EJC H2 Promo 2023 (Qn)

Duration: 3 hrs Marks: 100 2023 EJC H2 Maths Promo Paper Attempt all questions. 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 e quations 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 t he 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