Irish Leaving Certificate Ordinary Level Mathematics Paper 1 2009

Page 1 of 5 2009. M27 Coimisiún na Scrúduithe Stáit State Examinations Commission _____________________________ LEAVING CERTIFICATE EXAMINATION, 2009 _____________________________ MATHEMATICS ─ ORDINARY LEVEL PAPER 1 ( 300 marks ) _____________________________ FRIDAY, 5 JUNE – MORNING, 9:30 to 12:00 _____________________________ Attempt SIX QUESTIONS (50 marks each). _____________________________ WARNING: Marks will be lost if all necessary work is not clearly shown. Answers should include the appropriate units of measurement, where relevant. _____________________________ Page 2 of 5 1. (a) Conor and Alice share 50 apples in the ratio 3 : 7. (i) How many apples does Conor get? (ii) How many apples does Alice get? (b) Barbara works 35 hours a week and she is paid €12·60 per hour. (i) Find her total weekly pay. (ii) Barbara pays tax at the rate of 20% on all her income and has weekly tax credits of €53. Calculate her weekly take-home pay. (iii) In one particular week, Barbara worked 4 additional hours at the same rate of pay. By how much did her take-home pay increase that week? (c) €7500 was invested for 2 years at r%per annum compound interest. (i) The amount of the investment at the end of the first year was €7860. Find the value of r. (ii) At the start of the second year €X was withdrawn from the account. The interest earned during the second year was €252. Find the value of X. 2. (a) Find the value of 5 3x − 2y −1 when x = 13 and y = 14 . (b) (i) Find the value of 36 . (ii) Write 27 in the form 3k , where k ∈ N. (iii) Find the value of x for which . 729 27 × 3x = 1 (c) Let f (x) = x3 + x2 − 4x − 4 . (i) Verify that f (−2) = 0 . (ii) Solve the equation x3 + x2 − 4x − 4 = 0 . Page 3 of 5 3. (a) Simplify x(2x + 7) − 3(x − 4). (b) (i) Solve for x and y 29. 7 2 + 2 =+ = x yx y (ii) Which one of the values of y in (i) above satisfies the inequality 6 − 2y < 0 ? Justify your answer. (c) A rectangle has length 2 x cm and width x cm. The length of a diagonal of the rectangle is 45 cm. (i) Find the area of the rectangle. (ii) The area of a square is twice the area of the rectangle. Find the length of a side of the square. 4. (a) Given that i2 = −1, simplify 2(3 − 5i)+ 7i(2 + 3i) and write your answer in the form x + yi , where x, y ∈ R. (b) Let u = 3 + 5i . (i) Show that u is a solution of the equation z 2 − 6z + 34 = 0 . (ii) Express u 17 in the form x + yi . (c) Let z = 3 − 4i . (i) Calculate | z | . (ii) Find the real numbers p and q such that | z | (p + qi) + (q − pi) = 17 + 7i . Page 4 of 5 5. (a) The first term of a geometric sequence is 2 and the common ratio is 3. Find the second term of the sequence. (b) The first term of an arithmetic series is − 2 and the second term is 4. (i) Find d, the common difference. (ii) Find 10 T , the tenth term of the series. (iii) The kth term of the series is 292. Find k. (iv) Find 20 S , the sum of the first 20 terms of the series. (c) The first two terms of a geometric series are − 6 +12 + K. (i) Find r, the common ratio. (ii) Find 7 T , the seventh term of the series. (iii) Starting with the first term, how many terms of the series must be added to give a sum of 30? 6. (a) Let g(x) = 4 − kx . Given that g(−5) = 34 , find the value of k. (b) Let h(x) = x(1− x2 ), where x ∈ R. (i) Verify that h(3) + h(−3) = 0 . (ii) Find the values of x for which h′(x) = −11, where h′(x) is the derivative of h(x) . (c) Let f (x) = x3 − 6x2 +9x − 3, where x ∈ R. (i) Find the co-ordinates of the local maximum point and of the local minimum point of the curve y = f (x) . (ii) Draw the graph of the function f in the domain 0 ≤ x ≤ 4 . (iii) Use your graph to estimate the range of values of x for which x < 3 and f (x) ≥ 0 . Page 5 of 5 7. (a) Differentiate 3x5 − 7x2 + 9x with respect to x. (b) (i) Given that y =(x2 − 4x)5 , find the value of dx dy when x = 2. (ii) Differentiate 11 22 +− xx with respect to x. Write your answer in the form x n kx ( 2 +1) , where k, n ∈ N. (c) A ball is fired straight up in the air. The height, h metres, of the ball above the ground is given by h = 30t − 5t 2 where t is the time in seconds after the ball was fired. (i) After how many seconds does the ball hit the ground? (ii) Find the speed of the ball after 2 seconds. (iii) Find the maximum height reached by the ball. 8. (a) Let g(x) = 2(6 − 3x) , where x ∈ R. Find the value of x for which g(x)=0. (b) Differentiate 2x2 − 5x with respect to x from first principles. (c) Let 1 ( ) 1+ = x f x , x ∈ R, x ≠ −1. (i) Find f ′(x) , the derivative of f (x). (ii) Find the two values of x at which the slope of the tangent to the curve y = f (x) is −1. (iii) One of these tangents intersects the positive y-axis. Find the equation of this tangent. Blank Page Blank Page Blank Page