RVHS H2 MATHS P1 QP

1 2015 RVHS H2 Maths Prelim Paper 1 1 A graphic calculator is not to be used in answering this question. (a) Solve 2 74 223 x xx   . [3] (b) Hence solve    2 2 1 2 023 xx xx   . [2] 2 In the triangle ABC, 1AB , 1 2BC  and angle 2ABC  radians. Given that angle ABC is a sufficiently small angle, show that 1 221 24AC  . Hence, express AC in the form 24a b c , where a, b and c are constants to be determined. [5] 3 (a) The sum of the first n terms of a sequence is given by nnSn 32 2  . Prove that the given sequence is an arithmetic progression. Find the least value of n such that the sum of the first n terms exceeds 2015. [4] (b) In an extreme cake-making competition, Kimberley is tasked to make a multi-layered cake in which each layer is in the shape of a cylinder, starting with the base layer of height 10 cm. For each subsequent layer above, it must be smaller than the previous layer. I n particular, the he ight of any layer above the base must be )5( k % of the height of the previous layer, where k is a positive integer. The layers are joined together using whipped cream. (i) Show that the height of an n-layer cake is given by              n k k 20120 200 cm. [1] (ii) It is found that a cake with height exceeding 1.2 m will become structurally unstable. Using k = 19, find the maximum number of layers that Kimberley’s cake can have before it becomes structurally unstable. [2] (iii) State a possible assumption used in part (ii). [1]

2 2015 RVHS H2 Maths Prelim Paper 1 4 (a) Show, by using the substitution tanxu ,     21 2 tan1 2 tan dd1 x uxe x ue ux     . Hence find the given integral. [3] (b) The region R is bounded by the curve     21tan1 2 tan 1 x xe y x     , the line 2π 32π e8y and the y-axis. (i) Sketch the curve and identify the region R. [1] (ii) Find the exact volume generated when region R is rotated 2π radians about the x-axis. [4] 5 (a) Given that   2 1 1 1! nn nnuu n    and 0 1u  , prove by induction that 1 for 0! n nun n  . [5] (b) Using the recurrence relation in part (a), find 2 1 ( 1) ( 2)! N r rr r   . [4] 6 (i) For two non-zero and non-parallel vectors a and b, give the geometrical meaning of ab . [1] (ii) The position vectors of the points A and B are a and b respectively. Another point C on OB produced is such that 2:1: BCOB and the point D on line AC is such that 1:2: DCAD . The point M is the point of intersection between AB and OD. Find the following vectors in terms of a and/or b (a) OD ; [2] (b) OM . [3] (iii) It is given further that b =2 and the area of triangle OAC is 12 square units, find the shortest distance from A to OC. [3]

3 2015 RVHS H2 Maths Prelim Paper 1 7 A graphic calculator is not to be used in answering this question. (i) Find the roots of the equation 2 8iz  in the form iab . [4] (ii) Hence, sketch on a single Argand diagram, the roots of 4 64w  . [3] (iii) Find the roots of the equation   2 2 2i 4i 0zz    . [3] 8 Given that  )1ln(sin 1   xy , show that 1 1 d dcos  xx yy and 2 2 2 2 )1( 1 d dsind dcos      xx yyx yy . [2] (i) By further differentiation, find the Maclaurin series for y, up to and including the term in 3x . [3] (ii) Find the set of values of x for which the value of y is within 0.1 of the value found by its Maclaurin series. [3] (iii) Deduce the series expansion for 2))1(ln(1)1( 1  xx up to and including the term in 2x . [2] 9 A plane 1 contains points A, B and C with coordinates 3) 0, ,2( , 4) ,1 ,1(  and 0) 1, ,5( respectively. (i) Show that the vector i – 3j is perpendicular to the plane 1 . [3] (ii) Given a point M with position vector k, find the posit ion vector of the foot of perpendicular from M to plane 1 . Hence, find the distance of M to the plane 1 in the form b a , where a and b are positive integers to be determined. [4] Another plane 2 contains the point M and has equation 5 bzyax . (iii) Given that plane 2 is perpendicular to the y-z plane, find the acute angle between 1 and 2 . [4]

4 2015 RVHS H2 Maths Prelim Paper 1 10 (a) A curve )(f xy is transformed by a stretch with scale factor 2 parallel to the y-axis, followed by a reflection about the x-axis. The equation of the resultant curve is      2)( 1ln axy , where xa and 1a . Find f(x) and sketch )(f xy . [4] (b) The diagram shows the graph of )(g xy . The graph passes through the origin and has a turning point (3, 4)A . The asymptotes of the graph are 2x and 42  xy . On separate diagram, sketch the following graphs indicating the points corresponding to the axial intercepts, turning point and asymptotes where necessary. (i) )(g2 xy  [2] (ii) )(g' xy [3] (iii) )(g 1 xy [3] x = 2 A ● 0 y = 2x – 4 x y

5 2015 RVHS H2 Maths Prelim Paper 1 11 The curve C is defined by 21, ln ,x y tt for 2 2, 0tt    . Sketch C. [2] Find the equation of the normal at point P where tp . [3] Find the coordinates of the point on C where the tangent does not intersect the normal at P. [2] Use 1 2p in the following. Find the exact area bounded by the curve C, the x-axis and the line 2x . [3] By using cross product or otherwise, find the distance between the non-intersecting tangent and normal. [3]  End of Paper 