2023 VJC Promo (Qn)

2023 VJC H2 Maths Promo Paper Duration: 3 hrs Marks: 100 Attempt all questions. [You may skip Q4 & 6 for now, as they are on Integration. Remaining total marks is 84, Duration is 2 hr 31 mins.] 1 (a) Find an expression for d d y x in terms of x if 12tan 1yx  . [3] (b) It is given that cosxy x for 0x . By taking logarithm first, find an expression for d d y x in terms of x. [3] 2 The diagram shows the curve fy x . The curve passes through the points  1, 0A  ,  0, 2B ,  1, 0C and  3, 3D  . On separate clearly labelled diagrams, sketch the graphs of (a) 1 f2yx , [2] (b) f1y x . [2] 3 Given that a and 1a , use an algebraic method to solve the inequality   2 11 ax a xa x   . [5] Hence find the set of values of x such that   2 1 1 ax a xa x    . [2] 4 (a) Differentiate tane x with respect to x. [1] (b) Hence find tan 4es e cdx x x . [4] 5 The curve C has equation 22 44xy . (a) Sketch C, stating the equations of any asymptotes and the coordinates of any points of intersection with the axes. [3] (b) Describe a pair of transformations which transforms the graph of C onto the graph of  2 224xy  . [2] 0, 2B  1, 0A   1, 0C  3, 3D  1y fyx O y x

6 (a) Find 2 d 54 32 x x xx   . [4] (b) Show that 4 2 d32 xa bx  , where a and b are constants to be determined. [3] (c) Using the substitution sinx u , find the exact value of 3 22 22 2 1 d 1 x x x     . [4] 7 Do not use a calculator in answering this question. (a) One of the roots of the equation 3233 0xp xq x   , where p and q are real, is 12 i . Find the other roots and the values of p and q. [5] (b) The complex numbers w and z, with wz , satisfy the simultaneous equations *3iwz  and *2iwz  . Find w and z. [5] 8 It is given that     2fl n 1 2x x . (a) A polynomial  2p xa xb x c  is used to approximate  f x for 12 .x Given that    1f 1p  ,    1.5 f 1.5p  and    2f 2p  , find the values of the coefficients a, b and c. [3] (b) Using standard series from the List of Fo rmulae (MF26), find the Maclaurin series for  f x in ascending powers of x, up to and including the term in 6x . [2] (c) Find the set of values of x for the expansion in part (b) to be valid. [2] (d) Use 2 1 6x  in your series from part (b) to show that 4ln 31 6 2 m  , where m is an integer to be determined. [2] (e) Using your series from part (b), find the series expansion for 2 2 12ln 12 x x x     , up to and including the term in 4x . Give the coefficients in exact form. [2]

9 Do not use a calculator in answering this question. (a) It is given that 3i   and ππcos isin33  . Find 5 *   in the form  cos isinr   , where 0r and ππ  . [4] (b) The complex number z is given by cos isinz   , where ππ 22  . Show that 2 1* 2cos1 z z   . Hence find the modulus and argument of 2 1 1 z in terms of . [4] 10 The curve C has equation 233 1 xa xy x   , where a is a constant, 6a  . (a) Find the set of values of a such that C does not cut the x-axis. [2] (b) It is given that the equation of C is 236 3 1 xxy x   . Sketch C and give the equations of any asymptotes. Also, state the coordinates of any points where C crosses the axes and of any turning points. [4] (c) Let k be a positive constant. By sketching a suitable graph in the same diagram in part (b), find an inequality satisfied by k such that the equation  22222 36 311 2 1 xxkx k x      has 2 real and distinct roots. [2]

11 A company manufactures a closed container made of glass as shown in the figure below. The closed container, of negligible thic kness is made up of two components. The bottom component is a cylinder of base radius r cm and a height of h cm. The top component is a hemisphere of radius r cm. The company requires the volume of the container to be fixed at 108π cm3. The external surface area of the container is denoted by A cm2. The company wants the value of A to be as small as possible to reduce the cost of production. [The volume of a sphere of radius r is given by 34 π3 r and its surface area is given by 24πr .] (a) Show that 252 1 6 π3 rA r   . [3] (b) Using differentiation, find the exact value of r that gives the minimum value of A, proving that A is a minimum. [4] (c) Sketch the graph showing the external surface area of the container as the radius of the hemisphere varies, stating the coordinates of the end point of the graph. [3] The company decides to produce the glass container with 3r . To use the container as a decorative piece, the glass container is filled completely with a viscous liquid. (d) A small crack at the bottom of the container causes the viscous liquid to leak out of the container at a constant rate of 5 cm 3 per second. Find the rate of decrease of the height of the viscous liquid in the container 14 seconds after the container cracked. [3] h r r

12 The diagrams below show a sequence of patterns formed by squares. Stage 0 is represented by a square of length 1. At each successive stage, squa res with half the length of the smallest square in the previous stage are added to the unoccupied vertices of the squares in the previous stage. Stage 0 Stage 1 Stage 2 Let 0A be the area of the square in Stage 0 and nA be the area of each new square added in Stage n. For example, 0 1A  and 1 1 4A  . (a) Find an expression for nA , giving your answer in terms of n. [2] (b) Show that the total area of new squares added in Stage n is given by 1 3 4 n   . [2] (c) Show that the total area of all the squares in Stage n is given by 354 4 n   . [3] A square fractal is formed when this proce ss of adding new squares at each stage continues indefinitely. (d) Give a reason why the area of the square fractal converges and write down its value. [2] (e) The total area of the squares in Stage m first exceeds 90% of the area of the square fractal. Find the value of m and find the total number of squares at Stage m. [3]