RI 2020 Y5 H2 Math Promo (Qn)

MATHEMATICS 9758 3 hours Question(s) marked with * has/have been modified for the 9758 [Revised] syllabus first examined in 2025. 100 marks 2 Referred to an origin O, points A and B have position vectors given respectively by 48ij k and 623ij k . The point P on AB is such that :1 :AP PB  . (i) Show that the exact area of triangle OAB is 5 1012 . [1] (ii) Find the exact value of  for which OP is perpendicular to AB. [3] (iii) Find the exact value of  for which angles AOP and POB are equal. [3] 3 Given that 2f( ) e s i nxxx , show that f() 4 f () 5 f ()xx x   . [2] Hence find the first three non-zero terms in the Maclaurin series for f( )x . [3] Verify that the same result is obtained if the standard series expansions for 2e and sinx x are used. [2] 1 Mary runs a fruit stall and keeps a record of the daily sales revenue. The weights of fruits and total amounts received from the sale s of three particular types of imported fruits in the past three days are shown in the following table. Black seedless grapes (kg) Premium red cherries (kg) Premium black plums (kg) Total amount received from sales in $ Day 1 17.5 8.1 15.5 580.94 Day 2 12.1 7.5 13.1 431.93 Day 3 18.5 9.6 12.6 602.75 On Day 1, all the fruits were sold at their original prices. On Day 2, Mary gave a special one-day discount of 6% for all purchases. On Day 3, Mary increased the price per kg of premium red cherries by $0.40 due to a price increase by the supplier. Find the price per kg of each fruit on Day 1. [4] RAFFLES INSTITUTION 2020 YEAR 5 PROMOTION EXAMINATION

4 Do not use a calculator in answerin g this question. (i) Given that x is real, prove that 2 2x x is always positive. [1] (ii) Solve the inequality 32 2 8 132xx xxx  . [4] (iii) Hence solve the inequality . [2] 5 It is given that 2 1 11 5 1 1 43 2 6 2 3 n r rr n n     . (a) Explain why 2 1 1 43r rr    is a convergent series, and state the value of the sum to infinity. [2] (b) Find the smallest value of n for which is within of the sum to infinity. [2] (c)* Find   2 3 1 35 n r rr  in terms of n. [4] 6 A curve C has equation   22 223 2x yx y k  , where k is a positive constant. (i) Show that d4 .dy x yx x y  [2] Find, in terms of k, the coordinates of (ii) the two points at which the tangent is parallel to the y-axis, [3] (iii) the two points at which C cuts the y-axis. [2] Given that the four points found in parts (ii) and (iii) are the vertices of a quadrilateral, state its geometric shape and find its area. [2] 3 2 2 8 132xx xxx   2 1 1 43 n r rr  210

7* (a) Solve the equation  2 1 2i 1 7i 0.zz   [3] (b) It is given that one of the roots of the equation    32 14 i 39 i 1 42 i 0zzz     is purely imaginary. Find all the roots of the equation, giving your answers in cartesian form iab . [7] 8 The graphs of  fyx and f( )yx are shown in the diagram below, with a maximum point (4, )a on both graphs, and 0x and 6x are tangents to both graphs.  fyx f( )yx It is also given that the graph of  fyx has only one oblique asymptote. Sketch the following graphs, stating the equations of the asymptote(s), coordinates of turning point(s) and axial intercept(s) where possible. (i) 1fy x , [3] (ii) f( )yx , [2] (iii) f(2 1 )yx , [3] (iv) . [3] 9 The plane p passes through the points with coordinates  2, 2, 4 , 0, 6, 8 and  2, 2, 3 . (i) Find a cartesian equation of p. [4] The line l has equation 44 , 9xy z  . (ii) Show that the coordinates of the point of intersection of l and p is 2, 6, 9 . [2] (iii) Find an e quation of the line which is a reflection of l in p. [6] f( )yx  y x y x

10 The function f is defined by   22f( ) 2 2 1 e , 1 . xxx x x    (i) Use the first derivative test to show that f has a stationary point of inflexion at 0x . [2] (ii) Sketch the graph of f( )yx , giving the coordinates of any axial intercept(s). Explain why the inverse function of f exists. Sketch the graph of 1f( )yx  on the same diagram as f( )yx , giving the coordinates of any axial intercept(s). [5] (iii) The function g is defined by . Show that the composite function exists, and find the range of . Find the exact value of a, where [5] 11 A jewellery maker intends to make charms for pendants and accessories. She designed each charm such that its cross section includes an equilateral triangle inscribed in a circle with radius r cm and all three vertices of the tr iangle touch the circumference of the circle, as shown in the diagram below. In her design, the portion of the charm with cross section as shaded in the above diagram is filled with glitter resin. It is given that the thickness of each charm she makes is 1 cm. (i) Show that the volume of glitter resin needed for the charm is 2333 cm .4 r  [3] After making the first charm with 3r , she continues to make charms where each successive charm she makes has a value of r which is 5% less than that of the preceding charm. She then places the first two charms made in the first basket, and places the subsequent five charms made in the second basket, and continues in this manner such that the number of charms placed in each bask et after the first is three more than the number of charms placed in the previous basket. (ii) Show that the radius of the k th charm made is 60 0.9519 k cm. [1] (iii) Find, in terms of n, the total number of charms in the first n baskets. [2] (iv) Find an expression, in terms of n, for the volume of glitter resin needed to make the biggest charm in the n th basket. (You need not simplify your answer.) Hence, find the volume when 6.n [4] (v) One day, the jewellery maker would like to make a piece of fashion accessory using a charm with an approximate diameter of 0.306 cm. This charm can be found in the m th basket. Find m. [3] 2g( ) 2 (1 ) , xx x   1gf 1gf 1gf ( ) 7.a 