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This document consists of 5 printed pages and 1 blank page. [Turn Over YISHUN INNOVA JUNIOR COLLEGE JC 1 PROMOTIONAL EXAMINATION Higher 2 CANDIDATE NAME CG MATHEMATICS Paper 1 Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) 9758/01 30 September 2024 3 hours READ THESE INSTRUCTIONS FIRST Write your CG, index number and name on the work you hand in. Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. 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 100.

2 ©YIJC 9758/01/JC1PE/24 1 A concert sells tickets under three different age categories: ‘Senior Citizen’, ‘Adult’ and ‘Child’. Three groups of people A, B and C went to the concert on 3 different days. For Group A, all tickets were purchased at the original price s. For Group B, ‘Senior Citizen’ tickets were purchased at a 20% discount, while tickets under the other two categories were purchased at the original prices. For Group C, ‘Child’ tickets were purchased at half the ‘Adult’ original price, while ‘Senior Citizen’ and ‘Adult’ tickets were purchased at the original prices. The numbers in each age category for each group and the total price of the tickets for each group, are given in the following table. Group Senior Citizen Adult Child Total Price A 5 12 6 $2440 B 15 10 5 $2660 C 8 10 12 $2560 (a) Express this information as 3 linear equations and hence find the original price of a ticket in each of the age categories. [3] (b) The Lee family, consisting of 2 senior citizens, 2 adults a nd 3 children , went to the concert. Assuming their tickets were purchased at the original prices, find the total price they paid. [1] 2 Given that ( )23 1 2 14 n r nrn = =+ , find an eppression in terss of n for ( ) 2 1 34 n rn rn =+ + . eeave oour answer in fullo factorised fors. [4] 3 (a) By first expressing 2 23xx−+ in completed square form, show that 2 23xx−+ is positive for all real values of x . [2] (b) Without the use of a calculator, solve the inequality ( )( )2 2 2 3 1 0 2 x x x xx − + −  −− . [3] (c) Hence solve the inequality ( )( )2 3 1 0 2 x x x xx − + −  −− . [3] 4 (a) Given that cos xyx= , find d d y x in terms of x only. [3] (b) A curve has equation 22 6x y xy−= . (i) Find d d y x . [2] (ii) Find the exact coordinates of the point on the curve where d 1d y x = . [5]

3 ©YIJC 9758/01/JC1PE/24 [Turn Over 5 The diagram below shows a closed prism with a rectangular base ABCD. Triangles ABF and DCE are equilateral, congruent, and perpendicular to the base. The lengths of AB and AD are x cm and y cm respectively. The total surface area of the prism is 2300 cm . (a) Show that the volume of the prism, 3cmV , is given by 3125 3 8V x x=− . [4] (b) Hence find the maximum possible volume of the prism. [4] 6 A curve C has parametric equations ( )2x a t t=− , ( )212y a t=− , for 23 t−  , where a is a positive constant. (a) Sketch the graph of C, stating the coordinates of the end-points. [2] (b) Find, in terms of a, (i) the equation of the normal at the point where 1t=− , [4] (ii) the exact coordinates of the point where the normal cuts C again. [4] 7 The functions f and g are defined by f : e 4, , 1, g : sin , , π π. xx x x x x x x +    −   (a) Find ( )1f x− and state the domain and range of 1f.− [4] (b) Determine whether 1g− exists. [1] (c) Explain why the composite function fg exists. [2] (d) Find ( )fg x and state the domain of fg. [2] (e) Determine the exact range of fg. [1] A B C E D F y x

4 ©YIJC 9758/01/JC1PE/24 8 (a) Describe a series of transformations that will transform the curve with equation 2yx= onto the curve with equation 2(3 2) 1yx= + + . [3] (b) The diagram shows a sketch of the curve f ( )yx= with asymptotes 1x=− , 3x= and 2y= . The curve has a maximum point P(a, b) and passes through Q(0, c). On separate diagrams, s ketch the following graphs indicating the coordinates of the points corresponding to P and Q, and the equations of the asymptotes, if any. (i) ( )fyx= [3] (ii) ( )fyx = [3] 9 (a) Find 2sin 3 d . [2] (b) Find ( )2ln dx x x . [3] (c) Evaluate 3 2 31 d 21 x x x −    epactlo. [3] (d) Use the substitution exu= to evaluate ln 3 3 2 0 de e1 x x x +   epactlo. [4] x y O P(a, b) Q(0, c) x = −1 x = 3 y = 2

5 ©YIJC 9758/01/JC1PE/24 [Turn Over 10 (a) The line 1l has cartesian equation 12 423 xz y+−= − = . The plane 1p has cartesian equation 45xz−= . (i) Find a vector equation of 1l . [2] (ii) Find the acute angle between 1l and 1p . [2] (iii) Find the position vector of the point of intersection of 1l and 1p . [2] (b) The plane 2p contains the line 2l with vector equation 2 (2 3 ),= + − + − + r i j k i j k , and the point ( )1, 2, 2R − . (i) Find the shortest distance from R to 2l . [3] (ii) Find a cartesian equation of 2p . [3] (iii) Find a vector equation of the line of intersection between 1p and 2p . [1] 11 (a) Zack saves $x on 1 January 2024. On the first day of each subsequent month, he saves $10 more than in the previous month, so that he saves ( )$ 10x+ on 1 February 2024, $( 20)x+ on 1 March 2024, and so on. Find the value of x if he saves $25 000 in total on 1 December 2026. [3] (b) Dan opens a savings account. He deposits $200 on 1 January 2024 and continues to deposit $200 on the first day of each subsequent month. The interest rate of the savings account is 0.2% per month, so that on the last day of each month the amount in the account on that day is increased by 0.2%. (i) Find how much the $200 deposited on 1 January 2024 is worth at the end of 31 December 2024. [1] (ii) Show that, at the end of the nth month (where Jan uary 2024 is the first month, February 2024 is the second month, and so on), the total amount in the account is $ ( )100200 1.002 1n− . [4] (iii) In which sonth and oear will the asount in Dan’s account first epceed $4500? Epplain whether this occurs on the first or last day of the month. [4]

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