Question 7:
7
Total
PhysicsAndMathsTutor.com
1. f(x) = x 3 + ax2 + bx – 10, where a and b are constants.
When f(x) is divided by (x – 3), the remainder is 14.
When f(x) is divided by (x + 1), the remainder is −18.
(a) Find the value of a and the value of b. (5 marks)
(b) Show that (x – 2) is a factor of f(x). (2 marks)
2. (a) Write down the first four terms of the binomial expansion, in ascending powers of x, of
(1 + ax)n, where n > 2. (2 marks)
Given that, in this expansion, the coefficient of x is 8 and the coefficient of x2 is 30,
(b) find the value of n and the value of a, (4 marks)
(c) find the coefficient of x3. (2 marks)
3. A population of deer is introduced into a park. The population P at t years after the deer have
been introduced is modelled by
2000at
P = ,
4+at
where a is a constant. Given that there are 800 deer in the park after 6 years,
(a) calculate, to 4 decimal places, the value of a, (4 marks)
(b) use the model to predict the number of years needed for the population of deer to increase
from 800 to 1800. (4 marks)
(c) With reference to this model, give a reason why the population of deer cannot exceed
2000.
(1 marks)
4. Given that f(x) = (2x2 3 −3x − 2 3 )2 +5, x > 0,
(a) find, to 3 significant figures, the value of x for which f(x) =5. (3 marks)
B
(b) Show that f(x) may be written in the form Ax3 + +C, where A, B and C are constants to
x3
be found. (3 marks)
2
⌠
(c) Hence evaluate  f(x) dx. (5 marks)
⌡
1
PhysicsAndMathsTutor.com
5. Figure 1
y
B C
A 14 M 14 D x
Figure 1 shows the cross-section ABCD of a chocolate bar, where AB, CD and AD are straight
lines and M is the mid-point of AD. The length AD is 28 mm, and BC is an arc of a circle with
centre M.
Taking A as the origin, B, C and D have coordinates (7, 24), (21, 24) and (28, 0) respectively.
(a) Show that the length of BM is 25 mm. (1 marks)
(b) Show that, to 3 significant figures, ∠BMC = 0.568 radians. (3 marks)
(c) Hence calculate, in mm2, the area of the cross-section of the chocolate bar. (5 marks)
Given that this chocolate bar has length 85 mm,
(d) calculate, to the nearest cm3, the volume of the bar. (2 marks)
3 Turn over
PhysicsAndMathsTutor.com
6.
y
R
A B
y = 2
O x
y = 5 + 2x − x2
Fig. 1
Figure 1 shows the curve with equation y = 5 + 2x − x2 and the line with equation y = 2. The
curve and the line intersect at the points A and B.
(a) Find the x-coordinates of A and B. (3 marks)
The shaded region R is bounded by the curve and the line.
(b) Find the area of R. (6 marks)
7. Find all the values of θ in the interval 0 ≤ θ < 360° for which
(a) cos(θ – 10°) = cos 15°, (3 marks)
(b) tan 2θ = 0.4, (5 marks)
(c) 2 sin θ tan θ = 3. (6 marks)
END
4 Turn over