Math IB Revision Algebra

1.

Find the sum of the arithmetic series 17 + 27 + 37 +...+ 417.

2.

 1  x–1 Solve the equation 9 =  3 

3.

Find the coefficient of x in the expansion of (3x – 2)

4.

The second term of an arithmetic sequence is 7. The sum of the first four terms of the arithmetic sequence is 12. Find the first term, a, and the common difference, d, of the sequence.

6.

Given that (1 + x) (1 + ax) ≡ 1 + bx + 10x + ............... + a x , find the values of a, b ∈

7.

6 13 The ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is . If each term of this sequence is positive, and the product of the first term and the third term is 32, find the sum of the first 100 terms of this sequence.

9.

Using mathematical induction, prove that the number 2 – 3n – 1 is divisible by 9, for n = 1, 2,..

10.

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

11.

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

12.

Solve the equation 4

13.

An arithmetic sequence has 5 and 13 as its first two terms respectively.

2x

.

5

5

6

8

2

6 11

*.

2n

3 4

3x–1

7

–2

= 1.5625 × 10 .

(a)

Write down, in terms of n, an expression for the nth term, an.

(b)

Find the number of terms of the sequence which are less than 400.

16.

In an arithmetic sequence, the first term is 5 and the fourth term is 40. Find the second term.

17.

If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for

18.

2 − 4 + 8 − 16 + ... Find the sum of the infinite geometric series 3 9 27 81

19.

Find the coefficient of a b in the expansion of (a + b) .

20.

The Acme insurance company sells two savings plans, Plan A and Plan B.

5 7

(a)

log2 5; (b)

loga 20.

12

For Plan A, an investor starts with an initial deposit of $1000 and increases this by $80 each month, so that in the second month, the deposit is $1080, the next month it is $1160 and so on. For Plan B, the investor again starts with $1000 and each month deposits 6% more than the previous month. (a)

Write down the amount of money invested under Plan B in the second and third months. 1

Give your answers to parts (b) and (c) correct to the nearest dollar. (b)

Find the amount of the 12th deposit for each Plan.

(c)

Find the total amount of money invested during the first 12 months (i)

under Plan A;

(ii)

under Plan B. 7

10

22.

Find the coefficient of x in the expansion of (2 + 3x) , giving your answer as a whole number.

23.

The sum of the first n terms of an arithmetic sequence is Sn = 3n – 2n. Find the nth term un.

24.

 2  The probability distribution of a discrete random variable X is given by P(X = x) = k  3  , for x = 0,1, 2, ......Find the value of k.

2

x

25.

Mr Blue, Mr Black, Mr Green, Mrs White, Mrs Yellow and Mrs Red sit around a circular table for a meeting. Mr Black and Mrs White must not sit together. Calculate the number of different ways these six people can sit at the table without Mr Black and Mrs White sitting together.

27.

$1000 is invested at the beginning of each year for 10 years. The rate of interest is fixed at 7.5% per annum. Interest is compounded annually. Calculate, giving your answers to the nearest dollar (a)

how much the first $1000 is worth at the end of the ten years;

(b)

the total value of the investments at the end of the ten years.

28.

Find the sum of the positive terms of the arithmetic sequence 85, 78, 71, ....

29.

 1  7 x + 2  ax   is 3 . Find the possible values of a. The coefficient of x in the expansion of

7

30.

The sum of an infinite geometric sequence is Find the first term.

13 12

, and the sum of the first three terms is 13.

32. In how many ways can six different coins be divided between two students so that each student receives at least one coin? 2

 P   log10  QR 3   33. Let log10P = x , log10Q = y and log10R = z. Express in terms of x , y and z. 34.

Each day a runner trains for a 10 km race. On the first day she runs 1000 m, and then increases the distance by 250 m on each subsequent day.

(a) (b)

On which day does she run a distance of 10 km in training? What is the total distance she will have run in training by the end of that day? Give your answer exactly. 9

35.

36.

2   x– 2  . x  Determine the constant term in the expansion of 

Find the sum to infinity of the geometric series

38.

(b)

– 12 + 8 –

16 + .... 3

(a)

x The function f is defined by f : x  e – 1 – x

(i)

Find the minimum value of f.

x

Prove that e ≥ 1 + x for all real values of x.

(ii)

1 1 1 (1 + 1) ( 1 + ) (1 + ) ....(1 + ) = n + 1 2 3 n Use the principle of mathematical induction to prove that for all integers n ≥ 1. (1 +

(c)

Use the results of parts (a) and (b) to prove that e

(d)

1 1 1 1 + + + ..... + 2 3 n > 100 Find a value of n for which

1 1 1 + + ....+ ) 2 3 n

4

> n.

4

3

39.

Use the binomial theorem to complete this expansion. (3x +2y) = 81x + 216x y +...

40.

The nth term, un, of a geometric sequence is given by un = 3(4)

44.

n+1

,n∈

+

.

(a)

Find the common ratio r.

(b)

Hence, or otherwise, find Sn, the sum of the first n terms of this sequence.

The first three terms of an arithmetic sequence are 7, 9.5, 12. st

(a)

What is the 41 term of the sequence?

(b)

What is the sum of the first 101 terms of the sequence?

45.

1 Solve the equation log9 81 + log9 9 + log9 3 = log9 x.

46.

 4  4  4 (1 + x) 4 =1 +   x +   x 2   x 3 + x 4 .  1  2  3 Consider the binomial expansion

(a)

 4  4  4   +   +   . 1 2 3 By substituting x = 1 into both sides, or otherwise, evaluate       3

(b) 47.

 9  9   9  9  9  9  9   9   +   +   +   +   +   +   +   1 2 3 4 5 6 7 8 Evaluate                 .

Portable telephones are first sold in the country Cellmania in 1990. During 1990, the number of units sold is 160. In 1991, the number of units sold is 240 and in 1992, the number of units sold is 360. In 1993 it was noticed that the annual sales formed a geometric sequence with first term 160, the 2nd and 3rd terms being 240 and 360 respectively. (a)

What is the common ratio of this sequence?

Assume that this trend in sales continues. (b)

How many units will be sold during 2002?

(c)

In what year does the number of units sold first exceed 5000?

Between 1990 and 1992, the total number of units sold is 760. (d)

What is the total number of units sold between 1990 and 2002?

During this period, the total population of Cellmania remains approximately 80 000. 49.

(e) Use this information to suggest a reason why the geometric growth in sales would not continue. Consider the infinite geometric series 2

3

 2x   2x   2x  1 +   +   +   + .....  3   3   3  (a)

For what values of x does the series converge?

(b)

Find the sum of the series if x = 1.2.

50.

How many four-digit numbers are there which contain at least one digit 3?

53.

In an arithmetic sequence, the first term is –2, the fourth term is 16, and the n term is 11 998.

th

(a)

Find the common difference d.

(b)

Find the value of n. 9

54.

 2 1  3x –  . x  (a) Consider the expansion of  (b)

How many terms are there in this expansion?

Find the constant term in this expansion.

55.

Solve the equation log27 x = 1 – log27 (x – 0.4).

57.

Ashley and Billie are swimmers training for a competition. (a)

Ashley trains for 12 hours in the first week. She decides to increase the amount of time she spends training by 2 hours each week. Find the total number of hours she spends training during the first 15

weeks. (b)

58.

Billie also trains for 12 hours in the first week. She decides to train for 10% longer each week than the previous week. (i)

Show that in the third week she trains for 14.52 hours.

(ii)

Find the total number of hours she spends training during the first 15 weeks.

(c) In which week will the time Billie spends training first exceed 50 hours? Consider the arithmetic series 2 + 5 + 8 +.... (a)

Find an expression for Sn, the sum of the first n terms.

(b)

Find the value of n for which Sn = 1365. 3

5

61.

Find the coefficient of x in the expansion of (2 – x) .

62.

The diagram shows a square ABCD of side 4 cm. The midpoints P, Q, R, S of the sides are joined to form a second square. Q

A

B

P

R

D (a)

(i)

S

C

Show that PQ = 2 2 cm.

(ii) Find the area of PQRS. The midpoints W, X, Y, Z of the sides of PQRS are now joined to form a third square as shown. A

B

Q X

W P

R Y

Z D (b)

(i)

S

C

Write down the area of the third square, WXYZ.

5

(ii)

Show that the areas of ABCD, PQRS, and WXYZ form a geometric sequence. Find the common ratio of this sequence.

The process of forming smaller and smaller squares (by joining the midpoints) is continued indefinitely. (c)

(i) (ii)

th

Find the area of the 11 square.

Calculate the sum of the areas of all the squares. 8

63.

 1  1 – x  3 Find the coefficient of x in the binomial expansion of  2  .

∑ ln( 2 ) 50

r

, giving the answer in the form a ln 2, where a ∈

64.

Find

65.

A sequence {un} is defined by u0 = 1, u1 = 2, un+1 = 3un – 2un–1 where n ∈ (a)

r =1

+

.

Find u2,u3,u4.

(b)

(i) (ii)

68.

.

Express un in terms of n.

Verify that your answer to part (b)(i) satisfies the equation un+1 = 3un – 2un – 1.

(a) At a building site the probability, P(A), that all materials arrive on time is 0.85. The probability, P(B), that the building will be completed on time is 0.60. The probability that the materials arrive on time and that the building is completed on time is 0.55.

(b)

(i)

Show that events A and B are not independent.

(ii)

All the materials arrive on time. Find the probability that the building will not be completed on time.

There was a team of ten people working on the building, including three electricians and two plumbers. The architect called a meeting with five of the team, and randomly selected people to attend. Calculate the probability that exactly two electricians and one plumber were called to the meeting.

(c) 69.

The number of hours a week the people in the team work is normally distributed with a mean of 42 hours. 10% of the team work 48 hours or more a week. Find the probability that both plumbers work more than 40 hours in a given week. Gwendolyn added the multiples of 3, from 3 to 3750 and found that 3 + 6 + 9 + … + 3750 = s.Calculate s.

70.

Find the term containing x in the expansion of (5 + 2x ) .

71.

The number of hours of sleep of 21 students are shown in the frequency table below

10

2 7

Hours of sleep

Number of students

4

2

5

5

6

4

7

3

8

4

72.

75.

79.

12

1

log5x

1   (b) log5  x 

2

(c)

the interquartile range.

log25x

A geometric sequence has all positive terms. The sum of the first two terms is 15 and the sum to infinity is 27. Find the value of (a) the common ratio; (b) the first term.

x

Find the exact value of x satisfying the equation (3 )(4 where a, b ∈

77. 78.

2

Find: (a) the median; (b) the lower quartile; (c) Given that log5 x = y, express each of the following in terms of y.

(a) 73.

10

2x+1

ln a . Give your answer in the form ln b

x+2

)=6

. 4

Complete the following expansion. (2 + ax) = 16 + 32ax + … Arturo goes swimming every week. He swims 200 metres in the first week. Each week he swims 30 metres more than the previous week. He continues for one year (52 weeks). (a)

How far does Arturo swim in the final week?

(b)

How far does he swim altogether?

The diagrams below show the first four squares in a sequence of squares which are subdivided in half. The 1 area of the shaded square A is 4 . A

A

B

D

i a g

r a m

1

D

i a g

B

B C

(a)

(i)

i a g

2

A

A

D

r a m

r a m

C 3

D

i a g

r a m

4

Find the area of square B and of square C. 7

(ii)

Show that the areas of squares A, B and C are in geometric progression.

(iii)

Write down the common ratio of the progression.

(b)

(i)

th

(ii) (c) 81.

Find the total area shaded in diagram 2.

Find the total area shaded in the 8 diagram of this sequence.Give your answer correct to six significant figures.

The dividing and shading process illustrated is continued indefinitely.Find the total area shaded.

The first four terms of an arithmetic sequence are 2, a – b, 2a +b + 7, and a – 3b, where a and b are constants. Find a and b. 3

100 – x 2 =

1 2

82.

Solve log16

83.

A committee of four children is chosen from eight children. The two oldest children cannot both be chosen. Find the number of ways the committee may be chosen.

86.

The diagram below shows two circles which have the same centre O and radii 16 cm and 10 cm respectively. The two arcs AB and CD have the same sector angle θ = 1.5 radians. Find the area of the shaded region.

.

A

B D

C

O

87. The following table shows four series of numbers. One of these series is geometric, one of the series is arithmetic and the other two are neither geometric nor arithmetic. (a)

Complete the table by stating the type of series that is shown. Type of series

Series (i)

1 + 11 + 111 + 1111 + 11111…

(ii)

3 9 27 1 + 4 + 16 + 64 …

(iii)

0.9 + 0.875 + 0.85 + 0.825 + 0.8…

(iv)

1 2 3 4 5 + + + + 2 3 4 5 6…

(b) 88.

The geometric series can be summed to infinity. Find this sum.

The table below shows the marks gained in a test by a group of students. Mark

1

2

3

4

5

Number of students

5

10

p

6

2

The median is 3 and the mode is 2. Find the two possible values of p. 89.

93.

The three terms a, 1, b are in arithmetic progression. The three terms 1, a, b are in geometric progression. Find the value of a and of b given that a ≠ b. x2 x2 x2 + ln 2 + ln 3 + … 2 y y Find an expression for the sum of the first 35 terms of the series ln x + ln y xm n giving your answer in the form ln y , where m, n ∈

.

3

8

94.

Find the term containing x in the expansion of (2 – 3x) .

95.

 x2 y     z3   in terms of a, b and c. Let a = log x, b = log y, and c = log z. Write log 

96.

A company offers its employees a choice of two salary schemes A and B over a period of 10 years. Scheme A offers a starting salary of $11 000 in the first year and then an annual increase of $400 per year. (a)

(i) (ii)

Write down the salary paid in the second year and in the third year.

Calculate the total (amount of) salary paid over ten years.

Scheme B offers a starting salary of $10 000 dollars in the first year and then an annual increase of 7 % of the previous year’s salary. (b)

(i) (ii)

(c)

97.

Write down the salary paid in the second year and in the third year.

Calculate the salary paid in the tenth year.

Arturo works for n complete years under scheme A. Bill works for n complete years under scheme B. Find the minimum number of years so that the total earned by Bill exceeds the total earned by Arturo. 2

The sum of the first n terms of a series is given by Sn = 2n – n, where n ∈ (a) (b)

99.

.

Find the first three terms of the series. th Find an expression for the n term of the series, giving your answer in terms of n.

(a) (b)

+

5

Find the expansion of (2 + x) , giving your answer in ascending powers of x. 5

By letting x = 0.01 or otherwise, find the exact value of 2.01 .

9

102. A theatre has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and each successive row of seats has two more seats in it than the previous row. th

(a)

Calculate the number of seats in the 20 row.

(b)

Calculate the total number of seats.

103. A sum of $5 000 is invested at a compound interest rate of 6.3% per annum. (a)

Write down an expression for the value of the investment after n full years.

(b)

What will be the value of the investment at the end of five years?

(c)

The value of the investment will exceed $10 000 after n full years,

(i) Write down an inequality to represent this information. (ii) Calculate the minimum value of n. 104. A sum of $ 5 000 is invested at a compound interest rate of 6.3 % per annum. (a)

Write down an expression for the value of the investment after n full years.

(b)

What will be the value of the investment at the end of five years?

(c)

The value of the investment will exceed $ 10 000 after n full years. (i)

Write an inequality to represent this information.

(ii)

Calculate the minimum value of n.

106. There are 30 students in a class, of which 18 are girls and 12 are boys. Four students are selected at random to form a committee. Calculate the probability that the committee contains (a)

two girls and two boys;

(b)

students all of the same gender.

109. 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. 2

5

110. Consider the expansion of (x – 2) . (a)

Write down the number of terms in this expansion.

(b)

The first four terms of the expansion in descending powers of x are x – 10x + 40x + Ax + ...

10

Find the value of A. 2x

(1–x)

111. Find the exact solution of the equation 9 = 27

.

8

6

4

112.

(a) Given that log3x – log3(x – 5) = log3A, express A in terms of x. otherwise, solve the equation log3x – log3(x – 5) = 1.

(b)

Hence or

114. A team of five students is to be chosen at random to take part in a debate. The team is to be chosen from a group of eight medical students and three law students. Find the probability that (a)

only medical students are chosen;

(b)

all three law students are chosen.

116.The sum of the first n terms of an arithmetic sequence {un} is given by the formula 2

Sn = 4n – 2n. Three terms of this sequence, u2, um and u32, are consecutive terms in a geometric sequence. Find m. 2

117.The function f is defined for x > 2 by f(x) = ln x + ln (x – 2) – ln (x – 4).

118.(a)

 x  Express f(x) in the form ln  x + a  .

(b)

–1

Find an expression for f (x).

11