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# 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 1 of 19]

## AS Math Top 71 Questions

Level LS Chapter 1 Algebra
1.

a)

b)

2.

Solve
a)

x 2 y 8
2

x 3 y 5

b)

y x 1
2

x xy 3

c)

y 2 x 1
2

2
x y 3

3.

## Given that the equation k 1 x 2 4 x k 1 0 where k is a constant, has equal roots.

Find he possible values of k.

Level MS Chapter 6
4.

a)

b)

## State the value of the common difference.

n

c)

Show that

3r 2 2 3n 7
r 1

Level MS Chapter 1
5.

06/MH

## Find the set of values of x for which

a)

4 2 x 3 3 4 x

b)

3x 2 7 x 2 0

c)

4 2 x 3 3 4 x and 3x2 7 x 2 0

d)

x2 9 x 10 0

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## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 2 of 19]

Level J Chapter 2
6.

b)

## Calculate the coordinates of point T the point of intersection of the line

the line

c)

with slope

1
and passing through the point 8, 3 .
2

a)

with

## with the y-axis. Calculate the area of OCT .

Level LS Chapter 1
7.

a)

Write

b)

Express

c)

Simplify

3 3 2
4 2

in the form a c

6 x5 3
3x

8.

3
f x

x 2
x

logarithm.

## Grade 11 S ,MS Book ,Chapter 3

dy
2
of the curve (C) is given by 2 x 3
dx

9.

(a)
(b)
(c)

06/MH

Find the equation of the normal to (C) at P(1, 2), P is on the curve.
Find an equation for the curve (C)
Show that there is no point on (C) at which the tangent is parallel to the line
y = 4 3x.

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## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 3 of 19]

Level I Chapter 8
10.

a)

Factor completely: x3 5x 2 4 x

b)

## Given the equation x2 k 1 x 1 0

Find the value of k for which
i) the equation has two distinct roots
ii) the equation has two equal roots
iii) the equation has no real roots

## Grade 11 S ,MS Book ,Chapter 4

11.

Given f x x 1 x 2 9 . The curve cuts the x-axis at the points S, (1, 0) and T.
a)

b)

Show that

c)

## Sketch the graph (C) of f x .

d)

Find the equation of the tangent to the curve (C) at the point x 1

e)

Find the exact coordinates of the point R on (C) where the tangent is parallel to the

dy
3x 2 2 x 9
dx

## tangent found in (d).

06/MH

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## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 4 of 19]

## Grade 11 S ,MS Book Section 4.4

12.

2a
a
4a
Given that the capacity of the cuboid given above is 1000 cm3,
(a) express x in terms of a,
(b) show that the surface area, A cm2, of the cuboid is given by

A 16a 2

1500
.
a

(c) Find the value of a for which A is a minimum and calculate this minimum value and
prove that it is minimum.

## Grade 10 S ,LS Book ,Chapter 4,

Grade 11S ,MS Book, Chapter 4
13.

Given that
g(x) = x2 8x + 25, x 0,
(a) express g(x) in the form (x a)2 + b, where a and b are integers.
(b) Sketch the graph of C.
(c) Find the x-coordinates of the points of intersection of y 50 with g x .

06/MH

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## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 5 of 19]

Level J chapter 2
14.
The point D 10, 4 is the midpoint of the point of AC where A 2,8 and C(a, b).
a)

Find a and b.

b)

## Point B has coordinates B 2, 4 . Find the x-coordinate of the point of intersection

of the perpendicular to AC through B with the line AC.

Level J Chapter 2
15.

and

1

and

## with the line y a, a 1 respectively, find the area of PCD in terms of a.

Level MS Chapter 6
16.

## A sequence of numbers is given by an a 2n1 2an1 1 , a2 2 , n 2 . Find the values

of a3 , a4 and a5 and deduce the values of a20 and a21 and an .

Level MS Chapter 6
17.

a)

Sn
b)

## Prove that the sum of the first n terms of an arithmetic series is

n
2a1 n 1 d , where a1 is the first term and the common difference is d.
2
The first term and the second term of an arithmetic progression are respectively
150 and 147.
i)

ii)

## Form an equation in n and solve it.

Level MS Chapter 6
18.

A plant was observed growing for 10 consecutive days. The amount the plant grew in cm
in each of the 10 days form an arithmetic sequence with the first term to be a cm and the
common difference d cm. the plant grew 1.2 cm during the 10th day and grew a total of
13 cm over the 10 days. Find the values of a and d.

06/MH

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## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 6 of 19]

Level MS Chapter 6
19.

a)

b)

## Show that 3an an1 2an1

c)

Find

ar
r 1

Level MS Chapter 3
20.

The curve (C) has equation y x3 3x 2 . The points A and B on the curve (C)have
x-coordinates 3 and 1 respectively.
a)

Find the equations of the tangents to (C) at A and B. what is your conclusion

b)

Find the equation for the normal to (C) at A, giving your answer in the form
y mx b .

c)

## Find the length of AI where I is the midpoint of AB.

Level MS Chapter 10
21.

Find, giving your answer to 3 significant figures where appropriate, the values of x for
which
a)

5x 7

b)

log3 3x 2 log3 x 4

c)

d)

e)

## Solve for x: 3log 4 x log 4 3x 1

Level LS Chapter 2
22.

f x x3 ax 2 bx 30 .
When f x is divided by x 1 , the remainder is 16.
When f x is divided by x 1 , the remainder is 36.

06/MH

a)

b)

## Show that x 2 is a factor of f x .

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## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 7 of 19]

## Grade 11 S ,MS Book ,Chapter 8

23. Solve 4 cos2 x = 7(1 + sin x)

, 0 x 360

Grade 11 S ,MS Book ,Chapter 2
24. (a)

## Find the first 3 terms, in ascending powers of y, of the binomial expansion of

(1 + ry)6,

where r is a constant.
The first 3 terms are 1, 24y and my2, where m is a constant.
(b)

## Find the value of r and the value of m.

Level MS Chapter 2
25. Find the first 3 terms, in ascending powers of x in the binomial expansion of 3 x

## Grade 11 S ,MS Book ,Chapter 4

26.
4a m

bm

The figure shows a rectangle joined to a semicircle. The perimeter of the figure is 60 m.
(a)
(b)
(c)

06/MH

## Find an expression for the area of the figure in terms of a.

Find the value of a at which the area has a stationary value.
Prove that the value of a found in part (b) gives the maximum value of the area and
find this value.
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## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 8 of 19]

27.
(a)

## Find the points of intersection of y x 2 6 x 10 and y 5x 12

(b)

Calculate the area between the line and the curve given in part(a).

## Grade 11 S ,MS Book ,Chapter 10

28.

a)

Sketch the graph of y 2 x , showing clearly the intersection with the y-axis.
1

b)

Use the trapezium rule, with 5 intervals to find the approximate value of 2 x dx to 3
0

decimal places
Level MS Chapter 4
29.

Given f x 3x3 8x 2 4 x
a)
b)

## Find the coordinates of the turning points of the curve of f x

Use the second derivative to determine the nature of these turning points.

## Grade 11 S ,MS Book ,Chapter 12

30.

Given the points A(2,2) and B(4, 6) are the end points of a diameter of the circle M.
Find
(a)

(b)

(c)

## Level MS Chapter 3 & 10

31.

Given y x3 6 x
a)

06/MH

dy
dx

1
x

b)

1
, x 0 , find
x

d2y
dx 2

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c)

ydx

## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 9 of 19]

## Grade 11 S ,MS Book ,Chapter 6

32.

The sum to infinity of a geometric series is 9. If the first term is a, the common ratio r and the
second term of the series is 2.
(a) Find the values of r and the two corresponding values of a.
(b) Show that the sum, Sn, of the first n terms of the series is given by Sn = 9(1 rn).
(c) If r takes the larger of its two values, find the smallest value of n for which Sn exceeds 7.

## Grade 11 S ,MS Book ,Chapter 5

33.

Given f x 2 x 1, 0 x 30

10

f x

5.57

31.98

15

20

25

30

1024.00

(a) Complete the table, giving the values of f to 2 decimal places & use the trapezium rule to
30

evaluate

## 2 x 1 , giving your answer to 3 decimal places.

Level MS Chapter 7
34.

a)

## b) Solve for : tan 2 9 , 0 360

Level MS Chapter 9
c)

## Calculate the value(s) of x

A

x
C

06/MH

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## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 10 of 19]

Level MS Chapter 5
35.

## Evaluate each of the following integrals

27

a)

b)

dx
x

6 x

4x

6 3

dx
x x3

Level MS Chapter 7
36.

## Solve for x, 0 x 180

a)

cos x 50

b)

sin 2 x 0.6

1
2

Level MS Chapter 7

7

## Sketch the graph of y cos x , x

, showing clearly where the graph meets the
4 4
4

coordinate axes.

37.

38.

## (a) Prove that

n
1 cos 2a
tan a, a
, n .
sin 2a
2
(b) Solve 3(1 cos 2a ) = tan a ,

0<a <.

Level MS Chapter 7
39.

a)

Calculate BOA

c)

e)

## Find the perimeter of OACB

06/MH

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## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 11 of 19]

40.

## Find the coordinates of the centre of the circle C1 with equation x2 + y2 4x + 2y + 1 = 0

a)

Circle C2 has centre at the point (2, 1) and radius 2.
(b) Sketch the two circles on a single diagram and find the coordinates of their point(s) of
intersection.
Level MS Chapter 12 Algebra
41.

## Given the points A 6, 2 and B 12,12 respectively.

Find the coordinates of the midpoint of AB and the equation of the circle of diameter AB

Level MS Chapter 6
42.
a)

Find the common ratio of a geometric series of first term 150 and sum to infinity
450.

b)

Calculate the sum of the first 10 terms of the series in part (a).

c)

Calculate the smallest value of n if the sum of the first n terms of the series is
greater than 300.

## Level LS Chapter 2 & Level MS Chapter 10

43.

f x x3 4 x 2 11x 30
a)

b)

## Level MS Chapter 12 and 10

44.

a)

Write the equation of the circle with centre m, n , n 0 and radius 6 touching the
x-axis at 3, 0

b)

A tangent through the point K 5,12 touches the circle at P. Calculate the length
of KP.

06/MH

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## 1617-AS Math Top 71 Questions

45.

Updated on 7/10/2016

[Page 12 of 19]

## On a separate diagram, sketch the graphs of

a)

y f x

e)

y f x 1 f)

i)

y 3 f x

y f x

b)

c)

y f x 1 g)

y f x

1
f x

d)

y f x

h)

y f 3x

Level MS Chapter 12
46.

AB is a chord of a circle with center I. the point A has coordinates 2, 1 and the point
B has coordinates 4,9 . Given that M is the midpoint of AB .
a)

b)

c)

## Write the equation of the circle.

Level MS Chapter 12
47.

## The line 4 y 3x 20 0 is tangent to the circle C at the point T 4, 2 . The center I of C

belongs to the line whose equation is y 2 x 0

06/MH

a)

b)

c)

## Find an equation for C

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## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 13 of 19]

Level MS Chapter 12
48.

a)

Find the coordinates of the center M and the radius of the circle C whose equation
is x2 y 2 8x 14 y 4 0 .

b)

## Find the x-intercepts of the circle C

c)

The slope of the line tangent to C at a point P on the circle is 2. Find the equation
of the line PM

Level MS Chapter 6
49.

The third and the fifth terms of a positive geometric series are 0.5 and 0.125 respectively.
For this series,
a)

b)

## the sum to infinity,

c)

the difference between the sum to infinity and the sum of the first 7 terms, giving

Level MS Chapter 6
50.

a)

## Prove that the sum of the first n terms of a geometric series is a

1 rn
where a
1 r

and r are the first term of the series and the common ratio respectively.
b)

Mr. Marc paid in the year 2007 an amount of \$35,000 for his new house. The
dealer told him that the price increases 5% every year, the first increase being due
in 2008
i)

ii)

## find, price of the house in 2020.

Level MS Chapter 6
51.

a)

1

90

06/MH

b)

c)

## Find the sum to infinity of the geometric series

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2 2
2

.
3 21 147

## 1617-AS Math Top 71 Questions

Updated on 7/10/2016

[Page 14 of 19]

Level MS Chapter 5
52.

Find the exact area of the region bounded by the line with equation y 4 x 19 and the
curve with the equation y x 2 3x 11 .

Level MS Chapter 5
53.

## The points A and B of abscissas x 1 and x 2 respectively lie on the curve C of

f x 2x

16
x2

, x 0.

Find the equation of the line AB and the area bounded by AB and the curve C.

Level MS Chapter 5
54.

a)

## y 2 x3 9 x 2 12 x distinguishing between them.

The perpendicular to the x-axis through N meets the x-axis at T.
b)

Find the area of the region R bounded by the curve, the x-axis and the line MT.

Level MS Chapter 5
55.

Find the area of the finite region that is bounded by a curve (C) whose equation is

56.

## The figure shows the graph of the curve with equation y f x

In separate diagrams, sketch the curves with equations
a)

y f x

b)

y f x2

1
3

57.

r 1

## Find the least value of n for which Sn 210

06/MH

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3 n
3 1 .
2

## 1617-AS Math Top 71 Questions

58.

Updated on 7/10/2016

[Page 15 of 19]

If the 20th term of an arithmetic sequence is 50.25 and the sum of the first 20 terms is
1,050, find the first term and the common difference, giving exact values.

59.

2

a)

## Find the equation of the perpendicular bisector of AB

b)

Find the distance from the center of the circle to the chord AB

c)

Find the area of the sector AMB where M is the center of the circle.

60.
A man invested \$260,000 at the end of the year 2002 at a fixed interest rate of 3% per
year,

61.

a)

## Show that the amount of money at the end of 2004 is \$275,834

b)

At the end of which year will the amount of money exceed \$2,000,000?

a)

## Find the first 4 terms of the binomial expansion, in ascending powers of x, of

8

x
1 , giving each term in its simplest form.
3
b)

8

decimal places.

06/MH

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## 1617-AS Math Top 71 Questions

62.

Updated on 7/10/2016

point M.

1
x
4e 2

[Page 16 of 19]

## 5 x 2 and its minimum

M
x

O
a)

Show that the x-coordinate of M can be written in the form ln a , where the value of a
is a fraction to be stated.

b)

Find the exact value of the area of the region enclosed by the curve and the lines
x 0 , x 3 and y 0 .

63.

4
1

The line 1 has vector equation r 7 1 and the line
4
3

1
3

r 2 9 where and are parameters.
3
1

The lines

and

a)

## Find the position vector of B

b)

Find the value of cos where is the angle between the two lines.

c)

64.

The line

## has vector equation r 8i 4 j 10k i 2 j 3k where is a parameter.

The point A has coordinates 7, 6, m where m is a constant. The point B has coordinates

a)
06/MH

## Find the values of m and n.

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## 1617-AS Math Top 71 Questions

Point P lies on

65.

Updated on 7/10/2016

[Page 17 of 19]

c)

a)

## Find the value of m if the given pair of lines are intersecting

r 2i 3j k 2i 7 j k and r i 3j t mi 6 j k
b)

## Referred to a fixed origin, the points A and B have position vectors i 2 j 3k

and 5i 3j respectively. Find, in vector form, an equation of the line

which

66.

a)

1,

## passing through the two points A 1,1,1 and

B 2,3,5 .

b)

The line

1 2

of equation r 2 t 1 intersects
3 2

## at point C, find the

position vector of C.
c)

67.

Point M 1,3,5 is on

2.

## Find the shortest distance from M to

1.

5
1 3

Show that the two lines r 1 t 5 and r 6 do not intersect and find the
0
0 2

shortest distance between them.

68.

The lines

and

1 2
0
3

are of equations r 1 t 1 & r 2 1 respectively,
3 3
0
3

a)

## Find the position vector of the point C the intersection of

b)

If A is a position on

and

2.

## corresponding to t 1 and B is a point on

corresponding to 1 , calculate the area of triangle ABC to the nearest whole number.

06/MH

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## 1617-AS Math Top 71 Questions

69.

Updated on 7/10/2016

and

[Page 18 of 19]

## are given by the

equations

0
1

1 : r 3 2
5
1

5 q

2 : r 13 t 1 , where and t are parameters and p and q are constants.
10 2

Given that

and

a)

show that q 0

b)

## prove that A 5,14,8 is a point on

center C cuts

c)
70.

are perpendicular,

2.

at A and B.

## find the position vector of B

0
0

With respect to a fixed origin O, the line l has equation r 1 1 , is a scalar
2
1

parameter.
a)

## Show that the point N 0, 1, 4 lies on line l.

2p

The point M has position vector 0 relative to O, where p is a constant and MN is
p

perpendicular to l,
b)

c)

06/MH

## find the coordinates of the two possible positions of B.

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## 1617-AS Math Top 71 Questions

71.

Updated on 7/10/2016

[Page 19 of 19]

2

Relative to the origin O, the position vectors of points C and D are OC 3 and
0

1

OD 0 .
4

i)

## Find angle COD in degrees.

ii)

Find the area of triangle COD giving your answer to two decimal places.

iii)

iv)

v)

m 2

vi)

06/MH

## If OM and ON have the equal lengths, find the value(s) of m.

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