EJC 2023 JC1 Promos 9649

2023 JC1 H2 Further Mathematics Promotional Examination Paper 1 [Turn over EUNOIA JUNIOR COLLEGE JC1 Promotional Examination 2023 General Certificate of Education Advanced Level Higher 2 FURTHER MATHEMATICS Paper 1 9649/01 28 September 2023 3 hours Additional Materials: 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. Answer all 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing 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.

2 2023 JC1 H2 Further Mathematics Promotional Examination Paper 1 1 The curve C has polar equation 2s i 0 2 πn3 ,r   . (a) Sketch C, indicating clearly all key features and symmetries of the curve. [3] (b) Given that r attains a minimum value at the point with 11 π6  , find the exact cartesian equation of the tangent to C at this point. [3] 2 (a) Given that y is a function of x, find expressions for d d xyx ,  2 2 d d xyx and  3 3 d d xyx in terms of x and derivatives of y. Hence conjecture an expression for d d n n xyx in terms of x and derivatives of y, for positive integers n. [2] (b) Use mathematical induction to prove th e correctness of your conjecture. [4] (c) Hence find an expression for d ed n n xxx  in terms of x. [2] 3 Consider the system of equations 51 21 1 4 21 6 1 1 xyz xy a z xa y z     where a is a real constant. (a) Determine the values of a such that the system of equations is consistent. [3] (b) Given that infinitely many solutions exist, find the solutions of this system. [3] (c) Hence state the solutions of the system of equations 52 21 1 4 21 6 5 xyz x ya z a xa y z     where a is a constant such that infinitely many solutions exist. [2]

3 2023 JC1 H2 Further Mathematics Promotional Examination Paper 1 [Turn over 4 The curve T has equation f( )yx , where 3f( ) 3 1x xx . (a) Show that T has exactly one root  in the interval [1, 2]. [2] (b) Use one stage of the linear interpolatio n process to find an approximation,  , to the root  . Explain why this  will be an underestimate. [2] (c) (i) Write down the Newton-Raphson formula in the form 1 g( )nnx x  , where /g3nx is the n-th approximation using the formula. [1] (ii) A student proposes to use either one of th e end values from the interval as his initial approximation, 1 1x  or 1 2x  . Explain why one of these cases will fail and evaluate  correct to 3 decimal places using the other initial approximation. [3] (d) Given that the linear interpolation process alwa ys produces an underestimate over the interval  ,2nu , where 0 1u  and nu is the approximated root after the n-th iteration of the process, write down an expression for this iterative process in the form 1 h( )nnuu  . Hence comment on the efficiency of this process in comparison to the Newton-Raphson process for finding this root of the curve T. [2] 5 Let 3 be the set of vectors of the form a b c    , where a, b and c are real numbers. (a) S is defined as the subset of 3 for which 24 0ab c  . Show that S is a linear space. [2] The transformation T, from 33 , is defined by 3 T: 6 4 2 aa c ba b c ca b c                . (b) Show that T is a linear transformation. [2] It is given that T can be represented in the form T: aa bb cc         A , where A is a 33 matrix with real entries. (c) Determine the matrix A, and hence find a basis for the null space of A. [3] (d) State the rank of matrix A. [1] (e) Hence, or otherwise, show that the linear space S is the range space of T. [2] (f) Find the subset of 3 whose image under T is the point with position vector 2 0 1    . [2]

4 2023 JC1 H2 Further Mathematics Promotional Examination Paper 1 6 A sequence of real numbers  nu is defined by 0 1u  , 1u  where  is a constant, and the recurrence relation 124nnnuuu  , 0n . (a) Solve the recurrence relation to determine an expression for nu in terms of  . [4] (b) Hence state the value of  for which the sequence  nu converges, justifying your answer. [2] (c) It is now given that 0  . Another sequence  ns is defined by the expression 12 2 nn n n uus u  , 2n . (i) Show that 1 11 4(1 ) n n s s    . [2] (ii) Given that the sequence  ns converges, find the exact value of the limit. [2] (iii) State the exact value of 8s , and deduce a rational approximation for the value of 5 . [2] 7 Points P and Q lie on the x-axis with coordinates (, 0 )c and (, 0 )c respectively, with 0c  . The conic section curve L is defined as the locus of points R for which the difference between the distances, 2PR RQ a , where 0 ac . It is given that PR r and RPQ  . (a) For 2PRR Q a , show that  1c o sru v  , giving the constants u and v as fractions in terms of a and c. [4] (b) Hence write down an expression for the eccentricity, e, of L in terms of a and c. [1] (c) Using  ,x y as the coordinates of R and suitable substitutions for cosr  , sinr  and 2r , show that the cartesian equation of the curve L is   22 2 2 2 2 22ac xa ya ac  . [2] Given now that 3a and 2c  , we let the line S with equation ym xk , where 0k  , be a tangent to L. (d) Show that 22 31km . [4] Let T be another tangent to L with equation 1 f( )yx m m  , where T is perpendicular to S. (e) Write down the expression for f( )m . [1] (f) Using S and T, show that the intersection points between two perpendicular tangents lie on the circle with equation 22 2xy . [3]

5 2023 JC1 H2 Further Mathematics Promotional Examination Paper 1 [Turn over 8 A student, Ian, from the Makerspace Club plans to use the curve  222 3 2 72x yy x y  to form the shape of a crest for a club logo. (a) Show that the polar equation of the curve, ()fr  can be written in the form 35sin 2sinr   . [2] It is known that the area bounded by the curve,  1 π 22 0 )df(A   . Another student, Rian, from the same club suggests to get an estimate of A using the Simpson’s rule with 3 ordinates. (b) Evaluate Rian’s estimate, leav ing the answer in the form of πp q , where ,pq   . [3] A third student, Brian, believes that the estimate is unlikely to be re liable and suggests the following method to evaluate the exact value of A. (c) Use integration by parts to show that for integers 2n , ππ11 222 00 1si d si dnnnn n n    . [4] (d) By applying the formula in part (c), evaluate the exact value of A. [3] (e) Determine the perc entage error of the estimate in part (b) and comment on Brian’s claim. [2]

6 2023 JC1 H2 Further Mathematics Promotional Examination Paper 1 9 The National Population Commission wants to use a population model to study the migration patterns between two cities connected by a brid ge, city B and city E. The population, in thousands, of city B and city E at the end of the thn year after 2022 is defined as Bn and En respectively. In the model, it is assumed that every year 2% of city B’s p