Paper 5 Mathematical Formulae

1. ALGEBRA

Quadratic EquationFor the equation ,

Binomial expansion

,

where n is a positive integer and

2. TRIGONOMETRY

Identities

cosec 2 A = 1 + cot 2 A

Formulae for

1. Find all angles between 0˚and 360˚which satisfy the equation

. [5]

2. (a) Prove the identity

. [3]

(b) Given that and is measured in degrees, express in

terms of q. [3]

(c) Given that , evaluate . [4]

3. Evaluate . [5]

4. Find the range of values of k for which the line intersects the curve

at two distinct real points. [5]

5. The expression , where p and q are constants, is exactly divisible by

. Find the remainder when the expression is divided by . [6]

6. Solve

(a) , [2]

(b) . [5]

7. (a) Find the first 3 terms, in ascending power of x, in the expansion of

.

Hence, find the coefficient of in the expansion of

. [4]

(b) Find the coefficient of in the expansion of . [4]

8. The line intersects with the circle at two

points A and B. If A is ( − 4, 3) find

(i) the coordinates of B, [3]

(ii) the centre and the radius of the circle. [3]

9. A curve has an equation .

(i) Find the gradient of the curve at x = 0. [4]

A point moves along the curve such that the y-coordinate is increasing at a rate of 0.5

units per second.

(ii) Find the corresponding rate of change of the x-coordinate when x = 3. [2]

10. A pond of a shape of a sector with radius, r m, and arc length, s m, is to be constructed in an

upcoming park. The pond was fenced along the perimeter in the initial stage of

construction. The length of the fence was 60 m.

(i) Express s in terms of r. [1]

(ii) Show that the area of the pond is given by

. [3]

(iii) Given that r may vary, find the value of r when A is stationary. [2]

(iv) Find the corresponding value of A and determine the nature of this value. [2]

11. (a) The figure below shows part of a straight line obtained by plotting values of the

variables against x. Express y in terms of x. [3]

`

(b) The table below shows experimental values of the variables x and y which are

related by an equation of the form .

x 0.1 0.2 0.3 0.4 0.5 0.6

y 5.9 6.9 8.1 10.5 11.2 13.1

x2y

x

(2, 5)

(6, −1)

(i) Express in a form suitable for drawing a straight line and state the variables whose values should be plotted. [2]

(ii) Using these variables, plot on a graph paper the points corresponding to the values obtained from the table. Hence, state the value of y that has been recorded incorrectly. [3]

(iii) Ignoring the incorrect point, draw a suitable straight line to estimate the values of a and of b. [4]

(iv) Use your graph to estimate a value of y to replace the incorrect value. [2]

END OF PAPER

Answers Paper 51. x = 11.5o or 168.5o

2. (b) tan = (c)

3.

4. k < -59 or k > 1

5. -20

6. (a) 6.54 (b) x = 0 or x =

7. (a) 1230848 (b)

8. (i) B(4, 7) (ii) Centre of circle = (1, 3) and radius is 5

9. (i) 5 (ii) 0.661 units/s

10. (i) s = 60 – 2r (iii) 15 m (iv) A= 225 m2 ; maximum value

11. (a)

(b)(i) Plot lg y against x (ii) 10.5 (iii) a 5.01, b 1 (iv) correct y = 9.55