Edexcel past paper questions. Core Mathematics 4. Binomial Expansions

Transcription

1 Edexcel past paper questions Core Mathematics 4 Binomial Expansions Edited by: K V Kumaran kvkumaran@gmail.com C4 Binomial Page

2 Binomial Series C4 By the end of this unit you should be able to obtain the expansion of (ax+b) n where n can be negative and/or fractional. You will also be expected to apply the idea to functions expressed as partial fractions. Example Expand ( 6x) in ascending powers of x up to and including the term in x. State the set of values for which the expansion is valid. The following expansion was used in C to write binomial expansions; unfortunately we have a negative fractional power. n (a b) a a b n a b... n a b...b r n n n n nr r n Therefore we have to use the following one instead. Therefore: n(n ) n(n )(n )!! n ( x) nx x x... = 5 ( 6x) 9x -6x! 5 9x x For the expansion to be valid the modulus of the ax term in the bracket (+ax) n must be less than one. The reason for this is that if the higher powered terms are going to be ignored then the terms (-6x) r must tend to zero very quickly. Therefore: 6x x 6 C4 Binomial Page

3 Example Expand (8 4x) in ascending powers of x up to and including the term in x. Give each coefficient as a simplified fraction. We have to start by factorising the 8 out. x (8 4x) 8 Don't forget to multiply by the 8 at the end. 8 4 x x x 6 x x 6 8 Therefore (8 4x) x x 6 The final type to be considered uses a binomial expansion in a subsequent multiplication. C4 Binomial Page

4 Example a) Expand ( x) 5 x < in ascending powers of x up to and including the term in x simplifying each term. (x 4) b) Hence or otherwise find the first three terms in the expansion of 5 ( x) series in ascending powers of x. as a a) Using: n(n ) n(n )(n )!! n ( x) nx x x... ( x) x! 5 6 7! x x 5x 5x 945x b) Using the expansion in part (a) (x 4) ( x) 5 (x 4)( x) 5 (x 4)( 5x 90x 70x ) x 4 5x 5x 945x We are only asked for the first three terms therefore: x 4 5x 5x 945x 4 59x 55x C4 Binomial Page 4

5 Example 4 Expand ( 6x) in ascending powers of x up to and including the term in x. State the set of values for which the expansion is valid. ( x) = ( 6x) = [ + ( 6x) +! 5 ( 6x) ] = + 8x + 5x For the expansion to be valid the modulus of the ax term in the bracket (+ax) n must be less than one. The reason for this is that if the higher powered terms are going to be ignored then the terms (-6x) r must tend to zero very quickly. Therefore: 6x < x < 6 Partial Fractions can be used to give approximations of functions that can be split up into their quotients. Example 5 Expand (6x+) (+x)(5x+) We can split this up using partial fractions into: + x + 5x + Now expand ( + x) - and (5x + ) - as described above and add the expansions together. C4 Binomial Page 5

6 . Use the binomial theorem to expand C4 Binomial past paper questions (4 9x) x < 9 4 in ascending powers of x up to and including the term in x simplifying each term. (5) (C4 June 005 Q) x 6. f(x) = ( x)( x) = A ( x) + B ( x) C ( x) + x <. (a) Find the values of A and C and show that B = 0. (4) (b) Hence or otherwise find the series expansion of f(x) in ascending powers of x up to and including the term in x. Simplify each term. (7) (C4 Jan 006 Q5) x. f(x) = ( x) x <. x Given that for x ( x) = A ( x) B ( x) + where A and B are constants (a) find the values of A and B. () (b) Hence or otherwise find the series expansion of f(x) in ascending powers of x up to and including the term in x simplifying each term. (6) (C4 June 006 Q) 4. f(x) = ( 5x) x < 5. Find the binomial expansion of f(x) in ascending powers of x as far as the term in x giving each coefficient as a simplified fraction. (5) (C4 Jan 007 Q) 5. f(x) = ( + x) x <. Find the binomial expansion of f(x) in ascending powers of x as far as the term in x. Give each coefficient as a simplified fraction. (5) (C4 June 007 Q) C4 Binomial Page 6

7 6. (a) Use the binomial theorem to expand ( 8 x) x < 8 in ascending powers of x up to and including the term in x giving each term as a simplified fraction. (5) (b) Use your expansion with a suitable value of x to obtain an approximation to (7.7). Give your answer to 7 decimal places. () (C4 Jan 008 Q) 7. (a) Expand where x < 4 in ascending powers of x up to and including the term in x. (4 x) Simplify each term. (5) (b) Hence or otherwise find the first terms in the expansion of powers of x. x 8 (4 x) as a series in ascending (4) (C4 June 008 Q5) 8. f(x) = 7x x 6 x <. (x ) ( x) Given that f(x) can be expressed in the form f(x) = A ( x ) B ( x ) + + C ( x) (a) find the values of B and C and show that A = 0. (4) (b) Hence or otherwise find the series expansion of f(x) in ascending powers of x up to and including the term in x. Simplify each term. (6) (c) Find the percentage error made in using the series expansion in part (b) to estimate the value of f(0.). Give your answer to significant figures. (4) (C4 Jan 009 Q) 9. f(x) = x < 4. (4 x) Find the binomial expansion of f (x) in ascending powers of x up to and including the term in x. Give each coefficient as a simplified fraction. (6) (C4 June 009 Q) C4 Binomial Page 7

8 0. (a) Find the binomial expansion of ( 8x) x < 8 in ascending powers of x up to and including the term in x simplifying each term. (6) (b) Show that when x = the exact value of ( 8x) is. () 00 5 (c) Substitute x = into the binomial expansion in part (a) and hence obtain an approximation to 00. Give your answer to 5 decimal places. () (C4 Jan 00 Q). x 5x 0 ( x )( x ) A + B C +. x x (a) Find the values of the constants A B and C. (4) x 5x 0 (b) Hence or otherwise expand in ascending powers of x as far as the term in x. Give ( x )( x ) each coefficient as a simplified fraction. (7) (C4 June 00 Q5). (a) Use the binomial theorem to expand ( x) x < in ascending powers of x up to and including the term in x. Give each coefficient as a simplified fraction. (5) a bx f(x) = ( x) x < where a and b are constants. In the binomial expansion of f(x) in ascending powers of x the coefficient of x is 0 and the coefficient of x is 6 9. (b) the value of a and the value of b (5) (c) the coefficient of x giving your answer as a simplified fraction. () (C4 Jan 0 Q5) C4 Binomial Page 8

9 . f (x) = (9 4x x <. ) Find the first three non-zero terms of the binomial expansion of f(x) in ascending powers of x. Give each coefficient as a simplified fraction. (6) 4. (a) Expand (C4 June 0 Q) ( 5x) x < 5 in ascending powers of x up to and including the term in x giving each term as a simplified fraction. (5) kx Given that the binomial expansion of ( 5x) 7 + x 4 x < 5 is + Ax +... (b) find the value of the constant k () (c) find the value of the constant A. () (C4 Jan 0 Q) 5. f(x) = 6 9 x <. (9 4x) 4 (a) Find the binomial expansion of f(x) in ascending powers of x up to and including the term in x. Give each coefficient in its simplest form. (6) Use your answer to part (a) to find the binomial expansion in ascending powers of x up to and including the term in x of (b) g(x) = (c) h(x) = 6 (9 4x) 9 x < 4 () 6 (9 8x) 9 x <. 8 () (C4 June 0 Q) C4 Binomial Page 9

10 6. Given f(x) = ( + x) x < find the binomial expansion of f(x) in ascending powers of x up to and including the term in x. Give each coefficient as a simplified fraction. (5) (C4 Jan 0 Q) 7. (a) Use the binomial expansion to show that (b) Substitute x into 6 to obtain an approximation to. x x x x < (6) x x x x x Give your answer in the form a b where a and b are integers. () (C4 June 0 Q) 8. (a) Find the binomial expansion of (8 9 x) x < 8 9 in ascending powers of x up to and including the term in x. Give each coefficient as a simplified fraction. (6) (b) Use your expansion to estimate an approximate value for 700 giving your answer to 4 decimal places. State the value of x which you use in your expansion and show all your working. (C4 June 0_R Q4) C4 Binomial Page 0

11 9. Given that the binomial expansion of ( + kx) 4 kx < is 6x + Ax + (a) find the value of the constant k () (b) find the value of the constant A giving your answer in its simplest form. () (C4 June 04 Q) 0. (a) Find the binomial expansion of 9 0x 9 x 0 in ascending powers of x up to and including the term in x. Give each coefficient as a simplified fraction. (5) (b) Hence or otherwise find the expansion of x 9 0x 9 x 0 in ascending powers of x up to and including the term in x. Give each coefficient as a simplified fraction. () (C4 June 04_R Q). (a) Find the binomial expansion of 4 5 ) x < 5 ( 4 x in ascending powers of x up to and including the term in x. Give each coefficient in its simplest form. (5) (b) Find the exact value of ( 4 5x) when x =. 0 Give your answer in the form k where k is a constant to be determined. () (c) Substitute x = 0 into your binomial expansion from part (a) and hence find an approximate value for. Give your answer in the form q p where p and q are integers. () (C4 June 05 Q) C4 Binomial Page