Yishun Junior College 2010 Preliminary Exam H2 Maths 9740 Paper 1

1 A curve has equation given by y = , where f(x) is a quadratic function. Given

that the curve passes through the points (1, 4), (3, 12) and (2, 14), find the equation of the curve.

Hence sketch the curve, showing clearly the coordinates of the turning points and the equations of the asymptotes. [5]

2 (i) Show that . [2]

(ii) Find , in terms of N. [3]

(iii) Hence, find the value of [2]

3 The curve C has equation , where a and b are positive constants.

(i) Find, by differentiation, the coordinates of the turning point(s) of C. [3]

(ii) Sketch the graph of C, showing clearly the coordinates of any turning point(s) and the equations of the asymptotes. [2]

(iii) Hence, find the range of values of k, where k is a constant, for which the equation has no real root. [2]

(iv) On separate diagrams, draw sketches of the graphs of

(a) , [2]

(b) . [2]

4 The functions f and are defined by

, , ,, .

(i) Show that exists and define , giving its rule and domain. [4]

(ii) Determine, with a reason, whether the composite function exists or not.[2]

(iii) If and are real numbers such that < ≤ 3 and gf() > gf(), show that < 6 . [3]

5 The nth term of a sequence is given by un = n2n for n ≥ 1 and the sum of the first n terms is denoted by Sn.

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Yishun Junior College 2010 Preliminary Exam H2 Maths 9740 Paper 1

(i) Write down the values of S1, S2, S3, S4, S5. [2]

(ii) By considering Sn – 2, find a conjecture for the general term Sn in the form of Sn = p2(p + 2) + 2 , where p is in terms of n. [2]

(iii) Prove the conjecture by mathematical induction for all positive integers n. [4]

6 (i) By using the substitution , find, in terms of a and ,

[4]

(ii) Find, in terms of n and e, where . [4]

(iii) Hence, find the exact area of the region R bounded by the line x = 1 and by the curves and . [2]

7 A curve C has equation . The region R is enclosed by C and the line x

= 2. Another region S is enclosed by C and the lines x = 2 and x = k (k > 2). The volume of solid formed by region R is equal to the volume of solid formed by region S when both R and S are rotated completely about the x-axis. Find the exact value of k. [4]

8 Adam has many marbles that he wants to put in boxes.

(i) If he puts 13 marbles in the first box and for each subsequent box, he puts double the number of marbles he puts in the previous box, he will need

boxes for all his marbles. Given that he has 104 marbles in the kth

box, find the number of marbles he has. [4]

(ii) If he puts 13 marbles in the first box and for each subsequent box, he puts 13 marbles more than what he puts in the previous box, how many boxes will he need and what is the number of marbles in the last box? [4]

9 (a) The complex numbers z and w are such that z = 1 + i p, w = 1 + i q, where p and q are real and p is positive. Given that zw = 3 – 4i, find the exact values of p and q. [4]

(b) A complex number a is given by a = 1 + i . By using De Moivre’s theorem, express in the form x + i y, where x and y are exact values to be determined. [4]

10 It is given that .

(i) Find in terms of x and show that . [2]

(ii) By further differentiation of the result in (i), find the Maclaurin’s series for y, up to and including the term in x3. [3]

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Yishun Junior College 2010 Preliminary Exam H2 Maths 9740 Paper 1

(iii) Deduce the Maclaurin’s series for , up to and including the term

in x3. [3]

11 The rate at which a substance evaporates is proportional to the volume of the substance which has not yet evaporated. The initial volume of the substance is A m and the volume which has evaporated at time t minutes is x m . Given that it takes (2ln2) minutes for half of its initial volume to evaporate, show that .Find the additional time needed for three quarters of the substance to evaporate, giving your answer in exact form. [6]

12 The diagram shows a prism with the horizontal rectangular base PQRS. The triangular planes APS and BQR are vertical and AB is horizontal.

Given that PQ = SR = AB = 3 units, PS = QR = 2 units and each of the planes ABQP

and ABRS is inclined at an angle to the horizontal, where .

The point P is taken as the origin for position vectors, with unit vectors i and j parallel to PQ and PS respectively and unit vector k perpendicular to the plane PQRS.(i) Find . [1]

(ii) Find the exact value of the cosine of the angle PAR. [3]

(iii) Find a vector equation of the line AR. [1]

(iv) Show that the foot of perpendicular from P to the line AR has coordinates

. [3]

13

3

h

r

S

A B

PQ

R

i

jk

3

2

Yishun Junior College 2010 Preliminary Exam H2 Maths 9740 Paper 1

The diagram shows the cross-section of a cone of radius r and height h which is inscribed in a sphere of fixed radius R. Show that

,

where V is the volume of the cone.

Prove that, as r and h varies, the maximum value of V is obtained when . [8]

~ End of Paper ~

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