2020 J2 VJC H2 Math prelims (Paper 1)

[Turn Over 1 Water is poured at a constant rate of 300 cm3 per second into a conical container, as shown in the diagram. The conical container has a base radius of 30 cm and semi-vertical angle of 30o. At time t seconds, the water in the conical container ha s a volume of V cm3, depth h cm and the radius of the water surface r cm. (i) Show that   3 9000 3 30 3 .9Vh    [3] [The volume of cone with radius r and height h is 21 3 rh .] (ii) Find the rate of change of depth of the water in the conical container when 53h . [2] 2 Solve the inequality 2 2 23 0,( 1) 1 xx ax a x     for , 0, 1.a a a   You need to distinguish the cases where a is positive and a is negative. [5] 3 It is given that     1f ln 2 rr   , where  is a constant and 11    . By considering    f 1 frr , find 2 1 1 2ln 2 rn r r          in terms of n and  . [3] Hence, give a reason why the series 4 5 6 3 4 5 2 2 2ln ln ln ...2 2 2                             converges. Given that 0.7  , find the value of the sum to infinity. [4] h 30 cm 30 o r 300 cm3 per second

2 4 (i) It is given that  1 ln 1 2yx   . Show that d1 .d 1 2 yy xx  [2] (ii) By repeated differentiation of this result, find the Maclaurin expansion of y in ascending powers of x, up to and including the term in 2x . [3] (iii) Verify that the same result is obtained using the standard series expansions given in the List of Formulae (MF26). [3] 5 (i) One of the roots of the equation 3 30x x c   , where c is real, is 2 3i . Without the use of a calculator, find the value of c and the other roots of the equation. [5] (ii) If c is a non -zero purely imaginary number , explain if it is possible for 3 30x x c   to have real roots. [1] (iii) It is given instead that c is a real number. By considering the graph of 3 3y x x , find the range of values of c for which the equation 3 30x x c   has only real roots. [2] 6 The sequence 1 2 3, , ...a a a is a geometric progression A with common ratio 3 4 , and th e sequence 1 2 3, , ...b b b is an arithmetic progression B. The sum to infinity of A is equal to the sum of the first ten terms of B. Given that 19 13ab and the sum of a2 and b2 is 35.825, find 1 2 3 25 ...a a a a    , giving your answer correct to 2 decimal places. [8] 7 With reference to the origin O, the points A, B, P, Q and R have position vectors a , b , 2ab , 23ab and 2 ab respectively, where a and b are non -parallel vectors and 0ab . Given that a is a unit vector and the area of triangle PQR is equal to the magnitude of b, show that 1sin 4  , where θ is the angle between a and b. Hence, find the value of  . [5] M lies on PR such that 1 2PM MR . Given that PR is perpendicular to OM, find the magnitude of b, giving your answer correct to 3 decimal places. [5] 8 (a) The curve C with equation 2 2 xy x   , 4  , where  is a real constant, has a positive gradient at any point on the curve. (i) Find the range of values of . [2] (ii) Sketch C, stating the equations of any asymptotes and the coordinates of the points where C crosses the axes. [3] (b) The transformations A, B and C are given as follows: A: A translation of 3 units in the negative x-direction. B: A reflection about the x-axis. C: A stretch parallel to the y-axis with a stretch factor of 4, with x-axis invariant.

3 [Turn Over A curve undergoes in succession, the transformations A, B and C and the equation of the resulting curve is 4( 3) 2 xy x   . Determine the equation of the curve before the transformations were effected. [3] (c) It is given that 2 for 0 1, f ( ) ( 1)1 for 1 3,4 xx x x x      and that    f f 3xx for all real values of x. On separate diagrams, sketch for 2 4,x the graphs of (i) f ( ),yx [3] (ii) f (| |)yx . [2] 9 The position of a particle P, moving along a curve C, at any time t is given by the parametric equation sin , 3 cos ,2 tx t y t    for 30 2t  , where x and y are measured in metres and t in seconds. (i) Show that d 2sin d 1 2cos yt xt  . [2] (ii) Find the exact equation of the tangent to C at which the tangent is parallel to the y- axis. [3] (iii) Sketch the graph of C. Give in exact form the coordinates of the points where C meets the y-axis, and also give in exact form the coordinates of the end points and maximum point on the curve. [4] (iv) The speed of the particle at time t is given by the formula 22 dd dd xy tt           . Given that the particle is moving at maximum speed when t  , find its speed at this instance. [1] (v) The distance between two points along a curve is the arc length. The arc length between two points on C, where t  and t  , is given by the formula 22 dd ddd xy ttt                . Find the distance, in metres, travelled by the particle from the start to the instant when it attains maximum speed, giving your answer correct to 3 decimal places. [2]

4 0 The velocity, v of an object is given by d d xv t , where x is its displacement from a point O at time t. Given that the acceleration, a of the object is given by d d va t , prov e that d d vav x . [1] Newton’s second law states that the nett force (in N) acting on an object is equal to the product of its mass (in kg) and its acceleration (in 2ms ). In military exercises, parachutists jump from stationary helicopters and their motion is tracked by sensors tagged to their bodies. One parachutist of mass 80 kg falls vertically from a Chinook helicopter. When the parachutist is x m below the helicopter (when the parachute is not opened), his v elocity is 1 msv  and the nett force acting on him is 2800 0.4 v . Show that his motion can be modelled by the differential equation 2d 10 0.005d vvv x  . [2] Solve the differential equation and sketch v against x where 0v . [8] For an object falling through the atmosphere, the object is said to have reached terminal velocity when the object’s acceleration is zero. State the terminal velocity of the parachutist. [1] 11 Let  f t  be the outdoor temperature, in degree Celsius, of a typical day in May in a small town, t hours after 12 midnight. It can be shown that  f cos , 0 24 and and are positive constan ts.24 2t a t b t a b    It is given that on a typical day in May, the outdoor temperature is 25 C at 12 midnight and the maximum temperature of the day is 38 C at 12 noon. (i) Find the value b and show that 13a . [2] (ii) It is given  and  , where  are such that f ( ) f( ). Show that k , where k is a constant to be determined. [2] The rate of absorption of nutrients,  g  1mgs , of a plant is influenced by the outdoor temperature  , and can be modelled by the function g where g : 50ln 150, 23.    (iii) Write, in context of the question, what the composite function gf represents and show that this function exists. [3] (iv) Determine if gf has an inverse. [2] (v) Find the range of the rate of absorption between (12 – s) am and s pm on a typical day in May. [3]