2025 EJC Promo (Qn)

1 2025 EJC H2 Maths Promo Paper Duration: 3 hrs Marks: 100 Attempt all questions. 1 Four housewives bought three different kinds of organic vegetables at a Goodprice supermarket. They cannot remember the individual prices per kilogram, but three of them can remember the total amount that they each paid. The weights of vegetables purchased and the total amounts paid are shown in the following table. Amy Betty Carol Denise Pumpkins (kg) 1.20 1.50 1.30 2.30 Carrots (kg) 2.50 2.70 2.60 3.10 Broccoli (kg) 1.25 1.35 1.10 2.10 Total amount paid ($) 21.86 25.20 21.90 ? Given that, for each variety of vegetable, the price per kilogram paid by each of the housewives is the same, find the amount that Denise paid for her purchase. [4] 2 The diagram shows the sketch of the curves 2 5yx= − and 2yx= + . Solve the inequality 2 52xx−>+ exactly. [4] 3 (a) Given that ,×=ab 0 what can be deduced about the vectors a and b? [2] (b) Find a unit vector n such that ( )23.× −+ =n ij k 0 [1] (c) Find the cosine of the acute angle between 23−+ij k and the y-axis. [2] 5 2 -2

2 4 The diagram below shows the curve f( )yx= . The curve crosses the x-axis at point ( ,0)Aa , where 1a<− and has a maximum point 1 10,29B−− . The curve also has a horizontal asymptote 1y=− and vertical asymptotes 1x=− and 0x= . On separate diagrams, sketch the graphs of (a) 3f ( 1)yx= − , [2] (b) 1 f( )y x= , [3] showing clearly in each case, the equations of any asymptotes, the coordinates of any stationary points and axial intercepts in terms of a. The curve f( )yx= undergoes the following transformations in succession: P: Reflection in the y-axis Q: Scaling parallel to the x-axis by a scale factor of k (c) Determine the coordinates of the x-intercept of the transformed curve. [2] y x 0

3 5 The sequence 123, , , ...xxx satisfies the recurrence relation ( ) 2 1 2 for 1nn nxx x n+ = −≥ (a) Prove algebraically that, if the sequence converges, then it converges to either 1 or 0. [2] (b) Determine the behaviour of the sequence for each of the following cases. (i) 1 0.5x = [1] (ii) 1 1.5x = [1] (iii) 1 1x =− [1] (c) By considering the graph of 232y xxx= −− , show that if 01 nx<< , then 1 0nnxx+ −< . State briefly what this result tells us about the sequence in part (b)(i). [2] 6 A curve C is defined parametrically by 26,x ty t= = . (a) Show that the equation of the normal at the point ( ) 26,P pp is ( ) 3 18 3 0p yp x+ − −= . [3] (b) The normal to the curve at the point P meets the curve again at the point ( )54, 81Q − . Find the possible values of p. [2] (c) Sketch the graph of C, indicating clearly the coordinates of Q and the possible points of P. Sketch also the normals at each of these points P. [3]

4 7 [It is given that a sphere of radius r has surface area 24 rπ and volume 34 3 rπ .] A model of a toy is made up of three parts. • The top of the toy is modelled by a circular disc of radius r cm. • The sides of the toy are modelled by the curved surface of a cylinder of radius r cm and height h cm. • The base of the toy is an empty space which is modelled by the curved surface of a hemisphere of radius r cm. The three parts are joined together as shown in the diagram. The model is made of material of negligible thickness. It is given that the volume of the toy is a fixed value k cm3, and the external surface area is a minimum. Use differentiation to find the values of r and h in terms of k. Simplify your answers. [7] 8 The function f is defined by 1f : , , 1 1x xx x ∈≠− . (a) Explain why 2f does not exist. [2] The function g is defined by 1g : , , 0,11x xx x ∈≠− . (b) Given that 2g exists, find 2g() x . State its domain and range. [3] (c) Show that 3g() xx= and hence solve the equation 2025g () g () 0xx+= . [3]

5 9 The curve 1C has equation ( ) ( ) 22 3 4 1 16xy+− += . (a) Sketch the curve 1C , stating the equations of any asymptote(s) and other key features clearly. [3] Another curve 2C has parametric equations 1 2xt t= − − and 5yt= −+ where 22 t−≤< . (b) Find the coordinates of any point(s) of intersection between 1C and 2C . [2] (c) Sketch the curve 2C on the same diagram as part (a), showing clearly any asymptote(s) and the coordinates of any axial intercept(s). [4] 10 It is given that ( )tan ln 1 2yx=+  , for 0.5x>− . (a) Show that ( ) ( ) 2d1 2 21 d yxy x+= + . [2] (b) Hence find the first three non-zero terms of the Maclaurin expansion of ( )tan ln 1 2 x+ . [4] (c) It is given that the three terms found in part (b) are equal to the first three terms in the series expansion of ( )21 b x ax+ . Find the exact values of the constants a and b. [3] 11 (a) Find 1 1 e d x x x − − ⌠ ⌡ . [2] (b) Find 2 1 d4 4 17 xxx ++ ⌠⌡ . [3] (c) Use the substitution 21u x= + to find 2 1dx xx+∫ . [4]

6 12 Lines 1l and 2l have equations 1 : 32 11 ab l λ      = −+       r and 2 72:2 35 xzly −+ =−= − respectively, where λ is a parameter and a, b are constants. (a) It is given that lines 1l and 2l intersect each other at a right angle at point A. Find the values of a and b, and show that point A has coordinates ( )4, 1, 3 . [5] Plane 1p has equation 2 4.xz+= − (b) Find the shortest distance from point A to plane 1.p [2] Plane 2p contains both lines 1l and 2l , and has equation 11 8 5 51xyz− +−= − . (c) Find the acute angle between planes 1p and 2 .p [2] (d) Hence, or otherwise, find the shortest distance of point A to line 3 ,l the line of intersection between 1p and 2 .p (Give your answer to three significant figures.) [2] 13 The effective range of an electric car refers to the distance that an electric car can travel on a single full charge. Electrica, an electric car manufacturer, releases the Model LX with an initial effective range of 460 km to celebrate its 10th anniversary. The effective range decreases over time due to battery degradation. In the first year of ownership, Model LX’s effective range decreases by 5.6 km. In each subsequent year, the decrease in effective range is 0.75 km more than the decrease in the previous year. For instance, the effective range decreases further by 6.35 km in the 2 nd year. (a) Show that the decrease in effective range in the 7th year is 10.1 km. [2] (b) A battery replacement is recommended once the effective range falls below 75% of the initial range. Determine the year in ownership when a battery replacement is first recommended. [4] On 1 January 2025, Ben purchased a brand- new Model LX and took up a car loan of $150 000 from a bank. The loan is such that an interest of 0.2% is added to the outstanding loan amount at the start of every month, with the first interest amount added on 1 January 2025. Ben makes a monthly repayment of $ x on the 12 th of every month, with the first repayment on 12 January 2025. (c) Show that the outstanding loan amount, in dollars, at the end of n months is ( ) ( )150 000 1.002 500 1.002 1nn x−− . [3] (d) (i) Suppose Ben made the last monthly repayment of $x on 12 Dec 2029 to fully pay off the loan. Find the value of $x to the nearest cent. [2] (ii) Use your answer in part d(i) to calculate the total interest paid on the loan. [1]