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 cuts the y-axis, giving your answer in exact form. [2] (b) Show that 2 d cd 1 ot 2y x = . Find the coordinates of the points and the gradient of C where 4  = . What can be said about the tangent to C as 0→ ? [5] (c) Hence sketch C, labelling the exact coordinates of axial intercepts and endpoints, and showing clearly the features of the curve at the points where 0= and 4  = . [3] (d) Find a cartesian equation of C. [2]

4 © ASRJC 2024 9 (a) Find 2 2 d23 x xxx + +−  . [2] (b) Find 2 sin dx x x . [2] (c) Using the substitution 3sinx = , find the exact value of 3 22 0 9d xx− . [5] 10 For positive real numbers a, and b, the lines l1 and l2 have equations given by l1: 11 20 53  −       =+          r for   ℝ and l2: 1 97 9 a b         =+          r for   ℝ. It is given that the two lines intersect each other. (i) Show that 3a = b + 2. [2] (ii) Find the value of b if the angle between the two lines is 1 11cos 660 −   radians. [3] The equation of the plane 1 that contains both lines is now given by 3 16 1  −= r . (iii) A second plane 2 meets 1 in the line l1 and contains the point (2, 1, 3). Find an equation for 2 in scalar product form. [2] (iv) Hence find the angle between the planes 1 and 2 . [2] (v) The plane 3 is parallel to the plane 1 . Given that the distance between both planes is units and that 3 is closer to the origin than 1 , find a cartesian equation for 3 . [3] 7 11

5 © ASRJC 2024 [Turn over 11 (a) A sphere has radius r cm, surface area S cm2 and volume V cm3. (i) Show that d d2 Vr S = . [2] (ii) Given that the surface area of the sphere is increasing at a constant rate of 3 cm2/s, find the radius when the rate of increase of the volume is 9 cm3/s . [2] [V olume of sphere, 3 3 4 rV = ; Surface area of sphere, 24 rS = ] (b) CarsExtreme is hosting a competition where competitors must drive from Town A to Town C in the shortest time possible. Town A is located 250 km to the west of Town B, while Town B is positioned 80 km away from Town C at a bearing of 330 . A straight road connects Town A and Town B, as shown in the picture above. The three towns are in a desert and competitors must traverse this desert to reach their destinations. To ensure safety, a speed limit of 110 km/h is enforced within the desert, except on the straight road where the speed limit is 130 km/h. It is assumed that competitors will always drive at the maximum permitted speed and will take the shortest possible route when crossing the desert. Competitor P and Q have different strategies for the competition. Competitor P plans to drive along the straight road from Town A to a point x km before reaching Town B, and then cut diagonally across the desert to Town C. On the other hand, Competitor Q intends to drive directly from Town A to Town C, traveling entirely through the desert. (i) Show that the time taken, T hours, for Competitor P is ( ) 21 2750 11 13 80 64001430T x x x= − + + + [3] (ii) Use differentiation to find the minimum time that Competitor P will take, giving your answer correct to the nearest minute. (You need not show that your answer gives a minimum.) [4] (iii) However, prior to the start of the competition, weather changes in the desert prompts the organisers to adjust the speed limit to M km/h for desert driving. Competitor P then decides to drive to Town B before heading to Town C, while Competitor Q sticks to his original plan. Find the range of values of M if Competitor P arrives at Town C before Q. [3] Desert 250 km 80 km Town A Town B Town C North straight road

6 © ASRJC 2024 12 At a particular bank, a savings account pays an interest of 0.9% per month on the last day of each month. On 1 January 20 24 Mr M onie put an initial deposit of $10000 into the savings account and continues to make a deposit of $810 at the start of each subsequent month. (a) Find the total amount in Mr Monie’s savings account at the end of the 2 months. [1] (b) Show that the total amount in the savings account at the end of the nth month is given by ( ) 1 $100900 1 .009 $90810 n− − . [3] (c) On what date did the value of Mr Monie’s account first exceed $30000? [4] Mr Pey has $30000 and is considering whether to keep this amount (with no further deposit) in the savings account to earn interest or invest this amount in a nother financial product. This financial product gives a disbursement of $100 at the end of the first month. At the end of every subsequent month, the financial product gives $10 more than the previous month. For example, at the end of first month, Mr Pey will have $30100 and at the end of two months, he will have $30210, and so on. (d) Find