practise problems

NATIONAL UNIVERSITY OF SINGAPORE

SEMESTER 1, 2013/2014

MA1301 Introductory Mathematics Problem Set

This problem set comes from the textbook

W. S. Ng, Introductory Mathematics (2nd edition), McGram-Hill, 2008,

with small modifications.

The suggested answers and hints are provided by Wang Fei, Department of Mathe-

matics, National University of Singapore.

CONTENTS

1. Sets and Venn Diagrams 2

2. Quadratic Expressions and Equations 3

3. Surd, Indices and Logarithms 3

4. Functions 4

5. Polynomials 5

6. Partial Fractions 5

7. Inequalities 6

8. Trigonometry 6

9. Sequences and Series 8

10. Binomial Theorem 10

11. Mathematical Induction 101

MA1301 INTRODUCTORY MATHEMATICS 2

12. Differentiation 11

13. Applications of Differentiation 12

14. Curves Sketching 16

15. Integration 16

16. Applications of Integration 18

17. Differential Equations 19

18. Complex Numbers 20

Answer and Hint 22

1. SETS AND VENN DIAGRAMS

Exercise 1. In a school hostel, all the 70 students take lunch or dinner or both meals

at the hostel. 30 take lunch and 50 take dinner. Draw a Venn diagram to illustrate the

information. Find the number of students who take only lunch or dinner but not both.

Exercise 2. Let U be an universal set, and let A and B be two subsets of U. With the

aid of a Venn diagram, or otherwise, find in each case the greatest and least possible

value of |A ∩ B|.

(i) |A| = 52, |B| = 37, and |U| = 100.

(ii) |A| = 40, |B| = 45, and |U| = 70.

(|S| denotes the number of elements in S; for instance, |{a, b, c}| = 3.)

Exercise 3. Let A = {x ∈ R | −2 ≤ 4 − x < 3} and B = {x ∈ R | −2 ≤ x < 5}.

Express each of the following sets by intervals.

(i) A ∩ B;

(ii) A′ ∩ B;

(iii) A ∪ B′.

(A′ is the complement of A in R; that is, A′ = {x ∈ R | x /∈ A}.)


Exercise 4. Given that U = {x | x is a triangle}, A = {x | x is an isosceles triangle},

B = {x | x is an equilateral triangle} and C = {x | x is a right-angled triangle}, draw

a single, clearly-labeled Venn diagram to illustrate these sets.

2. QUADRATIC EXPRESSIONS AND EQUATIONS

Exercise 5. Find the range of values of the constants m for which the equation

2x2 − 3x + m = 0

in x has two distinct real roots.

Exercise 6. It is given that the graph of y = 2x + c − x2 lies entirely below the x-axis.

Find the range of values of c.

Exercise 7. The the diagram below, the line y = mx is tangent to the curve y2 = x − c.

Show that c =1

4m2.

O x

y

y2 = x − c

y = mx

3. SURD, INDICES AND LOGARITHMS

Exercise 8. Without using calculators, evaluate the following:

(i)1

3 −√

7+

1

3 +√

7;

(ii)9x

(3x−1 + 3x+1)2

;

(iii)(

log125 49)

·(

log1/7 16)

·(

log8

1

25

)

;

(iv)1

log4 30+

1

log9 30+

1

log25 30.


Exercise 9. Solve the following equations in x:

(i) 32x+5 = 27x+1;

(ii) 32x+5 = 28x+1;

(iii) (log4 x) · (6 logx 2 + log2 x) = 5.

4. FUNCTIONS

Exercise 10. Determine the range of each of the following functions.

(i) f : (−2, 3] → R, f (x) = 4 + x2;

(ii) g : [1, 3] → R, g(x) = 5 − x3;

(iii) h : (3, ∞) → R, h(x) =6

x − 2;

(iv) k : [−2, 1] → R, k(x) =√

x + 3;

(v) m : {−2,−3, 1, 3, 7} → Z, m(x) = remainder when x2 is divided by 4.

Exercise 11. The functions f and g are defined by

f : {x ∈ Z : x > 5} → R, f (x) = 2x − 70,

g : (0, ∞) → R, g(x) = 3x + 7.

(i) Determine whether the following functions exist. Justify your answers.

(a) f ◦ g; (b) g ◦ g.

(ii) Show that the composite g ◦ f does not exist. Find the largest subset S of Z such

that g ◦ f is defined.

A function f is said to be one-to-one if f (x1) = f (x2) implies x1 = x2. If f is a one-to-

one function, then its inverse f−1 is the function whose domain is the range of f such

that

f−1(b) = a ⇔ f (a) = b.

Exercise 12. Determine which of the following functions are one-to-one. Justify your

answers. For functions which are one-to-one, determine the corresponding inverse

functions.

(i) f : (−3, 0] → R, f (x) = 25 − x2;


(ii) g : {−3,−1, 1, 5} → Z, g(x) =x4 + 1

2;

(iii) h : {−2.4, 3.1, π, 4.44} → R, h(x) = ⌊x⌋, where ⌊x⌋ denotes the greatest integer

not exceeding x.

Exercise 13. The function f is defined by f : (−∞, 5] → R, f (x) = x(8 − x).

(i) Show that f is not one-to-one.

(ii) Determine the smallest value of c for which the function f is one-to-one when the

domain of f is changed to (∞, c]. In this case, find the inverse of f .

5. POLYNOMIALS

Exercise 14. Solve the following cubic equations.

(i) 2x3 + 3x + 4 = 9x2;

(ii) x2(3 + x) = 40 + 18x.

Exercise 15. Solve the equation

27x − 10 · (9x) + 3x+1 + 54 = 0.

6. PARTIAL FRACTIONS

Exercise 16. Convert the following rational functions into the partial fraction forms:

(i)12

x(x − 2)2;

(ii)2x + 4

x2 + 2x − 3;

(iii)2x3 + 5x2 − 1

(x + 2)(x − 1).

Exercise 17. Factorize the cubic expression 3x3 − 5x2 + x + 1. Hence, convert

12

3x3 − 5x2 + x + 1

into its partial fraction form.


7. INEQUALITIES

Exercise 18. Solve the following inequalities.

(i) 3x(5 − x) ≥ 2(3 − 2x);

(ii)12

2x − 3< 1 + 2x;

(iii)2x − 3

1 + x≤ 3.

Exercise 19. Show that

x2 + 8x + 17 > 0

for all real values of x. Hence, find the solution set of the following inequality

2 +37

2(x − 2)≤ 5

2(x + 2).

Exercise 20. Sketch the graph of y = |3x+ 1| − 4, indicating clearly the points at which

the graph meets the axes. Write down the range of values of x for which y > 0.

Exercise 21. Sketch the graphs of

y = 4 − |2x − 1| and y = x

in the same xy-coordinate system. Use your graphs to to solve the inequality

x + |2x − 1| < 4.

Exercise 22. Solve graphically the inequality

|x − 1| ≤ −2

x.

8. TRIGONOMETRY

Exercise 23. Prove the following identities.

(i)cos θ

1 − tan θ+

sin θ

1 − cot θ= sin θ + cos θ;

(ii)sin θ

1 + cos θ+

1 + cos θ

sin θ= 2 csc θ;

(iii) (1 − sin θ + cos θ)2 = 2(1 − sin θ)(1 + cos θ);

(iv) cos θ · csc θ + sin θ · sec θ = sec θ · csc θ;

(v) sec θ · csc θ = tan θ + cot θ.


Exercise 24. Use the identity csc2 A = 1 + cot2 A to show that if

x = 3 cot θ − 1 and y = 3 + 2 csc θ,

then

9y2 − 4x2 − 54y − 8x + 41 = 0.

Exercise 25. Prove the identities:

(i) cos 3α = 4 cos3 α − 3 cos α;

(ii) tan 3β =3 tan β − tan3 β

1 − 3 tan2 β.

Exercise 26. Prove the identity

2(sin θ + cos θ)

1 − cos 2θ + sin 2θ= csc θ.

By replacing θ with (π/2 − θ), deduce the identity

2(sin θ + cos θ)

1 + cos 2θ + sin 2θ= sec θ.

Exercise 27. Sketch the graphs of the following functions:

(i) f (x) = 3 − 2 sin 2x, x ∈ [0, 2π];

(ii) f (x) = 4∣

∣

∣sin

x

2

∣

∣

∣, x ∈ [−2π, 2π].

(iii) f (x) = |3 cos x| − 2, x ∈ [0, π/2].

Exercise 28. Use the identity tan 2α =2 tan α

1 − tan2 αto show that tan(22.5◦) =

√2 − 1.

Exercise 29. Prove the following identities:

(i)sin 9θ + sin 11θ

cos 9θ − cos 11θ= cot θ;

(ii)sin(α − β) + 2 sin α + sin(α + β)

cos(α − β) + 2 cos α + cos(α + β)= tan α;

(iii) sin 2θ sin 3θ + sin 4θ sin 9θ = sin 6θ sin 7θ;

(iv) cos γ − cos 3γ = 4 cos γ · sin2 γ.

Exercise 30. Show that if sin(α − β) = k sin(α + β), where k 6= 1, then

tan α =

(

1 + k

1 − k

)

tan β.

Exercise 31. Show that sin(15◦) =

√6 −

√2

4.


Exercise 32. In the triangle △ABC, let a, b, c be the lengths of the the sides opposite

A, B, C, respectively. Prove that

(i) a(sin B − sin C) + b(sin C − sin A) + c(sin A − sin B) = 0;

(ii)a sin C

b − a cos C= tan A;

(iii) a(b cos C − c cos B) = b2 − c2.

c

ab

BA

C

9. SEQUENCES AND SERIES

Exercise 33. The third and seventh terms of an arithmetic sequence are 175 and 105 re-

spectively. Find the first term and common difference of this series. Hence, determine

the number of terms that must be taken so that the sum is zero.

Exercise 34. Three consecutive terms of a geometric sequence have a sum of 28 and

and a product of 512. Find these numbers.

Exercise 35. The sum of the first four terms of a positive geometric sequence is 3.75

and the sum to infinity is 4. Find the second term.

Exercise 36. An arithmetic sequence has a common difference of 1, and a geometric

sequence has a common ratio of 3. A new sequence is formed by adding the corre-

sponding terms of these two progressions. It is given that the second and fourth terms

of the new sequence are 12 and 86 respectively. Find, in terms of n

(i) the nth term of the new sequence;

(ii) the sum of the first n terms of the new sequence.

Exercise 37. Find the possible values of x for which 3x + 6, x + 2 and 34 − x are suc-

cessive terms of a geometric sequence.


Exercise 38. An arithmetic sequence with first term a and common difference d ( 6=0) is such that the third, sixth and tenth terms are successive terms of a geometric

sequence.

Show that a = 7d. Hence, calculate the ratio of the sum of the first 30 terms to the sum

of the first 10 terms of the arithmetic sequence.

Exercise 39. Express the recurring decimal 0.345 345 345 · · · as a rational number.

Exercise 40. The first two terms of an infinite geometric sequence are x + 2 and x2 −2x − 8. Find the range of values of x for which the sum of the sequence has a finite

sum, and find the sum in terms of x.

Exercise 41. Express the following series in Σ notation and simplify

(i) lg 2 + lg 4 + lg 8 + lg 16 + · · ·+ lg 2048;

(ii) 3 + 5 +25

3+

125

9+

625

27+ · · · to 2n terms.

Exercise 42. Find the smallest n for which the sum to n terms of the series

5 + 8 + 11 + · · ·

exceeds 1500.

Exercise 43. Find the least number of terms that has to be taken for the sum

64 + 32 + 16 + · · ·

to exceed 127.95.

Exercise 44. Find the sum of the following telescoping series.

(i)100

∑r=1

lg

(

r + 1

r

)

;

(ii)N

∑r=2

1

4r2 − 1.

Exercise 45. Show that for any positive integer r,

(r + 1)! − r! = r(r!).

Hence, show thatm

∑r=1

r(r!) = (m + 1)! − 1.


10. BINOMIAL THEOREM

Exercise 46. Obtain the binomial expansion, in ascending powers of x, of (2 − 3x)6.

Hence, or otherwise,

(i) determine the coefficient of x3 in the expansion of (2 − 3x)8;

(ii) find the value of the constant a for which the coefficient of x in the expansion

(1 + ax)(2 − 3x)6

is zero.

Exercise 47. Find the constant term in the binomial expansion of

(i)

(

3x − 1

x

)12

.

(ii)

(

x

2+

4√x

)9

.

Exercise 48. Use the binomial theorem to show that(

n

0

)

+

(

n

1

)

+

(

n

2

)

+ · · ·+(

n

n

)

= 2n.

Exercise 49. Expand√

1 − x up to and including the term in x2. By taking x =1

64,

deduce that√

7 ≈ 10837

4096.

Exercise 50. (i) Convertx

x2 − 3x + 2into the partial fraction form.

(ii) Show that, if x is small that x3 and higher powers of x can be neglected, then

x

x2 − 3x + 2≈ 1

2x +

3

4x2.

State the range of x for which this expression is valid.

Exercise 51. Show that for sufficiently small x,√

4 − x

1 + x≈ 2 − 5

4x +

55

64x2.

11. MATHEMATICAL INDUCTION

Exercise 52. Prove the following results by mathematical induction.


(i)n

∑r=1

r2 =1

6n(n + 1)(2n + 1);

(ii)n

∑r=2

3

(r + 1)(r + 2)=

n − 1

n + 2.

Exercise 53. The sequence of numbers {un} is defined by

u1 = 2, and un+1 = 2(

un +un

n

)

for n ≥ 1.

Write down the values of u2, u3 and u4. Prove by induction that

un = n · 2n

for all positive integers n.

12. DIFFERENTIATION

Exercise 54. Differentiate the following functions with respect to x. Simplify your

answers as far as possible.

(i) tan5(√

x);

(ii)2x2 + 3x + 15

x2 + x + 5;

(iii) cos−1(ln x);

(iv)√

x sin(ex);

(v) csc(ln x)− esec 2x;

(vi) cot5

(

1√x

)

;

(vii)1 + sin x

1 − sin x;

(viii)e2x − e−2x

e2x + e−2x.

Exercise 55. For each of the following equation, finddy

dxin terms of x and y:

(i) x2 + 4xy + y2 = 20;

(ii) sin(x2y) + ex−2y = 7;

(iii) ln x + ln y + xy = 3.

Exercise 56. Differentiate the following function with respect to x:


(i) y = xln x;

(ii) y = (sin x)tan x;

(iii) y = ln

(√e−2x sin 4x

(x2 + 2)3

)

.

Exercise 57. Find the first and the second derivatives of y with respect to x in terms of

t. Simplify your answers.

(i) x = et + 1, y = tet.

(ii) x = sin t, y = 1 − 2 cos t.

Exercise 58. If x = t − sin t and y = 1 − cos t, show that

dy

dx=

sin t

1 − cos t,

and hence show that

d2y

dx2= − 1

(1 − cos t)2.

13. APPLICATIONS OF DIFFERENTIATION

Exercise 59. For each of the following, find the the equation of the tangent and normal

at the indicate point.

(i) y = 4x − x3 + 2 at x = −1;

(ii) x2 + y2 − 6x + 2y = 0 at the origin (0, 0).

Exercise 60. Use the method of linear approximation to show that

(i) 3√

8.01 ≈ 2401

1200;

(ii) sin−1(0.49) ≈ π

6−

√3

150.

Exercise 61. Water is poured into an inverted conical container of base radius 5 m and

height 15 m at a rate of 12 m3 per min. Calculate the rate at which the water level is

rising when the radius of the water surface is 2 m.


15 m

5 m

Exercise 62. The volume of a cube is increasing at a constant rate of 5 cm3 per second.

Find the rate at which the total surface of the cube is increasing at the instant when the

volume is 216 cm3.

Exercise 63. The volume of spherical balloon is increasing at a rate of 10 m3 per second.

Find the rate at which its surface area is increasing at the instant when the radius is

5 m.

Exercise 64. Find two non-negative numbers whose sum is 20 such that the sum of

their cubes is a minimum.

Exercise 65. A solid cylinder with a volume of 128π cm3 is to be manufactured with

minimum total surface area. Show that the height of such a cylinder is twice of its base

radius.

Exercise 66. A solid cylinder of base radius r cm is surmounted by a solid hemisphere

of the same radius. If the volume of this solid is to be fixed at 576π cm3, determine to

the value of r for which the solid has the least surface area.

Exercise 67. A closed rectangular box that has a square base is to be constructed using

the least amount of materials. Show that the box is a cube.

Exercise 68. A right circular cone has a base radius of 6 cm and a height of 9 cm. Find

the volume of the largest cylinder that can be fitted in the cone.


6 cm

9 cm

Exercise 69. A rectangle of maximum area is to be inscribed in the ellipse whose equa-

tion is given by

x2 + 4y2 = 72.

Find the area of such a rectangle.

O x

y

x2 + 4y2 = 72

Exercise 70. A piece of wire of length 32 cm is bent to form the pentagon ABCDE

shown below. △ABC is a right-angled triangle with ∠ABC = 90◦, AB = 3x cm and

BC = 4x cm.


A

B

C

DE

3x 4x

(i) Express the lengths of AC and CD in terms of x.

(ii) Show that the area, A cm2, enclosed by the wire is given by

A(x) = 80x − 24x2.

(iii) Find, as x varies, the stationary value of A.

(iv) Show that the stationary value of A in (iii) is a maximum.

Exercise 71. The diagram shows a line passing through the point P(8, 1) and cutting

the x- and y-axis at A and B, respectively. The angle ∠BAO, where O is the origin, is

denoted by θ.

bP(8, 1)

θ

O x

y

B

A

(i) Prove that L, the length of the line segment PQ, is given by

L =1

sin θ+

8

cos θ.

(ii) Prove that L has a stationary value when tan θ = 1/2.

(iii) If θ is increasing at a constant rate of 1 radius per second, calculate the rate at

which L is changing at the instant when θ = π/4.


14. CURVES SKETCHING

Exercise 72. Sketch the following curves. In each case, indicate clearly, where applica-

ble,

(i) the coordinates of the stationary points;

(ii) the coordinates of the points at which the curve meets the axes;

(iii) the equations of the asymptotes.

(a) y =2x2 + 3x − 5

x + 2;

(b) y =50

x2 + x − 6;

(c) y =2 + x + x2

x + 2;

(d) y =x2 − 4

2x2 + 6x + 5.

15. INTEGRATION

Exercise 73. Evaluate the following indefinite integrals.

(i)∫

(

x +2

x2

)

dx;

(ii)∫

(

tan2 x − sec2 2x)

dx;

(iii)∫

2 sin2 x + 1

cos2 xdx;

(iv)∫

3

2x2 − 8x + 58dx;

(v)∫

1√6x − x2 − 5

dx;

(vi)∫

(3 cos x − sin x)2 dx.

Exercise 74. Find the equation of each of the following curves.

(i) Its gradient function is 3x2 − 4x and it passes through the point (1, 0);

(ii)d2y

dx2= 6x − 4

x2and the tangent at the point (1, 6) is parallel to y = 10x + 20.


Exercise 75. Verify that

d

dx

(

tan2 x + 2 ln cos x)

= 2 tan3 x.

Hence, or otherwise, evaluate∫ π/4

0tan3 x dx.

Exercise 76. By means of an appropriate substitution, evaluate each of the following

integrals.

(i)∫

cos x

(2 sin x + 5)2dx;

(ii)∫ ln 2

0

√4 − e−x

exdx;

(iii)∫ e/2

1

√ln 2x

2xdx;

(iv)∫ 9

0

1√

x + x√

xdx;

(v)∫ π/4

0

2 + tan x

cos2 xdx;

(vi)∫

1

t2e2/tdt;

(vii)∫

tan−1 x

1 + x2dx;

(viii)∫

2

x√

4 ln x − (ln x)2dx.

Exercise 77. Find the following indefinite integrals:

(i)∫

cos x cos 9x dx;

(ii)∫

8 sin2 4x

1 + cos 8xdx;

(iii)∫

7 + 4x − x2

(x + 2)(x2 + 1)dx.

Exercise 78. Evaluate the following integrals using integrations by parts:

(i)∫ π

0x cos

x

2dx;

(ii)∫ 1

0

2x − 1

e2xdx;

(iii)∫ e

1x3 ln x dx;


(iv)∫ π/4

0x sec2 x dx;

(v)∫ π/2

0e3x cos 2x dx;

(vi)∫ e2

ex ln(x4) dx;

(vii)∫ 1/4

0sin−1(2x) dx.

Exercise 79. Evaluate the following indefinite integrals:

(i)∫

x tan2 2x dx;

(ii)∫

x sin x cos x dx;

(iii)∫

x sin2 3x dx.

16. APPLICATIONS OF INTEGRATION

Exercise 80. Calculate the area of the region bounded by the following curves/lines.

(i) y = 4x(1 − x) and y = 0;

(ii) y = 1 − x and y2 = 1 + x;

(iii) y = 2 sin x + 1 (0 ≤ x ≤ π), the y-axis and the line y =x

π, x = 0;

(iv) y =√

x, the x-axis and the line y = 6 − x.

Exercise 81. Consider the curve C defined by y = 5 − ex.

(i) Calculate the coordinates of the points at which C cuts the axes.

(ii) Find the equation of the asymptote of C.

(iii) Sketch the curve C.

(iv) Find the equation of the tangent to the curve C at the point where C cuts the

y-axis.

(v) The region R is bounded by the curve C, the tangent in (iv) and the x-axis. Calcu-

late the volume of the solid generated by rotating R completely about the x-axis.

Exercise 82. On the same xy-coordinate system, sketch the graphs of

y = x2 and y = 2 − x2.


Calculate the volume of the solid formed by rotating the region bounded between

these two curves completely about the x-axis.

Exercise 83. Sketch the graph of

y = 2 sin x + 1 for 0 ≤ x ≤ π.

Find the volume of the solid formed by completely revolving the region bounded by

the curve y = 2 sin x + 1, 0 ≤ x ≤ π, and the line y = 1 about the x-axis.

Exercise 84. In the diagram below, the region R is bounded by the graph of y = tan2 x,

the y-axis and the line y = 3. The region S lies within the rectangle OBAC but outside

the region R, where O is the origin.

y = 3

O x

y

AC

B

R

S

(i) Show that the region S is√

3 − π

3.

(ii) By considering the area of the region R, deduce from (i) that

∫ 3

0tan−1(

√y) dy =

4π

3−√

3.

(iii) Show that

d

dx

(

tan3 x − 3 tan x + 3x)

= 3 tan4 x.

(iv) The region R is revolved completely about the x-axis. Find the exact volume of

the solid generated.

17. DIFFERENTIAL EQUATIONS

Exercise 85. Solve the following differential equation.

(i) (1 − x)dy

dx= 6;

(ii) xdy

dx+ (x2 + 1) = 0, y = 1/2 when x = 1;


(iii)dy

dx=

3

yey−1, y = 1 when x = 0;

(iv) ey+1 dy

dx− e1−2y = 0, y = 0 when x = 1.

Exercise 86. Use the substitution y = v − 2x to solve the following differential equa-

tiondy

dx+

4y + 8x + 1

2y + 4x + 1= 0.

18. COMPLEX NUMBERS

Exercise 87. Given that z = 2 − 3i and w = 1 + 2i, express each of the following

complex umbers in the form a + bi, where a, b ∈ R.

(i) zw;

(ii) z∗ − w∗;

(iii) (iz − 1)∗;

(iv)w

z;

(v) w3 − (w∗)3.

(If z = a + bi, then z∗ = a − bi is the complex conjugate of z, where a, b ∈ R.)

Exercise 88. Find the modulus and arguments of the following complex numbers.

(i) −6i;

(ii) −3 + 3√

3 i;

(iii) −2√

3 − 2i;

(iv)1√2− 1√

2i;

(v) −3√

3 − 3i;

(vi) (1 + 5i)(3 − 2i).

Exercise 89. Find the square roots of 5 − 12i. Hence, solve the equation

z2 − 4z = 1 − 12i.

Exercise 90. Solve the following equations.

(i) (3 + 2i)z = 2z + 3 − 4i;


(ii)2z − i

2z + i= (i + 1)2;

(iii) z2 + 6z + 10 = 0.

Exercise 91. Solve the following pair of simultaneous equations:

{

2w + iz = 3,

(3 − i)w − z = 1 + 3i.

Exercise 92. In an Argand diagram, ABCD is a square in which the vertices are taken

in the anti-clockwise direction. If A and C represent the complex numbers 4 + 2i and

7 + 3i respectively, determine the complex numbers represented by B and D.

Exercise 93. Sketch the following loci in separate Argand diagrams.

(i) arg(z + 2i − 1) = −π/2;

(ii) arg(z + 2i − 1) = arg(1 + i);

(iii) |3 − 2i + z| = 2;

(iv) 1 < |z − 4 − 3i| ≤ 2;

(v) π/4 < arg(z + 2i − 1) ≤ π/2;

(vi) 3 Re(z) + Im(z) = 6.

Exercise 94. Use De Moivre’s theorem to prove the identities

(i) cos 4θ = 8 cos4 θ − 8 cos2 θ + 1;

(ii) sin 4θ = 4 sin θ (2 cos3 θ − cos θ).

Exercise 95. Find the first 3 positive integers N for which (1 − i√

3)N is a real number.

Exercise 96. Show that

(

1 − i

1 + i

)1234

= −1.

Exercise 97. Show that for all θ ∈ R,

eiθ + e−iθ = 2 cos θ.

Hence, show that

e4iθ + 4e2iθ + 6 + 4e−2iθ + e−4iθ = 16 cos4 θ.

Deduce that

cos 4θ + 4 cos 2θ + 3 = 8 cos4 θ.


Exercise 98. Show that for all real values of p, q and θ,

1

p + qeiθ=

p + q cos θ + (q sin θ)i

p2 + q2 + 2pq cos θ,

provided that p + qeiθ 6= 0.

ANSWER AND HINT

1. Answer: 60.

Hint: First find the number of students who take both lunch and dinner.

2. Answer: (i) 0, 37; (ii) 15, 40.

Hint: For the greatest number of |A ∩ B|, note that A ∩ B ⊆ A and A ∩ B ⊆ B.

For the least number of A ∩ B, note that |A ∩ B| = |A|+ |B| − |A ∪ B|. The greatest

possible value of |A ∪ B| is |A|+ |B| if |A|+ |B| does not exceed |U|, and it is |U| if

|A|+ |B| exceeds |U|.

3. Answer: (i) (1, 5); (ii) [−2, 1]; (iii) (−∞,−2) ∪ (1, ∞).

Hint: First write A and B as intervals: A = (1, 6] and B = [−2, 5).

4. Hint: Find the pairwise relations among A, B, C:

B ⊆ A, B ∩ C = ∅, A ∩ C 6= ∅.

5. Answer: m < 9/8.

Hint: A quadratic equation ax2 + bx + c = 0 (a 6= 0) has two distinct real roots ⇔the discriminant ∆ = b2 − 4ac > 0.

6. Answer: c < −1.

Hint: Write the function in the standard form y = ax2 + bx + c.

The graph lies below the x-axis ⇔ ax2 + bx + c = 0 has no real root ⇔ ∆ < 0.

7. Hint: In order to find the points of intersection, substitute the line into the parabola

to obtain a quadratic equation in x.

The line is tangent to the parabola ⇔ the equation has a repeated root ⇔ ∆ = 0.

8. (i) Answer: 3.

Hint:1√

a +√

b=

√a −

√b

(√

a +√

b)(√

a −√

b)=

√a −

√b

a − b.


(ii) Answer:9

100.

Hint: Let y = 3x. Then 9x = y2.

(iii) Answer:16

9.

Hint: loga b =ln b

ln a, where ln a = loge a; and ln(ar) = r ln a.

(iv) Answer: 2.

Hint:1

loga b=

1

ln b/ ln a=

ln a

ln b= logb a; and logb x + logb y = logb(xy).

9. (i) Answer: 2.

Hint: If 3a = 3b, then a = b. Note that 27 = 33.

(ii) Answer: − log3 28 − 5

log3 28 − 2≈ 1.90387.

Hint: If 3a = A, then a = log3 A. Also note that log3(xr) = r log3 x.

(iii) Answer: 4 or −1/4.

Hint: Let y = log2 x. Then x = 2y. Note that

logab c =1

bloga c and loga b =

1

logb a.

10. (i) Answer: [4, 13].

Hint: f is decreasing on (−2, 0] and increasing on [0, 3].

(ii) Answer: [−22, 4].

Hint: g is decreasing on [1, 3].

(iii) Answer: (−∞, 6).

Hint: h is decreasing on (3, ∞).

(iv) Answer: [1, 2].

Hint: k is increasing on [−2, 1].

(v) Answer: {0, 1}.

Hint: Since the domain is a finite set, check the images one by one.

11. (i) Answer: (a) No; (b) Yes.

Hint: For f ◦ g to be defined, we need Rg ⊆ D f .


(ii) Answer: {x ∈ Z | x ≥ 7}.

Hint: For g ◦ f to be defined, we need R f ⊆ Dg; that is, for x ∈ Z, f (x) > 0.

12. The domain of f−1 is the range of f .

(i) Answer: Yes; f−1 : (16, 25] → R, f−1(x) = −√

25 − x.

Hint: To find the domain of f−1, i.e., the range of f , note that f is increasing.

In order to find the expression of f−1, first set y = f (x), then solve x in terms

of y, say x = f−1(y), then interchange x and y.

(ii) Answer: No.

Hint: Evaluate all the images and check if any number appears twice (or more).

(iii) Answer: No.

Hint: Evaluate all the images and check if any number appears twice (or more).

13. (i) Hint: Note that the graph of f is symmetric about x = 4.

(ii) Answer: c = 4, f−1 : (−∞, 16] → R, f−1(x) = 4 −√

16 − x.

Hint: Verify that f is increasing on (−∞, 4] and increasing on [4, ∞). Then find

the inverse (both the domain and expression as in Exercise 12(i).

14. Hint: For a polynomial f (x), if f (a) = 0, then x − a is a factor of f (x). If f (x) is a

cubic function, then

g(x) =f (x)

x − a

is a quadratic function. Factorize g(x) = (x − b)(x − c). Then

f (x) = (x − a)(x − b)(x − c).

The equation f (x) = 0 thus has roots x = a, b, c.

(i) Answer: −1/2, 1, 4.

Hint: Let f (x) = 2x3 + 3x + 4 − 9x2. Then f (1) = 0.

(ii) Answer: −5,−2, 4.

Hint: Let f (x) = x2(3 + x)− (40 + 18x). Then f (4) = 0.

15. Answer: x = 1, 2.

Hint: Set y = 3x. Then the equation in x becomes a cubic equation in y:

y3 − 10y2 + 3y + 54 = 0.


Solve the equation to get y = −2, 3, 9.

16. (i) Answer:3

x− 3

x − 2+

6

(x − 2)2.

Hint: Write12

x(x − 2)2=

A

x+

B

x − 2+

C

(x − 2)2and compare coefficients.

(ii) Answer:3

2(x − 1)+

1

2(x + 3).

Hint: Factorize x2 − 2x + 3 = (x − 1)(x + 3). Set2x + 4

x2 − 2x + 3=

A

x − 1+

B

x + 3.

(iii) Answer: 2x + 3 − 1

x + 2+

2

x − 1.

Hint: First divide 2x3 + 5x2 − 1 by (x + 2)(x − 1) to obtain

2x3 + 5x2 − 1 = (2x + 3)(x2 + x − 2) + (x + 5).

Then writex + 5

(x + 2)(x − 1)=

A

x + 2+

B

x − 1and compare coefficients.

17. Answer: 3x3 − 5x2 + x + 1 = (3x + 1)(x − 1)2.27

4(3x + 1)− 9

4(x − 1)+

3

(x − 1)2.

Hint: Let f (x) = 3x3 − 5x2 + x + 1. Then f (1) = 0.

Write12

3x3 − 5x2 + x + 1=

A

3x + 1+

B

x − 1+

C

(x − 1)2and compare coefficients.

18. (i) Answer: 1/3 ≤ x ≤ 6.

Hint: f (x) ≥ g(x) ⇔ f (x)− g(x) ≥ 0.

(ii) Answer: −3/2 < x < 3/2 or x > 5/2.

Hint:f (x)

g(x)< 0 ⇔ f (x)g(x) < 0.

(iii) Answer: x ≤ −6 or x > 1.

Hint:f (x)

g(x)≤ 0 ⇔ f (x)g(x) ≤ 0 and g(x) 6= 0.

19. Answer: −2 ≤ x ≤ 2.

Hint: Complete the square. Note that if f (x) > 0, thenf (x)

g(x)≤ 0 ⇔ g(x) < 0.

20. Answer: x-axis: (−5/3, 0) and (1, 0), y-axis at (0,−3); x < −5/3 or x > 1.

Hint: Set x = 0 to obtain the intersections with y-axis; and set y = 0 to obtain the

intersections with x-axis. Note that |a| = b ⇔ a = b or a = −b.


21. Answer: −3 < x < 5/3.

Hint: First find the points of intersections. Consider |a| = a or |a| = −a.

22. Answer: −1 < x < 0.

Hint: First find the points of intersections. Note that if |x − 1| = −2/x, then x < 0,

and thus x − 1 < 0.

23. (i) Hint: Write tan θ =sin θ

cos θand cot θ =

cos θ

sin θ.

(ii) Hint: Note that sin2 θ + cos2 θ = 1.

(iii) Hint: Note that sin2 θ + cos2 θ = 1, and 1 − a + b − ab = (1 − a)(1 + b).

(iv) Hint: Note that sec θ =1

cos θ, csc θ =

1

sin θand sin2 θ + cos2 θ = 1.

(v) Hint: Note that tan θ + cot θ = sin θ sec θ + cos θ csc θ. Then use (iv).

24. Hint: Substitute x = 3 cot θ − 1 and y = 3 + 2 csc θ, and expand.

25. (i) Hint: Use cos(α + β) = cos α cos β − sin α sin β.

Hint: Use tan(α + β) =tan α + tan β

1 − tan α tan β.

26. Hint: Use cos 2θ = 1 − 2 sin2 θ and sin 2θ = 2 sin θ cos θ.

Note that sin(π/2 − θ) = cos θ, cos(π/2 − θ) = sin θ, cos(π − 2θ) = − cos 2θ and

sin(π − 2θ) = sin 2θ.

27. Hint: N.A.

28. Hint: Set α = 22.5◦. Then use the identity to obtain a quadratic equation in x =

tan(22.5◦). Note that tan(22.5◦) > 0.

29. (i) Hint: Use

sin α + sin β = 2 sinα + β

2cos

α − β

2, cos α − cos β = −2 sin

α + β

2sin

α − β

2.

(ii) Hint: Use

sin(α ± β) = sin α cos β ± cos α sin β, cos(α ± β) = cos α cos β ∓ sin α sin β.

(iii) Hint: Use sin α sin β =cos(α − β)− cos(α + β)

2.


(iv) Hint: Use cos α − cos β = −2 sinα + β

2sin

α − β

2.

30. Hint: Use sin(α± β) = sin α cos β± cos α sin β. Then divide both sides by cos α cos β.

31. Hint: Note that 15◦ = 60◦ − 45◦. Then use sin(α − β) = sin α cos β − cos α sin β.

32. (i) Hint: Use law of sine: Let R be radius of the circumscribed circle. Then the law

of sine states that a = 2R sin A, b = 2R sin B and c = 2R sin C.

(ii) Hint: Use law of sine, and note that sin B = sin(π − B) = sin(A + C).

(iii) Hint: Use law of cosine: cos C =a2 + b2 − c2

2ab.

33. Answer: a = 210, d = −17.5 and n = 25.

Hint: Let a be the first term and d the common difference of an arithmetic sequence.

Then the nth term is a + (n − 1)d, and the sum from the 1st term to the nth term is

[a + a + (n − 1)d]n/2.

34. Answer: 4, 8 and 16 (or 16, 8 and 4).

Hint: Set the three numbers be aq−1, a and aq. Solve that a = 8 and q = 1/2 or 2.

35. Answer: 1.

Hint: Let the first term and the common ratio of the geometric sequence be a and

q, respectively. The sum from the 1st term to the nth term isa(1 − qn)

1 − qand the sum

from the 1st term to infinity isa

1 − q. Solve that a = 2 and q = 1/2.

36. Answer: (i) n + 1 + 3n; (ii)n2 + 3n − 3 + 31+n

2.

Hint: Let a be the first term of the arithmetic sequence and b be the first term of the

geometric sequence. Then the nth term of the arithmetic sequence is a+ (n− 1) and

the nth term of the geometric sequence is 3n−1b.

37. Answer: 25.

Hint: Note that a, b, c form successive terms of a geometric sequence ⇔ b2 = ac and

none of them is zero. Solve a quadratic equation in x and check the roots.

38. Answer:129

23.


Hint: The nth term of an arithmetic sequence is a + (n − 1)d. x, y, z form successive

terms of a geometric sequence if and only if y2 = xz and none of them is zero. The

sum of the first n terms of an arithmetic sequence is (a + a + (n − 1)d)n/2.

39. Answer:115

333.

Hint: 0.abc abc abc · · · = 0.abc + 0.abc · 10−3 + 0.abc · 10−6 + · · · .

40. Answer: x − 4, 3 < x < 5,x + 2

5 − x.

Hint: The common ratio q is the ratio of any two consecutive terms. The sum is

finite ⇔ |q| < 1. For such a case, the total sum isa

1 − q.

41. (i) Answer:11

∑n=1

lg(2n) = 66 ln 2.

Hint: lg(an) = n lg a.

(ii) Answer:2n

∑k=1

3

(

5

3

)k−1

=9

2

[

(

5

3

)2n

− 1

]

.

Hint: Note that this is a geometric series of common ratio 5/3.

42. Answer: 31.

Hint: Note that this is an arithmetic series with common difference 3. Obtain a

quadratic inequality 3n2 + 7n > 3000 and solve it.

43. Answer: 12.

Hint: Note that this is a geometric series with common ratio 1/2.

44. (i) Answer: 101.

Hint: log

(

r + 1

r

)

= lg(r + 1)− lg r.

(ii) Answer:1

2− 1

2(2N + 1).

Hint: Convert1

4r2 − 1into the partial fraction form

1

2(r − 1)− 1

2(r + 1).

45. Hint: (r + 1)! = r! (r + 1).

46. Answer: 64 − 576x + 2160x2 − 4320x3 + 4860x4 − 2916x5 + 729x6.

Hint: (a + b)n =

(

n

0

)

an +

(

n

1

)

an−1b +

(

n

2

)

an−2b2 + · · ·+(

n

n

)

bn.


(i) Answer: −48384.

Hint: (2 − 3x)8 = (4 − 12x + x2)(2 − 3x)6.

(ii) Answer: −9.

Hint: The coefficient of x is given by

1 × linear coefficient of (2 − 3x)6 + a × constant term of (2 − 3x)6.

47. (i) Answer: 673596.

Hint: A general term of the binomial expansion is

(

12

n

)

(3x)n

(

−1

x

)12−n

. Choose

n properly so that the power of x is 0.

(ii) Answer: 43008.

Hint: A general term of the binomial expansion is

(

9

n

)

(x

2

)

(

4√x

)9−n

. Choose

n properly so that the power of x is 0.

48. Hint: Expand (1 + 1)n.

49. Answer:√

1 − x ≈ 1 − 1

2x − 1

8x2.

Hint: (a + b)r = ar + rar−1b+r(r − 1)

2ar−2b2 + · · · . Find an approximation of

√

63

64.

Then note that

√

63

64=

3

8

√7.

50. (i) Answer: − 1

x − 1+

2

x − 2.

Hint: Writex

x2 − 3x + 2=

A

x − 1+

B

x − 2and compare coefficients.

(ii) Answer: −1 < x < 1.

Hint:1

1 − r≈ 1 + r + r2, and it is valid if |r| < 1.

51. Hint: Use (1 + a)r ≈ 1 + ra +r(r − 1)

2a2 to show that

√4 − x = 2(1 − x/4)1/2 ≈ 2 − x

4− x2

64,

1√1 + x

= (1 + x)−1/2 ≈ 1 − x

2+

3

8x2.

52. (i) Hint: First verify that the identity holds for n = 1. Then suppose it holds for k,

and use this fact to show that it also holds for k + 1.


(ii) Hint: First verify that the identity holds for n = 2. Then suppose it holds for k,

and use this fact to show that it also holds for k + 1.

53. Answer: u2 = 8, u3 = 24, u4 = 64.

Hint: First verify that the identity holds for n = 1. Then suppose it holds for k, and

use this fact to show that it also holds for k + 1.

54. (i) Answer:5

2

sec2(√

x) tan4(√

x)√x

.

(ii) Answer: − x2 + 10x

(x2 + x + 5)2.

(iii) Answer: − 1

x√

1 − ln2 x.

(iv) Answer:1

2√

xsin(ex) + ex

√x cos(ex).

(v) Answer: −1

xcsc(ln x) cot(ln x)− 2 sec 2x tan 2x esec 2x.

(vi) Answer:5

2x√

xcot4

(

1√x

)

csc2

(

1√x

)

.

(vii) Answer:2 cos x

(1 − sin x)2.

(viii) Answer:8

(e2x + e−2x)2.

55. Hint: Differentiate the equation both sides with respect to x. Then solvedy

dxin terms

of x and y.

(i) Answer: −x + 2y

2x + y.

(ii) Answer: −2 cos(x2y)xy + ex−2y

cos(x2y)x2 − 2ex−2y.

(iii) Answer: −y

x.

56. (i) Answer:2

xln x xln x.

Hint: Take logarithmic function both sides to get ln y = ln x · ln x. Then apply

the method of implicit differentiation.

(ii) Answer: [sec2 x ln(sin x) + 1] (sin x)tan x.


Hint: Take logarithmic function both sides to get ln y = tan x ln(sin x). Then

apply the method of implicit differentiation.

(iii) Answer: −1 +2 cos 4x

sin 4x− 6x

x2 + 2.

Hint: Use the property that ln(ab) = ln a + ln b and ln(ar) = r ln a to simplify

the function.

57. Hint:dy

dx=

dy/dt

dx/dtand

d2y

dx2=

1

dx/dt

d

dt

(

dy

dx

)

.

(i) Answer:dy

dx= 1 + t;

d2y

dx2= e−t.

(ii) Answer:dy

dx= 2 tan t;

d2y

dx2= 2 sec3 t.

58. Hint:dy

dx=

dy/dt

dx/dtand

d2y

dx2=

1

dx/dt

d

dt

(

dy

dx

)

.

59. Hint: If the derivative at point (a, b) is m, then the tangent is y = m(x − a) + b and

the normal is y = − 1

m(x − a) + b.

(i) Answer: tangent: y = x; normal: y = −x − 2.

Hint: Let x = −1 to obtain the corresponding y-coordinate. Finddy

dx, and then

substitute x = −1 to obtain the slope of the tangent.

(ii) Answer: tangent: y = 3x; normal: y = −1

3x.

Hint: Finddy

dxby implicit differentiation. Substitute (x, y) = (0, 0) to obtain

the slope of the tangent.

60. Linear approximation: f (x) ≈ f (0) + f ′(0)x for small x.

(i) Hint: Let f (x) = (8 + x)1/3 and use x = 0.01 in linear approximation.

(ii) Hint: Let f (x) = sin−1(0.5 + x) and use x = −0.01 in linear approximation.

61. Answer:3

πm per min.

Hint: Express the volume V as a function in height h, say V =π

27h3. Note that

r : h = 5 : 15 at any time, where r is the radius of the water surface.

Let r = 2. Find the corresponding h,dV

dhand use the identity

dh

dt=

dV/dt

dV/dh.


62. Answer:10

3cm2 per second.

Hint: If the side is x, then the volume is V = x3 and the surface is S = 6x2.

Let V = 216. Find x. Then use the identitiesdx

dt=

dV/dt

dV/dxand

dS

dt=

dS

dx

dx

dt.

63. Answer: 4 m2 per second.

Hint: Let r be the radius. Then the volume V =4π

3r3 and the surface S = 4πr2.

FinddV

drand

dS

drat r = 5. Then use the identities as in Exercise 62.

64. Answer: 10 and 10.

Hint: Let one number be x. Then minimize f (x) = x3 + (20 − x)3, 0 ≤ x ≤ 20. Find

the stationary point by solving f ′(x) = 0.

65. Hint: Let the radius and height be r and h. Then V = πr2h. Minimize the surface

area S(r) = 2πr2 +256π

r, r > 0, to obtain r = 4 and h = 8.

66. Answer:123√

5≈ 7.02.

Hint: It is given that πr2h +2

3πr3 = 576π. Minimize the surface area

S(r) = 3πr2 + 2πr

(

576

r2− 2

3r

)

, r > 0,

by solving S′(r) = 0.

67. Hint: Let a and h be the side of the base and the height respectively. Then a2h = V

is fixed. The surface S(a) = 2a2 + 4a

(

V

a2

)

. Find the stationary point a = 3√

V by

solving S′(a) = 0.

68. Answer: 48 cm3.

Hint: Let r and h be the radius and height of the cylinder respectively. Thenh

9=

6 − r

6. Maximize the volume V(r) = πr2h =

3π

2r2(6 − r).

69. Answer: 72.

Hint: Let (x, y) be the vertex in the first quadrant. Then x2 + 4y2 = 72.

Maximize the area of the rectangle A(x) = 4xy = 4x

√

72 − x2

4, 0 < x <

√72. Find

stationary point x = 6 by solving A′(x) = 0.


70. (i) Answer: AC = 5x.

Hint: Use Pythagoreans’ theorem.

(ii) Hint: The area of △ABC is 6x2, and that of ACDE is 5x(16 − 6x).

(iii) Answer:200

3.

Hint: Let A′(x) = 0 to find x = 5/3.

(iv) Hint: Evaluate A′′(5/3) and apply the second derivative test.

71. (i) Hint: Use L = AB = AP + PB.

(ii) Hint: Solve L′(θ) = 0.

(iii) Answer: 7√

2.

Hint:dL

dt=

dL

dθ

dθ

dt.

72. Hint: Let x = 0 to find y-intercept and let y = 0 to find x-intercept.

Setdy

dx= 0 to find the stationary points.

Convert the function into the partial fraction form. The polynomial part is the

asymptote. The vertical asymptotes are given by x = zero of denominator.

(a) Answer: Intersection with axes (0,−5/2), (1, 0) and (−5/2, 0); no stationary

point; asymptotes x = −2 and y = 2x − 1.

(b) Answer: Intersection with axes (0,−25/3); stationary point x = −1/2; asymp-

totes x = −3, x = 2 and y = 0.

(c) Answer: Intersection with axes (0, 1); stationary points at x = 0 and x = −4;

asymptotes x = −2 and y = x − 1.

(d) Answer: Intersection with axes (0,−4/5), (−2, 0) and (2, 0); stationary points at

x = −4/3 and x = −3; asymptote y = 1/2.

73. (i) Answer:x2

2− 2

x+ C.

Hint: Integrate term by term.

(ii) Answer: tan x − x − 1

2tan 2x + C.

Hint: Note that tan2 x = sec2 x − 1.


(iii) Answer: 3 tan x − 2x + C.

Hint: Note thatsin x

cos x= tan x and

1

cos x= sec x.

(iv) Answer:3

10tan−1 x − 2

5+ C.

Hint: Complete the square in the denominator.

Then use∫

1

x2 + a2dx =

1

atan−1 x

a+ C.

(v) Answer: sin−1 x − 3

2+ C.

Hint: Complete the square in the denominator.

Then use the formula∫

1√a2 − x2

dx = sin−1 x

a+ C.

(vi) Answer: 5x + 2 sin 2x +3

2cos 2x + C.

Hint: Use cos2 x =1 + cos 2x

2, sin2 x =

1 − cos 2x

2and 2 sin x cos x = sin 2x.

74. (i) Answer: y = x3 − 2x2 + 1.

Hint: Integratedy

dxwith respect to x to obtain y = x3 − 2x2 +C. Then determine

C = 1.

(ii) Answer: y = x3 + 4 ln x + 3x + 2.

Hint: Integrated2y

dx2with respect to x to obtain

dy

dx= 3x2 +

4

x+C and determine

C = 3.

Then integratedy

dxwith respect to x to obtain y = x3 + 4 ln x + 3x + D and

determine D = 2.

75. Answer:1 − ln 2

2.

Hint: If F′(x) = f (x), then∫

f (x) dx = F(x) + C and∫ b

af (x) dx = F(b) − F(a).

76. (i) Answer: − 1

2(2 sin x + 5)+ C.

Hint: Use u = sin x.

(ii) Answer:7

6

√14 − 2

√3.

Hint: Use u = e−x to obtain indefinite integral2

3(4 − e−x)3/2 + C.


(iii) Answer:1 − (ln 2)3/2

3.

Hint: Use u = ln 2x to obtain indefinite integral1

3(ln 2x)3/2 + C.

(iv) Answer: 4.

Hint: Use u =√

x to obtain indefinite integral 4√

1 +√

x + C.

(v) Answer:5

2.

Hint: Let u = tan x to obtain indefinite integral 2 tan x +tan2 x

2+ C.

(vi) Answer:1

2e−2/t + C.

Hint: Let u = −2

t.

(vii) Answer:1

2(tan−1 x)2 + C.

Hint: Let u = tan−1 x.

(viii) Answer: 2 sin−1 ln x − 2

2+ C.

Hint: Use u = ln x.

77. (i) Answer:1

20sin 10x +

1

16sin 8x + C.

Hint: Use cos α cos β =1

2[cos(α + β) + cos(α − β)].

(ii) Answer: tan 4x − 4x + C.

Hint: Use 1 + cos 2α = 2 cos2 α.

(iii) Answer: − ln |x + 2|+ 4 tan−1 x + C.

Hint: Convert into the partial fraction form of the formA

x + 2+

Bx + C

x2 + 1.

78. Hint:∫

u dv = uv −∫

v du. Evaluate the indefinite integral first. Then substitute at

the two end points.

(i) Answer: 2π − 4.

Hint: Indefinite integral 2x sin(x/2) + 4 cos(x/2) + C.

(ii) Answer: −e−2.

Hint: Indefinite integral −xe−2x + C.


(iii) Answer:1 + 3e4

16.

Hint:1

4x4 ln x − 1

16x4 + C.

(iv) Answer:π − 2 ln 2

4.

Hint: Indefinite integral x tan x − ln | cos x|+ C.

(v) Answer: − 3

13(1 + e3π/2).

Hint: Indefinite integral3

13e3x cos 2x +

2

13e3x sin 2x + C.

(vi) Answer: 3e4 − e2.

Hint: Indefinite integral1

2x2 ln(x4)− x2 + C.

(vii) Answer:

√3

4− 1

2+

π

24.

Hint: Indefinite integral x sin−1(2x) +1

2

√1 − 4x2 + C.

79. (i) Answer:1

2x tan 2x +

1

4ln | cos 2x| − x2

2+ C.

Hint: Use tan2 α = sec2 α − 1.

(ii) Answer: −1

4x cos 2x +

1

8sin 2x + C.

Hint: Use 2 sin α cos α = sin 2α.

(iii) Answer:1

4x2 − 1

12x sin 6x − 1

72cos 6x + C.

Hint: Use sin2 α =1

2(1 − cos 2α).

80. (i) Answer:2

3.

Hint: Let y = 0 to find the range of x. A =∫ 1

04x(1 − x) dx.

(ii) Answer:9

2.

Hint: View x as a function in y. Let 1 − y = y2 − 1 to find the range of y.

A =∫ 1

−2[(1 − y)− (y2 − 1)] dy.

(iii) Answer: 4 +π

2.


Hint: A =∫ π

0

(

2 sin x + 1 − x

π

)

dx.

(iv) Answer:22

3.

Hint: Let√

x = 6 − x to find the intersection. A =∫ 4

0

√x dx +

∫ 6

4(6 − x) dx.

81. (i) Answer: (ln 5, 0) and (0, 4).

Hint: Let x = 0 to find the y-intercept and let y = 0 to find the x-intercept.

(ii) Answer: y = 5.

Hint: Let x → −∞, y → 5.

(iv) Answer: y = 4 − x.

Hint: The tangent is y = m(x − a) + b, where b is the value at a and m is the

derivative at a.

(v) Answer: π

(

148

3− 25 ln 5

)

.

Hint: V = V1 − V2, where V1 =∫ 4

0π(4 − x)2 dx is the rotation of y = 4 − x

and V2 =∫ ln 5

0π(5 − ex)2 dx is the rotation of y = 5 − ex.

82. Answer:16

3π.

Hint: Let x2 = 2 − x2 to find intersection. V =∫ 1

−1π[(2 − x2)2 − (x2)2] dx.

83. Answer: π(8 + 2π).

Hint: V =∫ π

0π[(2 sin x + 1)2 − 12] dx.

84. (i) Hint: If y = 3, then x = π/3. A =∫ π/3

0tan2 x dx.

(ii) Hint: The area of R is the area of the rectangle minis that of S. On the other

hand, the area of R is also given by∫ 3

0tan−1 √y dy.

(iv) Answer:8

3π2.

Hint: V =∫ π/3

0π[32 − (tan2 x)2] dx.


85. (i) Answer: −6 ln |1 − x|+ C.

Hint:dy

dx=

6

1 − x.

(ii) Answer: −1

2x2 − ln |x|+ 1.

Hint:dy

dx= −x2 + 1

x.

(iii) Answer: x =1

3(y − 1)ey−1.

Hint:dx

dy=

1

3yey−1.

(iv) Answer: x =1

3e2y +

2

3.

Hint:dx

dy= e3y.

86. Answer: x + y + (2x + y)2 + C = 0.

Hint: Let y = v − 2x. Then 2x + y = v anddy

dx=

dv

dx− 2. The equation becomes

dx

dv= 2v + 1.

87. (i) Answer: 8 + i.

Hint: Note that i2 = −1.

(ii) Answer: 1 + 5i.

Hint: If z = a + bi, then z∗ = a − bi.

(iii) Answer: 2 − 2i.

(iv) Answer: − 4

13+

7

13i.

Hint:w

z=

wz∗

|z|2 , where |z|2 = a2 + b2 if z = a + bi.

(v) Answer: −4i.

Hint: (a + b)3 = a3 + 3a2b + 3ab2 + b3.

88. (i) Answer: 6, π.

Hint: It lies on the negative part of the imaginary axis.

(ii) Answer: 6,2

3π.


Hint: It lies on the second quadrant.

(iii) Answer: 4, −5

6π.

Hint: It lies on the third quadrant.

(iv) Answer: 1, −1

4π.

Hint: It lies on the fourth quadrant.

(v) Answer: 6, −5

6π.

Hint: It lies on the third quadrant.

(vi) Answer: 13√

2,1

4π.

Hint: It lies on the first quadrant.

89. Answer: Square roots: 3 − 2i and −3 + 2i; z = 5 − 2i or z = −1 + 2i.

Hint: To find the square roots of z, set (a + bi)2 = z. Then compare the coefficients

to find a2 − b2 and 2ab. Then

a2 + b2 =√

(a2 − b2)2 + (2ab)2.

Hence, evaluate a2 and b2. By noting the sign of ab, find a + bi.

90. (i) Answer: −1 − 2i.

(ii) Answer: −2

5− 3

10i.

(iii) Answer: −3 ± i.

91. Answer: w =1

6+

1

6i and z = −1

3− 8

3i.

92. Answer: B is 6 + i and D is 5 + 4i.

Hint: The center E of the square is1

2(A + C). Then

B − E = i(A − E) and D − E = −i(A − E).

93. (i) Answer: The ray down from 1 − 2i.

(ii) Answer: The ray from 1 − 2i of angleπ

4.

(iii) Answer: Circle with center −3 + 2i and radius 2.


(iv) Answer: Annulus with center 4 + 3i and radii 1 and 2, include the circle of

radius 2.

(v) Answer: The wedge with center 1 − 2i and angle fromπ

4to

π

2, include the ray

of angleπ

2.

(vi) Answer: The straight line 3x + y = 6.

94. Hint: Let z = cos θ+ i sin θ. Then z4 = (cos θ+ i sin θ)4 =4

∑k=0

(

4

k

)

(cos θ)k(i sin θ)4−k.

On the other hand, z4 = cos 4θ + i sin 4θ. Then compare the real part and imaginary

part to obtain the results.

95. Answer: 3, 6, 9.

Hint: First show that arg(1 − i√

3) = −π

3.

96. Hint: First show that1 − i

1 + i= −i and note that (−i)4 = 1.

97. Hint: First assertion: Use eiα = cos α + i sin α.

Second assertion: Find (2 cos θ)4 using the previous result and binomial theorem.

Third assertion: Evaluate e4iθ + e−4iθ and e2iθ + e−2iθ using the first assertion.

98. Hint: Write p + qeiθ = p + q cos θ + i sin θ.

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