H1 MATH (Pure Math & Probability)

H1 MATHEMATICS REVISION

2011 Mr Teo | www.teachmejcmath-sg.webs.com 1

Revision Exercise 1

Q1 The function f and g are defined as follows:

f : x 21 4x , , 1x x & g : x 2 e x , x

i) By considering the graph of y = f(x) , explain why the inverse function 1f exists.

Define 1f in similar form stating clearly its domain. [5]

ii) Sketch the graphs of y = f(x), y = 1f (x) and y = 1f f (x) on the same diagram showing clearly

how they are related. [3]

iii) Find the range of g.

Give a reason why the composite function fg does not exist . Find the restriction on the domain

of g so that the composite function fg can exist. [4]

[AJC 06 Promo Q10]

Q2 A function is given by

2

4 17f ( )

6 8

xx

x x

, 2, 4x .

Show that if f ( )x k then 2 (6 4) (8 17) 0kx k x k .

Hence, find the range of values of k for which this quadratic equation has real roots.

[HCI 06 Promo Q6] [8]

Q3 a) Solve the equation 5(2 ) 2(4 ) 2x x [5]

b) Given that y = log a x3 and w = log x a , evaluate yw exactly . [3]

[AJC 06 Promo Q4]

Q4 Sketch, on the same diagram, the graph of 4cos2 1y x and the graph of

2 | sin |y x for 0 2x . [2]

Hence write down the number of solutions in this interval of the equation

12cos 2 | sin |

2x x . [1]

The smallest positive value of x satisfying the equation 1

2cos 2 | sin |2

x x is . Write

down, in terms of , two values of x such that 2x and they satisfy the same equation.

[2] [HCI 06 Promo Q2]



Q5. Sketch y = 1 cos x for 0 x 2 , showing clearly the axial intercepts and turning points

where necessary. [3]

By inserting a suitable sketch on the same axes as the graph of 1 cosy x , state the number of

solutions to the equation ln(1 cos )x x in the interval 0 < x < 2. [3]

[AJC 06 Promo Q2]

Q6 Differentiate the following with respect to x, simplifying your answers :

a)

6

2e 4x x [2]

b) ln ( 2e2 1x

) [3] [AJC 06 Promo Q1]

Q7 The diagram below shows the graph of the gradient function d

d

y

x against x for the curve y =

f(x). The graph has axial intercepts at ( 0, 0 ) and ( 4, 0 ) .

i) Find the range of x for which the curve is a decreasing function. [2]

ii) State the x-coordinates of the stationary points of the curve of y = f(x). [2]

It can be deduced from the graph above that one of the stationary points y = f(x) is a

maximum point. Find the x-coordinate of this maximum point , explaining your reason

clearly.

Determine the nature of the other stationary point showing your working clearly. [4]

[AJC 06 Promo Q5]

4

dy

dx

x 0 4



Q8 a) Evaluate

3ln 2

20

3e 2ed

5e

x x

xx

exactly. [4]

b) Given that 2

3

x

x = A +

3

B

x , find the values of A and B .

Hence find 2

d3

xx

x . [4]

c) Evaluate the exact value of 5

1

4 d .

3 1x

x [3]

[AJC 06 Promo Q3]

Q9 Sketch the graph of 1e xy and 2 5e xy on the same diagram, indicating clearly the axes

intercepts and the point of intersection. [3]

Hence find the exact area enclosed by the two curves and the y-axis. [5]

[HCI Promo 06 Q7]

Q10 a) Events A and B are such that P(A) = 4

7, P(A' B) = 1

3, P(B'A) = 5

12. A' and B'are the

complements of A and B respectively. Find

(i) P(A B), [2]

(ii) P(B), [3]

State, with a reason, whether or not A and B are independent events? [1]

b) In a game of darts, two players Bill and Clive hit the bulls eye with independent

probabilities of 4

5 and 3

4 respectively. If they each threw two darts, copy and complete the

following probability distribution tables.

Number of bulls eyes 0 1 2

Bill 8

25

Clive 9

16

[2]

Calculate the probability that

(i) Bill scores two bulls eyes and Clive failed to hit the bulls eye for both dart. [2]

(ii) the players scored two bulls eyes and two misses between them. [2] (iii) the players scored two bulls eyes and two misses between them given that Clive scored at

least one bulls eye. [2]

[NJC Promo 06 Q12]



Revision Exercise 2

Q1 a) Functions f and g are defined by 2

f ( ) , 22

xx x

x

and g( ) , x mx c x .

(i) Obtain an expression for f -1. (ii) Sketch, on the same diagram, the graph of f and f -1. (iii) Given that g -1(2) = f(3) and gf -1(2) = 5, find the values of m and c.

b) Given the following functions below, investigate the existence of the composite functions fg,

gf, fh, hg and hh. If a composite function exists, give its definition and state its range.

2

2

f ( ) 1,

g( ) 1 ,

h( ) 1 , [ 1,1]

x x x

x x x

x x x

Q2 a) State the minimum value of 2

3 2x and the corresponding value of x . Sketch the curve

y 2

3 2x for 3 0x .

b) Find the range of values of k for which the expression 2 28 2 15 2 7x kx x k k is never

negative for all real values of x .

c) A piece of wire of length y cm is bent into the form of a rectangle of area 1

132

cm2. If the

length of one side of the rectangle is x cm, show that 27

2y xx

. Given that the length of the

wire is 15 cm, calculate the length of the shorter side of the rectangle.

Q3 i) Show that 2x2 + x + 1 > 0 for all x . [2]

ii) By expressing f(x) = 2x3 3x2 x 2 in the form (x 2 )( 2x2 + ax + b ),

show that f(x) = 0 has only one real root. [4]

iii) Sketch the graph of the curve y = f(x) showing clearly the axial intercepts and hence solve

the inequality f(x) > 0. [3]

[AJC Promo 06 Q6]

Q4 a) Solve the following equations:

(i) 3

101 xx (ii) 03727 122 xx

(iii) xx 232 57 (iv) 02log84log 2 xx



b) Sketch the graphs of 3 1y x and 2 4 5y x x . Hence or otherwise find the roots of the

equation, 2 4 5 3 1 0x x x .

c) Sketch the graphs of cos 1,0 2y x x and 1tan , 0y x x . Hence find the number

of roots of the equation tan cos 1x x .

Q5 Differentiate the following with respect to x.

(a) 22 1xe

(b) 5

32ln 3x x

(c) 8

ln3x

(d) 3

21

x

xe

e

(e) 210 x

(f) 2

ln5 4

x

x

(g)

3

6

ln 1

xex

(h) 2

5log 2 1x

(i) 3xe

Q6 a) The curve C has its equation defined as ln4 1

xy

x

.

(i) Find the gradient of the tangent to the curve at P where x = 1.

(ii) Find the equation of the tangent at P.

(iii) If the tangent to the curve at P meets the x-axis at Q, calculate the exact coordinates of

Q.

(iv) Calculate the value of the constant k for which x = k is the equation of a

normal to the curve.



b) Consider a square vanguard sheet with length 16 cm. A simple container is formed by cutting

a square of side x cm from each corner and folding the remaining sides as shown.

Show that the volume of the container formed, V, is 256x 64x2 + 4x3.

Determine the length, l, of the container formed such that it has the greatest possible volume.

[7]

[NYJC Prom 06]

Q7 (a) Find 2 3dd

exx

. Hence find 2 3 dex xx . [2]

(b) Find 23

63 ( 1) dx x

x . [3]

(c) Evaluate 4

1 d

e 3xx

x

x

, giving your answer in the exact form. [3]

[PJC 06 Promo Q10]

Q8 The line y = x and the curve y = 10 + 4x x2 cut at points P and Q.

(i) State the coordinates of P and of Q. [2]

(ii) Find the area bounded by the line y = x and the curve y = 10 + 4x x2. [SRJC 06

Promo Q5] [2]

Q9(a) Sketch, on the same diagram, marking clearly any intersections with the axes, the graphs of

(i) x2 + y

2 = 4, [2]

(ii) x2 + 9y

2 = 9. [2]

(b) The above graphs intersect at four points P, Q, R and S. By solving the equations

simultaneously, show that the coordinates of one of these points is )8

5,

8

27( . [3]

(c) Hence, find, correct to 3 significant figures, the area of rectangle PQRS. [2]

[SRJC 06 Promo Q11]

16 cm

x

l



Q10 A certain research department comprises 26 academic and administrative staff members. There are 14 females and half of them are academic staff. Two thirds of the males are

administrative staff.

Three members are chosen at random from the department, find the probability that

(i) all three members are males and only one of them is an academic staff, (ii) two members are female academic staff and one member is a male administrative

staff,

(iii) there are more males than females, given that none of the selected members is an

academic staff,

(iv) only one member is an academic staff, given that there are more females than males in

the selection.

Given that 90% of the academic staff members and 80% of the administrative staff members

own cars. A staff member, selected at random, is found to own a car. What is the probability

that this person is an academic staff?



Revision Exercise 3

Q1 The diagram shows the graph of f ( )y x , where the domain of the function f is , 3 .

(i) Show by means of a graphical argument that the inverse function

1f does not exist. [1]

(ii) For 1f to exist, the domain of f is restricted to the subset ,S p .

State the largest possible value of p. [1]

(iii) Define the domain of f to be , p where p is the value obtained in (ii).

(a) Sketch the graph of 1f ( )y x . [1]

(b) The functions g and 1gf are defined as follows:

g : ln 2 , , 2x x x x ,

1 2gf : ln , , 0 1

1

xx x x

x

.

Find 1f (x). Hence determine f ( )x for x p . [4]

[RJC 06 Promo Q8]

Q2 a) Express 23 6 4x x in the form 2( )a x b c where a, b and c are integers, and state the

maximum value of 2

16

3 6 4x x . [3] [HCI Promo 06 Q1]

b) Find the value of c for which the equation 2 3 0x x c has equal roots. [2]

Hence, deduce the value of k for which the line 3y x k is a tangent

to the curve 2 5y x . [2] [RJC 06 Promo Q4]

y

x 0

x = 3

y = 1

f ( )y x



Q3 (a) Find the solution set of the equation | 1| | 2 1| 2x x x . [2]

(b) Sketch, on the same diagram, the graphs of | 2 1|y x and 22 1y x .

Hence, find the exact solution set of the inequality 2| 2 1| 2 1x x [6]

(c) Sketch, on the same diagram, the graphs of 21y x and 2y x .

Hence, find the exact solution set of the inequality 22 1x x [5]

[HCI 06 Promo Q13]

Q4 a) Given the simultaneous equations

3 5 and 2x y x y ,

show that ln 25

ln15x . [4]

[RJC 06 Promo Q3]

b) Solve the equations, leaving your answers in 3 significant figures where necessary.

(a) 5lg)1lg(2 x , [3]

(b) yy 75 1 . [2]

[SAJC 06 Promo Q3]

Q5 Sketch the graph of sin sin3y x x , x with the aid of a graphic calculator, showing

the coordinates of the points of intersection with the axes (if any). [3]

(i) Hence, sketch the graph of sin sin3y x x and state the stationary points of the graph.

[5]

(ii) From your graph in (i), state if f ( ) sin sin3x x x is an odd or even function. [1]

[NYJC 06 Promo Q7]

Q6 a) The gradient function of a curve is given by

3

2

d 8

d

y kx

x x

. Given that there is a turning point at x =

2 , find the value of k . [2]

i) Given that the curve passes through (2 ,7) , find the equation of the curve. [3]

ii) Hence sketch the curve , showing clearly all asymptotes, axial intercepts and stationary points. [3]

iii) By using the graph drawn in ii), find the range of values of p for which the equation

3 161

2

xp

x

has three distinct roots . [2][AJC 06 Promo Q9]



b) Determine the centre and radius of the circle with equation 2 21 1

4 43 1x x y y .

Hence sketch the circle. [5] [NYJC 06 Promo Q3]

Q7 (a) Differentiate the following functions with respect to x, expressing your answers in simplified

form.

(i) 2 7(2 1) ,y x [2]

(ii) 112ln .y x [4]

[NYJC 06 Promo Q4]

(b) The curve C has its equation defined as ln4 1

xy

x

.

(i) Find the gradient of the tangent to the curve at P where x = 1. [3]

(ii) Find the equation of the tangent at P. [3]

(iii) If the tangent to the curve at P meets the x-axis at Q, calculate the exact coordinates of Q.

[2]

(iv) Calculate the value of the constant k for which x = k is the equation of a normal to the curve.

[2] [HCI 06 Promo Q12]

Q8

The figure above shows a square BCDE of sides with fixed length 2 units inscribed in an isosceles triangle

FGH in which FG = FH and angle HFG = 2. I is the foot of the perpendicular from F to HG.

(i) Express FI and HI in terms of x, where x = tan. Hence show that the area A of the triangle FGH

is given by 1

4 4A xx

. [5]

Find the minimum value of A as x varies. [5]

(ii) The value of is increasing at a rate such that d

d

x

t= 2 units/s, where t denotes time. Find the rate

at which the area A is changing when is 60o. [4][TJC 06 Promo Q9]

C

2 units

H E I D G

2 units



Q9 (a) Find the following definite integrals in exact form (without the use of the graphic calculator):

(i)

2 10

1

e 3d

e

x

xx

[4]

(ii)

32

22

1(2 1) dx x [4]

(b) The diagram below shows the curves 2 2y x and y x . Find the area in exact form bounded

by the two curves, y-axis and the line x = 2. [5]

[NJC 06 Promo Q11]

Q10 Two boats, A and B, compete in a sailing competition that consists of a series of independent races.

Every race is won by either A or B and the first boat to win three races wins the competition. The probability

of winning is influenced by wind conditions and the probability of having strong winds is 0.2. In strong winds,

the probability that A will win is 0.9; in light winds, the probability that A will win is 0.5. For each race, the

wind condition is either strong or light and the result for each race is independent of the result for any other

races.

(i) Show that the probability of A winning a race is 0.58.

(ii) Calculate the probability of A winning not more than two races out of three.

(i) Given that A won the first race, determine the conditional probability that A will win the

competition.

2 2y x

y x

y

x

0

x = 2

H1 MATH (Pure Math & Probability)

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