Paper 6Mathematical 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 A cup of hot tea was left to cool on the table so that, t minutes later, its temperature, degree

Celsius, is given by = 26 + 64 e .

(a) Find

(i) the temperature of the tea 4 minutes later, [1]

(ii) the time taken for the tea to reach half its original temperature. [4]

(b) State the value which approaches as t becomes very large. [1]

2 (a) The curve y = 3 sin 2x + 1 is defined for 0 ≤ x ≤ 2.

(i) State the amplitude and period of y. [2]

(ii) Sketch the curve of y. [2]

(iii) By adding a suitable straight line on the same graph, find the number of solutions of the equation 6 sin 2x + x = 4 for 0 ≤ x ≤ 2. [2]

(b) Sketch the graph of y = for 0o ≤ x ≤ 360o on a separate diagram. [2]

3 The equation of a curve is y = 2 ex4 + 1.

(i) Find an expression for . [3]

(ii) Find the coordinates of the stationary point and determine the nature of the stationary point. [4]

4 (i) Express in partial fractions. [5]

(ii) Hence evaluate . [3]

5 The equation 3x2 – 5x + 1 = 0 has roots and .

(a) State the value of + and of . [2]

(b) Hence find

(i) the value of 2 + 2, [2]

(ii) the quadratic equation in x whose roots are + 1 and + 1. [4]

6 (a) Express in the form a + b , where a and b are integers. [4]

(b) Given that = a 2n3, find the value of a. [4]

7

In the circle, B, E, C, F and D are points on the circle such that CD is a diameter. The straight lines ABC and FE intersect at the point H. ADG is a tangent to the circle at D.

(a) Show that CD2 = AC.BC [4]

(b) If BDE = FDG, prove that BD is parallel to EF. [2]

(c) State, with reasons, why angle BHF = 90o. [2]

8 A particle moves in a straight line so that, t seconds after leaving a fixed point O, its velocity, v m/s2, is given by v = 5t – 6 – t2.

Find

(i) its initial velocity, [1]

(ii) the accelerations of the particle when it is instantaneously at rest, [4]

(iii) the total distance travelled by the particle in the interval t = 0 to t = 3, [4]

(iv) the maximum velocity. [2]

9

B

E

C

F G

D

A

H

y

xC

D

B0

A(1, 3)

y =

The diagram shows part of the curve y = .

The tangent to the curve at the point A(1, 3) crosses the x-axis at B.

The line CD, with equation x = 3, is parallel to y-axis.

(i) Find the equation of the line AB. [5]

(ii) Calculate the area of the shaded region. [6]

10

The diagram shows a parallelogram ABCD in which A is (3, 7), B is (1, 3) and D is (p, 4), where p is a constant. The perpendicular bisector of AB cuts the x-axis at point C.

(i) Find the coordinates of C. [5]

(ii) Find the value of p. [2]

(iii) Calculate the area of the parallelogram ABCD. [2]

(iv) Hence, or otherwise, find the shortest distance from C to AD. [3]

11

S T

R

Q

P

7 cm

4 cm

A(3, 7)

B(1, 3)

C

D(p, 4)

x

y

In the figure, PQRS is a rectangle of length 7 cm and breadth 4 cm and RST is radians, where is acute.

Let h cm be the perpendicular distance from Q to the line ST.

(i) Show that h = 4cos + 7sin . [3]

(ii) Express h in the form R cos( ), where R is a positive constant and is an acute angle in radians. [6]

(iii) Find the value of when h = 8. [2]

(iv) Find the maximum value of (4cos + 7sin )2 and the corresponding value of .

[2]

Paper 6 (Answers)1 (a) (i) 54.8oC (ii) 6.07 minutes (b) 26

2 (a) (i) Amplitude = 3, period = (ii)

(iii) 4 solutions

(b)

3 (i) = 8x2ex4 (4x4 + 3) (ii) (0, 3) is a minimum point

4 (i) (ii) x – 2ln(x + 2) + 2ln(x – 2) + C

5 (a) + = ; = (b)(i) (ii) 3x2 – 11x + 9 = 0

y

-2

1234

x2

23 2

23 xy

y = 3sin 2x + 1

180o 360o x

y

0o

y =

6 (a) 14 – 6 (b) a = 63

8 (i) –6 m/s (ii) –1 m/s2 & 1 m/s2 (iii) m (iv) 0.25 m/s

9 (i) y = –12x + 15 (ii) units2

10 (i) C = (12, 0) (ii) p = 14 (iii) 50 units2 (iv) 4.39 units

11 (ii) cos( 1.05) (iii) = 1.18 rad (iv) 65; = 1.05 rad