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1. Find the coefficient of a3b4 in the expansion of (5a + b)7.
Working:||
|Answer:......................................................................|
(Total 4 marks)


2. If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for
(a) log2 5;
(b) loga 20.
Working:||
|Answers:(a) ..................................................................(b)..................................................................|
(Total 4 marks)



3. Find the sum of the infinite geometric series

Working:||
|Answer:......................................................................|
(Total 4 marks)


4. Find the coefficient of a5b7 in the expansion of (a + b)12.
Working:||
|Answer:......................................................................|
(Total4 marks)



5. Let log10P = x , log10Q = y and log10R = z. Express in terms of x , y and z.
Working:||
|Answer:....................................................................|
(Total 4 marks)


6. Use the binomial theorem to complete this expansion.
(3x + 2y)4 = 81x4 + 216x3 y +...
Working:||
|Answer:.......................................................................|(Total 4 marks)



7. Consider the binomial expansion
(a) By substituting x = 1 into both sides, or otherwise, evaluate

(b) Evaluate .
Working:||
|Answers:(a) ..................................................................(b) ..................................................................|
(Total 4 marks)



8. Consider the expansion of
(a) How many terms are there in thisexpansion?
(b) Find the constant term in this expansion.
Working:||
|Answers:(a) ..................................................................(b) ..................................................................|
(Total 6 marks)


9. Consider the following statements
A: log10 (10x) > 0.
B: –0.5 £ cos (0.5x) £ 0.5.
C: – £ arctan x £ .
(a) Determine which statements are true forall real numbers x. Write your answers (yes or no) in the table below.
Statement|(a) Is the statement true for all real numbers x? (Yes/No)|(b) If not true, example|
A|||
B|||
C|||

(b) If a statement is not true for all x, complete the last column by giving an example of one value of x for which the statement is false.
Working:|
(Total 6 marks)


10. Find the coefficient of x3 inthe expansion of (2 – x)5.
Working:||
|Answer:......................................................................|
(Total 6 marks)



11. Find the term containing x10 in the expansion of (5 + 2x2)7.
Working:||
|Answer:..................................................................|
(Total 6 marks)



12. Given that log5 x = y, express each of the following in terms of y.
(a)log5 x2
(b) log5
(c) log25 x
Working:||
|Answers:(a) ..................................................................(b) ..................................................................(c) ..................................................................|
(Total 6 marks)



13. Complete the following expansion.
(2 + ax)4 = 16 + 32ax + …
Working:|||Answer:..................................................................|
(Total 6 marks)


14. Let p = log10 x, q = log10 y and r = log10 z.
Write the expression log10 in terms of p, q and r.
Working:||
|Answer:…………………………………………..|
(Total 6 marks)



15. Let a = log x, b = log y, and c = log z.
Write log in terms of a, b and c.
Working:||
|Answer:…………………………………………........|
(Total 6 marks)



16. Let Sn bethe sum of the first n terms of an arithmetic sequence, whose first three terms are u1, u2 and u3. It is known that S1 = 7, and S2 = 18.
(a) Write down u1.
(b) Calculate the common difference of the sequence.
(c) Calculate u4.
Working:||
|Answers:(a) ..................................................................(b) ..................................................................(c)...
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