Revision 9709 Paper 3

No Objectives 1 ALGEBRA

Examples Solve the following inequality: a) |x – 2| < 2x

b) |x + 2| > 2x + 1

c) 4|x| > |x – 1|

d) Find the quotient and remainder for 3x 4 + x2 – 7x + 6 ÷ (x + 3) by long division.

e) The polynomial x4 + 4x2 + x + a is denoted by p(x). It is given that (x2 + x + 2) is a factor of p(x). Find the value of a and the other quadratic factor of p(x). 1

f)

(g)

(h)

No Objectives 2 LOGARITHMIC AND EXPONENTIAL FUNCTIONS

Examples Solve the following equations: (a) 32x + 1 – 82(3x) + 27 = 0 (b) 22x+3 + 2x+3 = 1 + 2x (c) 2log2y = 4 + log2(y + 5) (d) Solve the simultaneous equation, 3x = 9(27)y log27 – log2(11y – 2x) = 1

(e) The figure shows part of a straight line graph obtained by plotting values of the variables indicated. Express y in terms of x.

No Objectives 3 TRIGONOMETRY

Examples (a) Sketch the graph of the following from 0° < θ < 180° (i) y = 2 + sin θ (ii) y = 5cos 2θ – 1 (iii) y = sec 2θ (iv) y = 2 cosec θ (b) Prove that cos θ – cos 3θ = 4 sin2 θ cos θ. (c) Prove that sin (A + B) + sin (A – B) = 2 sinAcosB. (d) Prove the identity cot θ – tan θ = 2 cot 2θ. (e) (i) It is given that f(x) = 3sin2x + 2 cos 2x. Express f(x) in the form R sin (2x +α), where R > 0, and 0 < α < ½ π. (ii)Find the solution, in radians, of the equation f(x) = 1, given that 0 ≤ x ≤ π.

No Objectives 4 DIFFERENTIATION

Examples Differentiate with respect to x, (a) ln(sin32x) (b) e1+sinx

(d) 2x e3x 1  sin 2 x (e) (c) ln x - x cos 2 x (f) A curve is defined by the parametric equations x = 120t – 4t2, dy y = 60t – 6t2. Find the value of at each of the points where the dx curve crosses the x-axis. (g) The equation of a curve y4 + x2y2 = 4a3(x + 4a), where a is a constant. Find the gradient of the curve at (a, 2a).

5 INTEGRATION

Integrate the following: 2x + 5

(a) 3 e

(b) sin(3x – 2)

Integrate the following: (e) 2 cos2 x

2

(c) sec (2x + 1)

(f) sin2 2x

1 (d) 2 x  1

No Objectives 5

Examples Integrate the following: (g)

2 x 1   2 2x  1 1  x ( 2  x) 2

(h) tan x, (j)

(i) cot 2x,

3x x 2 2

Integrate the following by parts, (k)  ln(2 x  1)dx

(l)

x

2

e x 1dx

(m)Using the substitution z = 1 – x, or otherwise, evaluate 1

x 0

2

(1  x)dx .

(n) By making the substitution x = ½ (1 + sin θ), show that 3 4 1 4



x x  x2

dx 



1 6  (1  sin  ) d . Hence evaluate 2  6

3 4 1 4



x x  x2

dx .

(o) Use the trapezium rule with 3 ordinates to obtain an approximation for the integral decimal places.

 1   2 0



sin  d giving your answer to two

No Objectives 6 NUMERICAL SOLUTION OF EQUATIONS

Examples (a) (i) Sketch on the same diagram suitable graphs to illustrate clearly that the equation, e x cos ecx  20 has exactly one root. (ii) Show that the iterative formula,

x n 1  sin 1

7 VECTORS

e xn 20

gives the same root α as equation (i). (iii) Use the iteration formula, with the first approximation of x = 0.1, to find the root α correct to 3 decimal places. (a) The points A and B have position vectors 3i + 2j and i – j respectively, with respect to the origin. The line l has a vector equation r = 5i + 5j + t(2i – j). Find the angle between l and the line passing through A and B. (b) Show that the lines given by r = (5i + 2j + 4k) + λ(I + 3j + k) and r = (3i + j + k) + μ(4i + 7j + 5k) intersect, and find the point of intersection. (c) (i) The straight line l passes through the points A and B with position vectors 7i – 3j + 6k and 10i + 3k respectively. The plane p has equation 3x – y + 2z = 8. Show that l is parallel to p. (ii) The point C is the foot of the perpendicular from A to p. Find a vector equation for the line which passes through C and is parallel to l. (d) Referred to the origin O, the position vectors of points A and B are 4i – 11j + 4k and 7i + j + 7k respectively. (i) Find a vector equation for the line l passing through A and B. (ii) Find the position vector of the point P on l such that OP is perpendicular to l. [4] (iii) Find the equation of a plane that contain l and perpendicular to the plane 2x + y – z = 0. [6]

No Objectives 8

Examples (a) A rectangular water tank has a horizontal square base of side 1 metre. Water is being pumped into the tank at a constant rate of 400 cm3/s. Water is also flowing out fo the tank from an outlet in the base. The rate at which water flows out at any time t seconds is proportional to the square root of the depth, h cm, of water in the tank at that time. When t = 0, the depth of the water in the tank is 81 cm and the rate at which water is flowing out is 500 cm3/s. (i) Explain how the information given above leads to the dh differential equation  0.04  0.01 h [4] dt (ii) Show that the solution of the differential equation in part (i) is given by (iii)

t

100 4 h

dh.

[1]

Use substitution x  h  4 to find the time for the depth of the water in the tank to decrease from 81 cm to 64 cm.

9

(a) u  2  i, uv  1  2i . Find in the form of a + ib, (i) u(1- v) (ii) v  1  3i is a complex root of the equation 2 2x2 + 2x + 5 = 0. Hence, state the other root of the equation. (c) Sketch on an Argand diagram the complex numbers 1 + 2i, 1 – 2i, -3 + i and -3 - 3i. Describe the complex numbers geometrically.

(b) Show that

No Objectives 9

Examples (d) Find the modulus and argument of

u=

3  i , giving the

argument in terms of π. Write the complex number u, in the form of r(cos θ + i sin θ). [3] (e) The complex number x + iy is such that (x + iy)2 = i. Find the possible values of the real numbers x and y, giving your answers in exact form. [4] Hence find the possible values of the complex number w such that w2 = - i. [2] (f) Show by means of an argand diagram, the locus of z such that z satisfies the following inequalities, |z – u| < 4, and |z – u| > |z|, where u = i - 2