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 expressed as k de , where k is a constant to be determined. [2]

8 A curve C has parametric equations , ln aax y tt , where a is a positive constant and 0t  . (i) Find the equation s of the tangent and normal to the curve at the point P with parameter p. [3] (ii) The tangent at P and the normal at P meet the y-axis at point A and point B respectively. Find the area of triangle APB in terms of a and p. [2] (iii) Sketch curve C and state, in terms of a, the coordinates of any point (s) where the curve crosses the axes. [1] (iv) State the equation of the tangent to C when 1p . Hence state the range of values of m such that the line y mx a cuts curve C at two points. [2] 9 It is given that     2 2 2 f 6 5, for 1 5, g 6 5, for 1 3, ln for 0 1,h( ) 6 5 for 1 5, x x x x x x x x xxx x x x                and that h( ) h( 5)xx for all real values of x. (i) Find the range of f. [1] (ii) Show that -1g exists. [1] (iii) Find 1g ( )x and state the domain of 1g . [3] (iv) Find the exact value of 11h h( 2)2   . [2] (v) Sketch the graph of h( ) for 4 6y x x    , stating the equations of any asymptotes and the coordinates of any points of intersection with the axes. [3] 10 The plane 1P has the equation 6 4 2 4x y z    . (i) Find the vector equations of the planes such that the perpendicular distance from each plane to 1P is 10 units. [2] The plane 2P has the equation 2x y z k    , where k is a constant. (ii) Find the angle between 1P and 2P . [2]

(iii) The planes 1P and 2P intersect in the line L . Show that a possible vector equation of L is 2 2 3 2 3 5 01 k k                     r ,  . [3] The plane 3P has the equation 55x y z   , where ,  . (iv) Given that the line L is contained in the plane 3P , find  and μ, giving your answer in terms of k if necessary. [2] (v) Given instead that the line L does not intersect 3P , what can be said about the values of  and μ? [1] 11 Antonio used a credit card to buy a diamond ring to propose to his girlfriend. On 1 st October 2022, he charged $7500 to the credit card. The credit card charges a 2% monthly interest rate that is applied on the last day of every month. From 1 st November 2022, Antonio started paying off his credit card debt by paying $ x at the start of each month, where x is a constant. (i) Find the amount of Antonio’s credit card debt left after his first repayment, giving your answer in terms of x. [1] (ii) Show that the amount of Antonio’s credit card debt after his nth repayment is   7500 50 1.02 50 nxx . [3] (iii) Find the minimum amount that Antonio must consistently pay each monthly repayment for him to fully pay off his credit card debt with his 60th repayment. Find also the amount of credit card interest he would have paid in this case. [4] (iv) Suppose that up to and including 1 st October 2023, Antonio had faithfully repaid $500 a month to service his credit card debt. As a promotional reward, the credit card company decided to lower Antonio’s monthly interest rate to 1% with immediate effect from October 2023, provided he maintains his $500 monthly repayments. Find the date of Antonio’s last repayment to the credit card company with this new interest rate, and the amount of his last repayment. [4] Antonio and his fiancée are deciding between two venues for their wedding banquet by comparing the costs of the catering at each venue. At the Gomez Hotel, the catering is priced at $1500 for each table of 10 guests. At the Grande Hotel, the catering is priced as follows:

GRANDE HOTEL BANQUET PRICE LIST Number of Tables (Each table sits 10 guests, venue can hold up to 40 tables) Price For first 10 tables $2000 per table For more than 10 tables Each additional table will be $50 cheaper than the previous. (So the 11th table will cost $1950, the 12th table will cost $1900, and so on.) (iv) How many tables will the couple need at least, for it to be cheaper to hold their wedding banquet at the Grande Hotel? [3] 12 Mr See wants to b uild a fish tank with a fixed capacity  53 k 3m . The following diagram shows the dimensions of the fish tank he desires, with variables x and y. The cross-section of the fish tank ABCD is a right trapezoid with AB being 3x m long, DC being 2x m long, and 60ABC . The cross -section ABCD is perpendicular to both the rectangular base ABFE and the rectangular opening DCGH, with AE BF DH CG y    m. (i) Show that the total surface area A 2m of the fish tank in contact with water when it is fully filled is  2 2 5 3 53 k Ax x   . [4] 2x y y y y A B C D E F G H

(ii) Mr See would like to minimise the amount of material require d to construct the walls and base of the fish tank. Using differentiation, find the