1/x18) T7 = [10×9×8×7×81] / [4×3×2×x10] T7 = 17010 / x10 ∴ 7th term of the expression (3x2 – 1/x3)10 =...

Answer: 5th term from the end is the (11 – 5 + 1)th, is., 7th term from the beginning. T7 = T6+1 T7 = 10C6 (3x)10-6 (-1/x2)6 T7 = 10C6 (3)4 (x)4 (1/x12) T7 = [10×9×8×7×81] / [4×3×2×x8] T7 = 17010 /...

Answer: Consider, 8th term as T8 T8 = T7+1 T8 = 10C7 (x3/2 y1/2)10-7 (-x1/2 y3/2)7 T8 = -[10×9×8]/[3×2] x9/2 y3/2 (x7/2 y21/2) T8 = -120 x8y12 ∴ 8th term of the expression (x3/2 y1/2 – x1/2 y3/2)10...

Answer: Consider, 7th term as T7 T7 = T6+1 ∴  7th term of the expression (4x/5 + 5/2x) 8 = 4375/x4.

Answer: Consider, Tr+1 be the 4th term from the end. Tr+1 is (10 − 4 + 1)th is the 7th term from the beginning.

Answer: Consider, Tr+1 be the 4th term from the end of the given expression. Tr+1 is (10 − 4 + 1)th term The term is the 7th term from the beginning. T7 = T6+1 ∴ 4th term from the end =...

Answer: Consider, Tr+1 be the 4th term from the end of the expression. Tr+1 is (9 − 7 + 1)th term, The term is 3rd term from the beginning. T3 = T2+1 ∴ 7th term from the end = 4032...

Answers: (i)   x10 is in the (r + 1)th term in the expression. Tr+1 = nCr xn-r ar (ii) If x7 is at the (r + 1) th term in the expression. Tr+1 = nCr xn-r ar 40 − 3r =7 3r = 40 – 7 3r = 33 r = 33/3 r...

Answers: (i) If x−15 is at the (r + 1)th term in the expression. Tr+1 = nCr xn-r ar (ii) If x9 occurs at the (r + 1)th term in the expression. Tr+1 = nCr xn-r ar For this term to contain x9, 18 −...

Answers: (i)  If xm is at the (r + 1)th term in the expression. Tr+1 = nCr xn-r ar (ii) If x is at the (r + 1)th term in the expression. (1 – 2x3 + 3x5) (1 + 1/x)8 = (1 – 2x3 + 3x5) (8C0 + 8C1 (1/x)...

Answers: (i) If a5b7 is at the (r + 1)th term in the expression. Tr+1 = nCr xn-r ar (ii) If x is at the (r + 1)th term in the expression. (1 – 3x + 7x2) (1 – x)16 = (1 – 3x + 7x2) (16C0 + 16C1 (-x)...

Answer: Consider, Tr+1 th term in the given expansion contains x and y to one and the same power. Tr+1 = nCr xn-r ar

Answer: If x9 is at the (r + 1)th term in the expression. Tr+1 = nCr xn-r ar For this term to contain x9, 40 – 3r = 9 3r = 40 – 9 3r = 31 r = 31/3 It is not possible.  [r is not an integer] Thus,...

Answer: If x-1 is at the (r + 1)th term in the expression. Tr+1 = nCr xn-r ar For this term to contain x-1, 24 – 3r = -1 3r = 24 + 1 3r = 25 r = 25/3 It is not possible.   [r is not an integer]...

Answers: (i)  Given, (2/3x – 3/2x)20  [n = 20] Middle term - (n/2 + 1) = (20/2 + 1) = (10 + 1) = 11. The term is 11th term T11 = T10+1 T11 = 20C10 (2/3x)20-10 (3/2x)10 T11 = 20C10 210/310 ×...

Answers: (i) Given, (x2 – 2/x)10  [n = 10] Middle term - (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6. The term is 6th term T6 = T5+1 Middle term is -8064x5. (iI) Given, (x/a – a/x) 10  [n = 10] Middle term...

Answers: (i) Given, (3x – x3/6)9  [n = 9] Middle terms - ((n+1)/2) = ((9+1)/2) = 10/2 = 5 and ((n+1)/2 + 1) = ((9+1)/2 + 1) = (10/2 + 1) = (5 + 1) = 6 The terms are 5th and 6th. T5 = T4+1 Middle...

Answers: (i) Given, (3x – 2/x2)15  [n = 15] Middle terms - ((n+1)/2) = ((15+1)/2) = 16/2 = 8 and ((n+1)/2 + 1) = ((15+1)/2 + 1) = (16/2 + 1) = (8 + 1) = 9 The terms are 8th and 9th. Middle term are...