EJC 9758 2024 Promo

Error! Reference source not found. EUNOIA JUNIOR COLLEGE JC1 Promotional Examination 2024 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. MATHEMATICS Paper 1 9758/01 07 October 2024 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 7 printed pages and 1 blank page.

Error! Reference source not found. 1 A small furniture factory produces standardised stools, chairs and tables using both metal and plastic as raw materials. The table below shows the amount of each raw material required to make one unit of each type of furniture. Furniture Item Amount of raw material needed (in kg) Metal Plastic Stool 1 2 Chair 4 2 Table 10 4 Given 68 kg of metal and 32 kg of plastic, determine the number of each type of furniture that could be produced such that all three types of furniture are made, and all the raw materials are fully utilised. [4] 2 Solve the inequality 2 531 2 xx +− −  . [3] 3 The graph of ( )fyx= is given below. It has one vertical asymptote 1x=− and one oblique asymptote 4yx=− . It cuts the x-axis at 0x= and 3x= . It has turning points at ( )3, 9−− and ( )1, 1− . On separate diagrams, s ketch the graph s of the following, stating the equations of any asymptotes, and indicating any axial intercepts and turning points. (a) 1 f ( )y x= , [3] (b) f ( )yx = . [3] x y O 3

Error! Reference source not found. 4 In a triangle ABC, 5 cmAB= and 4 cmAC= . If angle BAC is decreasing at a constant rate of 0.2 radians per second, find the rate of change of the length BC at the instant where π 3BAC= radians, giving your answer in centimetres per second to 3 significant figures. [5] 5 a and b are vectors such that 2=a and 6=b . The angle between a and b is 5π 6 . (a) Find the exact length of projection of a onto b. [2] (b) By considering ( ) ( )3 2 3 2++a b . a b , find the exact value of 32+ab . [4] 6 (a) Write down the derivative of sin n with respect to  , where n is a constant and 1n . [1] (b) Using the substitution 22cosx = for 0 2  , show that 1d 2 xxx −=  ( ) 428 cos sin cos sin d    − . [4] (c) Hence find 1d 2 xxx −  . [3] 7 (a) The terms 1 2 3, , ,u u u form a geometric sequence with first term a and common ratio r, where 0a . It is given that the sum of the first 10 terms is 33 times the sum of the first 5 terms. (i) Explain why r cannot be equal to 1. [1] (ii) Show that 2r= . [2] (iii) If 7u is the first term in the sequence greater than 11, find exactly the range of possible values of a. [2] (b) Given that ( )( )1 11 2 4 2 1 2 1 2 1 n r rr n= =− −+ + , find ( )( )4 1 2 3 2 5 n r rr= ++ in terms of n. [3]

Error! Reference source not found. 8 It is given that ( )tan ln 1 2yx=  +  . (a) Show that ( ) ( ) 2d1 2 2 1 d yxy x+ = + . [2] (b) By differentiating the result in part (a), find the Maclaurin expansion of ( )tan ln 1 2 x +  up to and including the term in 2x . [3] (c) Hence, find the Maclaurin expansion of 1 2e y , up to and including the term in 2x . [2] (d) Using your answer in part (c), evaluate 1 2 0 e1lim 3 y x x→ − . [2] 9 A curve C has parametric equations 2 1xt=− , 32 23y t at=− for t where a is a positive constant. P is a point on C where ta= . (a) Find the equation of the tangent to the curve at P. [3] (b) Q is another point on C. The tangent to the curve at Q is parallel to the tangent to the curve at P. Find, in terms of a, the coordinates of Q. [4] (c) Find the range of values of a for which C intersects the positive x-axis. [2]

Error! Reference source not found. 10 The figure shows a sketch of the graphs ( )1 124y x x=− and 1 42yx=+ , for 0 8x . The function f is defined as ( ) 1 0, f 1 12 for4 4 for2 : 8, xx xc xc x x     −   + where the value of c satisfies 08 c . (a) State the domain of f. [1] (b) (i) Explain why the inverse function 1f− does not exist when 6c= . [2] (ii) Find the set of values of c for which the function 1f− exists. [1] (c) (i) State the range of f when 6c= . [1] (ii) Find the range of f when 1c= . [2] (iii) Find the set of values of c for which the composite function 2f exists. [2] (8, 8) (2, 5) y x (6, 9) (0, 4) O

Error! Reference source not found. 11 A sequence 1 2 3, , , ...u u u is such that 1 2u = , 1 3nnu ku+ =− for all 1n , where k is a constant. (a) Let 1 5k = . (i) Write down the values of 2u , 3u and 4u . [2] (ii) Given that the sequence converges to l, find the value of l. [1] (b) Let 1k= . (i) Given that the sequence 1 2 3, , , ...u u u is an arithmetic progression, find the value of 1 4 r r n u = + . [3] (ii) Show that the sequence 7 10 13 16 3 4, , , , ..., , ... nu u u u u + is also an arithmetic progression. [2] (iii) Hence, find the value of 100 34 1 r r u + =  . [2] 12 The curve C has equation ( ) 2 322 2 kxyx xk −= − + − , where 0k , 2k . (a) Without the use of a calculator, find the coordinates of any points of intersection of C with the axes. [3] (b) Write down the equations of the asymptotes of C in terms of k. [2] It is now given that 1k= . (c) Show that the x-coordinates of the turning points of C satisfy the equation 42 7 12 2 0x x x− + − = . Hence find these x-coordinates. [3] (d) Sketch C, labelling all essential features. [3]

Error! Reference source not found. 13 v and n are unit vectors in 3-dimensional space, with 0v.n . Let P be a plane with unit normal n. The reflection of v in plane P, w, is given by the formula ( )2=−w v v.n n . (*) (a) Calculate the vector w when 0 0 1  =− v and 3 1 05 4  =  n . [2] When the unit vectors v and n vary, the formula (*) will always result in a unit vector for w. (b) Verify that the vector w calculated in part (a) is a unit vector. [1] Before photovoltaic technology allowed for cost-effective direct conversion of sunlight to electrical energy in solar panels, solar power plants relied on arrays of mirrors to concentrate a large area of sunlight into a receiver. In one such plant (see diagram), the receiver is modelled as a single point C with coordinates (0, 0,10) , with mirrors surrounding it. One particular mirror has centre M with coordinates ( 10, 5, 0)−− , and its surface is modelled as part of a plane P passing through M with unit normal n. (c) Write down the unit vector w in the direction of MC . [1] At midday, incoming beams of sunlight can be modelled as travelling along the direction 0 0 1  =− v . The mirror is angled so that a beam of sunlight shining on M is reflected to pass through the receiver C. (d) Using the formula (*) or otherwise, derive that a unit normal for plane P is 2 5 1 30 1  =  n . [3] (e) Hence find a cartesian equation of plane P. [2] (f) Find the acute angle that plane P makes with the xy-plane. [2] (g) State, in the co