SAMPLE PAPERS SET B Paper 1 (100 marks) (Time: 2.5 Hours), Lecture notes of Music

Download SAMPLE PAPERS SET B Paper 1 (100 marks) (Time: 2.5 Hours) and more Lecture notes Music in PDF only on Docsity! Cork Institute of Technology Maths Qualifying Exam — SAMPLE PAPERS SET B Paper 1 (100 marks) (Time: 2.5 Hours) Answer QUESTION 1 and THREE other questions. Question 1 is worth 40 marks. Questions 2–5 are worth 20 marks each. Total marks available: 100 marks. • The standard Mathematics Tables booklet is available. • Marks will be lost if all necessary work is not clearly shown. • Answers should include the appropriate units of measurement, where relevant. [P.T.O.] 1 Q1 NB This question is compulsory. Answer any eight parts [5 marks each]. (a) (i) The scale on a map is 1:25000. The length of a wall on the map is 5 mm. Calculate, in metres, the actual length of the wall. (ii) Write 1.2 × 103 0.4 × 102 as a decimal number. (b) Maria wants to bring £200 with her on holiday to London. If the exchange rate is e1 = £0.94 and the bank charges a transaction charge of e2.50, what will she have to pay the bank in euro? (c) Norma travels from her home to her workplace in Dublin, a distance of 72 km. The journey usually takes 1 hour 10 minutes. (i) Find her average speed in kilometres per hour for the journey. (ii) During school holidays, the traffic is lighter and Norma finds that her average speed for the journey increases to 80 km per hour. How long does Norma’s journey to work take her on such days? (d) Solve each of the following equations for x: (i) x(x + 4) − 3(2x + 1) = 0 (ii) 23−x = 4x (e) (i) Evaluate t[t(2t− 3) + 10] + 5 when t = −0.5. (ii) Express (64 × 33) ÷ 12 in the form 2a × 3b. (f) Let f(x) = 3x − 2, where x ∈ R. (i) Find the value of f(−1); (ii) Find the value of x for which f(x) = 19; (iii) If g(x) = 1 x , find g(f(−1)) and f(g(−1)). (g) Given that i2 = −1, find the value of (i) i4 (ii) i8 (iii) i3 (iv) i9 [Q1 continued overleaf] 2 Q4 (a) L is the line with equation 4x − 3y − 6 = 0. (i) Find the slope of the line L. (ii) Show that the point (3, 2) lies on L. (iii) Determine the point at which L intercepts the x-axis. (iv) Hence sketch a graph of the line L. (v) Find the equation of the line which is perpendicular to L and which passes through the point (1, 3). [8 marks] (b) K1 is the line x−2y+1 = 0 and K2 is the line which passes through the points (0,−11) and (4, 0). (i) Show that the lines K1 and K2 are neither parallel nor perpendicular. (ii) Find the equation of the line K2. (iii) Find the point of intersection of the lines K1 and K2. [6 marks] (c) The circle C has centre (0, 0) and passes through the point (8, 6) (i) What is the radius of the circle C? (ii) Find the equation of the circle C. (iii) Verify, by calculation, that the point (7, 7) lies inside the circle C. (iv) Show the circle C on a co-ordinate diagram. Mark the four points at which the circle C intersects the axes and label them with their coordinates. [6 marks] 5 Q5 (a) A circle has centre o and radius 24 cm. An arc of length 30 cm subtends an acute angle A at o. Calculate A, correct to the nearest degree. [4 marks] (b) A candle is in the shape of a cylinder surmounted by a cone. The cone has perpendicular height 24 cm and the length of the radius of its base is 10 cm. The height of the cylinder is equal to the slant height of the cone. Find the volume of the candle, correct to two decimal places. [5 marks] (c) Two ships, A and B, leave a port k at noon. Ship A is travelling due East while ship B is travelling due South. Calculate, to the nearest km, the distance between the two ships when A is 6 km from k and B is 11 km from k. [5 marks] (d) A triangular lot has sides which measure 100 m, 150 m and 300 m respec- tively. (i) Find the measure of its largest angle. (ii) Hence, or otherwise, find the measure of each of the other two angles. [6 marks] Q6 (a) A fifth-year student has to choose three subjects from the following list: Biology, Accounting, Technical Drawing, Art, Music, German. (i) How many different choices are possible? (ii) How many of these choices include Music? [4 marks] (b) Four cards, numbered 2, 3, 4, 5 respectively, are shuffled and then placed in a row with the numbers visible. Find the probability that the first and second numbers are even. [4 marks] [Q6 continued overleaf] 6 (c) Twelve blood samples are tested in a laboratory. Of these, it is found that five blood samples are of type A, four are of type B and the remaining three are of type O. Two blood samples are selected at random from the twelve. What is the probability that (i) the two samples are of type A? (ii) one sample is of type B and the other sample is of type O? [4 marks] (d) The following is the break-down of left-handers and right-handers in a tennis club, which has 200 members. Left− handed Right − handed Male 24 67 Female 29 80 A member of the tennis club is chosen at random. Find the probability that the person is (i) a right-handed male; (ii) a female; (iii) left-handed. [5 marks] (e) The mean of the following set of numbers is 4.5. Find the value of x. 3, 3, 6, 1, x, 5 [3 marks] 7