2024 NJC Paper 1 H2 Math Prelim (Qn Paper)

* © NJC 2024 NATIONAL JUNIOR COLLEGE SENIOR HIGH 2 Higher 2 NAME CLASS 2ma2 REGISTRATION NUMBER MATHEMATICS 9758/01 Preliminary Examination 09 September 2024 Paper 1 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, class and registration number in the boxes above. Please write clearly and use capital letters. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use paper clips, glue or correction fluid. Answer all the questions. Write your answers in the spaces provided in the question paper. 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. The use of an approved graphing and/or scientific calculator is expected, where appropriate. All relevant working, statements and reasons must be shown in order to obtain full credit for your solution. You are reminded of the need for clear presentation in your answers. Up to 2 marks may be deducted for improper presentation. The number of marks is given in the brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. Question Number Marks Possible Marks Obtained 1 4 2 4 3 5 4 7 5 8 6 8 7 8 8 9 9 10 10 11 11 12 12 14 Presentation Deduction – 1 / – 2 TOTAL 100 This document consists of 7 printed pages.

2 © NJC 2024 1 A circular sector has radius r cm and angle  radians. This sector has area A cm2 and fixed perimeter k cm. (i) Show that d 2.d2 Ak rr =− [2] (ii) Given that r is increasing at a constant rate of 1cm s ,10 k − find in terms of k, the rate at which A is changing when the arc length of the sector is equal to the radius. [2] 2 Two of the roots of the equation 32 0z az bz c+ + + = are 2i π33e − and –2. Given further that a, b and c are integer constants, find the values of a, b and c. [4] 3 Do not use a calculator to solve this question. (i) Solve the inequality 2 6 1.45 x xx − +− [3] (ii) Hence solve the inequality 2 2 6 1.45 xx xx − +− [2] 4 Relative to the origin O, points A, B and C have position vectors a, b and c respectively, where a, b and c are non-zero vectors that are not parallel to one another. The points A, B and C are not collinear. A point of trisection is a point that divides a line segment internally in the ratio 1: 2 or 2 :1. Suppose another two points D and E are points of trisection of line segments AB and AC respectively and both points are nearer to A than to B and C respectively. The lines BE and CD meet at point F. (i) Show that t he vector equations of the lines BE and CD can be expressed as ( )21 133  = + − +r a b c and ( )21 133  = + + −r a b c respectively, where  and  are parameters. Hence, show that at point F, ,  + + =a b c 0 where ,  and  are constants, each to be expressed in terms of  and . [4] (ii) Given further that OACB is a parallelogram, find the position vector of F in terms of a and b. [3]

3 © NJC 2024 5 [The volume of a cone of base radius r and height h is given by 21 π .]3V r h= A manufacturer makes a funnel-shaped ornament from the same material which consists of two parts as shown in Figure 1. • a right cone of radius r cm, height h cm and a slant height of 4 cm, • a cylinder with radius ar cm and height r cm, where 01 a . Figure 1 Figure 2 Figure 3 From the original cone, a similar cone with radius ar cm is removed from the vertex as shown in Figure 2. The remaining part of the cone is joined to the cylinder to form the funnel as shown in Figure 3. It may be assumed that the thickness of the funnel is negligible. Given that the volume of the ornament is V cm3, find V in terms of a and r. [3] For the remainder of this question, assume that 0.25a= . (a) The manufacturer wants V to be a maximum. If 1rr= gives the maximum value of V, show that 1r satisfies the equation 42457 9664 50176 0.rr− + = [3] (b) Show that one of the positive roots to the equation in part (a) does not give a stationary value of V. Hence find the value of h for which V is stationary. [2] 6 (i) Show that i i e 1e  − can be expressed as icot 1 ,2k  − where k is a real constant to be determined exactly. [3] (ii) Express the complex number i in three equivalent ier  forms, where 0r and 3π 3π.−   [2] (iii) Hence find the roots of the equation 3 i 0,1 w w  −= + leaving your answers in the form ( )icot 1 ,k − where ππ .22 −   [3] h 4 r r ar ar

4 © NJC 2024 7 (i) Given that πcosec 2 4yx =+  , show that 2 3 2 d 8 4 .d y yyx =− [3] (ii) By further differentiation of the result in part (i), find the first four terms of the Maclaurin series for πcosec 2 4x + exactly. [3] (iii) Hence estimate the value of 13π 13πcosec cot ,50 50            giving your answer in the form ( ) 22 π π ,p q r++ where p, q and r are rational constants to be determined. [2] 8 (a) The sum, ,nS of the first n terms of a sequence of numbers 1 2 3, , , ,u u u is given by 21 2n nS An Bn += + + , where A and B are non-zero constants. It is also given that the third term is 21 and the fifth term is 53. Find a simplified expression for nu in terms of n. [4] (b) (i) Use the method of differences to show that ( ) ( ) 2 1 2 2ln ln ln 2 11 n r rr n nr=  + +=−  ++  . [3] (ii) Hence, find the exact value of ( ) 2 2 0 43ln 2 n r rr r=  ++ +  in terms of n. [2] 9 (a) Show that the curve with equation ( ) 2 e xy x cx −=+ has two stationary points for all real values of c. [3] (b) The curves 1C and 2C have equations 22 4 6 7 0x y x+ − − = and 23 1 xy x −= − respectively. Write the equation of 1C in the form ( ) ( ) 22 22 1.x p y q ab −− += Sketch, on the same diagram, both 1C and 2C , indicating clearly their key features as well as the coordinates of their points of intersection. [7]

5 © NJC 2024 10 The function f is defined by ( )f ( ) 1 1 , for , 4 2.x x x x x= + +  −   (i) Find 1f.− [4] (ii) On the same diagram, sketch the graphs of f ( ),yx= 1f ( )yx −= and 1ff ( ),yx −= labelling clearly the coordinates of the end-points. [4] (iii) Solve exactly the inequality 1f ( ) f ( ).xx − [3] 11 The diagram below shows the Gateway Arch, which is a monument in St. Louis, Missouri, United States. The arch stands at 192 metres tall and is 192 metres wide.

6 © NJC 2024 The arch can be modelled by part of a curve as shown in the diagram below. The highest point of the curve lies on the y-axis and the curve is symmetrical about the y-axis. The two endpoints both lie on the x-axis. It is known that the curve satisfies the differential equation 22 2 d 1 d 1dd yy akx k x =+  for some constants a and k. (i) Show that the substitution 1d d yp kx= reduces the differential equation to 2d 1.d p apx =+ [1] (ii) By using the substitution tan ,pu= where ππ ,22 u−   to solve the reduced differential equation in part (i), show that ee .2 ax ax p −−= [8] (iii) Given that 0.0329a=− and 0.701,k = find y in terms of x. [3]

7 © NJC 2024 12 The planes 1π and 2π, which meet in the line 1,l have equations 1 2 π : 3 25 1   − = r and 2π : 2 15,x ky z+ − =− where k is a constant. Another line 2l has equation 15 3 0 1 , . 01  −       = +        −    r (i) Determine the position vector