TJC 2020 IP4 IM EOY Questions with Answers

1 [Turn over TemasekJC/IP4/IM/EoY/2020 TEMASEK JC INTEGRATED PROGRAMME YEAR FOUR 2020 END-OF-YEAR EXAMINATION INTERMEDIATE MATHEMATICS Date: 1 October 2020 Duration: 2 hours 30 minutes No additional materials required. READ THESE INSTRUCTIONS FIRST. Write in dark blue or black pen in the spaces provided in the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. 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. The calculator value for π should be used unless the question requires the answer in terms of π . The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. Marks will be deducted for poor or unclear presentation. The number of marks is given in brackets [ ] at the end of each question or part question. The total score for this paper is 100 marks. This paper consists of 13 printed pages. Question Marks 1 2 3 4 5 6 7 8 9 10 11 12 Presentation Deduction – 1 Total Name: ________________________ Class: ________________________ ________________________ Parent’s Signature

2 [Turn over TemasekJC/IP4/IM/EoY/2020 MATHEMATICAL FORMULAE Compound Interest 1 100 n rAP =+  Mensuration Curved surface area of a cone = rl Surface area of a sphere = 24 r Volume of a cone = 21 3 rh Volume of a sphere = 34 3 r Area of triangle ABC = 1 sin2 ab C Arc length r= , where  is in radians Sector area 21 2 r = , where  is in radians Trigonometry sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + − Statistics Mean fx f=  Standard deviation 22fx fx ff  =−   

3 [Turn over TemasekJC/IP4/IM/EoY/2020 Answer all questions in the spaces provided. Show all working clearly. 1 (a) Express 4 2 3 3 3 5 + + in the form 3ab + , where a and b are integers. [4] (b) Express ( ) ( ) 55 3 5 0 52 2 x x x − + in the form 5cx d+ , where c and d are integers. [4] 2 Figure 1 (not drawn to scale) shows a solid metal paperweight with a uniform cross - section and height 3 cm. Figure 2 (not drawn to scale) shows the cross -section of the paperwei ght where ABC is the arc of a circle of radius 4 cm and centre O. AC is a straight line of length 4 cm. (i) Show that the uniform cross-sectional area of the paperweight is 240 π 4 3 cm3  + . [2] (ii) Find the total surface area of the paperweight , leaving your answer cor rect to 3 significant figures. [4] (iii) The paperweight is melted and recast into several identical solid cones. The cones have a base radius of 2 cm and height of 5 cm. Find the maximum number of cones that can be formed. [4] 4 cm 4 cm O A B C Figure 2 Figure 1 3 cm

4 [Turn over TemasekJC/IP4/IM/EoY/2020 3 In the following diagram (not drawn to scale), the points A, C and E lie on the lines BF, DB and FD respectively such that CDE is congruent to BAD. (i) Show that FB is parallel to EC. [2] (ii) Prove that CGD is similar to BAD. [2] The ratio of AG : GD is 1 : 1. (iii) If the area of CDG = 3 units2, find the area of the quadrilateral AGEF. [4] 4 In the diagram shown below (not drawn to scale), O , P and Q are three points on level ground. John stands at the top of a wildlife observation deck OT that is 38 metres tall. At the same time, a rescue truck at P, which is located 200 metres away from O , travels along PQ to save a trapped animal at Q, which is 800 metres due West of O. Given that 156POQ =  , find (i) PQ , [2] (ii) OPQ , [3] (iii) the greatest angle of depression of the rescue truck from John during its travel from P to Q . [3] A F E D C B G T O Q O 800 m P 200 m 38 m

5 [Turn over TemasekJC/IP4/IM/EoY/2020 5 The diagram shown below (not drawn to scale) shows a quadrilateral ABCD, with vertices A(8, 4), (3, )Bk , C(p, q) and D. It is given that the line BD intersects the x-axis at the point E and 90ADB =  . (i) Given that 50=AB units, find the value of k. [3] It is given that the equation of line AD is 3 28=− +yx . (ii) Find the coordinates of E. [4] (iii) The point C lies on the line AE produced such that : 1: 3AE EC = . Find the value of p and of q , showing your working clearly. [2] y x A(8,4) B(3, k) O C(p, q) D E

6 [Turn over TemasekJC/IP4/IM/EoY/2020 6 In the diagram shown below (not drawn to scale), A, B, C and D are points that lie on a circle. AC and BD intersect at E. FG is tangent to the circle at D. It is given that 24BCA =  , 68BDG =  and 38CAD =  . (i) Find ADB . [1] (ii) Prove that AD is not a diameter of the circle. [2] (iii) Find AEB . [1] (iv) Find BDC . [3] A B C D E F G

7 [Turn over TemasekJC/IP4/IM/EoY/2020 7 In the diagram (not drawn to scale), OA= a and OB= b . C is the point on AB such that 3AB CB= . The point D lies on OB produced such that 4OD BD= . (i) Express, as simply as possible, OC in terms of a and/or b . [2] (ii) Show that 12 33CD=− +ab . [2] E is a point on OA such that 2 3OE= a . (iii) Show that ED kCD= , where k is a constant. [3] (iv) Write down two facts about ED and CD. [2] O A B C D E a b

8 [Turn over TemasekJC/IP4/IM/EoY/2020 8 The speeds of 120 cars on road A were measured. The cumulative frequency curve for the speeds of the cars on road A is shown below. (a) Complete the grouped frequency table for the speeds of the 120 cars on road A. [2] Speed (x km/h) 30 40x 40 50x 50 60x 60 70x Frequency 24 6 (b) Using the grouped frequency table in (a), calculate (i) the mean speed, [1] (ii) the standard deviation. [1] 0 20 40 60 80 100 120 30 40 50 60 70 Cumulative Frequency Speed in km/h

9 [Turn over TemasekJC/IP4/IM/EoY/2020 (c) The box -and-whisker plot below was obtained using the data from the cumulative frequency graph above. (i) State the values of x, y and z. [3] (ii) Hence find the interquartile range. [1] (d) The speeds of cars on road B were measured. The data for road B has the same median as the data from road A but with a larger interquartile range. Describe how the box -and- whisker plot for road B will differ from the one given in (c). [1] 9 The quadratic curve ( )fyx= has x-intercepts 2− and 3 . (i) Given that the coefficient of 2x is 1, show that ( ) 2 1 25f 24xx = − − . [2] (ii) State the coordinates of the y-intercept of the curve of ( )f x . [1] (iii) State the solution to the inequality (a) ( ) 25f 4x − , [1] (b) ( ) 25f 4x − . [1] Another quadratic curve ( )gyx= has the same x-intercepts as the curve of ( )fyx= . However, the y-intercept of the curve of ( )gyx= is different from that of the curve of ( )fyx= . (iv) Write down a possible expression for ( )g x in the form 2ax bx c++ , where the curve of ( )gyx= concaves downwards. [1] 10 The curves 1C and 2C have equations 24xy=− and 2 4 3y x=− + respectively. (i) State the equations of the asymptotes of the curves 1C and 2C respectively. [3] (ii) Sketch the graphs of 1C and 2C on the same diagram, indicating the intercepts with the axes and equations of any asymptotes. [4] (iii) Hence, deduce the number of real roots of the equation 2 42 7 0x x+ − = , sho