VJC_H2_2020_Prelim_P1

[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 magnitud