C2 Sequences & Series: Binomial Expansion …

1 C2 Sequences & series : Binomial Expansion Edexcel Internal Review 1 1. (a) Find the first 4 terms, in ascending powers of x, of the Binomial Expansion of (1 + ax)7, where a is a constant. Give each term in its simplest form. (4) Given that the coefficient of x2 in this Expansion is 525, (b) find the possible values of a. (2) (Total 6 marks) 2. Find the first 3 terms, in ascending powers of x, of the Binomial Expansion of (3 x)6 and simplify each term. (Total 4 marks) 3. (a) Find the first 3 terms, in ascending powers of x, of the Binomial Expansion of (2 + kx)7 where k is a constant. Give each term in its simplest form. (4) Given that the coefficient of x2 is 6 times the coefficient of x, (b) find the value of k. (2) (Total 6 marks) 4. Find the first 3 terms, in ascending powers of x, of the Binomial Expansion of ()523x , giving each term in its simplest form.

2 (Total 4 marks) C2 Sequences & series : Binomial Expansion Edexcel Internal Review 2 5. (a) Find the first 4 terms, in ascending powers of x, of the Binomial Expansion of (1 + ax)10, where a is a non-zero constant. Give each term in its simplest form. (4) Given that, in this Expansion , the coefficient of x3 is double the coefficient of x3, (b) find the value of a. (2) (Total 6 marks) 6. (a) Find the first 4 terms of the Expansion of 1021 +x in ascending powers of x, giving each term in its simplest form. (4) (b) Use your Expansion to estimate the value of ( )10, giving your answer to 5 decimal places. (3) (Total 7 marks) 7. (a) Find the first four terms, in ascending powers of x, in the Binomial Expansion of (1 + kx)6,where k is a non-zero constant. (3) Given that, in this Expansion , the coefficients of x and x2 are equal, find (b) the value of k, (2) (c) the coefficient of x3.

3 (1) (Total 6 marks) C2 Sequences & series : Binomial Expansion Edexcel Internal Review 3 8. (a) Find the first 4 terms, in ascending powers of x, of the Binomial Expansion of (1 2x)5. Give each term in its simplest form. (4) (b) If x is small, so that x2 and higher powers can be ignored, show that (1 + x)(1 2x)5 1 9x. (2) (Total 6 marks) 9. Find the first 3 terms, in ascending powers of x, of the Binomial Expansion of (2 + x)6, giving each term in its simplest form. (Total 4 marks) 10. (a) Write down the Binomial Expansion , in ascending powers of x, of (1 + 6x)4, giving each coefficient as an integer. (3) (b) Use your Binomial Expansion to find the exact value of 6014. (2) (Total 5 marks) 11. (a) Find the first 3 terms, in ascending powers of x, of the Binomial Expansion of (1 + px)9, where p is a constant.

4 (2) These first 3 terms are 1, 36x and qx2, where q is a constant. (b) Find the value of p and the value of q. (4) (Total 6 marks) C2 Sequences & series : Binomial Expansion Edexcel Internal Review 4 12. (a) Write down the first three terms, in ascending powers of x, of the Binomial Expansion of (1 + px)12, where p is a non-zero constant. (2) Given that, in the Expansion of (1 + px)12, the coefficient of x is ( q) and the coefficient of x2 is 11q, (b) find the value of p and the value of q. (4) (Total 6 marks) 13. In the Binomial Expansion , in ascending powers of x, of (1 + ax)n, where a and n are constants, the coefficient of x is 15. The coefficient of x2 and of x3 are equal. (a) Find the value of a and the value of n. (6) (b) Find the coefficient of x3. (1) (Total 7 marks) 14. Find the first three terms, in ascending powers of x, of the Binomial Expansion of (3 + 2x)5, giving each term in its simplest form.

5 (Total 4 marks) 15. (a) Find the first four terms, in ascending powers of x, in the Binomial Expansion of 52 +xk, where k is a constant. (2) C2 Sequences & series : Binomial Expansion Edexcel Internal Review 5 Given that the third term of this series is 540x2, (b) show that k = 6, (2) (c) find the coefficient of x3. (2) (Total 6 marks) 16. For the Binomial Expansion , in descending powers of x, of 12321 xx, (a) find the first 4 terms, simplifying each term. (5) (b) Find, in its simplest form, the term independent of x in this Expansion . (3) (Total 8 marks) 17. (a) Write down the first 4 terms of the Binomial Expansion , in ascending powers of x, of (1 + ax)n, n > 2. (2) Given that, in this Expansion , the coefficient of x is 8 and the coefficient of x2 is 30, (b) calculate the value of n and the value of a, (4) (c) find the coefficient of x3.

6 (2) (Total 8 marks) C2 Sequences & series : Binomial Expansion Edexcel Internal Review 6 18. The Expansion of (2 px)6 in ascending powers of x, as far as the term in x2, is 64 + Ax + 135x2. Given that p > 0, find the value of p and the value of A. (Total 7 marks) (c) find the coefficient of x3. (2) (Total 8 marks) 19. The first three terms in the Expansion , in ascending powers of x, of (1 + px)n, are 1 18x + 36p2x2. Given that n is a positive integer, find the value of n and the value of p. (Total 7 marks) 20. (a) Expand (2 + 41x)9 in ascending powers of x as far as the term in x3, simplifying each term. (4) (b) Use your series , together with a suitable value of x, to calculate an estimate of ( )9. (2) (Total 6 marks) C2 Sequences & series : Binomial Expansion Edexcel Internal Review 7 21.

7 The first four terms, in ascending powers of x, of the Binomial Expansion of (1 + kx)n are 1 + Ax + Bx2 + Bx3 + .., where k is a positive constant and A, B and n are positive integers. (a) By considering the coefficients of x2 and x3, show that 3 = (n 2) k. (4) Given that A = 4, (b) find the value of n and the value of k. (4) (Total 8 marks) C2 Sequences & series : Binomial Expansion Edexcel Internal Review 8 21. (a) )..(71or ..71)1(7axaxax++=+ (Not unsimplified versions) B1 32)(6567)(267axax + + Evidence from one of these terms is enough M1 )(21or )(21or2122222xaaxxa+++ A1 )(35or)(35or3533333xaaxxa+++ A1 4 Note The terms can be listed rather than added. M1: Requires correct structure: a correct Binomial coefficient in any form (perhaps from Pascal s triangle) with the correct power of x Allow missing a s and wrong powers of a, 2267ax , 323567x However, 21 + a2x2 + 35 + a3x3 or similar is M0.

8 1+ 7ax + 21+ a2x2 + 35 + a3x3 = 57 + .. scores the B1 (isw). 27 and 37 or equivalent such as 7C2 and 7C3 are acceptable, but not 37or 27 (unless subsequently corrected). 1st A1: Correct x2 term. 2nd A1: Correct x3 term (The Binomial coefficients must be simplified). Special case: If (ax)2 and (ax)3 are seen within the working, but then .. A1 A0 can be given if 21ax2 and 35ax3 are both achieved. a s omitted throughout: Note that only the M mark is available in this case. (b) 525212=a M1 5 =a (Both values are required) A1 (The answer a = 5 with no working scores M1 A0) 2 Note M: Equating their coefficent of x2 to 525. An equation in a or a2 alone is required for this M mark, but allow recovery that shows the required coefficient, 21a2x2 = 525 21a2 = 525 is acceptable, but 21a2x2 = 525 a2 = 25 is not acceptable.

9 After 21ax2 in the answer for (a), allow recovery of a2 in (b) so that full marks are available for (b) (but not retrospectively for (a)). C2 Sequences & series : Binomial Expansion Edexcel Internal Review 9 [6] 2. ( )[]( )( )24566 263 633 3xxx + += M1 = 729, 1458x, +1215x2 B1 A1 A1 Note M1 for either the x term or the x2 term. Requires correct Binomial coefficient in any form with the correct power of x condone lack of negative sign and wrong power of 3. This mark may be given if no working is shown, but one of the terms including x is correct. Allow 26or,16 (must have a power of 3, even if only power 1) First term must be 729 for B1, (writing just 36 is B0) can isw if numbers added to this constant later. Can allow 729( Term must be simplified to 1458x for A1cao. The x is required for this mark.)

10 Final A1is and needs to be +1215x2 (can follow omission of negative sign in working) Descending powers of x would be ( )( ).. 463 634256+ + +xxx x6 18x5+135x4+.. This is M1B1A0A0 if completely correct or M1 B0A0A0 for correct Binomial coefficient in any form with the correct power of x as before Alternative NB Alternative method: (3 x)6 = 36(1 + 6 ( 3x) + 26 ( 3x)2+ ..) is M1B0A0A0 answers must be simplified to 729, 1458x, +1215x2 for full marks (awarded as before) The mistake ( )( )( )( )..) 26 61(3 13 3233636+ + +==xxxx may also be awarded M1B0A0A0 Another mistake 36(1 6x+ )= would be M1B1A0A0 [4] C2 Sequences & series : Binomial Expansion Edexcel Internal Review 10 3. (a) (7 .. x) or (21 .. x2) The 7 or 21 can be in unsimplified form. M1 (2 + kx)7 = 27 + 26 7 kx + 25 2227xk = 128; + 448kx, +672k2x2 [or 672(kx)2] B1; A1, A1 4 (If 672kx2 follows 672(kx)2, isw and allow A1) Note The terms can be listed rather than added.