9758 H2 Mathematics EJC Practice Paper

9758 H2 Mathematics ~~ JCT Revision Package 2021 ~I~ Practice Papers ~Ur'7 Instructions : After completing your topical revi sion, complete Practice Papers A to E under time constraint of 2.5 hours each . Practice Paper A (2.5 hours, 80 marks) 1 A musical was held in a conf crcncc hall ~t has a maximum capacity of 800 scats. Three categories of tickets Cat A, Cat Band Cat C were sold at $60, $45 and SIS respectively . It was known that 720 tickets were sold and a total of $19200 was collected from the ticket sales . Furthermore, the combined number of Cat A and Cat B tickets sold was half of the number of Cat C tickets sold. For each category of tickets, find the amount of money collected from their sales . [4] 2 Jason is considering to invest $50,000 in a fund at the end of 2018 . The fund will credit an amount of $300 at the end of 2019 . At the end of each subsequent year, the amount credited increases by $150 as compared to the amount credited in the previous year . (l) Find the amount of money that will be credited in the fund at the end of 2022. [2] (iJ) Given that the money in Jason's account first exceeds $70,000 at the end of year N, find the value of N. [4] 3 ex2 - l2ex - 30 . · The cwve Chas equation y = - ---- , where c 1s the exponcnttal constant. x-5 (i) Sketch C, indicating clearly the equations of the asymptotes and the coordinates of any points of intersection with the axes. [ 4] (ii) State the exact coordinates of the point of intersection of the asymptotes . [ 1] (iii) Deduce the exact range of values of m such that the equation ex2 -12ex-30 2e - - - ---,-- =m--- (x- 5)2 x - 5 has no real root. (21 4 The sum of the f arst n terms of a series is given by 20 - S ( t, } (i) Show that the series is a geometric series. (ii) Explain why this series converges and find its sum to infinity . [3] [2] A new geometric series is formed by using the even-numbered terms (i.e. u2 , u, ,u6 , ... ) of the above seri es. Find the sum of the first 20 terms of thi s new series . [3] ] 7

I I euno1a JUNIOlt COLLIGI (a) Engraving is the process of incisin a des. serial numbers arc commonly en J ~gn or text onto a surface. For instance, identify them individually. Today grav onto nems such as machine parts, in order to the job required a mechanical linkage kn computers arc used for engraving, but in the past own as a pantograph. A A pollllcr that conlrOls tbr cngravw,g bit mova Oliff lhe SIC!Kil. The diagram shows a simple pantograph, in the x-y plane, consisting of two pairs of parallel rigid rods, joined together at four joints. The rods are free to rotate at these joints. One end of the pantograph is fixed at the origin O represented by the point (0, 0). Points ( x, y) are defined relative to the fixed point 0. A pointer is attached to A, while the engraving bit is attached to B. The engraver controls the pantograph by moving the pointer over a stencil, which then causes the engraving bit to trace out a copy of the shape. (i) A pantograph is set up such that the engraving bit, point B, always lies on the same line as OA as the pointer moves. The ratio of the distances OA: OB is set at 5: I . Describe a sequence of geometric transformations which maps the shape of the stencil (at A) to the shape of the engraving (at B). [2] (ii) Suppose that the shape on the stencil is a circle with centre at ( h,k) and radius r. State the equation of the circle. Find the equation of the engraved shape, showing your working clearly. [3] (ill) State the area enclosed by the engraved shape. (b) The curve S undergoes the following transformations in order: A: Scaling parallel to the x-axis with scale factor 3 B: Translation of I unit in the negative y-direction C: Reflection about the y-axis [1] (i) The point P(-Jr, 3) lies on the curve S. Find the co-ordinates of the corresponding point after the transformations have been applied. [21 (ii) The equation of the resulting curve, after the transfonnations, is y = - 2 cos ( 1) . Find the equation of curve S, showing your working clearly. [4]

eu JUH 1oa C 8 The function f is defined as f : x H x2 - 2x + 3, x e R, x ~ 1 . (i) Define r-1 in a similar form. [3] (ii) Sketch the graphs off and f"1 in a single diagram and state the relationship between the graphs off and r-1 • [3] Another function g is defined as (ill) (iv) (v) 2 - x g:xH-- , x e R, x~3 . x - 3 Determine whether the composite function fg exists. Given that h : x H 2 - x , x e R, x > 3 and that fh exists, find the range of fh. x - 3 Find the exact solution of th ( x) = 7 . [I) [2) [3)

4 11..UIIO\ ~ ,u.,o. coec:c:. ~ Practice Paper E (2.5 hours, 80 marks) EJC 2020 June Common Test JCJ Hl Matbs(Modified \ 1 A curve c has equation y = ax2 + b✓x + c, where a, band c a.re constants . It is given that C crosses the x-axis at x = 3 and has a twning point at (I, - 2.5) . Find the values of a, band c. ( 4] 2 The function f, with domain the set of non-negative integers , is given by for n = 0, f ( n) = 2 f ( ½ n) for n > 0, n even, 2 - f ( n - 1) for n > 0, n odd. (i) Find f(3), f(4),and f(6) . (ti) Does fhavc an inverse ? Justify your answer . 3 On the same axes, sketch the graphs of [3J [2] (i) x+l h le l . di . Y = la+ 1 , w ere > , m eating clearly the asymptotes and the axial interc epts ; [3] (ii) 4x2 + l 6x + 15 + (y - I )2 = 0 , indicating clearly the centre and any other relevant features . [3] Hence, deduce the number of real roots of the equation 4 x2 + 16x + 15 + ( x + 1 - 1) 2 = O . [ l] h + l (a) W.th . cal ul 1 th . ua1· 6x2 + 3x+ 2 2 1 out using a c ator, so ve e meq 1ty ---- ~ 1 + x . 2- x [4) (b) On the same diagram, sketch the graphs of y=l - x2 + 3x - tl and y =sinx , where 3,r O~x~-. 2 Hence solve sin x < j-x2 + 3x - Ij, where O ~ x S J7r . 2 (5)

-............. ... JUNIO• COLLIGI The points A, Band C have position vectors a band ~. + .!. b . . . ' 2 2 respectively. The polDl Pon AB is such that AP : PB= A. · 1-A. d the · · · an polDt Pon OC as such that OP : PC = µ : 1- µ . (i) (ii) Express OP in terms of A., a and b. [I] B · · OP · Y expressing an terms of µ , a and b, find the values of A. and µ . Hence show that P is the midpoint of 0C . [3] It is given that the position vectors of the points A and Bare 2j + k and t 21 - 2j - 3k respectively. The point Q lies on OA such that PQ is pc:rpcndicular to OA. (iii) Find the position vector of the point Q. [41 6 The curve C, has parametric equations x = 3sin0+ 4, y=3cos0 - 3, where OS 8 S 2,r, and the curve C2 has equation x1 - y 2 = 4 . (i) Sketch C1 and C2 on the same diagram, stating clearly the co-ordinates of any point s of intersection with the axes and the equations of any asymptotes . {5) (il) Show algebraically that the points of intersection of C, and C2 satisfy the equation 3(sin 2 0 - cos2 0) + 8sin 8 + 6cos0 + l = 0 . (iil) Hence, find the co-ordinates of the point s of intersection of C1 and C2 . [2] [3] .... 7 The sum of the first n tcnns of a series G is given by S,. = e - ck" , where k is a positive constant. (i) (ii) (iiO (iv) Find the n111 term of the series G. [2] Hence, show that the series G is a geometric series . [21 Find the exact range of values of k for the sum to infinity to exist . [ l] The second term of the series G is ! . Show that le = 2e , and find the exact value of S 4 , the sum to infinity of the series G. [3] The first term of an arithmetic series A is equal to the first term of G. The common difference of A is .Ji.. (v) The sum of the first n tenns of series A is denoted by L,. . Find the least value of n such that - 1- L differs from S by more than I 0. [4] I 00 ~ 55

10 euno1a JUNIOR COLLIGI A painter is tasked to paint the interior of a HOB flat. First, the painter fills up paint cans each having a volume of2 litres. To prevent spillage , the tap used to fill the paint cans has a special mechanism that can control the volume