HIGHER 2 MATHEMATICS PAPER 1 - OpenStudy

YISHUN JUNIOR COLLEGE 2011 JC2 PRELIMINARY EXAMINATION

9740/1

HIGHER 2 MATHEMATICS PAPER 1

17 AUGUST 2011 WEDNESDAY 0800h – 1100h Additional materials : Answer paper Graph paper List of Formulae (MF15) YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE

TIME

3 hours

READ THESE INSTRUCTIONS FIRST Write your CTG and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question.

This question paper consists of 6 printed pages including this cover page.

Yishun Junior College ♦ 2011 Preliminary Exam ♦ H2 Mathematics 9740 1

y

α

0

The diagram shows the graph of y =

x

x 3 − x . The two roots of the equation 2x 4

x 3 − x = 0 are 0 and α . 2x 4 (i) Find the exact value of α .

[2]

A sequence of positive real numbers x1 , x2 , x3 , … satisfies the recurrence relation

xn +1 =

xn 1 + xn , for n ≥ 1 . 2 xn 4

(ii) By considering xn +1 − xn , show that xn +1 > xn if 0 < xn < α .

[2]

(iii) For the case 0 < x1 < α , use the result in (i) and (ii) to find the exact value that the sequence converges to. [3]

2

−

1

(a) (i) Expand (1 + 3x ) 2 in ascending powers of x, up to and including the term in x3 . State the range of values of x for which the expansion is valid. [3] (ii) Hence, use the substitution x =

1 to obtain an approximation of 9

3 , expressing

your answer as a fraction.

[2]

(b) A sequence u0 , u1 , u2 , … is such that u0 = 0 , and un +1 = un + 2 ( n + 1) , n ≥ 0 . Prove by induction that un = n 2 + n, for n ≥ 0 .

3

[4]

A spherical tank of radius 5 m is initially full of water. The base of the tank is then punctured so that its contents drain out at a constant rate. 2

(i) Considering the curve x 2 + ( y − 5 ) = 52 , show that when the depth of the water is 1   h m, the volume of the water in the tank is π  5h 2 − h3  m3 . 3  

(ii) Given that the tank is drained of water in 500 s, find the rate of decrease of the depth of the water when h = 3 m. 2

[3]

[4]

Yishun Junior College ♦ 2011 Preliminary Exam ♦ H2 Mathematics 9740

4

(i) Express

2

in partial fractions.

2

4r − 1 N

(ii) Hence find

∑ 4r

1 2

r =1

−1

, expressing your answer as a single algebraic fraction.

N

(iii) Using (ii), find

[1] [3]

1

∑ ( 2r + 1)( 2r + 3) , expressing your answer as a single algebraic r =1

fraction.

[5]

x+3

≤

x2 − 5

5

Without the use of a calculator, solve the inequality

6

(a) Describe a sequence of transformations which transform the graph of y = the graph of y =

( x − 2)

3

x ( x − 2)

3

.

[3]

3x + 4 to x−2

1 . x

[3]

(b) The diagram shows the graph of y = f ( x ) . The curve passes through the points A(0, 0), B(3, 0), C(4,3), and D(6, 2) with a gradient of 2 and 3 at A and B respectively. y

x=2

y=4 ×

C(4, 3) ×

×

× A(0, 0)

D(6, 2)

B(3, 0)

x

On separate diagrams, sketch the graphs of (i) y = − f ( x ) ,

[3]

(ii) y = f ′ ( x ) ,

[3]

labelling the coordinates of the corresponding points of A, B, C, and D (if applicable) and the equations of any asymptotes.

3

Yishun Junior College ♦ 2011 Preliminary Exam ♦ H2 Mathematics 9740 X

7

W

Y

Z

In the diagram above, WXYZ is a square denoted by S1 , whose sides are each of length a1 . The midpoints of the sides of S1 are then joined to form the square S2 , whose sides

are each of length a2 . Similarly, the midpoints of the sides of S2 form the square S3 , whose sides are each of length a3 and the process continues indefinitely.

(i) Given that a1 , a2 , a3 , ... form a geometric sequence, find a3 in terms of a1 .

[2]

(ii) Show that the areas of consecutive squares form a geometric progression. Hence find the sum to infinity of this geometric progression in terms of a1 . [4]

8

(a) A wire of length 40 cm is bent to form an isosceles triangle ABC with AB = AC and BC = x cm. (i) Show that the area z cm 2 of the triangle ABC can be expressed as z = x 100 − 5 x .

[2]

(ii) Hence prove algebraically that the area of the triangle is maximised when the triangle is equilateral. [5] (b) The equation of a curve C is x3 + 2 y 3 + 3 xy = k , where k is a constant. (i) Find

dy in terms of x and y. dx

[3]

(ii) It is given that C has a tangent which is parallel to the x-axis. Show that the x-coordinate of the point of contact of the tangent with C must satisfy 2 x6 + 2 x3 + k = 0 . [3] (iii) Hence, find the values of k when the line y = − 1 is a tangent to the curve C. [2]

4

Yishun Junior College ♦ 2011 Preliminary Exam ♦ H2 Mathematics 9740

9

Let y = e

2( cos x + sin x )

.

2

d y dy = 2 ( cos x − sin x ) − 2 y ( cos x + sin x ) . 2 dx dx (ii) Hence find the Maclaurin series for y, up to and including the term in x3 .

(i) Show that

[2] [4]

(iii) Hence show that when x is sufficiently small for x 4 and higher powers of x to be 1 7  2 cos x + sin x ) neglected, e x + 2 − e ( [2] ≈ xe 2  x 2 − x − 1 . 2 6 

10 The planes π 1 , π 2 and π 3 have equations r ⋅ ( 2i − k ) = 3 , r ⋅ ( −i + 3 j) = 2 and

r ⋅ ( λ j + 14k ) = µ respectively, where λ and µ are constants. The planes π 1 and π 2 intersect in a line l. (i) Find a vector equation of l.

[2]

(ii) Given that all three planes meet in the line l, find λ and µ .

[3]

(iii) Find the coordinates of the point Q which is the reflection of the point P ( 5, − 2, 7 ) in π 2 .

[4]

(iv) It is given that P lies on π 1 . Hence or otherwise, find a vector equation of the reflection of the plane π 1 in π 2 .

[2]

11 Three towns A, B, and C have business transactions with one another. Town A produces electricity and sells its surplus electricity equally to Towns B and C at $0.40 per kWh. Town B rears fishes and sells its surplus fishes equally to Towns A and C at $5 per fish. Town C grows vegetables and sells its surplus vegetables equally to Towns A and B. Town C earns $1.75 per kg of vegetables sold. However, Town C’s trade goes through a middleman, so Towns A and B each pay $2 per kg for the vegetables. Considering just the trade of these three commodities in a week, Town B has a trade surplus of $3260, while Towns A and C have deficits of $2770 and $740 respectively. Determine the quantity of each commodity sold in a week. [4]

5

Yishun Junior College ♦ 2011 Preliminary Exam ♦ H2 Mathematics 9740

12 The function f is defined as follows. f : x a 4 x 2 − 12 x + 23 for x ∈ R . (i) Prove that f does not have an inverse function.

[2]

(ii) If the domain of f is further restricted to x ≤ a , state the largest value of a ∈ Z for [1] which the function f −1 exist. In the rest of the question, the domain of f is x ∈ R , x ≤ a .

(iii) Find f −1 ( x ) . State the domain and range of f −1 .

[5]

The function h is defined as follows: h : x a −e x , x ∈ R .

(iv) Find hf in a similar form and determine the exact range of hf .

~ End of Paper ~

6

[4]