ACJC_9758_2022_Promo_QP

2022 ACJC JC1 H2 Promotional Exam 1 (i) If 3 eln cos x y x   , find d d y x in terms of x, where 0 2x  . [2] (ii) Given that 1 ln xxyx  , find d d y x in terms of x and y. [3] 2 (i) Without using a calculator, solve the inequality 8851 10 2 xx  . [3] (ii) Hence, without using a calculator, solve the inequality 28851 10 12 xx x . [2] 3 The curve C with equation 2 31 ax bx cy x   passes through the point with coordinates ( 1, 4) and has a turning point at ( 3, 2) . (i) Find the values of a, b and c. [3] (ii) Hence sketch the graph of curve C, stating the equations of any asymptotes and the coordinates of any points where the curve crosses the axes. [3] 4 The diagram below shows the graph of f ( )yx . There are two vertical asymptotes with equations xa and xa , where a is a positive real constant, 1a . There is an oblique asymptote with equation y x a . The point s A, B and C have coordinates  2,0 ,  0, 2 and  1.5,0 respectively, where B is also a maximum turning point. Sketch the following curves and state the equations of the asymptotes, the coordinates of the turning points and of points where the curve crosses the axes, if any. Leave your answers in terms of a where necessary. (i) 1 f ( )y x , [3] A B C x y xa

(ii) f (1 2 )yx . [3] 5 The line L has vector equation 62 : 0 1 11 L                    r ,  . (i) Find the position vector of the point N, the foot of perpendicular from the point A with coordinates (3, 6,1) to the line L . [3] (ii) Find the position vector of the point 'A , the reflection of point A in the line L . [2] (iii) Find the coordinates of the points on the line L that are 33 units away from point A. [2] 6 The function f is such that  f cosrr  . (i) Show that       f 2 1 f 2 sin 2 1 sinr r k r      , where k is a constant to be determined. [2] (ii) Using your result in (i), show that   2 1 sinsin 2 1 sin n r nr     using the method of difference. [3] (iii) Hence find   1 sin 2 1 n r r    in terms of n and  . [2] 7 (a) Referred to the origin O, points A and B have position vectors a and b respectively, where a and b are non-zero and non parallel vectors. Point C lies on OA, between O and A, such that OC : CA  2 : 1. Point D lies on OB produced such that 5BD OB . Find, in terms of a and b, the position vector of the point E where the lines AB and CD meet. [3] (b) The position vectors of the points D and E relative to the origin O are d and e respectively, where d and e are non-zero and non parallel vectors. It is given that the length OE is 2 units and 3de . The point F, with position vector f, is the reflection of the point D in the line OE. (i) Express f in terms of vectors d and e. [3] (ii) Show that the area of triangle ODF can be expres