RI 2022 Year 5 H2 Mathematics Common Test (Questions)

H2 MA 9758/2022 RI Year 5 Common Test MATHEMATICS 9758 2 hours Answer all the questions. 1 It is given that 3 2f ( )x ax bx bx c    and g( ) ,x ax d  where a, b, c and d are constants. Given that the curve with equation f ( )y x has a stationary point at  1,1 and it intersects the curve with equation g( )y x at  3,17 , determine the values of a, b, c and d. [5] 2 Referred to the origin O, the points A and B have position vectors a and b respectively. Point C lies on BA produced such that : 3: 2.AB AC  (i) Find OC   in terms of a and b and show that the area of triangle OAC can be written as k a b , where k is a constant to be found. [3] (ii) Given that a and b are unit vectors and the magnitude of a b is 3, find the cosine of the angle between a and b. [3] 3 The line 1L has equation 63, 3 yx z   and the line 2L has equation 2 6, 7x y z    . (i) Find the acute angle between 1L and 2L . [2] (ii) Show that 1L and 2L are skew lines. [2] (iii) Let 1P and 1Q be two points on 1L such that distance between them is 10 . Let 2P and 2Q be the two respective points on 2L such that 1 2P P and 1 2Q Q are minimum. Find the exact distance between 2P and 2Q . [2] 4 (a) Differentiate  2 3log 2 1x  with respect to x. [2] (b) Find d ,d y x given that 1 2 tan 1 xy x    [2] (c) Given that 1 cos , 2sin ,x y     where 0 ,    find an expression for d d y x in terms of . [2] RAFFLES INSTITUTION 2022 YEAR 5 COMMON TEST (MODIFIED)

H2 MA 9758/2022 RI Year 5 Common Test 5 (a) Without using a calculator, solve the inequality 3 22 5 4 1 x x x xx     . [4] Hence solve the inequalities (i) 3 22 5 4 1 x x x xx     , [1] (ii) 3 22 cos cos 5cos 4 cos1 cos x x x xx     . [2] (b) Show that there are no solutions to the equation 1b x x b   , where 1b is a fixed constant. [4] 6 The function f is defined by   2f : 3 2, , .x x x x k     (i) Given that the function 1f  exists, state the least value of k . [1] Use the value of k found in part (i) for the rest of this question. (ii) Without finding 1f  , sketch on the same diagram the graphs of f ( )y x , 1f ( )y x  and 1f f ( )y x  . You should label your graphs clearly. [3] (iii) Hence find the exact solution(s) of the equation 1f ( ) f ( )x x  . [3] The function g is defined by g : , , 0.x x h x x    (iv) Find the set of values of h such that composite function fg exists and write down an expression for fg( )x in terms of h . [3] 7 Do not use a calculator in answering this question. The complex number is such that  arg z  where 2 4      . (i) In an Argand diagram, mark and label clearly the points P, Q and R representing the complex numbers , and respectively. [2] (ii) State the geometrical representation of OPRQ and express  arg iz z in terms of  and  . [2] (iii) It is now given that 2 2 3 iz  . Show that [5] z z iz iz z π 3 1sin . 12 2 2     

H2 MA 9758/2022 RI Year 5 Common Test 8 (a) A manufacturer can produce q thousand units of a particular machine per week at the unit price $p according to the equation 2 2 3 5q qp p   , where 3p and 3.q Find the rate of change of the supply of the machines when the quantity produced is 4000 units, and the unit price is $11 which is increasing at a rate of $0.10 per week. [3] (b) Isocost lines and isoquant curves are studied in economics. A general class of isoquant curves are the Cobb -Douglas functions of the form , where ,x y k    and k are positive constants. (i) Show that d d x y y x        . [2] The point at which the isocost line is a tangent to the isoquant curve is called an equilibrium point. (ii) The isocost line 27 2 6x y  is tangential to the isoquant curve 1 2 3 3x y k  , where 0x , 0y and k is a positive constant. Find the coordinates of the equilibrium point and the value of k . [7]