MATHEMATICS EXAMINATION FOR ENTRANCE SCHOLARSHIPS AND EXHIBITIONS FEBRUARY Time allowed - 2 hours

Transcription

1 EXAMINATION FOR ENTRANCE SCHOLARSHIPS AND EXHIBITIONS FEBRUARY 2015 MATHEMATICS Time allowed - 2 hours All answers should be written in the answer books provided, including any diagrams, graphs or sketches. Answer all questions in Section A and two questions from Section B. Calculators are permitted, provided they are silent, self-powered, without communication facilities, and incapable of holding text or other material that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorised to remove any suspect calculators. Acceleration due to gravity should be taken as 9.8m/s 2. Statistical tables will be provided. c 2015 Aberystwyth University

2 MATHEMATICS 2 of 5 Section A 1. Differentiate each of the following with respect to x, simplifying your answer as far as possible: (a) 2x 1 2 ; (b) ln(2 + x 3 ); (c) x 2 e 4x ; (d) cos(2x + 1); (e) x [10 marks] x Simplify the following expressions: (a) log log 5 5; (b) ; (c) e4x e 2x e 2x e x. [7 marks] 3. Integrate each of the following with respect to x: (a) cos(3x); (b) e 5x ; (c) xe x ; (d) (e x + 1) 2. [10 marks] 4. (a) Sketch the curve C described by the equation x 2 + y 2 + 4(y x + 1) = 0 in the x-y plane. (b) Find the points where the curve C meets the x and y axes. (c) For each point (x, y) on C, determine its distance from the point (2, -2). [8 marks] 5. If x = 1 and x = 2 are roots of the cubic f(x) = 2x 3 + ax 2 + x + b, determine the values of a and b. Hence factorise f(x) completely into its linear factors. [10 marks] 6. (a) Show that sin 2θ 1 + cos 2θ = tan θ (b) Solve the equation tan(θ + π/4) = 1 2 tan θ, for 0 θ π/2 [7 marks] 7. A sequence is given by a 1 = 4, a r+1 = a r + 3. Write down the first four terms of this sequence. Find the sum of the first 100 terms of the sequence. [5 marks]

3 MATHEMATICS 3 of 5 8. (a) Consider the function Section B f(x) = 3x 5 + 5x 3. (i) Determine the coordinates of all local maxima and local minima of f, and sketch its graph. [6 marks] (ii) What is the greatest value and what is the smallest value that f(x) takes on the interval 0 x 2? (iii) Write down the equation of the tangent to the graph of f when x = a. (b) Find the length of the shortest line segment between the origin (0,0) and the parabola y = 2 x 2 /2. [10 marks]

4 MATHEMATICS 4 of 5 9. A ball is kicked from ground level over horizontal ground. It leaves the ground at a speed of 25ms 1 and at an angle θ to the horizontal such that cos θ = (a) Show that the height of the ball above the ground t seconds after being kicked is given by y = 7t 4.9t 2. Show also that the horizontal distance travelled by the ball in this time is given by x = 24t. (b) Calculate the maximum height reached by the ball. (c) Calculate the times at which the ball is at half its maximum height. Find the horizontal distance travelled by the ball between these times. [5 marks] (d) Determine the following when t = 1.25: (i) The vertical component of the velocity of the ball; (ii) Whether the ball is rising or falling; (You should give a reason for your answer.) (iii) The speed of the ball. [5 marks] (e) Show that the equation of the trajectory of the ball is y = 0.7x (240 7x). 576 Hence, or otherwise, find the furthest distance travelled by the ball. [5 marks]

5 MATHEMATICS 5 of (a) In a hat there are 100 balloons, 12 of which have a hole in. If 2 balloons are drawn from the hat: (i) What is the probability that the first balloon is defective? [1 mark] (ii) What is the probability that the second balloon is defective given that the first balloon is defective? (b) Suppose a barrel contains 8 red fish and 3 black fish. If 4 fish are selected at random from the barrel: (i) What is the probability that all of the fish are red? (ii) What is the probability that 2 fish are red and 2 fish are black? (iii) What is the probability that fewer than 3 fish are black? (c) There are 17 balls in a bucket: 5 white and 12 black. If the first ball drawn is black, what is the probability that the next 2 balls drawn are both black? (d) In a mark-recapture experiment to estimate the number of fish in a lake, 124 fish are caught and tagged with a non-harming tag. Later, 98 fish are caught, of which 32 are tagged. Give an estimate for the number of fish in the lake. (e) Consider a standard pack of 52 playing cards. If 7 cards are dealt, write down an expression for the probability of observing exactly 2 court cards (J,Q,K). (f) An engineering company buys batches of n components. Before a batch is accepted, m of the components are selected at random and tested. The batch is rejected if more than d of the components are defective. If n = 20, m = 5, d = 1, and the true number of defective components is 6, what is the probability that the batch is accepted?