7/27/2019 Unit 1 - Practice 1 Revised

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PURE MATHEMATICS

UNIT 1: Practice 1

1. (a) Simplify ( )( )5 2 5 2 .+ [1]

(b) Express 8 18+ in the form 2 ,n where n is an integer. [2]

2. The quadratic equation 2 ( 4) (4 1) 0,x m x m+ + + + = where m is a constant has equal roots.

Determine the possible values ofm. [4]

3. (a) Express 23 4 9x x + in the form ( )2

,p x q r + where p, q andrare real numbers. [3]

Hence, or otherwise, state the coordinates of the minimum point of the curve with equation23 4 9.y x x= + [2]

(b) The line L has equation 2 12y x+ = and the curve C has equation 2 4 9.y x x= + Find the

coordinates of the points of intersection ofL andC. [4]

4. The point ( 1, 5) is the centre of a circle of radius 5 units. Obtain the equation of this circle in the

form 2 2 0.x y px qy r+ + + + = [3]

A andB are the points of intersection of the circle with the line 2.x = Calculate the y-coordinates

ofA andB. [3]

A second circle, whose centre is at the point (10, 5), also passes through A andB. If the radius of this

circle is r, prove that 2 80.r = [2]

5. It is given that o o3cos 2sin cos ( ) , where 0 and 0 90 .R R = + > <

7/27/2019 Unit 1 - Practice 1 Revised

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Pure Mathematics Unit 1 (Practice 1)

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8. (a) Prove that the point (10, 12) is equidistant from the two lines whose equations are

4 3 8 and 3 4 6.y x y x = = [6]

(b) Prove that the line 3 4 0x y c+ + = is perpendicular to 3 4 6y x = where c is any constant. [2]

(c) A circle is drawn, centre (10, 12), to touch the two lines of part (a). Deduce, using part (b), the

equations of the two lines which are perpendicular to 3 4 6y x = and which touch this circle. [6]

9. (a) Using the identity sin ( ) sin cos cos sin ,A B A B A B+ + show that

sin ( ) sin ( ) 2sin cos .A B A B A B+ + [1]

Using the identity cos ( ) cos cos sin sin ,A B A B A B+ show that

2cos 2 2 cos 1.A A [2]

Hence show that

(i) sin sin 2sin cos ,2 2

P Q P QP Q

+ +

[2]

(ii) 2sin 8 2sin 6 sin 4 4sin 6 cos .x x x x x+ + [4]

(b) Find4 2

0

sin 6 sin d .x x x

[6]

10. The triangle ABC, shown in the diagram below, is such that 8cm, 12cmAC CB= = and

ACB = radians.

The area of triangle ABCis 220 cm .

(a) Show that 0.430= correct to three significant figures. [2]

(b) The point D lies on CB such that AD is an arc of a circle with centre C and radius 8 cm. The

region bounded by the arc AD and the straight lines DB andAB is shaded in the diagram below.

Calculate, correct to two significant figures:

(i) the length of the arc AD, [2]

(ii) the area of the shaded region. [3]

11. (a) Show that 2 2 3 3( ) ( ) .p q p pq q p q+ + + [2]

The equation 22 4 5 0 has roots and .x x+ + =

(b) Without solving the equation, find the value of

(i) 3 3 ,+ (ii) 3 3 . [5]

(c) Hence, form a quadratic equation, with integer coefficients, which has roots

(i) 3 32 and 2 , (ii)3 3

5 5and .

[7]