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IB Mathematics SL Topical Review All numbered problems are required and are to be worked ON YOUR OWN PAPER and turned in as specified on your assignment sheet. You must show your work. On questions marked “no calculator,” you much show sufficient to answer them without the use of a calculator. Diagrams are not necessarily to scale. Give any numerical answers exactly or to three significant figures unless otherwise specified in the problems. The lettered problems marked “Be Able To” are additional practice, and are not required.

Algebra, Functions, and Equations Sequences and Series 1. The Acme insurance company sells two savings plans, Plan A and Plan B. 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. 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. 2.

(no calculator) Given that 24, b, c, are the first three terms of an arithmetic sequence, with non-zero common difference, and that 24, c, b, are the first three terms of a geometric sequence, find b and c.

3.

(no calculator) In an arithmetic sequence, the first term is –2, the fourth term is 16, and the nth term is 11 998. a) Find the common difference d. b) Find the value of n.

Exponents and Logarithms 4. Solve for real x:

5.

1 8− x = 4

3

a)

log 8 x = 3–1

e)

1 log 9 81 + log 9 + log 9 3 = log 9 x 9

b)

c)

log 27 x = 1 − log 27 ( x − 0.4 )

d)

2x = 7 x – 1

The number of radioactive atoms N of a particular material present at time t years may be written in the form N = 5000 e–kt, where 5000 is the number of atoms present when t = 0, and k is a positive constant. It is found that N = 2500 when t = 5 years. a) Determine the value of k. b) At what value of t will N = 50?

Binomial Theorem (Pascal’s triangle, combinations) 6.

5

In one of the terms in the expansion of ( x3 − 3 y 2 ) , the powers of x and y will be identical. Find this term, giving your answer in its simplest form.

7.

(no calculator) Find the coefficient of y3 in the expansion of (3 – 2y)5, simplifying your answer as much as possible. 1

9

8.

1 Consider the expansion of 3x 2 − . x

a) b)

How many terms are there in this expansion? Find the constant term in this expansion.

Functions and Graphing 9.

(no calculator) Let f(x) = 2x, and g(x) = a)

10.

(g ° f ) (3);

x , (x ≠ 2). Find x−2

b)

g–1(5).

(no calculator) The diagram shows parts of the graphs of y = x2 and y = 5 – 3(x – 4)2. The graph of y = x2 may be transformed into the graph of y = 5 – 3(x – 4)2 by these transformations. A reflection in the line y = 0 followed by a vertical stretch with scale factor k followed by a horizontal translation of p units followed by a vertical translation of q units. Write down the values of a) k; b) p; c)

q.

Circular Functions and Trigonometry 1.

2.

3.

4.

The diagram at right shows a sector AOB of a circle of radius 15 cm and centre O. The angle θ at the centre of the circle is 2 radians. a) Calculate the length of arc AB. b) Calculate the area of the sector AOB. c) Calculate the area of the shaded region. d) Calculate the perimeter of the shaded region.

A

B

θ O

(no calculator) a) Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c. b) Hence solve the equation 3 sin2 x + 4 cos x + 4 = 0, 0 ≤ x ≤ 90°, showing your work. π 3 (no calculator) Determine the two solutions in the interval [0, π) to the equation sin2x + = , 6 2 giving your answer in terms of π. Show your work. (no calculator) The depth, y metres, of sea water in a bay t hours after midnight is represented by the function 2π y = a + bcos t , where a, b, and k are constants. k The water is at a maximum depth of 14.3 m at midnight and noon, and is at a minimum depth of 10.3 m at 06:00 and 18:00. Write down the values of a, b, and k.

2

5.

A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m in length; a second side, [AB], is 65 m in length; and the angle between these two sides is 60°. a) Use the cosine rule to calculate the length of the third side of the field. 3 b) Given that sin 60° = , express the area of the field in the form p 3 , 2 where p is an integer. 104 m The farmer divides the field into two parts by constructing a straight fence, [AD], of length x m, which bisects the 60° angle, as shown in the diagram. A 65x c) Show that the smaller area is given by and obtain a similar 4 expression for the larger area. d) Hence determine the value of x in the form q 3 , where q is an integer. ˆ and sin ADB ˆ ? e) i) What can be said about sin ADC BD 5 = . ii) Use the result of part (i) and the sine rule to prove that DC 8

6.

7.

8.

S is the base of a vertical pole TS. S lies on AB, where A and B are 92.5 meters apart on horizontal ground. ∠TAB = 20˚ and ∠TBA = 30˚. Calculate the length of the pole TS to the nearest tenth of a meter.

D

65 m B

A

The diagrams show two triangles both satisfying the conditions AB = 20 cm, AC = 17 cm, ABˆ C = 50°. Calculate the size of ACˆ B in Triangle 2. B

30° 30°

A

C B

Triangle 1

C

Triangle 2

C

T

A

S

B

In the triangle ABC it is given that BC = 9 cm, CA = 13 cm, AB = 10 cm and D is the midpoint of [AB]. By applying the cosine formula to each of two triangles, or otherwise, find CD.

Be Able To A. B.

θ 2 1 (no calculator) Solve the equation cos = , 0° ≤ θ ≤ 360°. 2 2 (no calculator) Let f (x) = 6 + 6sinx. Part of the graph of f is shown here. The shaded region is enclosed by the curve of f, the x-axis, and the y-axis. a) Solve for 0 ≤ x < 2π. (i) 6 + 6sin x = 6; (ii) 6 + 6 sin x = 0. b) Write down the exact value of the x-intercept of f, for 0 ≤ x < 2π. c) The area of the shaded region is k. Find the value of k, giving your answer in terms of π.

π 2

Let g(x) = 6 + 6sin x − . The graph of f is transformed to the graph of g. d)

Give a full geometric description of this transformation.

e)

Given that

∫

p+ p

3π 2

g ( x ) dx = k and 0 ≤ p < 2π, write down the two values of p. 3

C.

(no calculator) Let f (x) =

3e 2 x sin x + e 2 x cos x , for 0 ≤ x ≤ π. Given that tan

π 1 = , solve 6 3

the equation f (x) = 0.

Statistics and The Normal Distribution 1.

2.

(no calculator) A test which is marked out of 100 is written by 800 students. The cumulative frequency graph for the results of the test is given at right. a) How many students scored 40 marks or less on the test? b) The middle 50% of test results lie between the marks a and b, where a < b. Write down the values of a and b.

The table below represents the weights, W, in grams, of 80 packets of roasted peanuts.

Weight (W) Number of packets

3.

80

View more...
Algebra, Functions, and Equations Sequences and Series 1. The Acme insurance company sells two savings plans, Plan A and Plan B. 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. 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. 2.

(no calculator) Given that 24, b, c, are the first three terms of an arithmetic sequence, with non-zero common difference, and that 24, c, b, are the first three terms of a geometric sequence, find b and c.

3.

(no calculator) In an arithmetic sequence, the first term is –2, the fourth term is 16, and the nth term is 11 998. a) Find the common difference d. b) Find the value of n.

Exponents and Logarithms 4. Solve for real x:

5.

1 8− x = 4

3

a)

log 8 x = 3–1

e)

1 log 9 81 + log 9 + log 9 3 = log 9 x 9

b)

c)

log 27 x = 1 − log 27 ( x − 0.4 )

d)

2x = 7 x – 1

The number of radioactive atoms N of a particular material present at time t years may be written in the form N = 5000 e–kt, where 5000 is the number of atoms present when t = 0, and k is a positive constant. It is found that N = 2500 when t = 5 years. a) Determine the value of k. b) At what value of t will N = 50?

Binomial Theorem (Pascal’s triangle, combinations) 6.

5

In one of the terms in the expansion of ( x3 − 3 y 2 ) , the powers of x and y will be identical. Find this term, giving your answer in its simplest form.

7.

(no calculator) Find the coefficient of y3 in the expansion of (3 – 2y)5, simplifying your answer as much as possible. 1

9

8.

1 Consider the expansion of 3x 2 − . x

a) b)

How many terms are there in this expansion? Find the constant term in this expansion.

Functions and Graphing 9.

(no calculator) Let f(x) = 2x, and g(x) = a)

10.

(g ° f ) (3);

x , (x ≠ 2). Find x−2

b)

g–1(5).

(no calculator) The diagram shows parts of the graphs of y = x2 and y = 5 – 3(x – 4)2. The graph of y = x2 may be transformed into the graph of y = 5 – 3(x – 4)2 by these transformations. A reflection in the line y = 0 followed by a vertical stretch with scale factor k followed by a horizontal translation of p units followed by a vertical translation of q units. Write down the values of a) k; b) p; c)

q.

Circular Functions and Trigonometry 1.

2.

3.

4.

The diagram at right shows a sector AOB of a circle of radius 15 cm and centre O. The angle θ at the centre of the circle is 2 radians. a) Calculate the length of arc AB. b) Calculate the area of the sector AOB. c) Calculate the area of the shaded region. d) Calculate the perimeter of the shaded region.

A

B

θ O

(no calculator) a) Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c. b) Hence solve the equation 3 sin2 x + 4 cos x + 4 = 0, 0 ≤ x ≤ 90°, showing your work. π 3 (no calculator) Determine the two solutions in the interval [0, π) to the equation sin2x + = , 6 2 giving your answer in terms of π. Show your work. (no calculator) The depth, y metres, of sea water in a bay t hours after midnight is represented by the function 2π y = a + bcos t , where a, b, and k are constants. k The water is at a maximum depth of 14.3 m at midnight and noon, and is at a minimum depth of 10.3 m at 06:00 and 18:00. Write down the values of a, b, and k.

2

5.

A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m in length; a second side, [AB], is 65 m in length; and the angle between these two sides is 60°. a) Use the cosine rule to calculate the length of the third side of the field. 3 b) Given that sin 60° = , express the area of the field in the form p 3 , 2 where p is an integer. 104 m The farmer divides the field into two parts by constructing a straight fence, [AD], of length x m, which bisects the 60° angle, as shown in the diagram. A 65x c) Show that the smaller area is given by and obtain a similar 4 expression for the larger area. d) Hence determine the value of x in the form q 3 , where q is an integer. ˆ and sin ADB ˆ ? e) i) What can be said about sin ADC BD 5 = . ii) Use the result of part (i) and the sine rule to prove that DC 8

6.

7.

8.

S is the base of a vertical pole TS. S lies on AB, where A and B are 92.5 meters apart on horizontal ground. ∠TAB = 20˚ and ∠TBA = 30˚. Calculate the length of the pole TS to the nearest tenth of a meter.

D

65 m B

A

The diagrams show two triangles both satisfying the conditions AB = 20 cm, AC = 17 cm, ABˆ C = 50°. Calculate the size of ACˆ B in Triangle 2. B

30° 30°

A

C B

Triangle 1

C

Triangle 2

C

T

A

S

B

In the triangle ABC it is given that BC = 9 cm, CA = 13 cm, AB = 10 cm and D is the midpoint of [AB]. By applying the cosine formula to each of two triangles, or otherwise, find CD.

Be Able To A. B.

θ 2 1 (no calculator) Solve the equation cos = , 0° ≤ θ ≤ 360°. 2 2 (no calculator) Let f (x) = 6 + 6sinx. Part of the graph of f is shown here. The shaded region is enclosed by the curve of f, the x-axis, and the y-axis. a) Solve for 0 ≤ x < 2π. (i) 6 + 6sin x = 6; (ii) 6 + 6 sin x = 0. b) Write down the exact value of the x-intercept of f, for 0 ≤ x < 2π. c) The area of the shaded region is k. Find the value of k, giving your answer in terms of π.

π 2

Let g(x) = 6 + 6sin x − . The graph of f is transformed to the graph of g. d)

Give a full geometric description of this transformation.

e)

Given that

∫

p+ p

3π 2

g ( x ) dx = k and 0 ≤ p < 2π, write down the two values of p. 3

C.

(no calculator) Let f (x) =

3e 2 x sin x + e 2 x cos x , for 0 ≤ x ≤ π. Given that tan

π 1 = , solve 6 3

the equation f (x) = 0.

Statistics and The Normal Distribution 1.

2.

(no calculator) A test which is marked out of 100 is written by 800 students. The cumulative frequency graph for the results of the test is given at right. a) How many students scored 40 marks or less on the test? b) The middle 50% of test results lie between the marks a and b, where a < b. Write down the values of a and b.

The table below represents the weights, W, in grams, of 80 packets of roasted peanuts.

Weight (W) Number of packets

3.

80

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