2023 TJC VJC NYJC H3 Math Prelim QP

TEMASEK JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 3 H3 MATHEMATICS 9820/01 Paper 1 22 September 2023 3 hours Additional Materials: Answer Booklet List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. Write your Civics Group and name on all the work that you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all 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. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. 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. This document consists of 6 printed pages and 2 blank pages. [Turn over

2 1 For each 1, 2, ...,in= , let ia and ib be non-zero real numbers. (i) By considering 2 1 ( ) 0 n ii i a t b = − for all real t, prove the Cauchy Schwarz’s inequality 2 22 1 1 1 n n n i i i i i i i a b a b = = =                     . [3] (ii) Let p and q be positive real numbers. If for each 1, 2, ...,in= , i i bpq a , show that 22 ( ) 0i i i ipqa p q a b b− + +  and deduce that 22 1 1 1 () n n n i i i i i i i p q a b b pq a = = = +  +   . [3] (iii) Let ,mM + be such that im a M and im b M for each 1, 2, ...,in= . By using the result in part (ii), or otherwise, show that 22 22 1 1 1 1 4 n n n i i i i i i i Mma b a b mM= = =      +                 . [3] 2 A differential equation of the form d f ( , ) d y xy x = is said to be homogeneous if f ( , ) f ( , )tx ty x y= for any real value t. (i) Show that such a differential equation can be written in the form d g d yx xy =   , where g is a function of x y . [2] (ii) Show that 22 2 2 2d2 e ( 2 )e d xx yy xxy y y x y            = + + is a homogeneous differential equation. [2] (iii) By using an appropriate substitution, find the general solution of the differential equation, leaving your answer in the form h xy y =   . [4] A solution curve of this differential equation has a tangent at the point (4, 2)− that is perp