2017 YEAR 5 PROMOTION EXAMINATION MATHEMATICS 9758

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RAFFLES INSTITUTION 07 YEAR 5 PROMOTION EXAMINATION MATHEMATICS 9758 September/October 07 Total Marks: 00 3 hours Additional materials: Answer Paper List of Formulae (MF6) READ THESE INSTRUCTIONS

1 RAFFLES INSTITUTION 07 YEAR 5 PROMOTION EXAMINATION MATHEMATICS 9758 September/October 07 Total Marks: 00 3 hours Additional materials: Answer Paper List of Formulae (MF6) READ THESE INSTRUCTIONS FIRST Write your name and CT group on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 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. RI07 This document consists of 5 printed pages. RAFFLES INSTITUTION Mathematics Department [Turn over

2 The equation of a circle C is given by x + y + ax + by + c = 0, where a, b and c are real constants. The line y = x + 3 intersects C at the points where x = 3 and x =. Given further that the centre of C lies on the line y = x, find the values of a, b and c. [4] Referred to the origin O, points A and B have position vectors a and b respectively, such that a and b are non-parallel vectors. Point C lies on line AB, such that the length of projection of OC uuur uuur onto OB is 5 units. Given that b = and a b =, find the possible position vectors of C in terms of a and b. [6] 3 (a) A piece of paper in the form of a semi-circle of radius r is cut into twelve sectors such that the areas are in arithmetic progression, and the area of the biggest sector is three times that of the smallest sector. Find the exact area of the smallest sector in terms of r. [] (b) An arithmetic series A has first term a and common difference d, where a and d are non-zero. A convergent geometric series G has common ratio r. The first three terms of G are equal to the first, eleventh and seventeenth terms of A, respectively. (i) Find r. [4] Using your answer in part (i), find the exact ratio of the sum to infinity of G to the sum of the first four terms of G. [] 4 (i) Find the first three non-zero terms in the series expansion of ( x ). x Show that + can be written in the form ( a + bx)( x ) for real x constants a and b to be determined. Hence or otherwise, find the first four non- zero terms in the series expansion of + x. x By setting x = in your answer, obtain an approximation of 7 as a fraction in lowest terms. [] H MA 9758/07 RI Year 5 Term 4 Promotion Examination

3 3 5 The equation of a curve C is x + xy + y = k, where k is a constant and k. (i) Find d y dx in terms of x and y. [] It is given that C has a tangent which is parallel to the x-axis. Show that the y-coordinate of the point of contact of the tangent with C must satisfy y 4y + 4k = 0. Hence find the range of values of k. [4] 3 In the case where k =, find the equations of tangents to the curve that are 4 parallel to the x-axis. [] 6 Given that f ( r) = sin(r + ) θ, show that f ( r) f ( r ) = Acos Brθ sin θ, where A and B are constants to be found. [] Use the result above to show that sin(n + ) θ cos θ + cos 4θ + L + cos Nθ =. sinθ Hence find cos θ + cos 4θ + + cos Nθ. L 7 Functions f and g are defined by x f : xa, x R, x, x g : x a x, x R. (i) Sketch the graph of y = f ( x), stating the equations of the asymptotes and the coordinates of the points where the curve crosses the x- and y- axes. On the same diagram, sketch the graph of y = g( x). Find the exact values of the x-coordinates of the points of intersection of the graphs in part (i). Hence solve the inequality g( x ). f ( x ) > [Turn over H MA 9758/07 RI Year 5 Term 4 Promotion Examination

4 4 8 (a) The complex number u is such that u 4u = 7 + 4i. Find the possible values of u, giving your answer in the form a + bi, where a and b are real numbers. [] (b) The complex number v is such that vv v = 7 + 4i, where v is the complex conjugate of v. Find the possible values of v, giving your answer in the form c + di, where c and d are real numbers. (c) The complex numbers z and w are given by z = + i and w = 6 i. i Without using a calculator, find an expression for z and w in the form re θ, where r > 0 and π < θ π. Hence express w 3 z in the form of r(cosθ + i sin θ ), where r > 0 and π < θ π. 9 A curve C is defined by the parametric equations x = tan θ, y = secθ π for 0 < θ <. (i) Show that d y = sin θ. [] dx The tangent and normal at ( tan θ, secθ ) P meets the x-axis at Q and R respectively. Show that the area, A, of the circle passing through P, Q and R can be expressed as A = π tan θ +. [5] tanθ Using differentiation, find the minimum value of t + for 0, t t > proving that it is a minimum. [4] Deduce the minimum value of A. [] H MA 9758/07 RI Year 5 Term 4 Promotion Examination

5 5 0 The function f is defined as follows 8 f : x a for,,. ( )( ) x x x x x + (i) Sketch the graph of y = f ( x), giving the equations of any asymptotes, the exact coordinates of any points where the curve crosses the x- and y-axes, and the exact coordinates of any turning points. [4] If the domain of f is further restricted to x k, find the least value of k for which the function f exists and write down the domain of f. [] For the value of k found in, show that if f ( x) x, = then 3 x + x x 8 = 0. Hence use an algebraic method to find the exact value of x for which f ( x) = x. [4] In the rest of the question, the domain of f is x, x, x, as originally defined. The function g is defined as follows g : xa for x, x, x 0, x. x (iv) Solve the inequality fg(x) > 0. 4 h h + The planes p and q have equations r = 3 and r = 0 + λ + µ 3 7 respectively, where h is a constant and λ and µ are parameters. (i) h h + Find 3 in terms of h. [] Find the value of h such that p and q are perpendicular. [] Given instead that p and q are parallel, find the perpendicular distance between p and q. (iv) In the case where h = 0, p and q intersect in a line l. The line l cuts the xzplane at the point A and the yz-plane at the point B. Find the position vectors of the points A and B. [4] Hence find an equation that describes the set of all points which are equidistant from the points A and B. ******* End of Paper ******* H MA 9758/07 RI Year 5 Term 4 Promotion Examination

DUNMAN HIGH SCHOOL Preliminary Examination Year 6

Name: Inde Number: Class: DUNMAN HIGH SCHOOL Preliminary Eamination Year 6 MATHEMATICS (Higher ) 970/0 Paper 7 September 05 Additional Materials: Answer Paper Graph paper List of Formulae (MF5) 3 hours