BINOMIAL EXPANSIONS EXAM QUESTIONS

Transcription

1 BINOMIAL EXPANSIONS EXAM QUESTIONS

2 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 x... Question (**) Expand ( 3 x) 5 in ascending powers of x, up and including the term in 3 x. CA, x x 70 x +... Question 3 (**) Find the binomial expansion of ( 1 5x) 4 in ascending powers of x. CB, x + 150x 500x + 65x

3 Question 4 (**) Find, without any calculating aid, the first three terms in the expansion of ( 7x) 6 ascending powers of x., in x x +... Question 5 (**) Find the binomial expansion of ( 1 x) 6 in ascending powers of x x + 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 ( ) x. CH, x + 5x x +..., 041

4 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, Question 9 (**+) Find, without using a calculator, the coefficient of 4 x in the expansion of 1 ( ) 7 4x. 110

5 Question 10 (**+) Find, without using a calculator, the coefficient of 5 x in the expansion of ( ) 7 x 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

6 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 = x x x +..., CN, ( ) ( ) 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, x + 64x x x +..., , error

7 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. ( ) x 3 80x 80x 40x 10x x ( ) 5 + = , x = 3 80x + 80x 40x + 10x x

8 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) MP1-N, x + 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) x + 48x + 96x + 64, x + 88x + 144x + 3x 64

9 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 x x x 3 = , , error 0.4

10 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 x + 84x + 80 x +..., x + 56x + 11 x +...

11 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) x + 11x 448 x +..., 3 9x + 176x 560 x +...

12 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 + = , 3 x = 460

13 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) ( ) f x = + x + x + x + x, x + 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

14 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 x + 8x x x

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

16 Question 8 (***+) Find the binomial expansion of 5 1 x x, x 0, simplifying each term of the expansion 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

17 Question 30 (***+) Given that k is a non zero constant and n is a positive integer, then ( kx) n x k x 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 Find the value of k and the value of A. k = 3, A = 768 4

18 Question 3 (***+) Find the coefficient of ( ) ( 3 ) ( 1 4 ) 7 f x = x + x. x in the polynomial expansion of f ( x ) Question 33 (***+) ( ) 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

19 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 ( ) ( ) 3 f x 56 56x 11x 8 x... = , ( ) Question 35 (***+) a) Find the binomial expansion of ( x) 4. b) Hence, by using the answer of part (a) with a suitable value of x find the exact 4 value of You may not give the answer in standard index form x x x x, 1,00,150,500,65

20 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 ( ) = , g ( x) x x x f x x x x = , 976 Question 37 (***+) Find the binomial expansion of 6 x + x, x 0, simplifying each term of the expansion x + 1x + 60x x x x

21 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 x + 400x x +... Question 39 (***+) It is given that ( )( ) 5 x kx A Bx x , where k, A and B are constants. Determine the possible values of k CG, k = 1, 3

22 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 ( ) CJ, x + x + x 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) ( ) 3 4 f x = 1+ 18x + 135x + 540x x +...

23 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 ) No credit will be given for any other type of evaluation. ( ) x 16x 96x 16x 16x 81 ( ) x = 16x + 96x + 16x + 16x + 81, 4 4 = + +, ( ) ( ) = 1154

24 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 x. CR, x x x x = , [ ] 4

25 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) x x x +... Question 45 (****) n 3 ( ax) x x bx 1+ = , 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

26 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) = x 10

27 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

28 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 CZ, f ( x) 1 10x 40x 80x 80x 3x = , ( ) f x = x + x x + x x x = ± 1

29 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

30 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

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

32 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 z in the binomial expansion of ( z ) ( z ) +. CU, x 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 CV, 11 = x 9664

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

34 Question 56 (****) ( )( ) 7 ax bx x x + 1+ = , 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 ) SP-B, x = 03

35 Question 58 (****+) a) Find the binomial expansion of ( + x) 5. b) Hence solve the equation ( x) ( x) = ( ) x 3 80x 80x 40x 10x x + = , x = ± 1 Question 59 (****+) ( ) 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

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

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

38 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

39 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

40 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

41 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 n 1 n. SP-X, k =

42 Question 66 (*****) It is given that n ( a bx) x x + = , 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

43 Question 67 (*****) It is given that n ( a bx) x x + = , 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

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