2019 J2 ASRJC H2 Math Prelim (with ans)

1 1( a) Show that 12 12 x x   can be written in the form 2 f( ) 14 x x where f(x) is a polynomial to be determined. Hence, find 12 d12 x xx      [3] (b) Show that 2e e 1 dln xxx    = ln 2.[ 3] 2 It is given that 1tan 22 yx , where 22 y  . (i) Show that  2 d12 2 2 d yx x .[ 2] (ii) Use the result from part (i) to find the first two non-zero terms in the Maclaurin series for y, giving the coefficients in exact form. [ 3] (iii) Hence, using standards series fro m the List of Formulae (MF26), find the expansion of 12t a n 2 cos 2 x x  in ascending powers of x, up to and including the term in x3, giving the coefficients in exact form. [ 3] 3 The position vectors of A, B and C referred to a point O are , and ab c respectively. The point N is on AB such that AN:NB = 2:1. (i) If O is the midpoint of CN, prove that + 2 3ab c 0  . [2] (ii) Show that A, O and M are collinear, and find the ratio AO:OM [3] (iii) If the point P is such that NP AM K K , show that the ratio of the area of PNAM : area of PNOM = 12:7 [3] +0DWK3UHOLP$65-& www.KiasuExamPaper.com 45

2 4( a) The diagram shows the graph of f( )yx . The curve crosses the x-axis at 0x  and 3x  . It has a turning point at (4, 5) and asymptotes with equations 2y  and 2x  . Showing clearly the coordinates of turning points, axial intercepts and equations of asymptotes where possible, sketch the graphs of (i) fyx ;[ 2] (ii) f'yx [3] (b) The curve with equation y = f(x) is transformed by a stretch with scale factor 2 parallel to the x–axis, followed by a translation of 2 units in the negative x– direction, followed by a translation of 3 units in the positive y–direction. The equation of the resulting curve is  3ln eyx . Find the equation of the curve y = f(x). [3] y x y = f(x) 0 (4, 5) 3 y = 2 x = 2 www.KiasuExamPaper.com 46

3 5 A curve C has parametric equations 2sin , cos ,xa t ya t where 0 and 0.2ta   (i) Find the cartesian equation of C, stating clearly any restrictions on the values of x and y. [2] (ii) Sketch C, showing clearly the axial intercepts. [1] (iii) The region bounded by C, the line 5 4yx a and the y-axis is rotated through ʌUDGLDQVDERXW the y-axis. Show that the exact volume of the solid obtained is 3ʌka where k is a constant to be determined. [5] 6 A function f is defined by 2 6 for 2,4f( ) 1 for 2 1, 2 for 1. xxx xx xx x            (i) Sketch the graph of f. [3] (ii) Evaluate exactly 4 3 f( ) dxx  .[ 4] 7 Do not use a calculator in answering this question. The roots of the equation z2 + (2 – 2i)z =– 3–2 i a r e 1z and 2z . (i) Find 1z and 2z in cartesian form x + iy, showing clearly your working. [5] (ii) The complex numbers 1z and 2z are also roots of the equation 43 24 14 4 13 0zz zz    . Find the other roots of the equa tion, explaining clearly how th e answers are obtained. [2] (iii) Using your answer in part (i), solve z2 + (2 + 2i)z = 3 + 2i. [2] www.KiasuExamPaper.com 47

4 8( i) Show that    34 21 2 1 rA B C rr r r rr      where A, B and C are constants to be determined. [1] (ii) Find the sum to n terms of 71 0 1 3 ...321 432 543   (There is no need to express your answer as a single algebraic fraction). [4] (iii) Hence show that 3 5 35 4 (1 ) (2 ) (3 )3 n r r rr r      .[ 3] 9 The diagram above shows an object with O at the centre of its rectangular base ABCD where AB = 8 cm and BC = 4 cm. The top side of the object, EFGH is a square with side 2 cm long and is parallel to the base. The centre of the top side is vertically above O at a height of h cm. (i) Show that the equation of the line BG may be expressed as 43 21 , 0 t h      r where t is a parameter. [1] (ii) Find the sine of the angle between the line BG and the rectangular base ABCD in terms of h.[ 2] It is given that h = 6. (iii) Find the cartesian equation of the plane BCFG .[ 3] (iv) Find the shortest distance from the point A to the plane BCFG.[ 2] O A B C D E F G H i jk www.KiasuExamPaper.com 48

5 (v) The line l, which passes through the point A, is parallel to the normal of plane BCFG. Given that, the line l intersects the plane BCFG at a point M, use your answer in part (iv) to find the shortest distance from point M to the rectangular base ABCD.[ 2] 10 Commonly used in building materials, sand is the second largest world resource used by humans after water. To reduce the environmental impacts of s and mining, an alternative approach is to make “sand” by crushing rock. A machine designed for this purpose produces sand in large quant ities. Sand falling from the output chute of the machine forms a pile in the shape of a right circular cone such that the height of the cone is always equal to 4 3 of the radius of its base. [Volume of a cone = 21 3 rh and curved surface area of a cone = rl where r is the radius of the base area, h is the height of the cone and l is the slant length of the cone] A machine operator starts the machine. (a) (i) Given that V and A denote the volume and the curved surface area of the conical pile respectively, write down V and A in terms of r, the radius of its base. [2] (ii) Hence show that the rate of change of A with respect to V is inversely proportional to the radius of the conical pile. [3] An architect is tasked to design sand-lined walking paths in a large park. He decides to base his design of the paths on the shape of astroids, which are shapes with equations 222 333 xyk (k > 0). On a piece of graph paper, he sketches an astroid with the equation 222 333 xyk . (b) The tangent at a point P  11,xy on the curve meets the x-axis at Q and the y- axis at R. Show that the length of QR is independent of where P lies on the curve. [7] Sand Chute www.KiasuExamPaper.com 49

6 11 (a) Find the sum of all integers between 200 and 1000 (both inclusive) that are not divisible by 7. [4] (b) Snowflakes can be constructed by starting with an equilateral triangle (Fig. 1), then repeatedly altering each line segment of the resulting polygon as follows: 1. Divide each outer line segmen ts into three segments of equal length. 2. Add an equilateral tr iangle that has the middle segment from step 1 as its base. 3. Remove the line segment that i s the base of the triangle from step 2. 4. Repeat the above steps fo r a number of iterations, n. The 1st, 2nd and 3rd iteration produces the snowflakes in Fig. 2, Fig. 3 and Fig. 4 respectively. (i) If 0a denotes the area of the original triangle and the area of each new triangle added in the nth iteration is denoted by na , show that 1 1 9 nnaa  for all positive integers n.[ 2] (ii) Write down the number of sides in the polygons in Fig.1, Fig. 2 and Fig. 3, and deduce, with clear explanations, that the number of new triangles added in the nth iteration is 4n nTk  where k is a constant to be determined. [2] (iii) Find the total area of triangles added in the nth iteration, nA in terms of 0a and n.[ 2] (iv) Show that the total area of the snowflake produced after the nth iteration is 0 83 4 55 9 n a      units2.[ 3] The snowflake formed when the above steps are followed indefinitely is called the Koch snowflake. (v) Determine the least number of iterations needed for the area of the snowflake to exceed 99% of the area of a Koch snowflake. [3] Fig. 1 Fig. 2 Fig. 3 Fig. 4 www.KiasuExamPaper.com 50

[Turn Over Section A: Pure Mathematics [40 marks] 1 Given that the curve 2 f( )1 ax bx cyx x   passes through the points (1, 3), 13,22    and 335, 2    , find the values of a, b and c.[ 3] Hence find the exact range of values of x for which the gradient of y = f(x) is positive. [3] 2 The functions f and g are defined by: 2f: xa x x6 , x  Թ g: e 1 xxa  6 , x  0 where a is a positive constant and 5a  . (i) Sketch the graphs of y = g(x) and y = f(x) on separate diagrams. Hence find the range of the composite functi