SMSS 2024 AMP1 Prelim Exam v4

This document consists of 19 printed pages and 1 blank page. ST. MARGARET’S SCHOOL (SECONDARY) Preliminary Examinations 2024 CANDIDATE NAME CLASS REGISTER NUMBER ADDITIONAL MATHEMATICS Paper 1 Secondary 4 Express / 5 Normal (Academic) Candidates answer on the Question Paper. 4049/01 15 August 2024 2 hours 15 minutes Additional Materials: NIL READ THESE INSTRUCTIONS FIRST Write your name, registration number and class on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. If working is needed for any question, it must be shown with the answer. Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π, use either your calculator or 3.142, unless the question requires the answer in terms of π. 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. The total of the marks for this paper is 90.

2 SMS(S) Prelims 2024 [Turn Over . Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, a ac b bx 2 42 − ± −= Binomial expansion nr r nnnn n b b ar nb anb ana b a + +     + +     +     + = + −−− ......21) ( 2 21 , where n is a positive integer and ! ) 1 ( ... ) 1 ( )! ( ! ! r r n n n r n r n r n + − −=−=      2. TRIGONOMETRY Identities 1cos sin2 2 = +A A AA 22 tan1sec + = AA 22 cot1cosec + = B A B A B Asin cos cos sin)sin( ±= ± B A B A B Asin sin cos cos)cos( = ± B A B AB A tan tan1 tan tan)tan(  ±= ± A A Acos sin2 2sin = AA A A A 222 2 sin2 1 1cos2sin cos2cos − = − = − = A AA 2tan1 tan22tan − = Formulae for ∆ABC, C c B b A a sin sin sin= = A bc c b acos22 2 2− + = C absin2 1= ∆

3 SMS(S) Prelims 2024 [Turn Over 1 The equation of a curve is 22 12 1yx x= −−+ . (a) Express 22 12 1xx−−+ in the form 2()ax b c++ . Hence state the maximum value of y and its corresponding value of x. [4] (b) The line 21yx= −+ intersects the curve at points A and B. Find the value of k for which the distance AB can be expressed as k . [4]

4 SMS(S) Prelims 2024 [Turn Over 2 A pot of cocoa butter is cooled from its initial temperature to a temperature of T °C in x minutes is given by ( ) xT AB= , where A and B are constants. The freezing point of the cocoa butter is 16 °C. The table below shows the corresponding values of T and x recorded. x 5 10 15 20 25 T 35.4 20.9 12.4 7.3 4.3 (a) Plot lg T against x and draw a straight line graph to illustrate the information. [2]

5 SMS(S) Prelims 2024 [Turn Over 2 (b) Use your graph to estimate the value of each of constants A and B. [4] (c) Use your graph to explain whether the cocoa butter is frozen at 13 minutes. [2]

6 SMS(S) Prelims 2024 [Turn Over 3 (a) Express ( ) 2 2 31 21 x xx − − in partial fractions. [5] (b) Hence, integrate ( ) 2 2 31 21 x xx − − with respect to x. [3]

7 SMS(S) Prelims 2024 [Turn Over 4 The equation of a curve is 2( 1) 4 2y k x xk= − + ++ , where k is a constant. (a) Find the range of values of k given that the curve lies completely below x axis. [5] (b) Find the values of k for which the line 23yx= −+ is a tangent to the curve. [3]

8 SMS(S) Prelims 2024 [Turn Over 5 (a) By considering the general term in the binomial expansion of 8 23x x  − , explain why there are no odd powers of x in this expansion. [2] (b) Given that there is no term in 4x in the expansion of ( ) 8 2 213ax x x −−  , find the value of constant a. [4]

9 SMS(S) Prelims 2024 [Turn Over 6 A curve is such that 2 2 2 sin 2cosdy xxdx = + . The curve passes through point A 3,24 π − and the gradient of the curve at A is 2 π . Find the equation of the curve. [7]

10 SMS(S) Prelims 2024 [Turn Over 7 In the diagram above, R and S are midpoints of AB and AC respectively. B, C, R and S lie on the circumference of the circle. TR is a tangent to the circle at R. TBC is a straight line. (a) Prove that angle SRC = angle RCB. [2] (b) Prove that triangle TBR is similar to triangle TRC. [2] R T S A C B