Teacher s Name Student Number Kno Grammar School 0 Year Yearly Eamination Mathematics General Instructions Reading Time 5 minutes Working Time 3 hours Write using black or blue pen Board approved calculators may be used Answer in simplest eact form unless otherwise stated Show all necessary working in questions 6 Total Marks 00 Section I 0 Marks Attempt Questions 0 Allow about 5 minutes for this section. Section II 90 marks Attempt Questions 6 Eaminer: E. Choy Number of Students in Course: 30 This paper MUST NOT be removed from the eamination room
Section I Multiple Choice 0 Marks Attempt Question 0. Allow approimately 5 minutes for this section.. Evaluate 3.3 0.45. correct to 3 significant figures. (A).7 (B) 8.6 (C) 8.67 (D).708. If 3 a b then the values of a and b are: (A) 4 and (B) 4 and (C) 4 and 3 (D) and 3 3. The solutions to the equation 7 = 0are: (A) 7± 33 4 (B) 7± 65 4 (C) 7± 33 4 (D) 7± 65 4 4. The domain and range for the function y 7 is: (A) 7 ; y 0 (B) 0 7 ; y 0 (C) 0 ; y 0 (D) All real, All real y 5. A function is defined by f () = 4 + 5. The point on y = f () which has a gradient of is: (A) (3, ) (B) (, ) (C) (3, ) (D) (, 3)
6. The derivative of is : (A) (B) (C) 3 (D) 7. The calculation which would be used to find the value of is: (A) 5 tan 36 (B) 5 cos 36 (C) 5 tan 36 (D) 5 cos 36 8. The equation has roots and. The value of is : 3 0 (A) (C) 4 (B) 3 3 (D) 0 3 9. The equation circle are: y 6 6 0 represents a circle. The coordinates of the centre of the (A) (3, 3) (B) (3, 3) (C) (3, 0) (D) (6, 0) 0. For what values of k is the epression k negative definite? 4 (A) k < (B) k > (C) k < 4 (D) k > 4 End of Section I 3
Section II 90 marks Attempt Questions 6 Allow about hours 45 minutes for this section Answer each question in a separate writing booklet. Etra writing booklets are available. All necessary working should be shown in every question. Question (5 marks) Use a SEPARATE writing booklet (a) Evaluate 3 65 4 correct to two decimal places. (b) Simplify a 5 a 5 (c) (i) Draw a neat sketch of the parabola y 3 (ii) Hence, determine the values of for which 3 noting and y intercepts. < 0. (d) If f 3 and g 4, for what value of is f g (e) On separate number planes, sketch neatly each of the following curves:? (i) y (ii) y (iii) y 5 (f) Find the eact value of such that cos = where 0 90. (g) Solve the pair of simultaneous equations: y 7 y. (h) What are the domain and range of y? 4
Question (5 marks) Use a SEPARATE writing booklet y Q, R O k P 6, 8 Diagram not drawn to scale (a) A line whose equation is y 8 0 Another line k has gradient intersect at the point R. has the point, and passes through the point 6, 8 Q on it. P. The lines k and (i) Show that the equation of the line k is y 4 0. (ii) Show that the co-ordinates of R are,6. (iii) Show that the distance of QR is 3 5. (iv) Find the perpendicular distance from P to the line. Leave your answer in surd form. (v) Find the area of PQR. Question continues on page 6 5
Question continued (b) Shade the region in the plane for which the inequations hold simultaneously. y 4 and 0 y (c) Find the locus of the point P, y whose distance from the point A(4, 0) is always twice its distance from the point B(, 0). (d) The general form of a line passing through the point of intersection of y5 0 and y 0 is y 5 k y 0. (i) Show that the gradient of line is k k (ii) Hence find the equation of the line passing through the point of intersection of y5 0 and y 0 and has gradient. Leave your answer in general form. End of Question 6
Question 3 (5 marks) Use a SEPARATE writing booklet (a) Two ships set sail from a point O. The first ship sails for 45 nautical miles on a course bearing N3W to a point Y and the second ship sails for 63 nautical miles on a course bearing N58E to a point Z. (i) Draw a half-page diagram representing the above information. (ii) Find, to the nearest nautical mile, the distance YZ. (b) The adjacent sides of a parallelogram have lengths 4 cm and 6 cm and they include an angle of 60. Calculate the length of the shorter diagonal to decimal places. (c) Given that 5 tan and is obtuse, find sin and sec. (d) If sin, epress cos in terms of. sec (e) (i) Show that cos tan sin (ii) Hence solve 8sin cos tan cosec for 0 360. 3 (f) The distance, d km, between two ports P and P can be found from the calculation: sin 44 sin9 d 900. sin 6 sin (i) Determine the distance, d, from port P to port P, correct to two decimal places. (ii) Determine the time it will take to sail from P to P at an average speed of 0km/h. Give your answer to the nearest minute. 7
Question 4 (5 marks) Use a SEPARATE writing booklet (a) If g 3 5, show that g a g a a (b) Find lim 4 (c) Differentiate: (i) 4 (ii) 5 3 (iii) 5 7 (d) Find the equation of the tangent to the curve y 4 at the point where. (e) A function is defined by f (i) Find f 0 f.5, for, for (ii) Sketch the graph y f for. (iii) The horizontal line y kmeets the graph of y f eactly once. Find the range of values of k for this to be true. 8
Question 5 (5 marks) Use a SEPARATE writing booklet (a) If and are the roots of the equation 7 4 0, find the value of : (i) (ii) (iii) (b) For the parabola y 4 4, find: () (i) The co-ordinates of the verte. (ii) The co-ordinates of the focus. (iii) The equation of the directri. (iv) The length of the latus rectum. (v) Its and y intercepts. () Hence, draw a neat sketch of the parabola. (c) Find the values of m for which the quadratic equation distinct roots. 3m 9 0 has real and (d) If one of the roots of a b 0 is three times the other, prove that 6b 3 a. 9
Question 6 (5 marks) Use a SEPARATE writing booklet (a) A 5 4 B D C The above figure shows a triangle ABC such that AB 5cm, AC 4cm and BAC 60. D is a point on BC such that AD bisects BAC. (i) Epress the area of ABD in terms of. (ii) Hence, using triangles ADC and ABC, show that 0 3. 9 (b) For what values of m is y m 6 a tangent to the parabola y 3? 3 Question 6 continues on page 0
Question 6 continued (c) y 4y A y, 0, B y, O A straight line with gradient m passes through the point 0,. The line also cuts the parabola 4y at two points A, y and B, y as shown above. (i) Show that the equation of the line is y m. (ii) Line cuts the parabola at two distinct points. Show that and are the roots of the equation 4m 4 0. (iii) Show that 4. Hence, or otherwise find in terms of m. (iv) From (iii), show that AB 4 m. 3 End of Eamination