2017 SNGS Sec 4 EM Prelim P2

3 CHIJ SNGS Preliminary Examinations 2017 - Mathematics 4048/02 [Turn Over Answer all the questions. 1 (a) Solve the simultaneous equations .1943 ,82   yx xy [3] (b) Simplify 22 22 65 4 yxyx yx   . [3] (c) (i) Express 372  xx in the form .)( 2 bax  [2] (ii) Hence solve the equation ,0372  xx giving your answers correct to two decimal places. [2] 2 The diagram shows a road junction at O with PO perpendicular to OQ and OP = 60 km. A cyclist, X, starts from P and travels towards O at a constant speed of 10 km/h. At the same time, another cyclist, Y, starts from O and travels towards Q at a constant speed of 15 km/h. (a) Cyclist X leaves from P and cyclist Y leaves from O at noon. Given that t is the time in hours, write down, in terms of t, (i) the distance of X from O after t hours, [1] (ii) the distance of Y from O after t hours. [1] (b) After t hours, the cyclists are 80 km apart. Write down an equation in t to represent this information and show that it reduces to .01124813 2  tt [2] (c) Solve the equation .01124813 2  tt [3] (d) Find the distance of cyclist X from O when the cyclists are 80 km apart. [1] Q 10 km/h P X O 15 km/h Y

4 CHIJ SNGS Preliminary Examinations 2017 - Mathematics 4048/02 3 The diagram shows a circle ABCDE, with centre O. AEF is a straight line. Angle BAE = 100 and angle CEF = 120. AE = ED. EC is a diameter. (a) Show that triangle ACE and triangle DCE are congruent. [2] (b) Find, giving reasons for each answer, (i) angle ECD, [2] (ii) angle ABD, [1] (iii) angle ECB. [1] (c) Given that AC = 8 cm, find the diameter of the circle. [2] 4 The first four terms in a sequence of numbers, ..,.........,,, 4321 uuuu are given below. 1030 1 u 5231 2 u 13432 3 u 33633 4 u (a) Write down an expression for 5u and show that .895 u [1] (b) Write down an expression for 8u and evaluate it. [1] (c) Find an expression, in terms of ,n for the nth term, ,nu of the sequence. [2] (d) Explain why the value of nu must be odd for all values of n. [2] (e) (i) Show that   221 3)2()3(3   nnn . [2] (ii) Find , and simplify, an expression, in terms of n , for 1 nn uu . [2] B C 120 O D F E A 100

5 CHIJ SNGS Preliminary Examinations 2017 - Mathematics 4048/02 [Turn Over 5 (a) The diagram shows an open container A, made from a cylinder and a cone. The cylinder has a radius of 8 cm and a height of 16 cm. The cone has a height of 12 cm. The container is completely filled with water. Find the volume of water in the container. [3] (b) The diagram shows a pot which is part of a right circular cone of height h cm. The open end of the pot is a circle of radius 6 cm. The base of the pot is a circle of radius 4 cm. The height of the pot is 10 cm. Some water from the container A is poured into the pot. The height of the water in the pot is 8 cm. The top of the water surface is a circle of radius r cm. (i) Show that h = 30 cm. [2] (ii) Find the value of r. [1] (iii) Find the percentage of water from container A that has been poured into the pot. [3] 8 cm 16 cm 12 cm 6 cm 4 cm 10 cm 8 cm r cm h cm

6 CHIJ SNGS Preliminary Examinations 2017 - Mathematics 4048/02 6   PQIn the diagram, 2a and   PR b. QS is parallel to PR and QS = 2 3 PR. T is the point on QR such that QT : TR = 3 : 1. U is the midpoint of PQ. (a) Express, as simply as possible, in terms of a and b, (i)  QR , [1] (ii)  PT , [2] (iii)  US . [1] (b) Show that PT is parallel to US. [2] (c) The point X lies on QP produced such that XP = 2 1 PQ. Given that XTS is a straight line and XT = TS, find PQST XPT ralquadrilate of area triangleof area . [3] T S R Q 2a b U P

7 CHIJ SNGS Preliminary Examinations 2017 - Mathematics 4048/02 [Turn Over 7 Answer the whole of this question on a sheet of graph paper. The speed, v m/s, of an object at a time t seconds after it starts from rest is given by the equation 22540 ttv  . Some values of t and the corresponding values of v are given in the table below. (a) Calculate the value of p. [1] (b) Using a scale of 2 cm to represent 1 second, draw a horizontal axis for 0  t  6. Using a scale of 2 cm to represent 10 m/s, draw a vertical axis for 0  v  70. On your axes, plot the points given in the table and join them with a smooth curve. [3] (c) Use your graph to find (i) the value of t when the object comes to a stop, [1] (ii) the length of time when the object is moving faster than 40 m/s, [2] (iii) the maximum speed of the object. Hence state the acceleration of the object at this speed. [2] (d) (i) Draw the tangent to the curve at the point where the gradient is 7 . [1] (ii) Explain what the gradient represents. [1] (iii) Write down the equation of the tangent. [1] t (s) 0 0.5 2 3 4 4.5 5 5.5 v (m/s) 40 42 42 37 28 p 15 7

8 CHIJ SNGS Preliminary Examinations 2017 - Mathematics 4048/02 8 (a) In a Mathematics examination, 400 students each took two papers. Both papers were marked out of 100. The cumulative frequency curves show the distribution of the marks for the two papers. 400 300 200 100 0 10 40 20 30 50 60 70 80 90 100 Marks Cumulative Frequency