PJC H2 MATHS P1

Candidate Name: ________________________ Class: _____________ JC2 PRELIMINARY EXAM Higher 2 MATHEMATICS 9740/01 Paper 1 14 Sept 2015 3 hours Additional Materials: Cover page Answer papers List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your full name and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages.

2 @PJC 2015 [Turn Over] 1 The volume of water V in a filtration tank at time t satisfies the differential equation d 5d V kVt  , where k is a positive constant. Find V in terms of k and t, given that the tank is initially empty. [5] State what happens to V for large values of t. [1] 2 (i) Given that 1sin (2 )e xy   , show that   2 22 d1 4 4 d yxy x   . [2] (ii) By further differentiation of this result, find the first three te rms of the Maclaurin series for y in ascending powers of x. [3] (iii) Deduce the first three terms of the Maclaurin series for 1sin (2 )e cos x x  in ascending powers of x. [3] 3 The curve C has equation 24xy xq  , where q is a non-zero constant. It is given that C has a stationary point at 4x and an asymptote 4y x r , where r is a non-zero constant. (i) Find the values of q and r. [3] (ii) Sketch C, stating clearly the equations of its asymptotes, stationary points and the coordinates of any point(s) of intersection with the axes. [3] (iii) State the set of values that y can take. [1] (iv) Using the graph in part (ii), find the range of values of a such that the equation   22 2 42 16 xxa xq      has a negative real root. [2]

3 @PJC 2015 [Turn Over] 4 (a) State a sequence of transformatio ns which transform the graph of lnyx to the graph of  ln 1 2yx . [3] (b) It is given that  f2 x a x , where 0a . By considering the graphs of  fy x a and  fy x a , find the value of the constant k for which      42 02 f d f d aa ax a x k x a x    . [4] 5 Relative to the origin O, the position vectors of two points A and B are a and b respectively, where 2a  and the angle between a and b is 3 4  radians. Given that a and 2ab are perpendicular, find (i) the exact length of b, [3] (ii) the exact length of projection of a onto b. [1] The point P lies on OB such that the ratio : 3:5OP OB  . (iii) Find the exact area of triangle APB. [4] 6 An art exhibition features sculptures of Singa and Nila, which are the mascots for the SEA Games held in Singapore in 1993 and 2015 respectively. The number of Singa and Nila sculptures corresponds to the numb er of sports contested at the two editions of the games respectively. The sculptures are of varying heights. The Singa sculptures are displayed in a line such that the tallest Singa sculpture is in the middle. Starting from both ends of the line, the height of each subsequent Singa sculpture is 10 cm more than the preceding Singa sculpture, up to the middle Singa sculpture. (i) Given that 29 sports were contested at the 1993 SEA Games and the total height of all the Singa sculptures is 3120 cm, find the height of the tallest and shortest sculptures. [3] The Nila sculptures are displayed in order of descending height. The height of the tallest Nila sculpture is 210 cm. The height of each subsequent Nila sculptu re is 5% shorter than the height of the preceding Nila sculpture. (ii) Given that the shortest Nila sculpture is the only Nila sculpture to have a height of less than 35 cm, find the number of sports contested at the 2015 SEA Games. [2] (iii) Find, to 2 decimal places, the height of the shortest Nila sculpture and the total height of all the Nila sculptures. [3]

4 @PJC 2015 [Turn Over] 7 The polynomial P( z) has real coefficients. The equation P( z) = 0 has a root ier  , where r > 0 and 0  . (i) Write down a second root in terms of r and θ, and hence show that a quadratic factor of P(z) is 22 2 cosz rz r  . [3] (ii) Solve the equation 4 625z  , expressing the solutions in the form ier  , where r > 0 and      . [3] (iii) Use your answers in parts (i) and (ii) to express 4 625z  as the product of two quadratic factors with real coefficients, giving each factor in non- trigonometrical form. [3] 8 Fig.1 Fig. 2 Fig.3 Fig. 1 shows a card in the form of a square of fixed side a. A triangle is cut from each side, to give the shape shown in Fig. 2. The remaining card shown in Fig. 2 is folded along the dotted lines, to form the right pyramid with square base of side x as shown in Fig. 3. (i) Show that the volume V of the pyramid is given by 22 2 32 x a axV  . [4] (ii) Use differentiation to find in surd form the value of a x that gives a stationary value of V. [4] a x x

5 @PJC 2015 [Turn Over] 9 (a) (i) By considering the derivative of 2 ex , find 2 edxxx . [2] (ii) Hence, find 23edxxx . [3] (b) Use the substitution 2sinux to find 1 du uu   . [5] 10 Prove by mathematical induction that 1 cos cos(2 1)sin(2 ) 2sin n r nr    for all positive integers n. [6] Hence, find an expression for sin cos sin 2 cos2 ... sin cos nn        in terms of  and n . [2] 11 A sequence 1 2 3, , , ...u u u is defined by 1 1u  and 1 1 (ln ) rr ruu k   , where k is a positive constant and 1r . (i) Using the method of differences, show that 11 ln 11 ln n n ku k   . [4] (ii) Given that nu converges, (a) state the limit of the sequence in terms of k, [1] (b) find the range of values of k. [2]

6 @PJC 2015 [Turn Over] 12 Do not use a graphing calculator in answering this question. The planes 1p and 2p have equations 23xz and 33xy  respectively. The point A with position vector i j k , where  and  are constants, is in both 1p and 2p . (i) Find the values of  and  . [2] (ii) The planes 1p and 2p intersect in a line l. Find a vector equation of l. [2] (iii) A third plane 3p has equation  r i j , where  and  are constants. Given that the three planes have no point in common, find the value of  . What can be said about the value of  ? [3] (iv) The point B has position vecto