VJC 2026 Prelim P1 QP

1 2026/VJC/Math Dept 1 (a) Without using a calculator, solve the inequality 43 1x xx+ ≤ − . [4] (b) Hence solve the inequality 2 22 43 1 x xx + ≤ − . [2] 2 (a) The complex number p is given by 4i , 3i a + where a is a real number and 0.a< Without using a calculator, find p in terms of a, and ( )arg p . [3] (b) The point Q on the Argand diagram represents the complex number q. The scales on the Re- and Im- axes are the same. On the copy of the Argand diagram in the Printed Answer Booklet, plot the points R , S and T to represent the complex numbers q*, 2i q and * 2iqq + respectively. Show clearly the geometrical relationship between the points Q, R, S and T. [3] 3 An invasive aquatic plant is introduced into a reservoir. Ecologists assume that the rate at which the area covered by the plant increases is proportional to the area that has yet to be covered. The area covered by the plant, A hectares, at time t weeks after introduction can be modelled by ( )d 120dA kAt = − where k is a positive constant. Initially, the area covered by the plant was 3 hectares. After 4 weeks, it was 18 hectares. (a) Explain in context what happens as A approaches 120. [1] (b) Show that 120 e ktAB −= − where B and k are constants to be found. [4] (c) Sketch the graph of A against t. [2] Im Re O Q

2 2026/VJC/Math Dept 4 A movie scene features a drone following another drone along a straight outdoor track. Points A and B are the starting points, positioned 30 m apart on the track. Drone X launches from A and flies toward B while Drone Y launches at the same time from B and flies away from A . At time t s, the distance travelled by X and Y are x m and y m respectively where 0 30.x≤< A fixed camera is mounted on a pole at point C, positioned 30 m vertically above B, to capture the movement. The camera's tracking angle between the two drones, ,θ is given by the angle XCY. (a) By considering θ as the sum of two acute angles, show that 30(30 )tan 900 30 xyy xyθ −+= −+ . [3] The drones are programmed such that the distance of Y from B is always three times the distance of X from A, that is, y = 3x. A wider angle θ gives a more dramatic and dynamic footage. (b) Use differentiation to find the value of x when θ is maximum. [4] [You do not need to show that θ is maximum.] 5 (a) Show by integration that ( ) ( )π 2 sin d 2 π cos 2sin .x xx x x x c− = − −+∫ [2] (b) The function f is defined such that ( ) 2 π 2 sin for 0 f cos 1 for 2 π, π π, xx x x xx  −<= −< ≤ ≤ and that ( ) ( )f πf2xx= + for all real values of x. Find the exact area bounded by the curve ( )f,yx= the x-axis and the lines 0x= and 3π.x= [5] C A B X Y 30 m 30 m θ y m x m

3 2026/VJC/Math Dept 6 The equation of a circle C is given by 22 2 2 16x y ax by+= + − where a and b are non-zero real constants. The radius of C is 3 units and the line 35yx= −+ passes through the centre of C. (a) Show that the equation of C is ( ) ( ) 22 3 4 9.xy− ++ = [4] The equation of curve D is given by 3 7 , 2 6, for 0x t yt t= += −≥ . (b) Find the exact gradient of curve D at the point of intersection of circle C and curve D. [4] 7 The hyperbola H has equation 22( 4) ( 2) 1.yx− −− = (a) State the coordinates of the vertices of H. [2] (b) Write down the equations of the tangent and the normal to H at the point where 3.y= [2] (c) Find the equations of the tangents to H in exact form at the points where 6.y= [4] 8 It is given that ( )( )1 1 2 12 1 11 24 2 n r r nr= −+ = − +∑ . (a) Give a reason why the series ( )( )1 1 2 12 1r rr ∞ = −+∑ converges, and write down its value. [2] (b) Show that 11 11 ...4 9 16+++ + is less than 2. [3] (c) Find ( )( )10 1 2 12 3r N rr= ++∑ . (There is no need to express your answer as a single algebraic fraction.) [3]

4 2026/VJC/Math Dept 9 The curve C below has equation 1 . 11 y x = ++ The shaded region R is bounded by C, the x-axis and the lines 0x= and 3.x= (a) Find the volume of the solid formed when R is rotated completely about the y-axis. [4] (b) By using the substitution 1,ux= + find the exact volume of the solid formed when R is rotated completely about the x-axis. [5] 10 The planes 1π and 2π have equations 412 1 11 2 23 λµ −       =++         r and 1 2 10 2 − −=r respectively, where and λµ are parameters. Planes 1π and 2π intersect in 1.l (a) Find the acute angle between 1π and 2π. [3] (b) Find a vector equation of 1.l [2] (c) Find the position vector of the point on l1 which is closest to the point ( )4, 2,3 .− [2] Another line, 2l , has equation , 1 5x my z= +=− , where m is a constant. (d) Given that 2π and 2l have no point in common, what can we say about the value of m. [1] (e) Given that 2m> and the distance from 2l to 2π is 3 units, find the value of m. [2] O

5 2026/VJC/Math Dept 11 The diagram below shows the graph of f ( ).yx= The graph has a turning point at (0, )k and intersects the x-axis at x = 3k, where 1.k > The equations of the asymptotes are 2xk= and .2xy=− (a) Sketch the graph of 1 ,f( )y x= stating the equations of asymptotes and coordinates of the axial intercepts. [3] (b) The function g is given by ( )g : ln 2 3 , for , 1.5 0xx x x + ∈ − << . Find the range of fg in terms of k. [2] (c) The domain of f is further restricted to 2.xk> On the same diagram, sketch the graph s of 1f− and 1ff− , stating the equations of asymptotes and coordinates of the axial intercept. [3] The function h is defined as follows. h : , for , 11axx xx x− ∈ ≠−+ , where a is a real constant and 1a≠− . (d) Find ( )2h x in simplified form. Hence find the set of values of a such that ( )3h xx= has two distinct real roots. [4] y x x = 2k (0, k) O (3k, 0)

6 2026/VJC/Math Dept 12 Mrs Lin wants to save at least $20 000 for her daughter's education. On 1 January 2024, she sets aside $50. On the 1st of each subsequent month, she sets aside $a more than the previous month. (a) State the amount, in terms of a, that she sets aside on 1 December 2025. [1] (b) Find, to the nearest cent, the smallest value of a such that her total savings by the end of December 2025 is at least $20 000. [2] On 1 January 2026, Mrs Lin deposits $20 000 into an education fund that earns interest at 0.2% per month, compounded monthly. Interest is credited at the end of each month. She withdraws $180 on 1 February 2026 and on the first day of every subsequent month to pay for her daughter's school expenses. (c) Show that the amount in the fund at the end of February 2026 is $19 899.72. [1] (d) Taking January 2026 as the first month, show that the amount , in dollars, in the fund at the end of the nth month is given by ( ) 00 1.0070 0 2 n k− , where k is a constant to be found. [4] (e) Determine the month and year in which the fund is first exhausted. [2] (f) Find the amount of the final withdrawal. [2]