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 a