h2 Mathematics Practice Paper 1 for Prelim Exam 2011

H2 Mathematics Practice Paper 1 for Prelim Exam 2011 Compiled by Wee WS (wenshih.wordpress.com) Answer all the questions in 3 hours. Maximum marks: 100. Q1 Induction involving a sum (HCI MYE 2011/Q2) Prove by induction that 23 + 43 + 63 + ... + ( 2n ) = 2n 2 ( n + 1) . 3

2

n

Hence evaluate

∑ r3 .

[5]

r =1

Q2 Inequalities (MJC MYE 2011/P1/Q1) Explain why ln x 2 + 1 ≥ 0 for all real values of x.

(

)

Hence, without using a graphing calculator, solve

[1]

( 3 − x ) ln ( x2 + 1) 2 x 2 − x3

>0.

[4]

Q3 Roots of z n (NYJC MYE 2011/P2/Q4) Given that w = −1 − 3i , express w in the form reiθ . Find the roots of the equation ( z Im ( w ) ) − ( ww *) = 0 , giving each root in the form 6

6

re i θ . [4] Sketch the roots on an Argand diagram, showing clearly any relationship among them. Deduce the greatest and the least exact values of z − w . [2] Q4 Method of differences (TPJC SA2 2011/Q3)

r2 ( r − 1) 1 Given that f ( r ) = r , show that f ( r ) − f ( r + 1) = r +1 − r . 2 2 2 2 n  r −1 ( ) 1 Hence find ∑  r +1 − r  in terms of n. 2  r =1   2 ∞  r −1 2 ( ) 1 Deduce the value of ∑  r +1 − r  . 2  r =2   2 2

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[7]

Q5 Behaviour of a recurrence relation (MJC MYE 2011/P2/Q3) A sequence of real numbers x1 , x2 , x3 , … satisfies the recurrence relation

xn +1 = 0.3e xn − 0.03 for n ≥ 1 . (i) If the sequence converges, it converges to either α or β , where α < β . Find the values of α and β , each correct to 3 decimal places. [2] (ii) Use a calculator to determine the behaviour of the sequence for each of the cases x1 = 0 , x1 = 1 and x3 = 2 . [3] (iii) State the range of values of xn for which (a) the sequence is increasing, [1] (b) the sequence is decreasing. [1] Q6 Loci and its applications (NYJC MYE 2011/P1/Q5)  w−2 2  The complex numbers z and w are such that z − a = a and arg   = arg ( a ) , i   where a = 2 + 2i . Sketch on a single Argand diagram the locus of z and w and hence determine the complex number satisfied by the two relations. [2] Hence find

(i)

the least possible value of w + 4 2 ,

(ii)

the greatest possible value of arg z + 4 2 .

(

[2]

)

[4]

Q7 AP/GP involving bank deposits and withdrawals (SAJC MYE 2011/P1/Q5) John opens a Superb Savers current account with ABC bank and decides to deposit $x at the beginning of 2011 and then a further $x at the beginning of 2012 and each subsequent year. The bank does not pay any interest. (a) If he deposits $300 at the beginning of 2011, what is the minimum number of years for him to deposit at least a total of $50,000? [3] (b)(i) If he draws out 10% of the remaining balance from that account at the end of each year, express the total amount, in terms of x, in his account at the end of n years. [3] (ii) If he deposits $50,000 in 2011, what is the total amount in his account in the long run? [2] Q8 Inverse functions (TJC MYE 2011/Q5) 2x − a A function f is defined by f : x ֏ , x ∈ ℝ , x < a where a > 2 . x−a Show that f has an inverse function and define f −1 in a similar manner. [6] −1 On a single diagram, sketch the graphs of f and f , showing clearly the relationship between them. [3]

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Q9 Transformations of graphs (NYJC JC 1 MYE 2011/Q5 modified) The diagram shows the graph of y = f ( x ) having x-intercepts at x = −1 and x = 4 . It has the y-axis as a vertical asymptote and its oblique asymptote intersects the axes at (3, 0) and (0, 3).

Sketch, on separate diagrams, the graphs of (i) y = f ( 2x) ,

(ii)

y=

[2]

1 , f ( x)

[3]

(iii) y = f ( − x ) ,

[3]

indicating clearly any asymptotes and axial intercepts.

Q10 DE formulation and family of solution curves (AJC MYE 2011/P1/Q12) (a) After the civil war, de-mining was carried out by the United Nations peacekeepers to remove land mines planted in Country A. The land mines are de-mined at a rate inversely proportional to the number of land mines remaining. In addition, due to accidental detonation caused by the local community and wildlife, the number of land mines is decreasing at a rate of one-eighth of the land mines remaining. (i) If x (in thousands) is the number of land mines remaining at time t (in months)

(b)

1 − t Ae 4 ,

after the war ended, show that x + 8k = where k and A are constants and k > 0. [4] (ii) There are 3000 land mines before the de-mining work starts and after 1 month, 2000 mines are left. Find the time required to remove all the land mines in Country A. [3] dy Find the general solution of the differential equation y − 2 = x in the form dx 2 y = f ( x ) . Sketch, in a single diagram, the family of solution curves for the differential equation. [ It is not necessary to label the axial intercepts in your diagram. ] [5] 2

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Q11 Integration techniques (by substitution, by parts) and volume of revolution (HCI MYE 2011/Q8) 1 (a) Use the substitution u = x to find ∫ dx . [4] x ( x − 1) (b)(i) Given that I = ∫ e2 x cos 2 x dx , show by integration, 4 I = e2 x ( sin 2 x + cos 2 x ) + C , where C is an arbitrary constant. [3] (ii) The region R is bounded by the positive y-axis, the positive x-axis and the curve y = e x cos x . Find the exact volume of the solid formed when R is rotated through 4 right angles about the x-axis. [5]

Q12 Vectors (involving points and relationships between planes) (NYJC JC 1 MYE 2011/Q4 first part, HCI MYE 2011/Q6) (a) ABCD is a quadrilateral. If P and Q are the midpoints of AB and DC respectively,    [3] prove that AD + BC = 2 PQ . (b) The planes p1 , p2 and p3 have equations as follows:

 2 1 0       p1 : x + y + z = 3 , p2 : r =  0  + λ  1  + µ  1  , p3 : 2 x + ay + bz = c . 1 0 1       The planes p1 and p2 intersect in the line l. (i) Find a vector equation of l. [3] (ii) Given that the planes p1 , p2 and p3 have no point in common, what can be said about the values of a, b and c? [3] (iii) The plane p contains the point of origin. Find the Cartesian equation of p such that p1 and p2 are reflections of each other in p. [3] *** End of Paper 1 ***

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