SJI 2025 AMATH PRELIM P2 QP

ST JOSEPH’S INSTITUTION PRELIMINARY EXAMINATION 2025 (YEAR 4) CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS Paper 2 Candidates answer on the Question Paper. 4049/02 27 August 2025 2 hours 15 minutes (09:00 – 11:15) 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 19 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 (a) Given that 2xu , express   2 12 5 2 3x x   as an equation in u. [1] (b) Hence find the value of x for which   2 12 5 2 3x x   , giving your answer correct to 2 significant figures. [4]

4 2 (a) Factorise     3 3 2 1 2 1x x   completely. [2] (b) Hence show that the equation       3 3 2 1 2 1 2 1 0x x x        has only one real root. [2]

5 [Turn over 3 (a) It is given that  3 1y x x   . Find d d y x in the form 2 1 ax b x   , where a and b are integers. [4] (b) Hence find the value of 3 0 9 d 1 x x x  . [5]

6 4 A mover needs to deliver a refrigerator to a customer. The refrigerator can be modelled by a cuboid of dimensions 80 cm by 80 cm by 185 cm. To move the refrigerator into the kitchen through a door, the mover has to tilt the refrigerator such that AD makes an acute angle  with the horizontal ground. (a) Show that the height, h cm, of C above ground is given by 185sin 80cosh    . [2] A B D C

7 [Turn over (b) Express h in the form sin( )R   , where 0R and 0 90   . [3] (c) Find the value of  when 180h . [3]

8 5 The equation of a curve is 22 5y x  . (a) Find the set of values of x for which 23y x and represent this set on a number line. [4]

9 [Turn over The line 3y kx  is a tangent to the curve at the point P. (b) Find the possible values of k. [4] (c) Given that k > 0, find the coordinates of P. [3]

10 6 (a) Explain why the binomial expansion of 6 2ax x     contains only even powers of x. [2] (b) Write down, and simplify, the first three terms in the expansion of 3 1 x a     in ascending powers of x. [2]