Paper 7

1 Given that A = , B = and C = ,

find the values of p , q and r if BA + C -1 = . [5]

2 Solve the equation log 3 x2 = 1 + 3 log x 3, leaving your answer as a surd where necessary. [4]

3 Find in ascending powers of x, the first three terms in the expansion of(i) ( 2 + x )5,(ii) ( 1 + ax )6.

Given that the coefficient of x2 in the expansion of ( 2 + x )5 ( 1 + ax )6 is 2960 , find the possible value(s) of a.

[5]

4 Differentiate x2 ln x with respect to x.

Hence, evaluate , giving your answer correct to two decimal places.

[6]

5 Given that the quadratic equation has distinct real roots , find the range of values of k. [4]

Hence or otherwise, state the range of values of k for which for all real values of x. [2]

6 Given that f(x) = ,

(i) find the value of a and of b for which 2x2 – 5x – 3 is a factor of f(x), [4]

(ii) state the third factor, [1]

(iii) using the values of a and b obtained in part (i), solve the equation

16 y3 + 4a y2 – 36 y + b = 0. [2]

7 Find all the angles, between 0 and 360, which satisfy the equation

4 sin x cos x = cos 2 x – sin 2 x . [5]

8 Find the value of p and of q for which 4 cos A – sin 2 A = (cos A + p)2 + q.Hence, or otherwise, find the maximum value and minimum value of 4 cos A – sin 2 A.

[5]

9 C

5m

D

45 1m

A Bh m

The diagram shows a right-angled triangle ABC in which AB = h m and BC = 6 m.The point D lies on BC such that BD = 1 m and DC = 5 m. CAD = 45 andBAC = . By using the expansion of tan ( 45), or otherwise, find thepossible value(s) of h.

[5]

10 The equation of a circle, C, is x2 + y2 + 2x – 4y 20 = 0.

(a) Find the coordinates of the centre of C and the radius of C. [3]

(b) The straight line x + 2y = 13 intersects the circle C at A and B. (i) Find the coordinates of A and of B.

[4](ii) Find the length of the perpendicular from the centre of the circle to the

chord AB. [2]

11 A curve has the equation y = x + .

(a) Find

(i) an expression for ,

[1]

(ii) the value of k for which y = 2x + k is a tangent to the curve. [3]

(b) A point P( x , y ) moves along the curve y = x + . When P is at the point

where x = 4, the x – coordinate is increasing at a rate of 0.02 units per second. Find the corresponding rate of change of y coordinate of P at this instant. [2]

12 The volume of a container in the shape of an open right circular cylinder of radius r cm and height h cm is 500 cm3.

(i) Express h in terms of r. [1]

(ii) A hemispherical lid is attached to the container as shown in the diagram below. External surfaces of the container and the lid are painted. It costs 3 cents per cm2 to paint the cylindrical surface and 4 cents per cm2 to paint the base and the lid. Let $C be the total cost of painting the container.

Show that C = .

[3]

r

h cm

(iii) Find the value of r which gives the minimum value of C and find the minimum cost of painting the container and the lid, giving your answer to the nearest cent. [4]

13 Answer the whole of this question on a sheet of graph paper.

The table shows experimental values of two variables x and y.

x 0.7 1.5 2.1 2.9 3.8

y 10.3 5.8 3.9 2.0 0.2

It is known that x and y are related by the equation , where a and b

are constants.

(a) Using a scale of 2 cm to represent 1 unit on each axis, draw a straight line graph of against , for the values given. [3]

(b) Use your graph to estimate

(i) the value of a and of b, [2]

(ii) the value of x when . [2]

(c) State the value of the gradient of the straight line obtained when

is plotted against .

[2]

Answer (Paper 7)

Qn No Solution 1

C -1 =

p = 2 , q = 5 , r = 16

2 x = 13.3 , 103.3, or x = 13. 3 , 103.3, 3 p = 2 ; q = 5

Maximum 4 ,Minimum 44 (i) x + 2 x ln x

(ii) = 2.31

5 (i) k x2 + ( 3k – 1 ) x + k = 0

k < or k 1 , 0

(ii)< k < 1

6 (i) a = 1 , b = 9(ii) ( x + 3)(iii)

7x =

8 (i) 32 + 80 x + 80 x2 +……

(ii) 1 + 6ax = 15a2x2 + …… a = 3 or a = 2

9 h = 2 or h = 310 (a) Centre ( 1, 2) ; radius = 5 units

b(i)

b(ii)

A ( 1,7) ; B ( 3,5)Midpoint of AB = ( 1,6)Perpendicular distance = = 4.47 unit

11 a (i)

a(ii) k = 3b 0.0175 units/sec

12 (i)

(ii)

(iii) r = 5 cmMinimum value of C = $ 28. 00

13 (b) a 1.18 , b 9.3x 2.69

(c) Gradient = 9.3