SJI 2025 AMATH PRELIM P1 QP

[Turn over ST JOSEPH’S INSTITUTION PRELIMINARY EXAMINATION 2025 (YEAR 4) CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS Paper 1 Candidates answer on the Question Paper. 4049/01 25 August 2025 2 hours 15 minutes (10:50 – 13:05) READ THESE INSTRUCTIONS FIRST Write your class, index number and name on all the work you hand in. Write in dark blue or black pen in the space provided. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all 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  , use either your calculator value or 3.142, unless the question requires the answer in terms of  . 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. This document consists of 23 printed pages and 1 blank page.

2 1. ALGEBRA Quadratic Equation For the equation 02  cbxax , a acbbx 2 42  Binomial expansion   nrrnnnnn bbar nbanbanaba                  ......21 221 , where n is a positive integer and ! )1)...(1( )!(! ! r rnnn rnr n r n       2. TRIGONOMETRY Identities 1cossin 22  AA AA 22 tan1sec  cosec 2 A = 1 + cot 2 A BABABA sincoscossin)sin(  BABABA sinsincoscos)cos( ∓ BA BABA tantan1 tantan)tan( ∓  AAA cossin22sin  AAAAA 2222 sin211cos2sincos2cos  A AA 2tan1 tan22tan  Formulae for ABC C c B b A a sinsinsin  Abccba cos2222  Cabsin2 1

3 [Turn over 1 Without using a calculator, find the values of the integers a and b for which the solution of the equation 18 72 3x x   is 2 5 a b . [5]

4 2 Integrate 2 5 2x x     with respect to x. [3]

5 [Turn over 3 The curve 2 3 6x xy x   and the line 2 2y x  intersect at two points. Find the x-coordinates of the points of intersection. [3]

6 4 Express 29 6 2x x   in the form   2 a x b c  and hence explain why 29 6 2x x   is negative for all real values of x. [4]

7 [Turn over B P Q A D C 5 In the diagram, A, B, C and P lie on a circle. The line PQ is the tangent to the circle at P. The tangent meets CA produced at Q. The lines AC and PB intersect at D. AP bisects angle QPB. (a) Show that triangle ABP is isosceles. [3] (b) Prove that (i) triangle ACP is similar to triangle APD, [2] (ii) 2AB AD AC  . [1]

8 6 Express 4 3 2 2 2 9 7 3 ( 1)( 5) x x x x x x       in partial fractions. [6]

9 [Turn over 7 It is given that   3 2f 2 10x x ax bx    , where a and b are constants, has a factor of   2 1x . (a) Find the value of a and of b. [4] (b) It is given that    g fx x k  , where k is a constant. Using the values of a and b found in (a), find the value of k such that  g x is divisible by 2 1x . [2]

10 8 The diagram shows a solid object made from joining two hemispheres of radius x cm to a cylinder of height y cm. The volume of the object is 750 cm3. (a) Express y in terms of x. [2] (b) Show that the surface area, S cm2, of the object is given by 24 1500 3S x x  . [2] y x