2023 EJC Promo (Qn)

2023 EJC H2 Maths Promo Paper Duration: 3 hrs Marks: 100 Attempt all questions. [You may skip Q10 for now, as they are on Integration. Remaining total marks is 90, Duration is 2 hr 42 mins.] 1 A function f is defined by  32f xxa xb x c  , where a, b and c are constants. The graph of fyx passes through the points  2, 1 and 2, 3 . The point 2,1 lies on the graph of f1yx . Find the values of a, b and c. [4] 2 The diagram below shows the graph of fyx . The curve passes through the x-axis at  2, 0 and  1, 0 , and has a maximum point with coordinates 2, 2 . The lines 1x and 1y are asymptotes to the graph. Stating the equations of any asymptotes and the coordinates of any points of intersection with the axes and stationary points, where possible, sketch the graphs of (a) 3fyx a , where a is a positive constant such that 12 a , [3] (b) fyx  . [3] 3 (a) Sketch, on the same diagram, the curves with equations 1 2 xy x   and 1ln 1 22 xy   , stating the equations of any asymptotes and the coordinates of an y points of intersection with the axes. Label the two curves clearly. [5] (b) Hence solve the inequality 11 ln 122 2 xx x     . [1] y x O     ×

4 Referred to the origin O, let P, Q and R be distinct points with position vectors p, q and r respectively. (a) Show that       rp rq qqrr pp . [2] (b) Give the geometrical meaning of 1 2 qqrrpp . [2] (c) Given that qqrrp0p , 3PR QR , and that PQ PR , express r in terms of p and q. [3] 5 (a) Verify that   2321 3 1 1! ! ( 1 ) ! 1! rr rr r r    . [1] (b) Hence find  2 1 31 .1! n r rr r   [3] (c) Use your answer to part (b) to find 2 3 3 ! n r rr r    . [3] 6 (a) The first n terms of a series are given by 21log 3 log 27 log 243 ... log 3 n aa a a    , where a is a positive constant. (i) Show that the series is an arithmetic series. [2] (ii) Given that sum of the first 30 terms of the series is 300, find the value of a. [2] (b) A geometric series has first term c and common ratio r, where c and r are non-zero. An arithmetic series has first term b and common difference d, where b and d are non-zero. It is given that the 5th, 8th and 10th terms of the arithmetic series are equal to the 2nd, 3rd and 4th of the geometric series respectively. Show that r satisfies the equation 235 2 0rr  and hence find the sum to infinity in terms of c. [4] 7 A function g is defined by 2 2 if 0,g( ) 1 0i f 0 . xx x x      (a) Give a reason why g does not have an inverse. [1] y x y = g(x) O 2

(b) The function 1g exists if the domain of g is restricted to xk . State the greatest possible value of k. [1] In the rest of the question, the domain of g is xk , where k takes the value determined in part (b). (c) Find 1g( ) x and state the domain of 1g . [3] (d) Sketch, on the axes given below, the graphs of g( )yx and 1g( )yx  . Label the two graphs clearly. Write down the equation of the line in which the graph of g( )y x must be reflected in order to obtain the graph of 1g( )yx  . [3] 8 The curve C has equation  2 x yx y  . It is given that C has only one turning point. (a) Show that 2d1 1d2 2 y y xx  . [4] (b) Hence, or otherwise, show that 32 2 dd 1dd y y x x  . [3] (c) Hence state, with a reason, whether the turning point is a minimum or a maximum. [2] y xO

9 It is given that   2ln 2 e xy  . (a) Show that d 4e 2d yy x  . [2] (b) Hence find the Maclaurin series for y, up to and including the term in 2x . [3] (c) Using standard series from the List of Formulae (MF26), expand  2ln 2 e x as far as the term in 2x , and use this expansion as a check on the correctness of the series found in part (b). [4] 10 (a) Find sin 3 cos dxx x . [2] (b) Find 2 d41 3 x xxx  . [4] (c) Use the substitution 3sinx  to find 29d xx . [4] 11 [A sphere of radius r has surface area 24 r and volume 34 3 r .] A water fountain is to be constructed in the middle of Bishan East Park. It consists of a hemisphere with radius r m joined to an open cylinder with radius r m and height h m (see diagram). The thickness of the fountain is neglig ible. It is given that the fountain , when filled to the brim, can hold a fixed volume k m3 of water. (a) The interior of the fountain is to be painted with a layer of special reflecting paint. The cost of painting is $3 per m2 for the hemispherical surface and $2.50 per m2 for the cylindrical wall. Show that the total cost of painting, $C, is given by 28$ 3 5kr r  . [3] (b) Using differentiation, find the value of r, in terms of k, such that C is a minimum. [4] Keeping C at a minimum, it is now given that 50k  . (c) Find the numerical values of r and h. [2] (d) When the fountain is filled to the brim, a leak develops at the joint between the cylinder and the hemisphere. Water leaks at a constant rate of 0.002 m 3 per minute. Assuming that water is neither lost nor added to the fountain in any other way, find the rate at which the level of water is decreasing. [3] h r

12 Methane ( 4CH ) is a chemical compound with a tetrahedral structure. The 4 hydrogen (H) atoms form a regular tetrahedron, and the carbon (C) atom is in the centre. Let the centre of the C-atom be the point P, and the centres of the 4 H-atoms be the points Q, R, S and T. The coordinates of P, Q, R and S are  21, 0, ,  ,2, 1 1 ,  32, 1, and 0, 1, a respectively. The angle θ subtended by any two C-H bonds at the C-atom, such as angle QPR, is known as the H-C-H bond angle (see diagram above). (a) Find the bond angle, correct to 2 decimal places. [3] (b) By using the fact that QS RS , show that 3a . [2] (c) Find a cartesian equation for plane π, which contains the points P, Q and R. [3] (d) F is the point on  that is closest to the point S. (i) State a vector equation for the line SF. [1] (ii) Hence, show that the coordinates of F are 0, 0, 2 . [3] (iii) Given that the point T is the mirror image of the point S in π, find the position vector of T. [2] Q R S T P θ