2024 ASRJC_JC1_H2 Math_Promo Exam Questions

[Turn over 2024 JC1 Promotional Exam 1 A hyperbola has the equation ( ) ( ) 22 21a x b y c− − + = where a, b, and c are positive integers. It passes through the point ( )7,10 . A line 1yx=− intersects the hyperbola at 1x= . Find one possible value for each of a, b and c. [4] 2 It is given that tan3yx= . (a) Show that 2 2 dd dd yy Ayxx = , where A is a constant to be determined. [3] (b) Hence show that 4 2 3 4 2 3 d d d d d d d d y y y yB Cyx x x x=+ , where B and C are constants to be determined. [3] 3 The graph shows the curve C with equation 2yx= and the line L with equation 21yx=− . The line L is tangent to C at (1,1). The region R located in the first quadrant is bounded by the curve C, the line L and the x-axis. (a) Find the area of region R, giving your answer in exact form. [2] (b) The region R is rotated one complete revolution around the x-axis. Find the volume of this solid of revolution, giving your answer in exact form. [3] 4 (a) Sketch the graph of 2 4 2 3 2 xy xx −= +− . Give the equations of the asymptotes and the coordinates of the point where the curve intersects the x-axis. [4] (b) Solve the inequality 2 4 0.2 3 2 x xx − +− [1] (c) Hence solve the inequality 2 4 0.2 3 2 x xx − +− [2] y x O

2 © ASRJC 2024 5 In triangle PQR, the vectors a, b and c represent the vectors PQ → , QR → and PR → respectively. (a) State a geometrical interpretation of ˆca and hence show that the area of the triangle PQR can be written as 1 2 ca units2. [3] (b) With clear explanations, show that  = c a b a and deduce that sin sinPQR QPR= cb (Sine Rule). [2] 6 The diagram below shows the graph of curve C1, with equation f ( ),yx= defined for \{4},x with asymptotes 4,x= and 1 12yx=− + , and cuts the x -axis at the origin and (6,0) . (a) Sketch the graph of f ( )yx = , labelling the equations of asymptotes if any. [2] P R Q c b a y x

3 © ASRJC 2024 [Turn over (b) Given another curve C2 has equation ( ) ( ) 2 2 2 1 41y xk + − − = , where k is a positive constant. Describe a sequence of transformations that transform the graph of 22 1yx−= onto the graph of C2. [3] (c) Determine the range of values of k for which C1 and C2 intersect exactly 2 times. [1] 7 The function f is defined by f : 5 2xx + , for x , 5.2x− (a) Find the value of a such that 2f ( ) 5a = . [2] Another function g is defined by g: 12 xx x− , for x , xk . (b) State the largest value of k for g−1 to exist. [1] It is now given that 1k=− . (c) Find g−1(x) and state its domain. [3] (d) Explain why the composite function fg exists. State the exact range of fg. [2] 8 A curve C has parametric equations 1 2cos2x =− , sin 2 1y =− , for 30 8  . (a) Find the coordinates of the point where C c