2002 – 2011

Compiled & Edited By

Dr. Eltayeb Abdul Rhman

www.drtayeb.tk

First Edition

2011

ADDITONAL MATHEMATICS

CLASSIFIED BINOMIALS

3

0606/21/M/J/11

ForExaminer’s

Use

1 Without using a calculator, express (5 + 2 3 )2

2 + 3 in the form p + q 3 , where p and q are integers.

[4]

2 (i) Find the coefficient of x3 in the expansion of �1 – x2�

12. [2]

(ii) Find the coefficient of x3 in the expansion of (1 + 4x) �1 – x2�

12. [3]

4

2 The coefficient of x3 in the expansion of (2 + ax)5 is 10 times the coefficient of x2 in the

expansion of �1 + ax3 �

4 . Find the value of a. [4]

8

0606/12/O/N/11

ForExaminer’s

Use

6 (i) Given that (3 + x)5 + (3 – x)5 = A + Bx2 + Cx4, find the value of A, of B and of C. [4]

(ii) Hence, using the substitution y = x2, solve, for x, the equation

(3 + x)5 + (3 – x)5 = 1086. [4]

8 (i) In the binomial expansion of �x + �8, where k is a positive constant, the term independent of x

is 252.

Evaluate k. [4]

(ii) Using your value of k, find the coefficient of x4 in the expansion of �1 – � �x + �8. [3]

k––x3

x4––4

k––x3

4 (i) Find the first three terms, in ascending powers of u, in the expansion of (2 + u)5. [2]

(ii) By replacing u with 2x – 5x2, find the coefficient of x2 in the expansion of (2 + 2x – 5x2)5. [4]

5 Find the coefficient of x3 in the expansion of

(i) (1 + 3x)8, [2]

(ii) (1 – 4x)(1 + 3x)8. [3]

2 (i) Find the first 3 terms of the expansion, in ascending powers of x, of (1 + 3x)6. [2]

(ii) Hence find the coefficient of x2 in the expansion of (1 + 3x)6 (1 – 3x – 5x2). [3]

4

0606/23/O/N/11

3 (i) Find the coefficient of x3 in the expansion of (1 – 2x)7. [2]

(ii) Find the coefficient of x3 in the expansion of (1 + 3x2)(1 – 2x)7. [3]

7 Obtain

(i) the expansion, in ascending powers of x, of (2 – x2)5,

(ii) the coefficient of x6 in the expansion of (1 + x2)2(2 – x2)5.[6]

�9 (a) Calculate the term independent of x in the binomial expansion of . [3]

(b) In the binomial expansion of (1 ! kx)�n, where n ≥ 3 and k is a constant, the coefficients of x�2

and x�3 are equal. Express k in terms of n. [4]

�x −1

2x��5 �18

11 (a) (i) Expand (2 + x)5. [3]

(ii) Use your answer to part (i) to find the integers a and b for which (2 + )5 can be expressedin the form a + b . [3]

(b) Find the coefficient of x in the expansion of �x – �7. [3]4–x

33

5

0606/22/M/J/10© UCLES 2010

7 (i) Sketch the graph of y = ⏐3x + 9⏐ for –5 < x < 2, showing the coordinates of the points where the graph meets the axes. [3]

(ii) On the same diagram, sketch the graph of y = x + 6. [1]

(iii) Solve the equation ⏐3x + 9⏐= x + 6. [3]

8 (a) (i) Write down the first 4 terms, in ascending powers of x, of the expansion of (1 – 3x)7. [3]

(ii) Find the coefficient of x3 in the expansion of (5 + 2x)(1 – 3x)7. [2]

(b) Find the term which is independent of x in the expansion of �x2 + 2––x �9. [3]

7 The coefficient of x2 in the expansion of �1 + x5 �

n, where n is a positive integer, is 35 .

(i) Find the value of n. [4]

(ii) Using this value of n, find the term independent of x in the expansion of

�1 + x5 �

n �2 – 3

x � 2. [4]

4

0606/12/M/J/11

2 The coefficient of x3 in the expansion of (2 + ax)5 is 10 times the coefficient of x2 in the

expansion of �1 + ax3 �

4 . Find the value of a. [4]

5 (i) Expand (1 + x)5. [1]

(ii) Hence express (1 + 2 )5 in the form a + b 2 , where a and b are integers. [3]

(iii) Obtain the corresponding result for (1 – 2 )5 and hence evaluate (1 + 2 )5 + (1 – 2 )5. [2]

4 Find the coefficient of x4 in the expansion of

(i) (1 + 2x)6, [2]

(ii) �1 – x

4�(1 + 2x)6. [3]

4 (i) Find, in ascending powers of x, the first 4 terms of the expansion of (1 + x)6. [2]

(ii) Hence find the coefficient of p3 in the expansion of (1 + p – p2)6. [3]

3

0606/21/M/J/11

1 Without using a calculator, express (5 + 2 3 )2

2 + 3 in the form p + q 3 , where p and q are integers.

[4]

2 (i) Find the coefficient of x3 in the expansion of �1 – x2�12

. [2]

(ii) Find the coefficient of x3 in the expansion of (1 + 4x) �1 – x2�12

. [3]

2 Find the first three terms in the expansion, in ascending powers of x, of (2 + x)6 and hence obtain thecoefficient of x2 in the expansion of (2 + x – x2)6. [4]

5 The binomial expansion of (1 + px)n, where n > 0, in ascending powers of x is

1 – 12x + 28p2x2 + qx3 + ... .

Find the value of n, of p and of q. [6]

6 Given that the coefficient of x2 in the expansion of (k + x)(2 – x2)6 is 84, find the value of the

constant k. [6]

2002 – 2011

Compiled & Edited By

Dr. Eltayeb Abdul Rhman

www.drtayeb.tk

First Edition

2011

ADDITONAL MATHEMATICS

CLASSIFIED BINOMIALS