Question 7:
7
Total
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1. f(x) = px3 + 6x2 + 12x + q.
Given that the remainder when f(x) is divided by (x – 1) is equal to the remainder when f(x) is
divided by (2x + 1),
(a) find the value of p. (4 marks)
Given also that q = 3, and p has the value found in part (a),
(b) find the value of the remainder. (1 marks)
2. Figure 1
y
(a, b)
5
O 4 x
The circle C, with centre (a, b) and radius 5, touches the x-axis at (4, 0), as shown in Fig. 1.
(a) Write down the value of a and the value of b. (1 marks)
(b) Find a cartesian equation of C. (2 marks)
A tangent to the circle, drawn from the point P(8, 17), touches the circle at T.
(c) Find, to 3 significant figures, the length of PT. (3 marks)
3. (a) Expand (2√x + 3)2. (2 marks)
2
⌠
(b) Hence evaluate  (2√x+3)2 dx, giving your answer in the form a + b√2, where a and b
⌡
1
are integers. (5 marks)
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4. The first three terms in the expansion, in ascending powers of x, of (1 + px)n, are
1 – 18x + 36p2x2. Given that n is a positive integer, find the value of n and the value of p.
(7 marks)
5. Find all values of θ in the interval 0 ≤ θ < 360 for which
(a) cos (θ + 75)° = 0. (3 marks)
(b) sin 2θ ° = 0.7, giving your answers to one decima1 place. (5 marks)
6. Given that log x = a, find, in terms of a, the simplest form of
2
(a) log (16x), (2 marks)
2
x4 
(b) log  . (3 marks)
2 
 2 
(c) Hence, or otherwise, solve
x4  1
log (16x) – log   = ,
2 2 
 2  2
giving your answer in its simplest surd form. (4 marks)
 π
7. The curve C has equation y = cos x+ , 0 ≤ x ≤ 2π.
 4
(a) Sketch C. (2 marks)
(b) Write down the exact coordinates of the points at which C meets the coordinate axes.
(3 marks)
(c) Solve, for x in the interval 0 ≤ x ≤ 2π,
 π
cos x+  = 0.5,
 4
giving your answers in terms of π. (4 marks)
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8. Figure 2
y
B
R
x
O A
Figure 2 shows part of the curve with equation
y = x3 – 6x2 + 9x.
The curve touches the x-axis at A and has a maximum turning point at B.
(a) Show that the equation of the curve may be written as
y = x(x – 3)2,
and hence write down the coordinates of A. (2 marks)
(b) Find the coordinates of B. (5 marks)
The shaded region R is bounded by the curve and the x-axis.
(c) Find the area of R. (5 marks)
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9
A
P
9 cm
5 cm
Q
C B
10 cm
Fig. 3
Triangle ABC has AB = 9 cm, BC 10 cm and CA = 5 cm.
A circle, centre A and radius 3 cm, intersects AB and AC at P and Q respectively, as shown in
Fig. 3.
(a) Show that, to 3 decimal places, ∠BAC = 1.504 radians. (3 marks)
Calculate,
(b) the area, in cm2, of the sector APQ, (2 marks)
(c) the area, in cm2, of the shaded region BPQC, (3 marks)
(d) the perimeter, in cm, of the shaded region BPQC. (4 marks)
END
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