NYJC TJC VJC FM 2024 Prelim P1

1 TEMASEK JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 09 September 2024 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 7 printed pages and 1 blank page. [Turn over

2 1 Polar equations of the form cos ,r a a n =+ , , 2a n n are commonly called petal curves because of the shapes of their graphs. The number and sizes of the petals are dependent on the values of a and n. (a) State the relationship between the number of petals and n. [1] (b) Show that the area bounded by the curve is independent of n. [3] 2 Show that ( ) ( )1 i tan sec cos isin k k kk   + = + . [1] Hence or otherwise, show that 1 0 cos sec cot sin sec , n kn k kn     − = = provided  is not an integer multiple of .2  [5] 3 Let 2 0 cos dn nI x x x  = . (a) Prove that for 2,n ( ) 212 n nnI n n I − = − − . [3] (b) R is the region enclosed by the graph of 2 1sin 2y x x= , the line 2x = and the x-axis. By using the result in (a), find the exact volume of the solid generated when R is rotated 2 radians about the x-axis. [5] 4 Let nP be the vector space of real polynomials of degree n or less. The transformation 43:T P P→ is defined by, ( )( ) ( ) ( ) ( ) 23f f 0 f(1) f 1 f 2 .T x x x x = + + − + Show that T is a linear transformation. [2] Find a basis for the null space of T. [7]

3 5 The half-line L with equation 8y mx=+ , 0x , 0m , is tangential to the locus 4z = at point Q represented by complex number 4z . (a) Sketch the locus 4z = and L on a single Argand diagram. [2] (b) Find the exact value of m. [2] (c) Sketch the locus of ( )4 5arg 6zz − =− on the same Argand diagram in part (a). [2] (d) The complex number iz x y=+ satisfies the following relations: 8y mx+ , 0x , ( )Im 0zz − , 4z  , ( )4 5 arg63 zz−  − − . Shade the region R that contains z. Hence find the least ( )arg 4iz+ , leaving your answer in radians, correct to 4 decimal places. [3] 6 The polar equations of two curves are given by 1 : 1 cosCr =+ and ( )2 : 3 1 cosCr =− for 02  . (a) Sketch 1C and 2C on the same diagram, indicating clearly the symmetries and the exact polar coordinates of the point(s) of intersection of the curves. [4] (b) Find the exact polar coordinates of the P on C2, where 0,  and P is the furthest away from the x-axis. [4] (c) Find in exact form, the shortest distance between the points P and A, where A is the point of intersection between 1C and 2C in the first quadrant. [2] (d) Express the equation of C1 in parametric form using the given  as the parameter. Hence find the area of the curved surface generated when the segment OA on C1 is rotated 2 radians about the x-axis. [4] [Turn over

4 7 (a) Let A be a square matrix with eigenvalue  . Show that  is also an eigenvalue for TA . [2] A square matrix is called a stochastic matrix if all the entries are non-negative and the sum of entries of each column is 1. Let 11 12 1 21 22 2 12 n n n n nn b b b b b b b b b   =   B be a stochastic matrix. (b) By considering TB and a suitable column vector, show that an eigenvalue of B is 1. [2] Let 12() T nx x x=x be an eigenvector of TB with corresponding eigenvalue  . (c) By considering TBx , find an expression for kx , 1 kn . [2] It is further given that kx is the entry in x with the largest absolute value. (d) By considering kx , show that 1  . [4] (You may apply without proof the result 12 21 nna a a a a a+ + +  + + + for any real numbers 1,, na a .)

5 8 Diseases such as smallpox is generally considered to impart immunity for life. To assess the effect of smallpox, an experimental group of individuals born on the same day in one specific year are closely monitored until they reach t years of age. Let • ( )N N t= denote the number in the group who survived to age t, • ( )S S t= denote the number who have not had the disease but are still susceptible to it at age t, • p denote the probability of a susceptible individual getting the disease, • 1 m denote the proportion of those who die due to the disease. To study the effects of smallpox, the Swiss mathematician Daniel Bernoulli proposed in 1760 the following differential equation: 2dd dd S S N SpS pt N t mN=− + + . (1) (a) By multiplying both sides of (1) by 2 N S , show that d d Rp pRtm=− (2) where NR S= , displaying your working clearly. [3] (b) Regarding p and m as constants, obtain the general solution of equation (2) for R in terms of t. [3] (c) Assuming that no individual died at birth and no one was born with smallpox, write down a relation between ( )0N and ( )0S and hence show that ( ) ( ) ( )1 1 e pt mN tSt m= +− . [3] (d) Bernoulli’s data states that 1 8p= and 8m= when 24t = . Estimate the proportion of individuals who would not have had smallpox by the time they reached 24 years of age, giving your answer to the nearest form 1 n where n is a positive integer to be determined. [3] [Turn over

6 9 Plankton are a collection of tiny organisms that live at and beneath the surface of lakes, rivers, ponds, and oceans across the planet. A group of plankton biologists are investigating plankton in a lake on a boat. At a depth of h metres, the density of plankton, in millions per cubic metre, is modelled by the function, ( ) 22 0.005p 0.15 e hhh −= for 0 30h and is modelled by a continuous function, f(h) for 30h which is not explicitly given. (a) Find ( )p 20 . Using correct units, interpret the meaning of ( )p 20 in the context of the problem. [2] (b) Consider a vertical column of water in this lake with a uniform horizontal cross sectional area of 2 square metres. Use Simpson’s rule with 6 equal intervals to estimate 30 0 p( ) dhh . Hence, estimate the number of plankton, to the nearest million, in this column of water from h = 0 to h = 30. [4] (c) It is given that there is a function u such that ( ) ( )0 f u hh for all 30h and 30 u( )d 105hh  = . The column of water in part (b) is H metres deep, where H > 30. Write down an expression involving one or more integrals that gives the number of plankton, in millions, in the entire column. By using your answer in (b), explain why the number of plankton in this ent