MIX 123 Pure Math

1.
A ball is such that when it is dropped from a height of 1 metre it bounces vertically from the ground to a height of
0.96 metres. It continues to bounce on the ground and each time the height the ball reaches is reduced. Two
different models, A and B, describe this.
Model A : The height reached is reduced by 0.04 metres each time the ball bounces.
Model B : The height reached is reduced by 4% each time the ball bounces.
(i) Find the total distance travelled vertically (up and down) by the ball from the 1st time it hits the ground until it
hits the ground for the 21st time,
(a) using model A,
(b) using model B.
(ii) Show that, under model B, even if there is no limit to the number of times the ball bounces, the total vertical
distance travelled after the first time it hits the ground cannot exceed 48 metres
1
2. (a) Show that the equation + 3 sin θ tan θ + 4 = 0 can be expressed as 3 cos2 θ − 4 cos θ − 4 = 0, and hence
cos θ
1
solve the equation cos θ + 3 sin θ tan θ + 4 = 0 for 00 ≤ 1 ≤ 3600.
(b) The diagram shows part of the graph of y = a cos x − b, where a and b are constants. The graph crosses the x-
axis at the point C (cos – 1 c , 0) and the y-axis at the point D (0 , d)
Find c and d in terms of a and b
3. The function f is defined by f(x) = 3x + 1 for x ≤ a, where a is a constant.

The function g is defined by g(x) = – 1 – x2 for x ≤ – 1
(i) Find the largest value of a for which the composite function gf can be formed. [2] For the case where a = −1
(ii) Solve the equation fg (x) + 14 = 0
(iii) find the set of values of x which satisfy the inequality gf (x) ≤ – 50
dy
4. A curve passes through the point A (4 , 6) and is such that dx = 1 + 2x – ½
A point P is moving along the curve in such a way that the x-coordinate of P is increasing at a constant rate of 3
units per minute.
(i) Find the rate at which the y-coordinate of P is increasing when P is at A.
(ii) Find the equation of the curve.
(iii) The tangent to the curve at A crosses the x-axis at B and the normal to the curve at A crosses the x-axis at C.
Find the area of triangle ABC
5. The function f is defined by f(x) = 2x + (x + 1) – 2 for x > – 1
(i) Find f’(x) and f”(x) and hence verify that the function f has a minimum value at x = 0
(ii) The points A (– ½ , 3) and B (1 , 2¼) lie on the curve y = 2x + (x + 1) – 2 , as shown in the diagram
Find the distance AB
(iii) Find, showing all necessary working, the area of the shaded region
a
6. It is given that 0
(3e3x + 5ex ) dx = 100 , where a is a positive constant
1
(i) Show that a = ln (106 – 5ea)
3
(ii) Use an iterative formula based on the equation in part (i) to find the value of a correct to 3 decimal places.
Give the result of each iteration to 5 decimal places
7. (i) Express ( 5) cos θ – 2 sin θ in the form R cos (θ + α) , where R > 0 and 00 < α < 900 .
Give the value of a correct to 2dp
(ii) Hence
(a) Solve the equation ( 5) cos θ – 2 sin θ = 0,9 for 00 < θ < 3600
(b) State the greatest and least values of : 10 + ( 5) cos θ – 2 sin θ , as θ varies
sin 2x
8. The equation of a curve is y = cos x + 1
dy 2(cos 2 x+cos x−1)
(i) Show that dx = cos x + 1
(ii) Find the x-coordinate of each stationary point of the curve in the interval −0 < x < 0. Give each answer correct
to 3 significant figures
9. The angles A and B are such that : sin (A + 450) = (2 2) cos A and 4 sec2B + 5 = 12 tan B
Without using a calculator, find the exact value of tan (A – B)
10. (i) Show that (X + 1) is a factor of 4x3 – x2 – 11x – 6

4x 2 + 9x − 1
(ii) Find 4x 3 − x 2 − 11x − 6
dx
11. A plane has equation 4x − y + 5z = 39. A straight line is parallel to the vector i − 3j + 4k and passes through the point
A (0 , 2 , – 8). The line meets the plane at the point B.
(i) Find the coordinates of B
(ii) Find the acute angle between the line and the plane
(iii) The point C lies on the line and is such that the distance between C and B is twice the distance between A and
B. Find the coordinates of each of the possible positions of the point C
12. (a) It is given that (1 + 3i)w = 2 + 4i

Showing all necessary working, prove that the exact value of |w2| is 2 and find arg (w2) corrct to 3sf
(b) On a single Argand diagram sketch the loci |z| = 5 and |z – 5| = |z|.
Hence determine the complex numbers represented by points common to both loci, giving each answer in the
form reiθ
13. Naturalists are managing a wildlife reserve to increase the number of plants of a rare species. The number of plants
at time t years is denoted by N, where N is treated as a continuous variable
(i) It is given that the rate of increase of N with respect to t is proportional to (N – 150) .
Write down a differential equation relating N, t and a constant of proportionality
(ii) Initially, when t = 0, the number of plants was 650. It was noted that, at a time when there were 900 plants,
the number of plants was increasing at a rate of 60 per year. Express N in terms of t
(iii) The naturalists had a target of increasing the number of plants from 650 to 2000 within 15 years
Will this target be meet ?

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1.

3. The function f is defined by f(x) = 3x + 1 for x ≤ a, where a is a constant.

10. (i) Show that (X + 1) is a factor of 4x3 – x2 – 11x – 6

12. (a) It is given that (1 + 3i)w = 2 + 4i

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