2014 H2 Maths Prelims 2 P1 Qn Paper

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2    IJC/2014/JC2 9740/01/S/2014 [Turn over 1 Adam bought a total of 50 fruits consisti ng of apples, oranges and pineapples. The apples, oranges and pineapples cost $0.80, $0.60 and $1.20 each respectively. The total cost of all the fruits he bought is $40. If the cost of apples is doubled and that of oranges is halved, then the total cost of all the fruits that he bought would be $53. Find the number of each type of fruit bought by Adam. [3] 2 It is given that x, y, z are the first three terms of a geometric progression. When the three terms are arranged in the order of z, x, y, they form three consecutive terms of an arithmetic progression. (i) Show that 2 20zz yy      . [4] (ii) Hence determine if the sum to infinity of the geometric progression exists. [2] 3 (i) Without the use of a calcul ator, solve the inequality    2 24 3 02 xx x x    . [5] (ii) Hence solve the inequality     2 24 3 02 xx x x    . [2] 4 (i) Solve the equation 4 3iz   , giving the roots in the form ier  where 0r and   . [ 4 ] (ii) Show the roots on an Argand diagram and state the cartesian equation of a geometrical shape that the roots lie on. [3] 5 (i) Express   2 2 5 21 x x x   in the form of   2 2 1 A B x x   , where A and B are constants to be found. [3] (ii) Hence, expand   2 2 5 21 x x x   as a series of ascending powers of x up to and including the 2x t e r m . [ 4 ] (iii) State the range of values of x for which the expansion is valid. [1]

3    IJC/2014/JC2 9740/01/S/2014 [Turn over 6 (a) The diagram shows the graph of  fy x with asymptotes ,yk 0x and the graph cuts the x-axis at  ,0a . On separate diagrams, sketch the graphs of (i)  1 fy x and (ii) fyx , giving the equations of any asymptotes and the coordinates of any points where the curves cross the x- and y-axes. [4] (b) The curve  gy x undergoes the transformations A, B and C in succession: A. A stretch parallel to the x-axis with scale factor 2, B. A reflection in the y-axis, and C. A translation of 1 unit in the direction of the y-axis. Find an expression for  g x if the equation of the resulting curve is 11y x . [3] 7 A sequence 1u , 2u , 3u , … is such that 1 3 4u  and 2 1 31 42 n nnuu   for all 1n . (i) Write down the values of 23 4, and uu u . [1] (ii) By considering 1 nu or otherwise, write down a conjecture for . nu Use the method of mathematical induction to prove the conjecture. [5] (iii) Hence find 2 2 31 42 rN r    in terms of N. [2] (iv) Find the smallest value of N such that 2 2 31 42 rN r    exceeds 3 50 . [2] y x y = k

4    IJC/2014/JC2 9740/01/S/2014 [Turn over 8 (a) Find the exact value of the constant p such that 19 4 250 11 dd 9 14 p x x x x     . [4] (b) Use the substitution 2cosx  to find the exact value of 1 2 0 d1 x xx . [5] 9 Relative to the origin O, the position vectors of points A and B are a and b respectively, where a and b are non-zero and non-parallel vectors. The point P on OA is such that OP : PA = 2 : 3. The point Q is such that OPQB is a parallelogram. (i) Find OQ   in terms of a and b. [3] (ii) Show that the area of the triangle OAQ can be written as k ab , where k is a constant to be found. [2] (iii) State the ratio of the area of triangle OPB to area of triangle OAB. [1] (iv) Given ab is a unit vector, 2a and the angle between a and b is 60 , find the exact value of b . [3] 10 (i) On a single Argand diagram, sketch the locus of points representing the complex number z such that 42 i 2z  and 35zz  . [3] (ii) Find the greatest and least possible values of (a) z , [ 4 ] (b) arg z . [ 3 ] 11 The curve C has equation cos 2yx x , where 0 x   . (i) Find the exact x-coordinates of the points where C crosses the x-axis. [3] (ii) Sketch C, stating the coordinates of any points where the curve crosses the x- and y-axes. [2] (iii) Find the exact value of 4 cos 2 dx xx   . [4] (iv) Find the volume of revolution when the region bounded by the curve C, the x-axis and the line x  is rotated completely about the x-axis. [2]

5    IJC/2014/JC2 9740/01/S/2014 [Turn over 12 [It is given that a cone with radius r, height h and slant height l has curved surface area rl .] (a) A drinking cup is manufactured in the shape of a cone. It has a volume of 350 cm . Show that 2 22 4 2 22500A rr   , where A is the curved surface area of the cone. Use differentiation to find the height cmh and radius cmr of the cup that will require the least amount of material. [8] (b) Another drinking cup of the same shape is manufactured. At the instant when the depth of water in the drinking cup is h cm, the volume V cm 3 of the water is given by 3 12 hV  . The cup is filled and it is discovered that there is a leak at the vertex of the cup and the volume of water in the cone is decreasing at the constant rate of 3 31cm s . Calculate, (i) the rate at which the depth is decreas ing at the instant when the depth is 2 cm, [3] (ii) the time taken in seconds for the depth to decrease from 6 cm to 3 cm.