(a)
Marks
Let f ( x ) = x 3 + 5 x 2 + 17 x − 10. The equation f ( x ) = 0 has only one real root.
4
(i) Show that the root lies between 0 and 2.
(ii) Use one application of the ‘halving the interval’ method to find a smaller interval containing the root.
(iii)
Which end of the smaller interval found in part (ii) is closer to the root?
Briefly justify your answer.

Consider the equation
4
x 3 + 6 x 2 − x − 30 = 0.
One of the roots of this equation is equal to the sum of the other two roots.
Find the values of the three roots.

1996

(a) ( x − 2) is a factor of the polynomial P( x ) = 2 x 3 + x + a . Find the value of a.

Marks
The function f ( x ) = x 3 − ln( x + 1) has one root between 0·5 and 1.
(i)
(b)
Show that the root lies between 0·8 and 0·9.
(ii) Hence use the halving-the-interval method to find the value of the root, correct to one decimal place.

1997

(b)
A particle is moving in simple harmonic motion. Its displacement x at time t is given by
4
x = 3 sin(2t + 5) .
(i) Find the period of the motion.
(ii) Find the maximum acceleration of the particle.
(iii)
(c)
Find the speed of the particle when x = 2 .
The polynomial P( x ) = x 3 + bx 2 + cx + d has roots 0, 3, and –3.
(i) Find b, c, and d.
(ii) Without using calculus, sketch the graph of y = P( x ) .
(iii)
Hence, or otherwise, solve the inequality x2 − 9 all over x > 0.

1998

Let α, β and γ be the roots of the polynomial 2x3 − 14x − 1 = 0. Find αβγ .

A particle moves in a straight line and its position at time t is given by x = 1 + sin 4t + 3 cos 4t .
(i) Prove that the particle is undergoing simple harmonic motion about x = 1.
(ii) Find the amplitude of the motion. When does the particle first reach maximum speed after time t = 0?

(b) The polynomial P(x) = 4x3 + 2x2 + 1 has one real root in the interval 6 –1 < x < 0.
(i) (ii) 1 Let x = – – be a first approximation to the root. Apply Newton’s method 4 once to obtain another approximation to the root.
(iii)
(c)
Sketch the graph of y = P(x) for x between –1 and 1. Clearly label any stationary points.
Explain why the application of Newton’s method in part (ii) was NOT effective in improving the approximation to the root.

1999

Let f(x) = x + loge x.

Use Newton’s method with a first approximation of x = 0·5 to find a second approximation to the root of x + loge x = 0.

The diagram shows the graph of the parabola x 2 = 4y. The tangent to the parabola at P(2p, p2 ), p > 0, cuts the x axis at A. The normal to the parabola at
P cuts the y axis at B.
(i) Derive the equation of the tangent AP.
(ii) Show that B has coordinates (0, p2 + 2).
(iii) Let C be the midpoint of AB. Find the cartesian equation of the locus of C.

(c)
The polynomial P(x) = x3 + px2 + qx + r has real roots
(i)
(d)
Show that kα = r.
(iii)
4
Explain why α + p = 0.
(ii)
k , − k and α .
Show that pq = r.

(b)
Show that aa pilot wishing to fly as far as possible in level flight should fly approximately 32% faster than the speed given in part (i). y 8
V
T r θ α O x The diagram shows an inclined plane that makes an angle of α radians with the horizontal. A projectile is fired from O, at the bottom of the incline, with a speed of V m s–1 at an angle of elevation θ to the horizontal, as shown.
With the above axes, you may assume that the position of the projectile is given by x = Vt cosθ y = Vt sin θ − 1 gt 2 ,
2
where t is the time, in seconds, after firing, and g is the acceleration due to gravity. 9
QUESTION 7 (Continued)
Marks
For simplicity we assume that the unit of length has been chosen so that
2V 2
=1 . g (i)
Show that the path of the trajectory of the projectile is y = x tan θ – x 2 sec2 θ .
(ii)
Show that the range of the projectile, r = OT metres, up the inclined plane is given by r= (iii) cos2 α
.
Hence, or otherwise, deduce that the maximum range, R metres, up the incline is
R=
(You may assume that
(iv)
sin(θ − α ) cosθ
1
.
2(1 + sin α )
2sin A cos B = sin( A + B) + sin( A − B) .)
Consider the trajectory of the projectile for which the maximum range R is achieved.
Show that, for this trajectory, the initial direction is perpendicular to the direction at which the projectile hits the inclined plane.