1. Solve the simultaneous equations 5 and 1. [5]. (i) Sketch the graph of, showing the coordinates of the points where our graph meets the coordinate aes. [] Solve the equation 10, giving our answer correct to significant figures. []. The equation of a line is k, where k is a constant, and the equation of a curve is 6. (i) In the case where k 8, the line intersects the curve at the points A and B. Find the equation of the perpendicular bisector of the line AB. [6] Find the set of values of k for which the line k intersects the curve 6 at two distinct points. []. B 5,11 C A 1, X, O The diagram shows a triangle ABC in which A has coordinates 1,, B has coordinates 5,11 and angle ABC is 90. The point X, lies on AC. Find (i) the equation of BC, [] the coordinates of C. [] Prepared b Mr Ang, Oct 01 1
5. A circle passes through the points 0,, 0, 9 and 0,. (i) Find the equation of the circle and the length of its radius. [7] Is the point 6, 0 inside, on the circumference of, or outside the circle? Give a reason for our answer and show all necessar working. [] 6. The table shows eperimental values of two variables and. 1 9.1 1.9 0. 69 1. 77 It is known that and are related b the equation constants. a b, where a and b are (i) A straight line graph is to be drawn to represent this information. Given that plotted on the vertical ais, state the variable to be plotted on the horizontal ais. [1] On a graph paper, draw this straight line graph. [] (iii) Use our graph to estimate the value of a and of b. [] (iv) Estimate the value of when is.7. [] is 7. The polnomial p is defined b 1 5 1 p. (i) Show that p 0 and factorise p completel. [] Given that state the value of 1 7 59 1 0, and hence find correct to significant figures. [] Prepared b Mr Ang, Oct 01
8. (a) Solve the equation sin sin sin 6 0, giving all values of between 0 and π inclusive. [5] (b) Prove that 1 cos sin cot. [] 1 cos sin Hence find the solutions for of the equation, for, 1 sin cos for which cos 0. [] 9. The function f is defined, for 60 0, b f sin 1. (i) State the amplitude and period of f. [] Sketch the graph of f. [] d 10. (a) Find in each of the following cases: d (i) ln1 sin, [] tan. [] (b) Resolve 11 in partial fractions. Hence evaluate 6 11 d. [7] 11. A watermelon is assumed to be spherical in shape while it is growing. Its mass, M kg, and radius, r cm, are related b the formula M kr, where k is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per da. On a particular da the radius is 10 cm and the mass is. kg. Find the value of k and the rate at which the mass is increasing on this da. [5] Prepared b Mr Ang, Oct 01
1. The diagram shows part of the curve 9, which meets the -ais at the origin O and at the point A. The line 18 0 passes through A and meet the -ais at the point B. 9 O B 18 0 A (i) Show that, for 0, 9 108. [] Find the area of the shaded region bounded b the curve, the line AB and the -ais. [6] 1. A particle P moves along the -ais such that its distance, m, from the origin O at time t s t is for t 0. t 1 (i) Find the greatest distance of P from O. [] Find the acceleration of P at the instant when P is at its greatest distance from O. [] Prepared b Mr Ang, Oct 01
Answer: 1 1 1., 5,, 6. i. 0,,, 0 ii.. 81. i. 6 ii. k or k. i. 1 7 ii. 1, 7 5. i..5 5.5. 5, r 5. 70 ii. On the circumference. 6. i. iii. 0. 6 b, a 10 iv. 1. 8 7. i. ii., 0. 69 8. a. 0,,,, ii. 0.6,. 68 9. i., 10 10. a. i. cos 1 sin sec tan ii. b. 1 5, 5 ln 11.. 10,. 096 0 kg/da 1. ii. 511 square units 1 1 1. i. m ii. m/s Prepared b Mr Ang, Oct 01 5