Higher Mathematics Objective Test Practice · PDF fileHigher Mathematics Objective Test Practice Book Page 3 y f x 7 The diagram shows a sketch of y f x ( ). Find an expression for

Higher Mathematics Objective Test Practice Book

Page 1

y

y

x ( )y f x

( )y f x

( )y f x

( )y f x

( )y f x

( )y f x

1 The diagram shows a sketch of part of

the graph of ( ).y f x

What are the domain and range of the

function f ?

2 The diagram shows part of the graph of ( ).y f x

The curve passes through the points

P(0, 8) and Q(2, 32).

Which of the following represents the

equation of the curve?

A 2 8y x

B 2 7xy

C 32xy

D 2log 7y x

3 The diagram shows a sketch of the cubic ( ).y f x

Which diagram shows a sketch of ( ) ?y f x

A

B

C

D

Domain Range

A -3 ≤ x ≤ 8

-3 ≤ f(x) ≤ 5

B -3 ≤ x ≤ 5 -2 ≤ f(x) ≤ 8

C -2 ≤ x ≤ 8 -3 ≤ f(x) ≤ 5

D 3 5x 3 ( ) 8f x

y

x

x

y

x

y

x

y

y

x

x

0

0

0

0

0

0

0

R (3, 8)

P ( 3, 0)

Q ( 2, 2) S (5, 3)

P (0, 8)

Q (2, 32)

2 1 a

2

a

a 2

2

2 1

a

a

1


Page 2

( )y f x

( )y g x

2

3

( )y f x

4 The diagram below shows a sketch of ( ).y f x

Which equation below represents

( )?y f x

A sin2y x

B sin(2 1)y x

C 1 2siny x

D 1 sin2y x

5 The diagram below shows a sketch of ( ).y g x

Which equation below represents ( )?y g x

A cosy x

B 2 cosy x

C 2cos2y x

D 1 cos3y x

6 The diagram shows a sketch of ( ).y f x

Which diagram below shows a sketch of ( ) 2 ?y f x

A

B

C

D

y

x

2

1

0 135

y

y

y

y

y

x

x

x

x

x

0

0

0

0

0

y

x 0

2


Page 3

( )y f x

7 The diagram shows a sketch of ( ).y f x

Which diagram below shows a sketch of 2 ( ) ?y f x

A

B

C

D

8 Functions f and g are defined on suitable

domains by

2 1( ) and ( )

1f x x x g x

x

Find an expression for ( ( )).f g x

A 2( 1)

x

x

B

2

1

x x

x

C 2 1

1x

x

D 2 1

1x x

x

9 When 211 8 2x x is written in the

form 22( ) ,x p q what is the value of

q ?

A 5

B 11

C 19

D 27

10 Here are two statements about the

graph of the curve with equation

2( 1) ( 1)y x x

(1) There is a turning point on the x-axis.

(2) The minimum value of y is at 1.x

A Neither statement is correct.

B Only statement (1) is correct.

C Only statement (2) is correct.

D Both statements are correct.

11 What is the minimum value of

( ) (4 2)(4 2) 3?f x x x

A 17

B 1

C 3

D 15

y

y

y

y

x

x

x

x 0

0

0

0

y

x 0


Page 4

y

x

( )y f x

12 A quadratic function f is given by

2( ) ,f x ax bx c where

20 and 4 0a b ac

Which graph below is a possible sketch

of ( ) ?y f x

A

B

C

D

13 The diagram shows part of the graph of ( ).y f x

The curve cuts the y-axis at (0, 2) and the

x-axis at (-1,0), (1, 0) and (2, 0).

Which of the following represents the

equation of the curve?

A cos 1y x

B 2cosy x

C 3 22 2y x x x

D 3 22 2y x x x

14 Here are two statements about the circle

with equation 2 2( 2) ( 3) 25x y and

the line with equation 4 3 1 0.x y

(1) The line intersects the circle at (6, 6)

(2) Part of the line is a diameter of the

circle.





15 A sequence is defined by the recurrence

relation 1 04 9, 5.n nu u u

What is the value of 2 ?u

A 13

B 1

C 11

D 35

y

y

y

y

x

x

x

x 0

0

0

0

0

1 1 2

2


Page 5


relation

1 2

5 2 , 3n n

u u u


A 0

B 2

C 3

D 5


relation

1 2

1

212, 20

n nu u u


A 8

B 12

C 18

D 23

18 Here are two statements about the limits

of sequences.

(1) The sequence generated by

1 0 75 6n nu u

has a limit 8 as .n

(2) The sequence generated by

1 4n nu u

has no limit.





19 A sequence is generated by the

recurrence relation 1 0 2 6.n nu u

What is the limit of this sequence as ?n

A 1

25

B 15

2

C 12

D 30

20 What are the roots of the quadratic

equation ( 1)(4 ) 2 ?x x

A 2 and 3

B 1 and 4

C 1 and 4

D 2 and 3

21 Solve the inequality

2 0x x

A 0x

B 1x

C 1 0x

D 0 1x

22 Solve ( 2)( 3) 0.x x

A 2 or 3x x

B 2 or 3x x

C 3 2x

D 2 3x

23 Solve ( 8)(2 1) 0.x x

A 1

28 or x x

B 1

28 or x x

C 1

28 x

D 1

28 x

24 What is the nature of the roots of the

quadratic equation 2 6 7 16 ?x x

A Two real equal roots

B Two real distinct roots

C No real roots

D Three real distinct roots


Page 6

25 For what values of p does

2 (2 1) 0px p x p

have equal roots?

A 1

B 1

2

C 1

4

D 0

26 What is the equation of the parabola

with roots 5 and 2 , passing through

the point (0, 30).

A 2 3 30y x x

B 2 3 30y x x

C 23 9 30y x x

D 23 9 30y x x

27 The diagram shows a sketch of the

curve with equation

( 1)( )( 1)y k x x a x

What are the values of a and k?

28 A quadratic equation is given by

8( )( ) 0x a x b

It has roots at 1

2x and

3

4x .

What is the y-intercept?

A 4

B 3

C 1

4

D 6

29 What is the remainder when

4 23 4 2 3x x x is divided by ( 2) ?x

A 47

B 2

C 31

D 33

30 Evaluate 3 3 3

log 9 log 6 log 18.

A 3

B 1

C 1

D 3

31 Express 10 10 10

log log 8 log 3y x in a

form without using logarithms.

A 8 3xy

B 3 8y x

C 24xy

D 24y x

a k

A 4

1

2

B 4 2

C 4 1

2

D 4 2

y

x 0 1 4 1

(0, 2)


Page 7

log ( )b

y x a

32 The expression 5log (2 ) 3log (4 )e ee e

can be written in the form P log Q.e

What are the values of P and Q?

33 Which diagram below shows the graph

of 101log ?x

y

A

B

C

D

34 The diagram shows the graph with

equation log ( ).by x a

What are the values of a and b?

35 What is the solution of 8

2

3log ?x

A 1

12

B 4

C 16

3

D 12

36 If 2

3log 64 ,a what is the value of a ?

A 16

B 2

342

C 96

D 512

37 Which of the following equations is

equivalent to 10 10 10

log 3log log 5 ?b a

A 3 5b a

B 8b a

C 35b a

D 3 5ab

P Q

A 2

2

B 1 4

C e 4

D 2 1

4

a b

A 2

1

B 1 2

C 2 5

D 3 6

y

x 0 2 1

(3, 1)

y

y

y

y

x

x

x

x

0

0

0

0


Page 8

38 Solve 3 3

log log 4 2.x

A 2

B 10

C 12

D 36

39 The diagram shows the graph of 5log y

plotted against x.

The graph is a straight line through the

origin with gradient 1.

Find an expression for y in terms of x.

A y x

B 5y x

C 5xy

D 5y x

40 The graph illustrates the law .np kv

What is the value of k?

A 2

B 1

2

C 2

D 100

41 The curve with equation

10log ( 3) 1y x cuts the x-axis at P.

What is the x-coordinate of P?

A 2 9

B 3

C 3 1

D 12

42 Given that 2( ) (4 3) 3,f x x x

find ( ).f x

A 8 3x

B 212 6x x

C 2 2x x

D 4 3 3x x x

43 Given that

5 1

2 24

52 ,y x x find .

dy

dx

A 3 1

2 26

5x x

B

3 3

2 22x x

C 7 3

2 22 4x x

D 7 1

2 28

354x x

44 What is the derivative of 2 4

4

x

x

?

A 1

2x

B 2x

C 21

4x x

D 21

4x

5log y

y

x 0

0

10log p

10log v

2

4


Page 9

45 A tangent to the curve with equation 2 3 5y x x has gradient 1.

What is the x-coordinate of the point of

contact of the tangent and curve?

A 0

B 1

C 2

D 3

46 Given that 2

33s t , find the rate of

change of s with respect to t, when

8.t

A 1

B 8

C 12

D 16

47 For what values of x is the function with

derivative 2( ) 3 4 1f x x x strictly

decreasing?

A 1

31 or x x

B 1

31 x

C 1

31x

D 1 3x

48 For what values of x is the function with

derivative 2( ) 3 3f x x neither

increasing or decreasing?

A 3 and 1

B 3 and 3

C 1 and 1

D 1 and 3

49 A function is given by 2( ) 3 5.f x x

Find the maximum value of f in the

interval 1 2.x

A 0

B 3

C 5

D 17

50 Given that

0, for 2

( ) 0, for 2

0, for 2

x

f x x

x

Which diagram shows the curve with

equation ( ) ?y f x

A

B

C

D

y

y

y

y

x

x

x

x

0

0

0

0

2

2

2

2


Page 10

51 A function ( )f x is such that

( ) ( 3)f x x x .

Which graph could be a sketch of ( ) ?y f x

A

B

C

D

52 Find 2 21

26x x dx

A 36 2x x c

B 3 51

10x x c

C 3 51

23x x c

D 3 55

29x x c

53 What is the integral of 2( 1)x with

respect to x?

A 2 2x c

B 31

3x x c

C 3 21

3x x x c

D 3 21

3x x c

54 Find 1

2, 0

xdx x

A 3

2x c

B 3

21

4x c

C x c

D 4 x c

55 Find 5

1

xdx

A 7

22

7x c

B 3

22

3x c

C 7

25

2x c

D 7

22

7x c

56 Find 2 6

2, 0

a ada a

a

A 51

5a a c

B 34a c

C 71

7a c

D 3 7

3

7 3

7

a ac

a

y

x 0

y

x 0 3

3

y

0

y

0 3

x

x

3

(1, 5)


Page 11

57 Find 1 1 3

2 2 22 3 x x x dx

A 2 6x c

B 3 3 5

2 2 22 4 6

3 3 5x x x c

C 2 3x x c

D 5 7

4 48 12

5 7x x c

58 Evaluate 0

3

2

4x x dx

A 8

B 4

C 4

D 8

59 Evaluate 1

4

1

2x x dx

A 2

B 2

5

C 2

5

D 8

60 The diagram shows the curve with

equation ( ).y f x

Which of the following gives the area

between the curve and x-axis?

A ( )

c

a

f x dx

B ( )

b

a

f x dx

C ( ) ( )

b c

a b

f x dx f x dx

D ( ) ( )

b c

a b

f x dx f x dx

y

x a b c 0


Page 12

61 The diagram shows the curve with

equation 3 3 .y x x

Which of the following gives the value

of the shaded area?

A

3

3

3

3x x dx

B

0 3

3 3

03

3 3x x dx x x dx

C

0

3

3

2 3x x dx

D

3

3

3

3x x dx

62 The diagram shows the curves with

equations 2 .y x and 24 .y x x

Which of the following gives the value

of the shaded area?

A

2

2

0

4 2x x dx

B

2

2

0

2 4x x dx

C

4

2

0

4 2x x dx

D

4

2

0

2 4x x dx

y

x

0 3 3

y

x 0 2

4

24y x x

2y x


Page 13

63

What is an expression for the shaded

area in the diagram above?

A ( ) ( )

c

a

f x g x dx

B ( ) ( )

c

a

f x g x dx

C ( ) ( )

d

b

f x g x dx

D ( ) ( )

c

a

g x f x dx

64

Find an expression for the total shaded

area in the diagram above.

A

0

( )

b

f x x dx

B

0

( ) ( )

a b

a

f x x dx x f x dx

C

0

( ) ( )

a b

a

f x x dx f x x dx

D

0

( ) ( )

a b

a

f x x dx x f x dx

65 The gradient of the tangent to a curve is

given by 23 .dy

x xdx

If the curve passes through the point (2,

11) find its equation.

A 6 1y x

B 23 1y x x

C 3 21

2y x x

D 3 21

25y x x

( )y f x

( )y g x

a b

y

x 0

( , )a b

( , )c d

y

x

( )y f x

0

y x


Page 14

66 Given that 4 1dy

xdx

and

5 when 1,y x find an expression for

y in terms of x.

A 22 3y x

B 22y x x

C y = 2x2 – x + 4

D 4y

67 Given that 2

0

64

3,

p

x dx find the value of

p.

A 4

B 8

3

C 32

3

D 8

68 If 6sin(2 3),y x find .dy

dx

A 6 cos(2x + 3)

B -6 cos(2x + 3)

C 18 cos(2x + 3)

D 12 cos(2x + 3)

69 If 3cos(5 2 ),y x find .dy

dx

A 2sin(5 2 )x

B 1

2sin(5 2 )x

C 3sin(5 2 )x

D 6sin(5 2 )x

70 If 4( ) (5 2) ,f x x find ( ).f x

A 320(5 2)x

B 320 16x

C 31

20(5 2)x

D 51

25(5 2)x

71 What is the derivative of 5sin x with

respect to x?

A 5cos x

B 45cos x

C 41

5sin x

D 45sin cosx x

72 Find 2

5 4x dx

A 5

3(5 4)x c

B 10(5 4)x c

C 31

15(5 4)x c

D 31

2(5 4)x c

73 Find 3

5 2x dx

A 3

25

22x x c

B 41

20(5 2)x c

C 41

3(5 2)x c

D 4(5 2)x c

74 Find 6cos 3 2x dx

A 2sin(3 2)x c

B 6sin(3 2)x c

C 18sin(3 2)x c

D 18sin(3 2)x c

75 Find sin(4 1)x dx

A 3cos(4 1)x c

B 1

4cos(4 1)x c

C cos(4 1)x c

D 4cos(4 1)x c


Page 15

76 What is the derivative of 2sin x with

respect to x?

A cosx

B sin2x

C 2sinx

D 2cos x

77 G has coordinates (3, 3, 1) and

H ( 2, 0, 3). What is the distance

between G and H?

A 14

B 26

C 30

D 50

78 P is the point (3, 2) , Q (5, 3) ,

R ( 4,1) and S (1, 1). Here are two

statements about PQ and RS.

(1) PQ 29 units

(2) PQ RS

A Neither statements is correct.




79 The point (5, 4) lies on the circle with

centre (1, 1) .

Find the length of the diameter of the

circle.

A 10

B 11

C 12 5

D 22

80 Calculate the length of the line joining

P ( 2,1) to Q (3,13) .

A 145

B 169

C 197

D 221

81 What is the distance between the points

A(1, 4, 3) and B( 3, 2, 2) ?

A 17

B 53

C 77

D 103

82

Find the gradient of the line in the

diagram above.

A 3

B 3

2

C 1

2

D 3

83 Find the gradient of the line with

equation 2 5 0x y .

A 1

B 1

2

C 1

2

D 1

84 The line with equation 2 5 6y x meets

the y-axis at the point C.

What is the gradient of the line joining C

to the point A (2, 3) ?

A 3

B 9

2

C 2

9

D 3

2

120

y

x 0


Page 16

85 The line through K(3, 10) and

L(7, )a has a gradient of 4.

What is the value of a ?

A 9

B 3 5

C 4

D 6

86 What is the equation of the line through

the points (3, 1) and (2, 3)?

A 1 2( 3)y x

B 1 2( 3)y x

C 1 2( 3)y x

D 1 2( 3)y x

87 Find the equation of the line through the

points P( 4, 0) and Q( 1, 9).

A 1 4

3 3y x

B 1

34y x

C 3 4y x

D 3 12y x

88 A line passes through the points

( 1, 5) and (2, 4).

What is the gradient and y-intercept of

this line ?

89 What is the equation of the line joining

the points (3, 1) and (6, 4) ?

A 3 5 18y x

B 3 5 2y x

C 5 3 2y x

D 5 3 18y x

90 A line, passing through the point

( 4, 2), makes an angle of 45 with the

positive direction of the x-axis.

What is the equation of the line?

A 2 6y x

B 2y x

C 2y x

D 2 6y x

91 Find the equation of the line through the

points (0, 2) and (4, 3).

A 5 8 0x y

B 4 5 2 0x y

C 5 4 2 0x y

D 5 4 8 0x y

92 What is the gradient of a line parallel to

the line with equation 3 2 7 0 ?x y

A 3

B 3

2

C 3

2

D 3

93 Here are two statements about the lines

PQ : 3 2 2x y and RS : 2 3 6 0x y

(1) PQ and RS are perpendicular

(2) PQ and RS have the same y-intercept

Which of the following is true?





m y-intercept

A 3

-2

B 3 10

C 1

3

10

3

D 1

3

2

3


Page 17

94 A line L is perpendicular to the line with

equation 6 2 5 0x y

What is the gradient of the line L?

A 1

3

B 1

6

C 1

3

D 1

2

95 A line L has gradient 2

3.

What is the equation of the line through

the point (1, 4) and perpendicular to

line L?

A 2 3 10x y

B 2 3 5x y

C 3 2 11x y

D 3 2 14x y

96 A line has equation 3 4 20 0.x y

Which of the following lines is

perpendicular to this line?

A 4 3 2x y

B 4 3 2x y

C 3 4 2x y

D 3 4 2x y

97 A line L is perpendicular to the line with

equation 4 6 7.x y

What is the gradient of line L?

A 3

2

B 2

3

C 2

3

D 3

2

98 P is the point (2, 3) and Q( 4, 7).

What is the gradient and the midpoint R

of the line joining P to Q?

99 What are the coordinates of M, the

midpoint of the line joining P (1, 7, 4)

and Q(5, 3, 2).

A (3, 5, 3)

B (3, 2, 3)

C (2, 5, 3)

D (3, 5, 2)

100 The diagram shows triangle PQR with

altitude PS.

P has coordinates (5,1) and the gradient

of QR is 1

2.

What is the equation of altitude PS?

A 2 11 0x y

B 2 9 0x y

C 2 3 0x y

D 2 7 0x y

m R

A 5

3

( 1,2)

B 3

5 ( 1,2)

C 5

3 (3, 5)

D 3

5 (3, 5)

y

x 0

Q

R

S

P(5, 1)


Page 18

101 The diagram shows triangle EFG with

median EH drawn.

EH has gradient 1

2.

F has coordinates (5,7) and G is (7,1) .

What is the equation of median EH?

A 2 2y x

B 2 7y x

C 2 5y x

D 2 8y x

102 The diagram shows triangle PQR with

altitude PS drawn.

P has coordinates (2,3) and the gradient

of QR is 1

2.

What is the equation of altitude PS?

A 2 7 0x y

B 1

22 0x y

C 2 1 0x y

D 1

24 0x y

103 The points P( 9, 3) , Q( ,1)x and R(5,4)

are collinear.

What is the value of x?

A 4

B 1

C 2

D 17

104 A circle has equation

2 2( 4) ( 3) 18 x y

What are the coordinates of its centre

and length of its radius?

105 A circle has equation

2 2 6 4 6 0 x y x y

What is the radius of this circle?

A 14

B 19

C 26

D 46

10 6 A circle has equation

2 2 12 4 7 0 x y x y

What is the radius of this circle?

A 3

B 15

C 47

D 111

Centre Radius

A (4,3)

18

B ( 4, 3) 18

C (4,3) 18

D ( 4, 3) 18

y

x 0

E

H

G(7, 1)

F(5, 7)

y

x 0

Q

R

S

P(2, 3)


Page 19

107 A circle with centre (3, 2) passes

through the point (6, 3).

What is the equation of the circle?

A 2 2( 3) ( 2) 34 x y

B 2 2( 3) ( 2) 34 x y

C 2 2( 3) ( 2) 10 x y

D 2 2( 3) ( 2) 10 x y

108 A circle with centre (4, 2) passes

through the point (3, 5).


A 2 2( 4) ( 2) 50 x y

B 2 2( 4) ( 2) 40 x y

C 2 2( 4) ( 2) 40 x y

D 2 2( 4) ( 2) 50 x y

109 A circle has centre ( 4, 0) and radius 6

units.


A 2 2 8 20 0 x y x

B 2 2 8 20 0 x y x

C 2 2 8 10 0 x y x

D 2 2 8 10 0 x y x

110 The point P(4, 1) lies on the circle with

equation 2 2 4 6 7 0. x y x y

What is the gradient of the tangent at P?

A 1

B 1

2

C 1

2

D 1

111 The point P(4,2) lies on the circle with

equation 2 2( 3) ( 5) 10. x y

What is the gradient of the tangent at P?

A 1

B 1

3

C 1

3

D 1

112 The point T(9, 3) lies on the circle with

equation 2 2 10 2 6 0. x y x y

What is the gradient of the tangent at T?

A 2

B 1

C 5

19

D 2

113 To find the points of intersection of a

line and circle the equation 2 2 3 0 x x k is formed.

What is the value of k for which the line

is a tangent to the circle?

A 1

B 1

3

C 1

3

D 1

114 To find the points of intersection of a

line and circle the equation 2 3 3 0 x x p is formed.

What is the value of p for which the line

a tangent to the circle?

A 2

B 3

4

C 3

4

D 2


Page 20

115 The circle with equation

2 2 5 6 0 x y x y

intersects the y-axis at the points

A ( 2, 0) and (3, 0)

B ( 3, 0) and (2, 0)

C ( 6, 0) and (1, 0)

D ( 1, 0) and (6, 0)

116 PQ is a diameter of the circle with

equation

2 2( 2) ( 1) 5 x y

P has coordinates (5, 7) , what are the

coordinates of Q?

A ( 1,3)

B (1, 3)

C (3, 8)

D (-1, -9)

117 The vectors and a b can be expressed in

the form 2 3 i j k and 2 3 i j k

respectively.

Find ?a b

A 6

B 14

C 18

D 42

118 The points C and D have coordinates

(6, 3, 1) and ( 2, 4, 2) respectively.

Find CD .

A 18

B 28

C 56

D 122

119 What is the magnitude of vector 2 AB

where AB has components

7

3 ?

11

A 2 15

B 4 18

C 2 161

D 4 179

120 The vector GH can be expressed as

7 7 4 i j k .

The coordinates of H are (4, 2, 3).

What are the coordinates of G ?

A ( 3, 9, 1)

B ( 3, 5, 1)

C (3, 5, 1)

D (11, 9, 7)

121 The points K and L have coordinates

(6, 3, 1) and ( 2, 4, 2) respectively.

Express KL in terms of , and ?i j k

A -8i + j – 3k

B -8i + 7j – k

C -8i + 7j – 3k

D 8i - 7j + 3k

122 If L is the point (7, 1, 4) and

2

KL 0

6

What are the coordinates of K?

A (5, 1, 10)

B (5, 1, 2)

C (9, 1, 2)

D (9, 1, 2)


Page 21

123 Vector u, where

1

2

1

2

k

u , is a unit

vector.

Given that 0,k find the value of k.

A 1

B 0

C 1

2

D 4

124 Given that 2 4 5 p i j k , find 3p .

A 9 5

B 3 11

C 18 10

D 3 40

125 Given that 3 2 u i j k and

2 v i j k , find u v .

A 7

B 20

C 32

D 34

126 Here are two statements about

3 6

4 and 8

5 10

u v

(1) u is parallel to v

(2) u and v have the same direction






127 Which of the following vectors is

parallel to vector u , where

2

4 ?

6

u

A

6

12

18

B

6

12

18

C

6

12

18

D

6

12

18

128 Given that

2

1

3

u and

1

3

1

v ,

Find the components of vector 2 .u v

A

3

5

7

B

3

2

2

C

6

4

4

D

2

8

8

4

8

10

4

6

12

6

4

2


Page 22

129 Given that

4

2

1

u and

3

1

0

v ,

What are the components of 2 ?u v

A

14

6

2

B

2

0

1

C

1

1

1

D

5

3

2

130 Here are two statements about

3

AB 4

6

and

6

DC 8

12

(1) AB 2 CD

(2) AB is parallel to DC






131 Vectors u and v are shown in the

diagram below.

BC 2 DE

Find AC in terms of u and v.

A u v

B 2u v

C 2 2u v

D 2 u v

132

Which of the following is equivalent to c?

A a b

B d e

C b a

D e d

a b

c

d

e

A

B

C

D

E u

v

v 10

4

1


Page 23

133

A is the point (3, 3, 0),

B(8, 3, 0) and G(8, 7, 2).

What are the components of AH ?

A

3

4

2

B

3

7

2

C

0

4

2

D

2

4

0

134 OPQRS is a pyramid as shown

SM is perpendicular to the plane OPQR.

SM 5

What are the components of SQ ?

A

8

6

5

B

5

3

4

C

4

3

5

D

4

5

3

135 The points A(3, 4, 5), B(5, 7, 9) and

C( , 13, 17)x are collinear.

What is the value of x?

A 3

B 2

C 7

D 9

136 The points P( 2, 1, 4), Q(4, 2, 16) and

R(2, 1, 12) are collinear.

Find the ratio PQ : QR.

A 1 : 2

B 1 : 3

C 3 : 1

D 4 : 1

A B

C D

E F

G H

P

Q

S

R

0

y

x 8

6 M

5

z


Page 24

137 The points P(1, 2, 4), Q(3, 4, 0) and

R(9, 22, 12) are collinear.

Find the ratio PQ : QR.

A 4 : 3

B 2 : 1

C 1 : 3

D 1 : 4

138 P is the point (2, 1, 3) and Q is (6, 5, 11).

R divides PQ in the ratio 3 : 1.

What are the coordinates of R?

A (2, 2, 4)

B (2, 1 5, 3 5)

C (3, 2, 5)

D (5, 4, 9)

139 P is the point (5, 5, -13) and Q is ( 1, 4, 2).

R divides PQ in the ratio 2 : 1.

What are the coordinates of R?

A ( 9, 16, 2)

B ( 3, 7, 1)

C (0, 5, 5)

D (1, 1, 3)

140 Given that

2

PQ 4

0

and P is (0, 2, 2).

What are the coordinates of R, the

midpoint of PQ?

A (1, 3, 1)

B (1, 2, 1)

C (1, 1, 1)

D (1, 0, 2)

141 Given that

1 4

2 and 3 ,

3 0

a b

find . .a b a a

A 20

B 8

C 12

D 13

142 Given that

3 6

2 and 4 ,

1 2

p q

find . .p q q q

A 6

B -84

C 17

D 104

143 Given that

2 4

3 and

1 1

x

a b , find x

if a is perpendicular to b.

A -1

B -2

C 1

D 2

144 Given that

2 4

3 and ,

1 1

x

a b find x

if a is perpendicular to b.

A 12

B 3

C 2

D 0

x

3

7


Page 25

145 u and v have components

4

2

1

and 3

4

x

respectively.

Find x if u and v are perpendicular.

A 6

B 9

4

C 5

2

D 13

146 u and v are the vectors with components

2 0

1 and 1

0 1

respectively.

What is the exact value of cos

where is the angle between the

vectors?

A ½

B 1

6

C 6

D 2

147

What is the angle between OP and OQ?

A 30

B 45

C 60

D 90

148

In the diagram, 3 r s and the angle is 3

.

What is the value of . ( )?r r s

A 0

B 9

C 18

D 27

2

149

In the isosceles triangle shown

1 and 2. a b c

Find . ( ).a b c

A 0

B 1

2

C 1

D 3

a

b c

4

3

r

s

O(0, 0, 0)

P( 1, 0, 1)

Q(0, 1, 1)

2

2

0


Page 26

150

In the equilateral triangle shown the

length of each side is 2 units.

Find . .q r

A 2

B 2

C 2 3

D 4

p

q

r

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Higher Mathematics Objective Test Practice · PDF fileHigher Mathematics Objective Test Practice Book Page 3 y f x 7 The diagram shows a sketch of y f x ( ). Find an expression for