A2 Mathematics Assignment 7 Due Date: Friday 28 th February 2014

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A Mathematics Assignment 7 Due Date: Friday 8 th February 0 NAME. GROUP: MECHANICS/STATS Instructions to Students All questions must be attempted.

1 A Mathematics Assignment 7 Due Date: Friday 8 th February 0 NAME. GROUP: MECHANICS/STATS Instructions to Students All questions must be attempted. You should present your solutions on file paper and submit them with this cover sheet. (Work submitted without a cover sheet complete with name will not be marked). All work is to be submitted either to your module teacher or the Faculty Office by.5pm on the due date above. SKILLS AUDIT PLEASE COMPLETE BELOW, YOUR TEACHER WILL CONFIRM THE SKILLS YOU HAVE MET IN THIS ASSIGNMENT SKILLS AUDIT YES NO NEEDS WORK I can convert an expression with linear factors in the denominator into partial fractions I can convert an expression with repeated linear factors in the denominator into partial fractions I can covert an improper fraction (into partial fraction form I can expand for any constant n I can expand for any constant a, b and n I can determine the range of values for which the expansion is valid I can use partial fractions to expand more complex fractional expressions REFERENCES Core Maths Edexcel textbook Partial fractions with linear denominators: page - Partial fractions with repeated linear denominators: page 6 Improper fractions into partial fractions: page 7 Using the binomial expansion with positive, negative or fractional powers: pages -8 Using the binomial expansion to expand : pages 9 0 Using partial fractions with the binomial expansion: page Students Feedback: Please comment on how you performed on this assignment and state the questions that you found difficult. TURN OVER FOR COMMENTS ABOUT YOUR WORK Haringey Sixth Form Centre Mathematics Department

2 TEACHER COMMENTS Strengths Areas to develop Haringey Sixth Form Centre Mathematics Department

3 . The function f is given by ( x ) f(x) =, x, x, x. ( x )( x ) (a) Express f(x) in partial fractions. Hence, or otherwise, prove that f(x) < 0 for all values of x in the domain. () () (Total 6 marks). x ( x) ( x) f x. Given that, for x, x ( x) A B ( x) ( x), where A and B are constants, (a) find the values of A and 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) (Total 9 marks). Given that 5x ( x)( x) A B, x x (a) find the values of the constants A and B. () Hence, or otherwise, find the series expansion in ascending powers of x, up to and including the term in x, of 5x. ( x)( x) (5) (c) State, with a reason, whether your series expansion in part is valid for x =. () (Total 0 marks) Haringey Sixth Form Centre Mathematics Department

4 . x 5x 0 B C A x x x x (a) Find the values of the constants A, B and C. x 5x 0 Hence, or otherwise, expand in ascending powers of x, as far as the term x x in x. Give each coefficient as a simplified fraction. (7) (Total marks) 5. 7x x 6 f ( x), x (x ) ( x) Given that f(x) can be expressed in the form A B C f ( x), (x ) (x ) ( x) (a) find the values of B and C and show that A = 0. 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 to estimate the value of f (0.). Give your answer to significant figures. (Total marks) 6. 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. (Total 6 marks) Haringey Sixth Form Centre Mathematics Department

5 7. (a) Find the first four terms in the expansion, in ascending powers of x, of x x, giving each term in its simplest form. Hence write down the first four terms in the expansion, in ascending powers of x, of 0000x, x. () (Total 6 marks) 8. The binomial expansion of ( x) in ascending powers of x up to and including the term in x is + 9x + px + qx, x<. (a) Find the value of p and the value of q. Use this expansion with your values of p and q together with an appropriate value of x to obtain an estimate of (.6). () (c) Obtain (.6) from your calculator and hence make a comment on the accuracy of the estimate you obtained in part. () (Total 8 marks) 9. When ( + ax) n is expanded as a series in ascending powers of x, the coefficients of x and x are 6 and 7 respectively. (a) Find the value of a and the value of n. (5) Find the coefficient of x. () (c) State the set of values of x for which the expansion is valid. () (Total 8 marks) Haringey Sixth Form Centre Mathematics Department 5

6 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. Show that, when x =, the exact value of ( 8x) is () (c) Substitute x into the binomial expansion in part (a) and hence obtain an 00 approximation to. Give your answer to 5 decimal places. () (Total 9 marks). (a) Expand, where x, in ascending powers of x up to and including the term ( x) in x. Simplify each term. (5) x 8 Hence, or otherwise, find the first terms in the expansion of as a series in ( x) ascending powers of x. (Total 9 marks). f(x) = ( + x), < x <. ( x) (a) Find the series expansion of f(x), in ascending powers of x, up to and including the term in x. (6) Hence, or otherwise, prove that the function f has a minimum at the origin. (Total 0 marks) Haringey Sixth Form Centre Mathematics Department 6

AS Mathematics Assignment 8 Due Date: Friday 15 th February 2013

AS Mathematics Assignment 8 Due Date: Friday 15 th February 2013 NAME GROUP: MECHANICS/STATS Instructions to Students All questions must be attempted. You should present your solutions on file paper and