Paper 8Mathematical Formulae

1. ALGEBRAQuadratic Equation

For the equation ax2 + bx + c = 0,

x = .

Binomial expansion

,

where n is a positive integer and

2. TRIGONOMETRYIdentities

sin2 A + cos2 A = 1.sec2 A = 1 + tan2 A.

cosec2 A = 1 + cot2 A.

sin 2A = 2 sin A cos A.

cos 2A = cos2 A – sin2 A = 2 cos2 A – 1 = 1 – 2 sin2 A

.

sin A + sin B = 2 sin ( A + B ) cos ( A – B )

sin A – sin B = 2 cos ( A + B ) sin ( A – B )

cos A + cos B = 2 cos ( A + B ) cos ( A – B )

cos A – cos B = –2 sin ( A + B ) sin ( A – B )

Formulae for ΔABC

.

.

Δ = .

1 (a) Express in partial fractions. [4]

(b) Evaluate , giving your answer correct to 2 decimal places.

[4]

2 (a) Given that and , find in terms of x and y,

(i) ,

(ii) . [5]

(b) Simplify and give your answer in form where

a and b are rational numbers. [3]

3 (a) The equation has roots and while the roots of the

equation are Find the value of p and of k. [4]

(b) The roots of the equation differ by 2, find the possible values of h. [4]

4 The equation of a function is defined by for all values of x.

(a) State the amplitude and period of f. [2]

(b) Find the values of x, for , where the curve cuts the x-axis. Give your answers in terms of . [4]

(c) Sketch the graph of [2]

5 The equation of a curve is Find an expression for Hence find the

equation of the normal to the curve at the point [4]

6 To measure the thyroid condition in a person, a small dose of a radioactive iodine is injected into the blood stream. The amount of iodine A remaining in the blood after x hours is given by the equation where D is a constant. If a dose of iodine is injected into the bloodstream at noon on Monday, find

(a) the percentage of iodine remaining in the bloodstream at 6 pm on the same day, [2]

(b) the number of hours it takes for the percentage level of iodine to drop to 50% of its original dose. [3]

7 In the diagram, PAQ is a tangent to the circle at point A and AC is a diameter of the circle.

(a) Prove that . Hence or otherwise, prove that [4]

(b) Prove that . [4]

A

B

O

C

P Q

8 Express in the form where R is positive and is acute. [4]

Hence find (a) the acute angle x for which , [3]

(b) the maximum value of and the angle x between and which gives this value, [2]

(c) the minimum value of and the angle x between and which gives this minimum value. [3]

9 The equation of a curve is .

(a) Find expressions for [2]

(b) Find the co-ordinates of the stationary points and determine the nature of the stationary points. [4]

10 (a) Evaluate giving your answer correct to 2 decimal places. [4]

(b) The diagram shows part of the graphs of and . Find

(i) the coordinates of A, B and C,[3](ii) the area of the shaded region giving your answer correct to 2 decimal

places. [4]

11 A particle moves in a straight line so that, at time t seconds after leaving a fixed point

O , its velocity , v m s1, is given by .

(a) Find (i) the initial acceleration of the particle, [2](ii) the value of t when the particle is instantaneously at rest, [3]

(iii) the distance of the particle from O when t = 2. [3]

x

y

A

C

B

12

xy

xy 2cosO

(b) Sketch the velocity-time curve for t 0, indicating the coordinates of the points of intersection with the axes. [3]

12 The diagram shows a quadrilateral ABCD where AC is parallel to the line and A is on the x-axis. The coordinates of P and D are and

respectively, PC=2AP, , units and APC is a straight lines. Find

(a) the equation of AC, [2](b) the coordinates of A, [1](c) the equation of AB, [2](d) the coordinates of C and B, [5](e) the area of the quadrilateral ABCD. [1]

END OF PAPER

Answers (Paper 8)

1. (a) (b) 48.49

2. (a)(i) (ii) (b)

A

C

D

P

x

y

O

B

3. (a) (b)

4. (a) amplitude=3, period= radian (b)

5. , .

6. (a) 95.3% (b) 86.6 hrs

7. (a) (b)

8. (a) (b) (c) 0;

9. (a) (b) (3, 36) max; (-3, -36) min

10. (a) 1.14 (b) (i) A(0, 1) , B( , C (ii) 0.45 unit

11. (a)(i) 1 (ii) t=2.77 sec (iii) 1.53 m

12 (a) (b) A(-6, 0) (c)

(d) C(3, 6), B(-2, -6) (e) 67.5 units

END