Add math Zon C

7/27/2019 Add math Zon C

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Paper 1

Functions1. In Diagram 1, the function f maps x to y and the function g maps y to z . Determine

(a) )2( f

(b) )3(1

g

(c) )2( gf [3 marks ]

2. Diagram 2 shows the function q p x

x f +: , where p and q are constants.

Find(a) the value of p and of q.(b) )4(1

f [3 marks ]

Diagram 1

g f z y x

3

-1

2

Diagram 2

f x p

+ q x

5

7

-2


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3. Given the function x f : 815 =+ x , find the values of x such that 8)( = x f .[3 marks ]

4. Diagram 4 shows the arrow diagram with function k x xm

x f

+ ,

31

: , where m is a

constant.

Find

(a) value of k

(b) value of m

(c) )3(1 f [3 marks ]

Quadratic Equations

1. Solve the quadratic equation 156 2 = y y . Give your answers correct to four

significant figures. [3 marks ]

2. A quadratic equation ( ) p x x x = 423 has two distinct roots. Find the range of

values of p. [3 marks ]

3. Express the quadratic equation 0342 =+ x x in the form ( ) 02 =++ cb xa , where a,

b and c are constants. Hence, state the values of a , b and c. [3 marks ]

4. The quadratic equation 02 2 =++ k hx x has roots 2 and -3. Find the values of h and k .

[3 marks ]

Quadratic Functions

h m+1

x-3 x

5

2


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1. Find the range of the values x for )1(2 x > ( )13 + x x . [3 marks ]

2. Diagram 2 shows the graph of a quadratic function ( ) q p x x f ++= 2)( , where p and q

are constants.

Diagram 2

0 x

y = f (x)

(3 , 2)

y

The curve )( x f y = has the minimum point (3, 2). State

(a) the value of p

(b) the value of q

(c) the value of k [3 marks ]

3. Find the range of the values x for ( ) ( )41 + x x 0. [3 marks ]

(0, k )


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4. Diagram 4 shows the graph of a quadratic function ( ) q p x x f ++= 2)( , where p and q

are constants.

Diagram 4

0 x

y = f (x)

(2 , -1)

y

The curve )( x f y = has the minimum point (2, -1). State

(a) the value of p

(b) the value of q

(c) the equation of the axis of symmetry. [3 marks ]

Progression

1) The first terms of a sequence are 4, 1+ x , 9.Find the positive value of x such that the sequence is

(a) an arithmetic progression,

(b) a geometric progression. [2 marks ]

2) The fourth term of an arithmetic progression is 12 + p and the fifth term of the

progression is 23 p , where p is a constant.

Given that the first term is 3, find the value of p. [3 marks ]

3) The first two terms of an arithmetic progression are 1 and 3.


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Find

(a) the common difference of the progression,

(b) the sum of the first 16 terms after the second term. [4 marks ]

4) The sum of the first two terms of a geometric progression is 6. The sum to infinity of

the progression is 8.

Given that the common ratio is positive, find

(a) the first term a and common ratio r of the progression,

(b) the sixth term. [4 marks ]

Paper 2

Section A

Simultaneous Equations

1. Solve the simultaneous equations 1= y x and 632 =+ y x . Give the answers correct

to three decimal places. [5 marks ]

2. Solve the simultaneous equations ( ) 11122 22 +== y x y x y x [5 marks ]

Trigonometric Function

1. (a) Sketch the graph of for 20 x . [4 marks ]

(b) Hence using the same axes, sketch the straight line to find the numbers of solutions

for the equation for 20 x . State the number of solutions.

[3 marks ]2. (a) Sketch the graph of for 20 x [3 marks ]

(b) Hence using the same axes, sketch a suitable graph to determine the number of

solutions for equation +3 for values of x between 0 and 2 .


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[4 marks ]

Progression

1. (a) Diagram 1 shows the vertical cross-section of the three smallest of n cylindrical

pipes. The external diameters of the pipes are shown

Diagram 1

The external diameter of each pipe is 4 cm longer than that of the preceding pipe.

If the external diameter of the biggest paip is 96 cm, calculate

(i) the value of n,

(ii) the total length, in cm, of the external diameters of all the pipes. [4 marks ]

(b) The sum of the first n terms, nS , of a progression is given by

=

n

nS 32

181

(i) Express the nth term, nT , in terms of n.

(ii) Determine which term in the progression is the first term less than 3. [4 marks ]

2. Two boys, John and Bob, start to save money at the same time.

a) John saves RM x in the first month and his savings increase constantly by RM y

every subsequent month. He saves RM60 in the 12 th month and his total savings

for the first 6 months are RM105. Calculate the values of x and y. [4 marks ]

16 cm 20 cm 24cm


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b) Bob saves RM17 in the first month and his savings increase constantly by RM3

every subsequent month. If John and Bob save the same amount of money in the

nth month, find the value of n. [4 marks ]

Integrations

1.

The diagram shows part of the curve ( )2134

=

x y . A region is bounded by the curve, the

x-axis and the straight line x = 1 and x = 3.

a) Find the area of the region. [3 marks ]

b) The region is rotated 360 about x-axis. Find the volume generated in terms of

.[4 marks ]

2.

Diagram shows the curve 3 x y = and the normal to the curve at the point A(1,1).

Calculate

( )2134

= x

y

x

y

0

x0

A(1,1)

3 x y =

y


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a) The equation of the normal to the curve at A. [3 marks ]

b) The area of the region enclosed by the curve, the normal and the x-axis, [3 marks ]

c) The volume of revolution, in terms of , if the region 2(b) is rotated 360 about

the x-axis. [4 marks ]

Coordinate Geometry

1. Solution by scale drawing is not accepted.

In diagram 1, the straight line AB has the equation 01832 =+ y x .

Diagram 1

The point T lies on the line AB such AT : TB = 1: 2

a) Find the coordinate of T . [3 marks ]

b) Show that the locus of a point P , which moves such that its distance from T is

always 3 units. [3 marks ]

2. Solution by scale drawing is not accepted.

Diagram 2

y

x

B

A

T

01832 =+ y x

0

y

x

R

P

Q

33 = x y

0


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In diagram 2, shows the straight line PQ has the equation 33 = x y . PQ

intersects the x-axis at point Q and intersects the y-axis at point P . Point R lies on

PQ such that PR : RQ = 4 : 1. Find

a) The coordinates of R. [3 marks ]

b) The equation of the straight line that passes through R and perpendicular to

PQ . [3 marks ]

Statistics

.1.

Table 1 shows the distribution of ages of a group of participants in a workshop. The mean

age of the data is 31.

a) Find the value of x. [4 marks ]

b) Without using an ogive, find the median of the data. [3 marks ]

2. Table 2 shows the scores of 60 contestants in a quiz.

Scores Number of contestants

15 24 1225 34 1735 44 x45 54 y55 64 8

Table 2

a) Given that the mean is 36.0, find the value of x and y.

b) Hence, state the modal class. [6 marks ]

Age (years) 20 24 25 29 30 34 35 39 40 44Frequency 12 x 14 10 6


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