UT2-DSE-MATH-CP 1 Solution

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1 UT-DSE-MATH-CP 1 Solution SECTION A (70 marks) 1. Factorize x 5x, (b) x y xy x 5x. ( marks) x 5x ( x 1)( x ) (b) x y xy x 5x ( x 1) xy (x 1)( x ) ( xy x )(x 1) 19 4x. Solve the inequality 17 x x (b) Find the number of integers satisfying 17 x and greater than 5. 5 (4 marks) 19 4x 17 x x 85 10x 10x 4x x 66 x 11 (b) The integers satisfying both conditions are 6, 7, 8, 9, 10 and 11. The required number is 6. 1

2 . The present value of a diamond is $ If the value of the diamond increases 0% per year, find the value of the diamond years later; (b) years ago. (Give the answers correct to the nearest dollar if necessary.) (4 marks) Value of the diamond years later $69 10 ( 1 0%) $ (cor. to the nearest dollar) (b) Let $x be the value of the diamond years ago. x ( 1 0%) x The value of the diamond years ago was $ Let L be the straight line passing through (, 1) and (0, ). Find the slope of L. (b) Find the coordinates of the point of intersection of L and the x-axis. (4 marks) Slope of L (b) Let (x, 0) be the coordinates of the point of intersection of L and the x-axis. 0 ( 1) x 1 1 x x The required coordinates are (, 0).

3 5. In Figure 1, A, B and C are three bus stops on the horizontal ground. A is due north of B and C is 50 m due east of B. The distance between A and C is 400 m. Find the bearing of A from C correct to the nearest degree. N A 400 m B 50 m Figure 1 C E ( marks) BC cos ACB AC ACB 51 (cor. to the nearest degree) The bearing of A from C is N(90 51)W N9W (or ). 6. Figure shows a solid formed by some identical cubes. Draw the front view, top view and side view of the solid. Top Front Figure Side ( marks)

4 Front view Top view Side view A 7. In Figure, ABCD is a parallelogram. AC and BD intersect at E. BD // AF and AC // FD. B A E F C D Figure Prove that DAE ADF. (b) If ABCD is a rhombus, is ADF a right-angled triangle? Explain your answer. (4 marks) In DAE and ADF, DAE ADF DA AD ADE DAF alt. s, AC // FD common side alt. s, BD // AF DAE ADF ASA Marking Scheme: Case 1 Any correct proof with correct reasons Case Any correct proof without reasons 1 (b) If ABCD is a rhombus, then DEA 90 (property of rhombus) Q AFD DEA (corr. s, s) AFD 90 Hence ADF is a right-angled triangle. 4

5 8. Figure 4 shows a solid consisting of a cube of side length k cm joined to the bottom of a right pyramid of height 15 cm and with a square base of side length k cm. It is given that the volume of the right pyramid and the volume of the cube are the same. Figure 4 Find k. (b) Find the volume of the solid. (4 marks) Volume of the cube Volume of the pyramid 1 k k (15) k 5 (b) Volume of the solid 1 5 (5) (15) cm 50 cm 9. The stem-and-leaf diagram below shows the distribution of ages of 0 teachers in a school. Stem (tens) Leaf (units) Find the mean, median and mode of the above distribution. ( marks) 5

6 Mean Median 1 (4 + 46) Mode The table below shows the distribution of the numbers of mobile phones owned by a group of people. Number of mobile phones 0 1 Number of people Find the median of the distribution. (b) A person is selected randomly from the group. Find the probability that the selected person has not more than 1 mobile phone. ( marks) Median 1 (b) P(have not more than 1 mobile phone)

7 11. The table below shows the distribution of the salaries of all employees in a department of a company. Salary ($x) x x x Employee Male 1 5 k Female If an employee is randomly selected from the department, the probability that the salary 7 of the selected employee is more than or equal to $ is. 9 Find the value of k. ( marks) (b) (i) If an employee is randomly selected from the department, find the probability that the selected employee is a female with salary less than $ (ii) If an employee is randomly selected from the male employees in the department, find the probability that the salary of the selected employee is less than $ (4 marks) Total number of employees 1 5 k k 5 k k 9 9(5 k) 7(7 k) 477 9k 511 7k k 4 k 17 (b) (i) P(a female with salary less than $15 000) (ii) P(salary less than $10 000)

8 1. Figure 5 shows a parallelogram ABCD. The diagonals AC and BD intersect at E. A(, 7) y D E B(0, 1) O Figure 5 C(6, 1) x Find the coordinates of E and D. (b) Is AC perpendicular to BD? Explain your answer. (c) Find the perimeter of ABCD. ( marks) ( marks) ( marks) Coordinates of E 6, 7 ( 1) (, ) Let (x1, y1) be the coordinates of D. By the mid-point formula, x 1 0 and y 1 1 x1 4 and y1 5 The coordinates of D are (4, 5). 8

9 (b) Slope of AC ( ) 1 Slope of BD Slope of AC Slope of BD AC is perpendicular to BD. (c) AB ( 0) (7 1) 40 BC ( 0 6) [1 ( 1)] 40 Perimeter of ABCD ( 40 40) 5. (cor. to sig. fig.) 1. In Figure 6, port B is 00 m due south of port A. Ship P leaves port B and travels 50 m at a bearing of NE to port C. N C A 00 m 50 m B Figure 6 Find the shortest distance between port A and ship P in its journey. ( marks) (b) Find the true bearing of port C from port A correct to the nearest degree. (4 marks) 9

10 Suppose ship P will be closest to port A at the point X on BC. Then AX BC. N C A 00 m 50 m X B AX sin ABX AB AX sin 00 m AX 74.9 m (cor. to sig. fig.) The required shortest distance is 74.9 m. 10

11 (b) Let D be the point which is due north of A and due west of C. D N C A 00 m 50 m B CD sin ABC BC CD 50 m sin CD 50 sin m BD cos ABC BC AD 00 m 50 m cos AD (50 cos 00) m CD tan AD 50sin m (50cos 00) m 46 (cor. to the nearest degree) The true bearing of port C from port A is

12 14. Figure 7 shows a solid in the form of an oblique pyramid with a rectangular base. The length and the width of the rectangular base are 1 cm and 8 cm respectively. The height of the solid is 16 cm. 16 cm 1 cm Figure 7 8 cm Figure 7(b) Find the volume of the solid. ( marks) (b) In Figure 7(b), the solid is cut into a smaller pyramid and a frustum along a rectangular plane which is parallel to the base of the solid. It is given that the base area of the smaller pyramid is 6 cm. (i) Find the volume of the smaller pyramid. (ii) If the total area of the lateral surfaces of the smaller pyramid is A cm, express the total area of the lateral surfaces of the frustum in terms of A. (5 marks) 1 Volume cm 51 cm (b) (i) Volume of the smaller pyramid Volume of the original pyramid Height of thesmaller pyramid Height of the originalpyramid Base area of the smaller pyramid Base area of the original pyramid Volume of the smaller pyramid 51 cm 6 cm 81 cm Volume of the smaller pyramid 8 cm 1

13 (b) (ii) Let B cm be the total area of the lateral surfaces of the frustum. A cm 6 cm ( A B) cm 96 cm B 15A The total area of the lateral surfaces of the frustum is15a cm. 15. The heights (in cm) of the members of football team A are shown as follows: Find the mean and the median of the heights of the members of football team A. ( marks) (b) The stem-and-leaf diagram below shows the distribution of the heights (in cm) of the members of football team B. It is given that the means of the heights of the members of football team A and football team B are the same. Stem (tens) Leaf (units) 15 a b (i) Find all possible values of a and b. (ii) If a team member is selected randomly from all the team members in the two teams, find the probability that the height of the member is less than the median of the heights of the members of football team A. (5 marks) Mean cm cm Arrange the heights (in cm) in ascending order: Median (16 16) cm 1 6.5cm 1

14 (b) (i) (150 a) (170 b) a b a b 9 Note that 0 a 6 and 8 b 9, where a and b are integers. a 1 b 8 or a 0 b 9 (ii) Required probability END OF PAPER 14

For all questions, E. NOTA means none of the above answers is correct. Diagrams are NOT drawn to scale.

For all questions, E. NOTA means none of the above answers is correct. Diagrams are NOT drawn to scale. For all questions, means none of the above answers is correct. Diagrams are NOT drawn to scale.. In the diagram, given m = 57, m = (x+ ), m = (4x 5). Find the degree measure of the smallest angle. 5. The