2024 TJC JC2 Prelim Exam H2 Maths Paper 1 (Questions)

2024 TJC Preliminary Exam H2 Mathematics Paper 1 1 A ball is rolling in a straight line such that its distance away from the starting point, s cm, can be modelled using the equation 4 bs at c t = + + + , where t is the time taken in seconds, and a, b and c are real constants. The ball is at the starting point when 0t= , and moved 10 cm in the first 5 seconds. It moved another 9 cm in the next 16 seconds. Find the ball’s distance away from the starting point when 50.t = [4] 2 On a single diagram, s ketch the graphs of 2 and y x p y qx= − = where the following conditions are satisfied, indicating the axial intercepts. • p and q are constants, p > 1 and q > 0, and • the graphs have only one point of intersection. [2] (a) State the least value of q. [1] (b) Solve the inequality 2,x p qx− leaving your answer in terms of p and q. [2] 3 Find (a) ( )2tan 1 dxx− , [2] (b) 1sin 2 dxx−  . [3] 4 Do not use a calculator in answering this question. (a) It is given 3iw=− + . (i) Find arg w. [1] (ii) Express 8iw in the form ier  where r > 0 and .  −   [3] (b) (i) It is given that ( ) 2 1 i 3 4ia+ =− − . Find the value of the real constant a. [2] (ii) Hence solve the equation ( ) ( ) 22 3 2i 1 i 0zz+ − + + − = . [3]

5 The points A and B have position vectors a and b respectively. C is the point on line OB such that AC is perpendicular to OB. (a) By using a suitable scalar product, or otherwise, show that ( ) 2OC = ab b b  . [3] (b) Give a geometrical interpretation of ab b  . [1] (c) It is given that 3 3 1 − =− a and 1 0 h  = b . Given also that the length of the line segment AB is 5 units and angle AOB is an obtuse angle, find the exact value of h. [4] 6 The curve C is defined by the parametric equations 1 cos , sin2 ,x t y t= − = where 0 2t  . (a) Sketch C, giving the exact coordinates of the points where C meets the x-axis. [1] (b) The normal to C at the point where 2t = cuts the y-axis at D. Show that the y-coordinate of D is 1 2− . [4] (c) Find the exact area of the region bounded by C, the normal in part (b) and the y-axis. [5] 7 (a) The diagram shows the curve with equation f ( ).yx= The curve crosses the x-axis at 1x= and 2.5,x= crosses the y-axis at 1y= and has a maximum point at ( )4, 6 . The equations of the asymptotes are 2x= and 3.y= Sketch the graph of f '( )yx= , giving the equations of asymptotes, coordinates of turning points and axial intercepts, where possible. [2] y = 3 y x 1 (4, 6) x = 2 1 2.5

(b) The curve C has equation 2 1 1 x kxy x +−= + , where k is a non-zero constant. (i) Find the range of values of k for which C has no stationary points. [4] (ii) Given that y = x + 3 is an asymptote of C, show that k = 4. [2] (iii) State a sequence of transformations which transform the graph of 1 4