presbyterian high school 2023 A math paper 1

Name: Index No.: Class: PRESBYTERIAN HIGH SCHOOL ADDITIONAL MATHEMATICS 4049/01 Paper 1 18 August 2023 Friday 2 hours 15 min PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL 2023 SECONDARY FOUR EXPRESS / FIVE NORMAL (ACADEMIC) PRELIMINARY EXAMINATIONS DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO. INSTRUCTIONS TO CANDIDATES Write your name, index number and class in the spaces provided above. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Write your answers in the spaces provided below 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. The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 90. For Examiner’s Use Qn 1 2 3 4 5 6 7 8 9 10 11 12 13 Marks Deducted Marks Setter: Mr Gregory Quek Vetter: Mr Tan Lip Sing This question paper consists of 17 printed pages and 1 blank page. Category Accuracy Units Symbols Others Question No. TOTAL MARKS 90

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0,ax bx c+ + = 2 4 2 b b acx a −  −= Binomial expansion 1 2 2( ) ... ... , 12 n n n n n r r nn n na b a a b a b a b b r − − −     + = + + + + + +           where n is a positive integer and ! ( 1)...( 1) ( )! ! ! n n n n n r r n r r r  − − +== − 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 1 cotAA=+ sin( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin 2 2sin cosA A A= 2 2 2 2cos2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2tantan 2 1 tan AA A = − Formulae for ABC sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + − 1 sin2 bc A=

3 1 The line 2 15yx=+ intersects the curve 2 63y x x= + + at points A and B. Find the value of p for which the distance AB can be expressed as 5.p [5] 2 A curve is such that 2 2 2 d 12e e . d xxy x −=+ The curve intersects the y-axis at (0, 5)P and the tangent to the curve at P is parallel to 4 3.yx=+ Find the equation of the curve. [6]

4 3 A function is defined by 2f ( ) 2 2 3x x kx k= + + + for all real values of x, where k is a constant. (a) Find the discriminant of f ( )x in terms of k. [2] (b) Show that the discriminant of f ( )x in part (a) can be expressed in the form ( ) 2 4, k a b−− where a and b are integers. [2] (c) Find the range of values of k for which f ( ) 0x = has no real roots. [3]

5 4 It is given that 32f ( ) 2 5 4 12x x x x= − − + . (a) Show that 23x+ is a factor of f ( ).x [2] (b) Factorise f ( )x completely. [2] (c) Hence find the roots of the equation ( ) ( ) ( ) 322 2 5 2 4 2 12 0y y y− − + = . [3]

6 5 (a) Using long division, show that 32 2 2 5 10 2. 5 x x x x x − + − =− + [2] (b) Hence, by first expressing the denominator as a product of two factors, express 2 32 21 2 5 10 x x x x + − + − in partial fractions. [5]

7 6 (a) Find the first 3 terms, in ascending powers of x, of the binomial expansion of 8 2, 4 ax + where a is a non-zero constant. Give each term in its simplest form. [2] (b) Given that the coefficient of 2x is –320 in the expansion of ( ) 8 2 3 2 , 4 axx −+  find the possible value(s) of a. [4]

8 7 The diagram shows a quadrilateral PQRS whose vertices lie on the circumference of a circle. The diagonals PR and QS intersect at U. The tangent at R meets PS produced at T. If QR = RS, prove that (a) // ,QS RT [3] (b) triangle PQR is similar to triangle QUR. [3] S Q T U P R

9 8 (a) The equation of a curve is ( ) 3ln . xy xe −= The normal to the curve at the point P has a gradient of 1 2 . Find the coordinates of P. [4] (b) The normal to the curve at P meets the x-axis at Q. Find the area of triangle OQP, where O is the origin. [3]

10 9 Atmospheric pressure is a measure of the force exerted by the mass of air on an object. Altitude is the vertical height above sea level. The atmospheric pressure, P millibars, exerted at the altitude h kilometres is related by the equation e,bhPA= where A and b are constants. The following table shows the mean atmospheric pressure at various altitudes. (a) Plot ln P against h and draw a straight line graph to illustrate the information. [2] h (kilometres) 2 4 6 8 10 P (millibars) 810 595 446 340 262 ln P h 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 0