BINOMIAL EXPANSIONS EXAM QUESTIONS

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BINOMIAL EXPANSIONS EXAM QUESTIONS

Question 1 (**) Find, without any calculating aid, the first three terms in the expansion of ( 5x) 5, in ascending powers of x. CL, 3 400x + 000 x... Question (**) Expand ( 3 x) 5 in ascending powers of x, up and including the term in 3 x. CA, 3 43 810x + 1080x 70 x +... Question 3 (**) Find the binomial expansion of ( 1 5x) 4 in ascending powers of x. CB, 3 4 1 0x + 150x 500x + 65x

Question 4 (**) Find, without any calculating aid, the first three terms in the expansion of ( 7x) 6 ascending powers of x., in 64 1344x + 11760 x +... Question 5 (**) Find the binomial expansion of ( 1 x) 6 in ascending powers of x. 3 4 5 6 1 1x + 60x 160x + 40x 19x + 64x Question 6 (**) a) Find the first four terms, in ascending powers of x, in the binomial expansion of ( 1+ 3x) 8. b) Determine the coefficient of 6 x in the binomial expansion of ( ) 8 1+ 3x. CH, 3 1+ 4x + 5x + 151 x +..., 041

Question 7 (**+) Find, without using a calculator, the coefficient of 3 x in the expansion of ( ) 6 + 3x. 430 Question 8 (**+) Find the coefficient of 5 x in the binomial expansion of ( ) 9 + 3x. MP1-M, 489888 Question 9 (**+) Find, without using a calculator, the coefficient of 4 x in the expansion of 1 ( ) 7 4x. 110

Question 10 (**+) Find, without using a calculator, the coefficient of 5 x in the expansion of ( ) 7 x 3. 6048 Question 11 (**+) a) Find the first four terms, in ascending powers of x, in the binomial expansion of ( 1 x) 10. b) Use the answer of part (a) with a suitable value of x to find an approximate 10 value for 0.98, giving the answer correct to three decimal places. CF, 3 1 0x + 180x 960 x +..., 0.817

Question 1 (**+) a) Find, in ascending powers of x, the first four terms in the binomial expansion of ( + x) 9. b) By using the answer of part (a), or otherwise, find the first four terms in the binomial expansion of 1 ( ) 9 x. 4 + x = 51 + 304x + 4608x + 5376 x +..., CN, ( ) 9 3 1 ( ) 9 3 51 576 88 84... 4 x = x + x x + Question 13 (**+) a) Find the first five terms, in ascending powers of x, in the binomial expansion of ( 1+ x) 1. b) Use the answer of part (a) with a suitable value of x to find an approximate 1 value for 1.0. c) Determine the error in this approximation. CP, 3 4 1+ 4x + 64x + 1760x + 790 x +..., 1 1.0 1.6839, error 0.0000059

Question 14 (**+) In the binomial expansion of ( 1+ kx) 6, where k is constant, the coefficient of Find the value of k. 3 x is twice as large as the coefficient of x. CD, k = 3 Question 15 (**+) a) Find, in ascending powers of x, the binomial expansion of ( + x) 5. b) Use the answer of part (a) to find the binomial expansion of ( ) 5 x. ( ) 5 3 4 5 x 3 80x 80x 40x 10x x ( ) 5 + = + + + + +, 4 6 8 10 x = 3 80x + 80x 40x + 10x x

Question 16 (***) a) Find the first four terms, in ascending powers of x, of the binomial expansion of ( 1+ x) 7. b) Hence determine the coefficient of x in the expansion of ( 1 x) ( 3 x) 7 4 + +. MP1-N, 3 1+ 14x + 84x + 80 x +..., 1350 Question 17 (***) a) Find, without any calculating aid, the binomial expansion of ( x + 4) 3, in descending powers of x. b) Hence determine the expansion of ( x 1)( x 4) 3 +. 3 8x + 48x + 96x + 64, 4 3 16x + 88x + 144x + 3x 64

Question 18 (***) a) Find, in ascending powers of x, the first three terms in the binomial expansion of ( 3x) 10. b) Use the first three terms in the binomial expansion of ( 3x) 10 with a suitable value for x, to find an approximation for c) Find the error in this approximation. 10 1.97. ( ) 10 x x x 3 = 104 15360 + 103680 +..., 10 1.97 880.768 881, error 0.4

Question 19 (***) ( ) = ( 1+ ) 7 f x x a) Find, without any calculating aid, the first four terms in the expansion of f ( x ), in ascending powers of x. b) Hence, determine the first four terms in the expansion of ( 1 x)( 1 x) 7 +, giving the answer in ascending powers of x. 3 1+ 14x + 84x + 80 x +..., 3 1+ 1x + 56x + 11 x +...

Question 0 (***) ( ) = ( 1 ) 8 f x x a) Find the first four terms in the expansion of f ( x ), in ascending powers of x. b) Hence determine, in ascending powers of x, the first four terms in the expansion of ( 3x)( 1 x) 8 +. 3 1 16x + 11x 448 x +..., 3 9x + 176x 560 x +...

Question 1 (***) a) Find, in ascending powers of x, the first four terms in the binomial expansion of ( + x) 9. b) Hence find the coefficient of 3 x in the expansion of ( 1 1 ) ( ) 9 8 x + x. CC, ( ) 9 3 x x x x + = 51 + 304 + 4608 + 5376 +..., 3 x = 460

Question (***) ( ) = ( + ) 4 f x x a) Find the expansion of f ( x ), in ascending powers of x. b) Deduce the expansion of ( 3x) 4, also in ascending powers of x. c) Determine the coefficient of x in the expansion of ( x) ( 3x) 4 4 +. 3 4 ( ) 16 3 4 8 f x = + x + x + x + x, 3 4 16 96x + 16x 16x + 81x, 104 Question 3 (***+) It is given that if k is a non zero constant, then ( ) 6 3 kx a bx bx cx + + + + +... Determine the value of each of the constants a, b and c. CE, a = 64, b = 153.6, c = 81.9

Question 4 (***+) In the binomial expansion of ( x k ) 4 +, where k is a non zero constant, the coefficient of Find the value of k. x is 1 times as large as the coefficient of 3 x. CV, k = 16 Question 5 (***+) Find the binomial expansion of 4 x + x, x 0, simplifying each term of the expansion. 4 3 16 x + 8x + 4 + + 4 x x

Question 6 (***+) Find the term which is independent of x, in the binomial expansion of 4x x, x 0. 3 1 8 7 Question 7 (***+) ( ) 5 1+ a + b. Determine the value of each of the constants a and b. a = 41, b = 9

Question 8 (***+) Find the binomial expansion of 5 1 x x, x 0, simplifying each term of the expansion. 5 3 10 5 1 x 5x + 10x + x 3 5 x x Question 9 (***+) In the binomial expansion of 6 x k, where k is a positive constant, one of the terms is a) Find the value of k. 960x. b) Determine the coefficient of 3 x. k = 4, 160

Question 30 (***+) Given that k is a non zero constant and n is a positive integer, then ( kx) n x k x 1+ 1+ 40 + 10 +... Find the value of k and the value of n. CI, k = 5, n = 16 Question 31 (***+) Given that k and A are constants with k > 0, then ( ) 8 kx Ax x 56 + + 1008 +... Find the value of k and the value of A. k = 3, A = 768 4

Question 3 (***+) Find the coefficient of ( ) ( 3 ) ( 1 4 ) 7 f x = x + x. x in the polynomial expansion of f ( x ). 1017 Question 33 (***+) ( ) 5 5 4 3 + x = x + 10x + ax + bx + cx + 3. a) Find the value of each of the constants a, b and c. b) Hence, or otherwise, simplify ( ) 5, giving the final answer in the form p + q, where p and q are constants. a = 40, b = 80, c = 80, 3 164

Question 34 (***+) ( ) 1 ( ) 8 f x = + x. 4 a) Find the first four terms in ascending powers of x in the expansion of f ( x ). b) Use the expansion found in part (a) to find an approximation, to 3 significant figures, for 81 ( ) 8 40. ( ) 3 f x 56 56x 11x 8 x... = + + + +, ( ) 8 81 40 83 Question 35 (***+) a) Find the binomial expansion of ( 5 + 10x) 4. b) Hence, by using the answer of part (a) with a suitable value of x find the exact 4 value of 1005. You may not give the answer in standard index form. 3 4 65 + 5000x + 15000x + 0000x + 10000x, 1,00,150,500,65

Question 36 (***+) ( ) = ( 1 ) 6, g ( x) = ( + x) 7, h( x) f ( x) g ( x) f x x =. a) Find the first four terms in ascending powers of x in the binomial expansion of g x. f ( x ) and the binomial expansion of ( ) b) Hence determine the coefficient of x in the binomial expansion of h( x ). 3 3 ( ) = 1 1 + 60 160 +..., g ( x) x x x f x x x x = 18 + 448 + 67 + 560 +..., 976 Question 37 (***+) Find the binomial expansion of 6 x + x, x 0, simplifying each term of the expansion. 6 4 40 19 64 x + 1x + 60x + 160 + + + 4 6 x x x

Question 38 (***+) ( ) ( 3 ) ( 1 ) 6 f x = x + x. Find the binomial expansion of f ( x ) in ascending powers of x, up and including the term in 3 x. 3 9 + 96x + 400x + 768 x +... Question 39 (***+) It is given that ( )( ) 5 x kx A Bx x 1 + + + 40 +..., where k, A and B are constants. Determine the possible values of k CG, k = 1, 3

Question 40 (***+) a) Find the binomial expansion of 1 ( ) 10 including the term in 1+ x in ascending powers of x up and 4 3 x, simplifying fully each coefficient. b) Use the expansion of part (a) to show that ( ) 10 41 1.8. 40 CJ, 5 45 15 3 1 + x + x + x +... 16 8 Question 41 (***+) ( ) ( 1 3 ) 6 f x = + x. a) Find, in ascending powers of x, the binomial expansion of f ( x ) up and including the term in 4 x. b) Use the expansion found in part (a) to show that ( 1.003) 6 1.018135541. ( ) 3 4 f x = 1+ 18x + 135x + 540x + 115 x +...

Question 4 (***+) a) Find the binomial expansion of ( 3 + x) 4. b) State the binomial expansion of ( 3 x) 4. c) Use the answers of part (a) and (b) to find ( 3 8) ( 3 8 ) 4 4 + +. No credit will be given for any other type of evaluation. ( ) 4 4 3 3 x 16x 96x 16x 16x 81 ( ) 4 4 3 3 + x = 16x + 96x + 16x + 16x + 81, 4 4 = + +, ( ) ( ) 3+ 8 + 3 8 = 1154

Question 43 (***+) a) Find the first four terms, in ascending powers of x, of the binomial expansion of x 1+ 7, giving each coefficient in exact simplified form. b) Hence determine the coefficient of x in the expansion of 7 x 1+ 1+ x. CR, 7 1 35 3 1 x x x... 4 8 x = + + + +, [ ] 4

Question 44 (***+) a) Find the binomial expansion of ( 5x) 10 in ascending powers of x up and including the term in 3 x, simplifying fully each coefficient. b) Use the expansion of part (a) to show that ( 1.99) 10 974. 3 104 5600x + 88000x 190000 x +... Question 45 (****) n 3 ( ax) x x bx 1+ = 1 30 + 405 + +..., where a and b are constants, and n is a positive integer. Determine the value of n, a and b. CM, n = 10, a = 3, b = 340

Question 46 (****) ( ) ( 1 ) n f x + x, x R, n N. Determine showing a clear complete method the coefficient a) of the highest power of x in the binomial expansion of f ( x ), when n = 13. b) of the second highest power of x in the binomial expansion of f ( x ), when iii n = 14. CK, 1716, 3003 Question 47 (****) Find the coefficient of 5 x in the binomial expansion of ( 1 x) ( 1 x) 5 6 +. 5 = x 10

Question 48 (****) 1 f ( x) = 4x + kx, 7 where k is a positive constant. Given the coefficient of value of k. 3 x in the binomial expansion of f ( ) x is 1, determine the CY, k = 3

Question 49 (****) ( ) ( 1 ) 5 f x = + x, x R. a) Find the binomial expansion of f ( x ).. b) Hence state the binomial expansion of f ( x) c) Find the two non zero solutions of the equation f ( x) f ( x) = 64x. 3 4 5 CZ, f ( x) 1 10x 40x 80x 80x 3x = + + + + +, 3 4 5 ( ) 1 10 40 80 80 3 f x = x + x x + x x x = ± 1

Question 50 (****) In the binomial expansion of ( 1 ax) coefficient of x is 8 and the coefficient of n +, where a and k are non zero constants, the x is 30. a) Determine the value of a and the value of k. b) Find the coefficient of 3 x. a = 1, k = 16, 70

Question 51 (****) f ( x) ( k + x) n, x R, where k and n are constants such that k R, k 0, n N, n > 3. a) Given the coefficients of equal, show clearly that x and 3 x in the binomial expansion of f ( ) x are n = 3k +. b) Given further that k =, determine the coefficient of f x. expansion of ( ) 4 x in the binomial CO, 4 = x 110

Question 5 (****) a) Find the binomial expansion of ( 3 + x) 4, simplifying fully each coefficient. b) Hence solve the equation ( x) ( x) 4 4 3 + + 3 = 386. 3 4 81+ 108x + 54x + 1x + x, x = ±

Question 53 (****) a) Find the binomial expansion of ( x 4) 5, simplifying fully each coefficient. b) Hence find the coefficient of i. y in the binomial expansion of ii. 8 5 y + 16 4. 5 5 z in the binomial expansion of ( z ) ( z ) +. CU, 5 4 3 3x 30x + 180x 560x + 560x 104, = y 40, 8 = z 30 Question 54 (****) Find the coefficient of 11 x in the binomial expansion of 9 x x 3 13. CV, 11 = x 9664

Question 55 (****) If k > 0 and n is a positive integer, then n 3 ( kx) Ax Bx Bx 1+ 1 + + + +..., where A and B are non zero constants. a) By considering the coefficients of x and nk = k + 3. 3 x, show that b) Given that A = 7, find the value of n and the value of k. n = 14, k = 1 4

Question 56 (****) ( )( ) 7 ax bx x x + 1+ = 41 + 357 +..., where a and b are integers. Show that b = 3 and find the value of a. CQ, a = 1 Question 57 (****+) Find the coefficient of x in the binomial expansion of ( x 3 x 1 ) 7 + +. SP-B, x = 03

Question 58 (****+) a) Find the binomial expansion of ( + x) 5. b) Hence solve the equation ( x) ( x) 5 5 + + = 105.5. ( ) 5 3 4 5 x 3 80x 80x 40x 10x x + = + + + + +, x = ± 1 Question 59 (****+) ( ) 6 3 1+ x x = 1 + Ax + Bx + Cx +... Determine the value of each of the constants A, B and C. CX, A = 6, B = 1, C = 10

Question 60 (****+) It is given that if k is a constant then ( k ) 4 1+ 3 89 336 3. Determine the value of k. k = 3

Question 61 (****+) If A, k and n are constants, with n N, then 3 ( kx) n Ax x x 1+ = 1+ + 64 + 1760 +... a) Show that ( n ) k = 0. b) Determine the value of A. MP1-P, A = 4

Question 6 (*****) The binomial coefficients are given by n n! = k k! ( n k )! N., k N, n { 0} Show directly from the above definition that n n 1 n 1 = +. k k 1 k CS, proof

Question 63 (*****) It is given that where n is a positive integer constant. a) Evaluate f ( 1). n n r, r r= 0 n r ( ) = ( 1+ + ) f x x x x b) Find the value of n that satisfies the equation 178 f ( 3) f ( ) = 178. CT, f ( 1) = 0, n = 59

Question 64 (*****) The coefficients of n x, arithmetic progression. n 1 x + and n x + in the binomial expansion of ( ) 3 1+ x are in Determine the possible values of n. SP-S, n = 8, 13

Question 65 (*****) It is given that n n n n n + + +... + + = k 0 1 n 1 n where n and k are positive integer constants. a) By considering the binomial expansion of ( 1 x) n n, +, determine the value of k. b) By considering the coefficient of n x in ( 1 x) ( 1 x) ( 1 x) n n n + + +, simplify fully n n n n n + + +... + + 0 1 n 1 n. SP-X, k =

Question 66 (*****) It is given that n ( a bx) x x + = 819 + 6656 + 496 +..., where a, b and n are non zero constants. Use algebra to determine the values of a, b and n. No credit will be given to solutions by inspection and/or verification MP1-T, [ ] 1 a, b, n =,,13 8

Question 67 (*****) It is given that n ( a bx) x x + = 51 + 576 + 88 +..., where a, b and n are non zero constants. Use algebra to determine the values of a, b and n. No credit will be given to solutions by inspection and/or verification SPX-F, [ ] 1 a, b, n =,,9 4

Question 66 (*****) Find the term independent of x in the expansion of x x + 1 x 1 x + 1 x x 1 1 3 3 10. SP-U, 10

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