algebra paper 1 math ib

7/29/2019 Algebra Paper 1 Math Ib

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ALGEBRA

PAPER 1

120 min

1. Find the sum of the arithmetic series

17 + 27 + 37 +...+ 417.

Working:

Answer:

.........................................................................

(Total 4 marks)

2. Find the coefficient ofx5

in the expansion of (3x 2)8.

Working:

Answer:

......................................................................

(Total 4 marks)

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3. An arithmetic series has five terms. The first term is 2 and the last term is 32. Find the sum of

the series.

Working:

Answer:

......................................................................

(Total 4 marks)

4. Find the coefficient ofa3b4 in the expansion of (5a + b)7.

Working:

Answer:

......................................................................

(Total 4 marks)

5. Solve the equation 43x1

= 1.5625 102

.

Working:

Answer:

......................................................................

(Total 4 marks)


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6. If loga2 = x and loga 5 =y, find in terms ofx andy, expressions for

(a) log2 5;

(b) loga 20.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)

7. Find the sum of the infinite geometric series

...8116

278

94

32 ++

Working:

Answer:

......................................................................

(Total 4 marks)

8. Find the coefficient ofa5b

7in the expansion of (a + b)

12.

Working:

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Answer:

......................................................................

(Total 4 marks)

9. Let log10P=x , log10Q =y and log10R =z. Express

2

310log

QRP

in terms ofx ,y andz.

Working:

Answer:

....................................................................

(Total 4 marks)

10. Use the binomial theorem to complete this expansion.

(3x + 2y)4

= 81x4

+ 216x3y +...

Working:

Answer:

.......................................................................

(Total 4 marks)

11. Consider the binomial expansion

.3

4

2

4

1

41)1( 4324 xxxxx +

+

+

+=+

(a) By substitutingx = 1 into both sides, or otherwise, evaluate

.3

4

2

4

1

4

+

+

(b) Evaluate

+

+

+

+

+

+

+

89

79

69

59

49

39

29

19

.


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Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)

12. Consider the expansion of

.1

3

9

2

xx

(a) How many terms are there in this expansion?

(b) Find the constant term in this expansion.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 6 marks)

13. Consider the following statements

A: log10 (10x

) > 0.

B: 0.5 cos (0.5x) 0.5.

C: 2

arctanx 2

.

(a) Determine which statements are true for all real numbersx. Write your answers (yes or

no) in the table below.

Statement (a) Is the statement true for all

real numbersx? (Yes/No)

(b) If not true, example

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A

B

C

(b) If a statement is not true for allx, complete the last column by giving an example of one

value ofx for which the statement is false.

Working:

(Total 6 marks)

14. Find the coefficient ofx3

in the expansion of (2 x)5.

Working:

Answer:

......................................................................

(Total 6 marks)

15. Find the term containingx10

in the expansion of (5 + 2x2)7.

Working:


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Answer:

..................................................................

(Total 6 marks)

16. Given that log5x =y, express each of the following in terms ofy.

(a) log5x2

(b) log5

x

1

(c) log25x

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(c) ..................................................................

(Total 6 marks)

17. Complete the following expansion.

(2 + ax)4

= 16 + 32ax +

Working:

Answer:

..................................................................

(Total 6 marks)

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18. Letp = log10x, q = log10y and r= log10z.

Write the expression log10

zy

x2

in terms ofp, q and r.

Working:

Answer:

..(Total 6 marks)

19. Find the term containingx3

in the expansion of (2 3x)8.

Working:

Answer:

........(Total 6 marks)

20. Let a = logx, b = logy, and c = logz.

Write log

3

2

zyx

in terms ofa, b and c

Working:

Answer:


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........(Total 6 marks)

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21. Let Sn be the 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) ..................................................................

(Total 6 marks)

22. Given that ( )3

73 + =p + 7q wherep and q are integers, find

(a) p;

(b) q.

W o r k i n g :

A n s w e r s :

( a ) . . . . . . . . . . . . . . . . . . .

( b ) . . . . . . . . . . . . . . . . . . .

(Total 6 marks)


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23. (a) Let logc 3 =p and logc 5 = q. Find an expression in terms ofp and q for

(i) log c 15;

(ii) log c 25.

(b) Find the value ofdif log d6 = 21

.

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..............................................................................................................................................(Total 6 marks)

24. Let ln a =p, ln b = q. Write the following expressions in terms ofp and q.

(a) ln a3b

(b) ln

b

a

..............................................................................................................................................

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..............................................................................................................................................(Total 6 marks)

25. Given thatp = loga 5, q = loga 2, express the following in terms ofp and/orq.

(a) loga 10

(b) loga 8

(c) loga 2.5

..............................................................................................................................................

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..............................................................................................................................................(Total 6 marks)

26. Consider the arithmetic sequence 2, 5, 8, 11, .....

(a) Find u101.

(3)

(b) Find the value of n so that un = 152.

(3)

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..............................................................................................................................................(Total 6 marks)

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algebra paper 1 math ib

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