O Level Additional Mathematics Nov 2017

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NOT DC (KS) 155455 © UCLES 2017 The number of marks is given in brackets [ The total number of marks for this paper is 80. You are reminded of the need for clear presentation in your answers. are reminded of the need for clear presentation You all your work securely together. At the end of the examination, fasten Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of Give non-exact numerical answers angles in is expected, where appropriate. The use of an electronic calculator Do not use staples, paper clips, glue or correction fluid. Do not use staples, paper clips, glue DO Answer Write your Centre number, candidate number and name on all the work you hand in. candidate your Centre number, Write in dark blue or black pen. Write or graphs. may use an HB pencil for any diagrams You READ THESE INSTRUCTIONS FIRST Candidates answer on the Question Paper. Candidates answer on the Question Additional Materials are required. No ADDITIONAL MATHEMATICS ADDITIONAL Paper 2

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Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation ax2 + bx + c = 0, −bb2 − 4ac x = 2a

Binomial Theorem

n n n n–1 n n–2 2 n n–r r n (a + b) = a + (1 )a b + ( 2 )a b + … + ( r )a b + … + b , n n! where n is a positive integer and = ( r ) (n – r)!r!

2. TRIGONOMETRY

Identities

sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A

Formulae for ∆ABC a b c sin A = sin B = sin C

a2 = b2 + c2 – 2bc cos A 1 ∆ = bc sin A 2

1 (a) On each of the diagrams below, shade the region which represents the given set.

Ᏹ Ᏹ A B A B

C C

ͧA ʜ Bͨ ʝ C' ͧA ʝ B' ͨ ʜ C [2]

(b) Ᏹ P Q

7 2 4

8 1 3 5 6

The Venn diagram shows the number of elements in each of its subsets.

Complete the following.

n(P') = ......

n((Q  R)  P) = ......

n(Q'  P) = ...... [3]

2 Solve the equation 31xx-=5 + . [3]

p 1 3 Find integers p and q such that + =+q 33. [4] 31- 31+

4 Solve the simultaneous equations

+=+ logl33xy11og , ^h log xy-=2. [5] 3 ^h

5 D

O B C

The diagram shows points O, A, B, C, D and X. The position vectors of A, B and C relative to O are 3 OA = a, OB = b and OC = b. The vector CD = 3a. 2 (i) If OX = mOD express OX in terms of m, a and b. [1]

(ii) If AX = nAB express OX in terms of n, a and b. [2]

(iii) Use your two expressions for OX to find the value of m and of n. [3]

AX (iv) Find the ratio . [1] XB

OX (v) Find the ratio . [1] XD

6 The functions f and g are defined for real values of x by

f xx=+212 + , ^^hh x - 2 1 g x = , x ! . ^h 21x - 2 (i) Find f 2 -3 . [2] ^h

(ii) Show that gg-1 xx= . [3] ^^hh

8 (iii) Solve gf x = . [4] ^h 19

7 A particle moving in a straight line passes through a fixed point O. Its velocity, v ms-1 , t s after passing through O, is given by vt=-32cos 1 for t H 0.

(i) Find the value of t when the particle is first at rest. [2]

r (ii) Find the displacement from O of the particle when t = . [3] 4

(iii) Find the acceleration of the particle when it is first at rest. [3]

B C

–1 50 m 1 ms 3 ms–1

a A

A man, who can row a boat at 3 ms−1 in still water, wants to cross a river from A to B as shown in the diagram. AB is perpendicular to both banks of the river. The river, which is 50 m wide, is flowing at 1 ms−1 in the direction shown. The man points his boat at an angle a° to the bank. Find

(i) the angle a, [2]

(ii) the resultant speed of the boat from A to B, [2]

(iii) the time taken for the boat to travel from A to B. [2]

On another occasion the man points the boat in the same direction but the river speed has increased to 1.8 ms−1 and as a result he lands at the point C.

(iv) State the time taken for the boat to travel from A to C and hence find the distance BC. [2]

d ln x 13- ln x 9 (i) Show that = . [3] dxcmx34x

ln x (ii) Find the exact coordinates of the stationary point of the curve y = . [3] x3

Jln xN K O (iii) Use the result from part (i) to find yK 4 Odx . [4] L x P

sin x 1 + cos x 10 (a) Show that + = 2 cosec x. [3] 1 + cos x sin x

(b) Solve the following equations.

(i) cotc2yy+-osec 50= for 0°°GGy 360 [5]

r 3 (ii) cos 2z +=- for 0 GGz r radians [4] `j42

Question 11 is printed on the next page.

11 The cubic equation x3 + ax2 + bx − 36 = 0 has a repeated positive integer root.

(i) If the repeated root is x = 3 find the other positive root and the value of a and of b. [4]

(ii) There are other possible values of a and b for which the cubic equation has a repeated positive integer root. In each case state all three integer roots of the equation. [4]

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