Section A [12 marks]
Paper 10
1. (a) (i) Expand [pic]up to the first 3 terms. [2]
(ii) Hence, given that [pic],
find the values of h and k. [3]
(b) Evaluate the coefficient of x7 in the binomial expansion of [pic]. [3]
2. (a) If α and β are the roots of the equation [pic]and β = 2α, calculate
the values of h. [3]
(b) Given that α and β are the roots of the equation [pic], form another
equation whose roots are [pic] and [pic]. [4]
3. The line [pic]cuts the circle [pic]at two points, A and B. Find
(a) the coordinates of A and B, [5]
(b) the equation of the perpendicular bisector of AB and show that it passes
through the centre of the circle. [6]
4. The diagram shows part of the curves [pic] and [pic].
(a) Find the coordinates of P. [2]
(b) Calculate the area of the shaded region. [4]
[pic]
5. (a) Solve the following equations for 0( ( x ( 360(:
(i) [pic], [3] (ii) [pic]. [4]
(b) Prove the identity
[pic]. [3]
6. Differentiate the following with respect to x :
(a) (3 ( 2x3)10 [2]
(b) [pic] [3]
(c) [pic], leaving your answer to the simplest form. [4]
7. Variables x and y are related by the equation [pic], where a and b are constants. The table below shows measured values of x and y.
|x |1 |2 |3 |4 |5 |6 |
|y |5.7 |5.6 |5.9 |6.2 |6.6 |6.9 |
a) On a graph paper, plot [pic] against x, using a scale 2 cm to represent 1 unit on the x axis and 1 cm to represent 1 unit on the [pic] axis. Draw a straight line graph to represent the equation [pic]. [3]
b) Use your graph to estimate the value of a and of b. [3]
c) On the same diagram, draw the line representing the equation [pic] and hence find the value for which [pic]. [2]
8. (a) Given that [pic] and [pic], evaluate
(i) [pic], [2]
(ii) [pic]. [2]
(b) Evaluate each of the following :
(i) [pic] [2]
(ii) [pic] [2]
9. A particle moves along a straight line so that its displacement, s metres, from a fixed point P is given by [pic], where t is the time in seconds after passing P. Find the
(a) initial velocity and acceleration of the particle, [4]
(b) minimum velocity, [2]
(c) range of values of t for which the velocity is negative. [2]
10. The diagram shows the curve [pic] crosses the x-axis at P.
a) Find the coordinates of P. [1]
b) Find the equation of normal to the curve at P. [4]
11. The solution to this question by accurate scale drawing will not be accepted.
The diagram shows a rectangle ABCD. The coordinates of A and D are A(4, 1) and D(16, 5) and the equation of AC is y = x – 3.
Find
(a) the equation of CD, [2]
(b) the coordinates of C and B, [3]
(c) the length of AB, [1]
(d) the area of the rectangle ABCD. [2]
12. The diagram shows a prism such that each cross-section is a quadrant of a circle of radius x cm, with angle at the centre equal to 90(. The cross-sections are OAB and PDC where A, B, C, D lie on the curved surface of the prism and the vertical line OP is the intersection of the vertical plane faces OADP and OBCP. The cross-sections are horizontal and y cm apart.
a) Given that the volume of the prism is 20( cm3, express y in terms of x.
Hence show that the total surface area, A cm2, of the prism is given by [pic]. [4]
b) Find the value of x for which A has a stationary value, [2]
c) Find the stationary value of A, [1]
d) Determine if the stationary value of A is a maximum or a minimum. [2]
e) Given also that the total surface area, A, is increasing at a constant rate of 3 cm2 s-1, find the rate at which x is changing when x ( 4. [3]
----------------------------------------------END OF PAPER-------------------------------------------
-----------------------
Q
x
y
S
x cm
P
D
C
B
A
O
R
O
B
C
D(16, 5)
A(4, 1)
x
y
y cm
P
................
................
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