PRE-JUNIOR CERTIFICATE EXAMINATION, 2010 MATHEMATICS HIGHER LEVEL. PAPER 2 (300 marks) TIME : 2½ HOURS

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1 J.20 PRE-JUNIOR CERTIFICATE EXAMINATION, 2010 MATHEMATICS HIGHER LEVEL PAPER 2 (300 marks) TIME : 2½ HOURS Attempt ALL questions. Each question carries 50 marks. Graph paper may be obtained from the superintendent. The symbol indicates that supporting work must be shown to obtain full marks J.20 1/8 Page 1 of 7

2 1. (a) The area of a rectangle is 500 cm 2. The ratio of the length to the width of the rectangle is 4 : 5. Find the perimeter of the rectangle. (b) A pencil case in the shape of a cylinder with a hemisphere at each end as shown. The hemisphere has a diameter of 8 cm and the pencil case has a length of 22 cm. (i) Find the internal volume of the pencil case, correct to one decimal place. Find the total surface area of the pencil case, correct to the nearest cm 2. A solid cone has a base radius of 9 cm and a slant height of 15 cm. (i) Find the perpendicular height of the cone. Hence, find the volume of the cone in terms of π. The cone is melted down and recast into nine identical spheres. (iii) Calculate the radius of each sphere J.20 2/8 Page 2 of 7

3 2. (a) P(2, 1) and R(-1, 3) are two points. (i) Find the slope of PR. Find the equation of PR. (b) Prove that in a right-angled triangle, the square of the length of the side opposite to the right angle is equal to the sum of the squares of the lengths of the other two sides. A(-2, 1), B(4, 2) and C(3, 8) are three points. (i) Find AB and BC. Hence, or otherwise, prove that ABC is a right angle. (iii) Find the co-ordinates of D, the image of A under the translation BC. (iv) Find the area of ABCD J.20 3/8 Page 3 of 7

4 3. (a) PQR is an isosceles triangle, with PQ = PR and PRS = 130. Find RPQ. (b) (i) Prove that a diagonal bisects the area of a parallelogram. In the parallelogram ABCD, AB = 8 cm, BC = 5 cm and ACD = 29. ABC = 2 BCA. Find ABC and hence, calculate the area of the parallelogram ABCD. Give your answer correct to two decimal places. In the triangles PQR and QST, QRP = QST. (i) Prove that the triangles RPQ and STQ are equiangular. Given that QT = 2 5 cm, QR = 8 cm and PQ = QS, find PT. Give your answer correct to the nearest cm J.20 4/8 Page 4 of 7

5 4. (a) [AB ] is the diameter of a circle of centre O. CAB = 32. (i) Find ABC. Find CDA. (b) (i) Construct a triangle PQR in which PQ = 12 cm, PR = 8 cm and RPQ = 50. All construction lines must be clearly shown. Show how to bisect the angle PQR without using a protractor. A, B, C and D are points on a circle, of centre O, with [ CD ] produced to E. [BD ] intersects the chord [ AC ] at F. AD = DC. (i) Prove ABC = ADE. Prove that the triangles ABD and BCD are congruent. (iii) Hence, prove that [ BD ] is perpendicular to [ AC ] J.20 5/8 Page 5 of 7

6 5. (a) If cos A = - 1, find the two values for the angle A, where 0 A (b) In the triangle ABC, the line AD bisects CAB. ABC = 110, ADC = 130, AD = 10 3 cm and AC = 15 5 cm. (i) Find AB, correct to one decimal place. Find BC, correct to one decimal place. A kite is flown on a cord 12 5 m long. The cord, held taut by a boy with its end at a height of 1 5 m above ground level, makes an angle of 70 with the horizontal, as shown. (i) Find the height of the kite above ground level, correct to two decimal places. From a point P on level ground, the angle of elevation of the kite is 40. Calculate the distance from the boy to the point P, correct to one decimal place J.20 6/8 Page 6 of 7

7 6. (a) The table shows the results of a survey of the types of houses in a housing estate. House type Detached Semi-detached Terraced Number of houses Draw a pie chart to illustrate the above information. (b) A group of students were asked how much money they spent on mobile phone credit in a certain week. The results are shown in the histogram below: (i) Given that there are 2 students in the 0 5 interval, copy and complete the frequency table below. Amount spent on credit Number of students 2 [Note: 5 15 means 5 or more but less than 15, etc.] Taking mid-interval values, calculate the mean amount of money spent on phone credit by the group of students, correct to the nearest cent. The ages of guests who attended a wedding were recorded. The following were the results. Age Number of guests [Note: means 10 or more but less than 20, etc.] (i) Construct the cumulative frequency table. On graph paper construct the ogive. (iii) Use your graph to estimate the median age. (iv) Use your graph to estimate the number of guests who were aged 40 years or more J.20 7/8 Page 7 of 7

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