2022 YIJC H2 Math Promos (QP + Answers)

Promos Practice Paper 3 [YIJC 2022] 98marks 1 [It is given that the volume of a circular cone with base radius r and height h is 21 3 rh .] A cone-shaped paper drinking cup with height 9 cm and radius 3 cm is shown above. Water is poured at a constant rate of 5 cm3 per second into the cup. When the depth of water is h cm, the surface of the water has radius r cm (see diagram). Find the rate of increas e of the depth of water when 2r = . [4] 2 The diagram below shows the graph of ( )fyx= with asymptotes ya= and 3ya= , where 0a . The curve crosses the y-axis at the point A 0, 2 a  . Given that f is an increasing function, sketch the graphs of (a) ( )fy x a=+ , [2] (b) ( )fyx=− , [2] stating the equations of any asymptotes and the coordinates of the point corresponding to A after the transformation. h cm r cm 3 cm 9 cm

3 MI Promo 9758/2021/PU2/01/Q2 (i) Given that  is sufficiently small, show that 2sin 3 a b c    −  + + , where a, b and c are constants to be determined. [3] (ii) By u sing the substitution 10  = , find an approximate value for 4cos 15   , giving your answer correct to 5 decimal places. [2] (iii) By using a calculator to evaluate 4cos 15   , correct to 5 decimal places, explain why the approximation in part (ii) is not good. [1] 4 MI PU2 Promo 9758/2019/01/Q4 Do not use a calculator in answering this question. (i) Given that 2iz=+ is a root of the equation 32 2 19 30 0z z z+ − + = , find the other roots. [4] (ii) Hence find in cartesian form the roots of the equation 32i 2 19i 30 0w w w− + + = . [2] 5 A curve C has equation 22 3 144 0xy x y− + = . (i) Find d d y x and the coordinates of the turning point of C. [5] The point P on C has coordinates (4, k) for some constant k. (ii) Find the equation of the tangent at P. [3] 6 An arithmetic series has first term 3 and common difference d, where d is non-zero. A geometric series has first term a and common ratio r. Given that the 37 th, 7 th and 1 st terms of the arithmetic series are consecutive terms of the geometric series, find d. [3] Deduce that the geometric series is convergent. Find, in terms of a, the sum to infinity of the odd-numbered terms (i.e. the 1st, 3rd, 5th, …. terms) of the geometric series. [3] Given further that a = 3072, find the least value of n such that the sum of the first n terms of the arithmetic series exceeds the sum to infinity of the odd -numbered terms of the geometric series. [2]

7 A curve C has parametric equations 12 , , 1 1x t y t t= =  − . Show that the normal to C at the point with parameter t has equation ( ) ( ) ( )331 2 1 1 4 1 .t y t x t t− + − = + − [4] State the equation of the normal at the point P where t = 2. This normal cuts C again at the point Q. Find the exact coordinates of Q. [4] 8 Functions f and g are defined by 2 2f : 1 , , 0 2, 4 1g : , , 1. x x x x x x x x +    −  (i) Find ( )1f x− and state the domain of 1f − . [3] (ii) Sketch on the same diagram the graphs of ( )fyx= , ( )1fyx −= and ( )1ffyx −= , giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the x- and y-axes. [4] (iii) Solve the equation ( ) 2fg 5x = . [2] 9 On 1 January 2022, Jerald puts $500 into a savings account which pays interest at a rate of 0.1% per month on the last day of each month. On the first day of each subsequent month from February 2022, he puts another $x into the account. (i) Write down how much Jerald’s initial deposit of $500 will become on 30 November 2022 after adding compound interest earned. [1] (ii) Taking January 202 2 as the first month, show that the amount of money in Jerald’s account on the last day of the nth month is ( ) ( ) – 1 500 1 .001 1001 1 .001 1 nn x+−  . [3] (iii) Find the least integer value of x so that the interest earned for the first six months of the year 2022 exceeds $30. [2] (iv) Given that 300x= , how much will Jerald have in his account on 31 December 2025? Hence state the date on which the amount will first exceed $15 000. [3]

10 The curves C and D have equations 54 33yx x= − − − and ( )2 2 22 1 1 6 x y k − += respectively, where k is a positive constant. (i) Using an algebraic method, find the exact range of values of y that C can take. [4] (ii) On the same axes, sketch (a) the graph of C, stating the equations of any asymptotes and the coordinates of the turning points, [2] (b) the graph of D for the case where 2k = , stating the coordinates of the centre, the turning points and the points of intersection with the x-axis. [2] (iii) State the exact range of values of k such that C and D intersect at more than one point. [1] (iv) State the range of values of m such that the line with equation ( )4 33y m x+ = − does not intersect C. [1] 11 (a) Mr Tan buys 5 bottles of cooking oil, 4 packets of biscuits and 2 packets of rice. Based on the usual retail price in the supermarket, the total amount paid is $73.45. Currently, there is a “Buy 6 Get 1 Free” promotion for the biscuits. Under this promotion, Mr Suresh pays $53.30 and receives 2 bottles of cooking oil, 14 packets of biscuits and a packet of rice. Ms Siti receives a 5% membership discount on the total bill and pays $103.93 for 4 bottles of cooking oil, 2 packets of biscuits and 5 packets of rice. Write down and solve equations to find the usual retail price of a bottle of cooking oil, a packet of biscuits and a packet of rice. [4] (b) Without using a calculator, solve (i) 2 16 x x−  , [3] (ii) ( )( ) 22 4 2 4 3 1x x x x−  + − − . [3]

12 A closed cylindrical can with radius r cm and height h cm has fixed volume 20 cm3. The material for the top and bottom faces costs $0.50 per cm2 and the material for the curved surface costs $0.30 per cm2. It also costs $0.80 per cm to weld the top and bottom faces onto the cylinder and $0.60 per cm to weld the seam up the curved surface of the cylinder (see diagram). (i) The total cost of the can is $C. Show that 2 2 12 123.2C r r r r = + + + . [3] (ii) Use differentiation to find the values of r and h which give a minimum value of C, proving that C is a minimum. State this value of C. [7] (iii) It is given instead that 0.5 2 r . Find the corresponding range of values of C. [2] r h Welding

Answers 1 50.398 (or ) cm/s4 2 Graph 3 (i) 3 1 3,,2 2 4a b c = =− =− (ii) (ii) ( )0.66621 5 d.p 4 2iz=− and 6z=− 5 (i) ( ) ( ) 6d d 2 3 y x yy x x y x −= − , ( 2, 12)−− 4x= 6 2d = , 25Sum to infinity of odd-numbered terms =24 a , least 56n= 7 27yx+= , 3 , 4 .2Q   8 (i) ( )1 1f 21f 4 , D , 12x x − − = + = −  −  ( )1.22 3 s.f.x 9 (i) $505.53 (iii) 1798 $14968.22 , 1st Jan 2026 10 (i) 16 8or 33yy−  (iii) 8 3k  1m− 11 (a) Let $x, $y and $z be the usual retail price of a bottle of cooking oil, a packet of biscuits and rice respectively. 6.85x= , 2y = and 15.6z = (i) 3 or 2 6xx −   , (ii) 12 or 2xx − = 12 1.61 (3 s.f.) and 7.68 (3 s.f.)rh== . The least cost is $52.37. 52.37 129.21C

1 Promos Practice Paper 3 [YIJC 2022] Solutions Qn Solutions 1 Let V cm3 represent the volume of water in the cup when the depth of water is h cm. 2 2 2 3 3 9 --- (1) 3 1Substitute into ,33 1 3 1 = 33 1 = 27 r h hr hr V r h V r h h h h     = = == =   ( ) 2 2 2 2 d1 d9 d d d d d d 9 = 5 45 = When 2 , 2 3 6 d 45 45 5When 6 , 0.398d 36 4 6 V hh h h V t V t h h rh hh t     = =  = =  = = = = =  5The rate of increase of the depth of the water is 0.398 (or ) cm / s4 2a ( )fy x a=+ 2b ( )fyx=− Alternatively, Substitu