1) The diagram shows a water tower standing on horizontal ground. The height of the tower is 26.5 m. x m

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1 1) The diagram shows a water tower standing on horizontal ground. The height of the tower is 26.5 m. x m From a point on the ground the angle of elevation to the top of the tower is 28. On the diagram, show and label the angle of elevation, 28. alculate, correct to the nearest metre, the distance x m. (Total 4 marks) 2) The gradients of several lines are as follows: Line a b c d e f g h Gradient Find two pairs of lines that are parallel to each other. Find any two pairs of lines that are at right angles to each other. 3) The height of a vertical cliff is 450 m. The angle of elevation from a ship to the top of the cliff is 23. The ship is x metres from the bottom of the cliff. (Total 4 marks) Draw a diagram to show this information. alculate the value of x. (Total 4 marks) 4) The diagram represents a small, triangular field,, with = 25 m, angle = 55 and angle = 75.

2 Write down the size of angle. alculate the length of. alculate the area of the field. N is the point on such that N is perpendicular to. M is the midpoint of N. (d) alculate the length of NM. goat is attached to one end of a rope of length 7 m. The other end of the rope is attached to the point M. (e) Decide whether the goat can reach point P, the midpoint of. Justify your answer. (5) (Total 15 marks) 5) rectangular cuboid has the following dimensions. Length Width Height 0.80 metres (D) 0.50 metres (DG) 1.80 metres (D) alculate the length of G. alculate the length of F. Find the size of the angle between F and G. (Total 6 marks)

3 6) farmer has a triangular field,, as shown in the diagram. = 35 m, = 80 m and  =105, and D is the midpoint of. Find the size of Ĉ. alculate the length of D. (5) The farmer wants to build a fence around D. (d) (e) (f) alculate the total length of the fence. The farmer pays USD for the fence. Find the cost per metre. alculate the area of the triangle D. layer of earth 3 cm thick is removed from D. Find the volume removed in cubic metres. (Total 18 marks) 7. path goes around a forest so that it forms the three sides of a triangle. The lengths of two sides are 550 m and 290 m. These two sides meet at an angle of 115. diagram is shown below. alculate the length of the third side of the triangle. Give your answer correct to the nearest 10 m. (4) alculate the area enclosed by the path that goes around the forest.

4 Inside the forest a second path forms the three sides of another triangle named. ngle  is 53, is 180 m and is 230 m. 986 m alculate the size of angle 625 m 102 Ĉ. (4) (Total 11 marks) 8. The diagram shows a pyramid VD which has a square base of length 10 cm and edges of length 13 cm. M is the midpoint of the side. alculate the length of VM. alculate the vertical height of the pyramid. alculate the angle between a sloping face of the pyramid and its base. (Total 6 marks) 9. On a map three schools, and are situated as shown in the diagram. Schools and are 625 metres apart. ngle ˆ = 102 and = 986 metres. 400 m 30º

5 Find the distance between and. Find the size of angle Â. 10. farmer wants to construct a new fence across a field. The plan is shown below. The new fence is indicated by a dotted line. 75 Diagram not to scale 410 m 40 alculate the length of the fence. (5) The fence creates two sections of land. Find the area of the smaller section of land, given the additional information shown below. Diagram not to scale m (Total 8 marks) 11. D 4.5 cm 3 cm 25 3 cm In the diagram, = = 3 cm, D = 4.5 cm, angle ˆ = 90 and angle ĈD = 25. alculate the length of. alculate the area of triangle D. alculate the area of quadrilateral D. otal 8 marks)

6 500 m 12. The following diagram shows a sloping roof. The surface D is a rectangle. The angle DE is 55. The vertical height, F, of the roof is 3 m and the length D is 7 m. 3 m 7 m 55 E F D alculate D. alculate the length of the diagonal D. (Total 8 marks) 13. cross-country running course is given in the diagram below. Runners start and finish at point O. O Not to scale 1500 m 800 m 110 Show that the distance is 943 m correct to 3 s.f. Show that angle is 58.0 correct to 3 s.f. (i) alculate the angle O. (ii) alculate the distance O. (5)

7 (d) alculate the area enclosed by the course O. (4) (e) Gonzales runs at a speed of 4 m s 1. alculate the time, in minutes, taken for him to complete the course. (Total 16 marks) 14. Three right pyramids ndal, atsu and artos were discovered in the dense jungle of Marhartmasol. Each pyramid has a square base with centres, and respectively. ndal artos atsu Diagram not to scale surveying team was lowered from a helicopter to the top of ndal to take measurements of the area. ndal is 40 metres high. The angle of elevation from the top of ndal to the top of atsu is 3. The horizontal distance from, the centre of the base of ndal, to, the centre of the base of atsu is 600 metres. Use the diagram below to find the height of atsu. Diagram not to scale 3º 40 m ndal atsu 600 m artos is found to be 92 metres high and the angle of elevation from the top of ndal to the top of artos is 4. (i) Draw a diagram similar to the diagram in part to show the relationship between ndal and artos. (ii) What is the horizontal distance from to? (4) The diagram below represents measurements relative to the centres of the bases of the pyramids. The surveyors determined the angle at to be 110, and the distance to be 600 m.

8 Diagram not to scale 110º 600 m (i) (ii) (iii) What is the distance between and? Give your answer to the nearest metre. What is the size of angle? What is the area of the land inside triangle? (8) (Total 15 marks) 15) The base of a prism is a regular hexagon. The centre of the hexagon is O and the length of O is 15 cm. Write down the size of angle O. Find the area of the triangle O. The height of the prism is 20 cm. Find the volume of the prism. 16) Tennis balls are sold in cylindrical tubes that contain four balls. The radius of each tennis ball is 3.15 cm and the radius of the tube is 3.2 cm. The length of the tube is 26 cm. Find the volume of one tennis ball. alculate the volume of the empty space in the tube when four tennis balls have been placed in it. (4) 17) The weights in kg, of 80 adult males, were collected and are summarized in the box and whisker plot shown below.

9 (d) Write down the median weight of the males. alculate the interquartile range. Estimate the number of males who weigh between 61 kg and 66 kg. Estimate the mean weight of the lightest 40 males. 18) Tony wants to carry out a χ 2 test to determine whether or not a person s choice of one of the three professions engineering, medicine or law is influenced by the person s sex (gender). State the null hypothesis, H 0, for this test. Write down the number of degrees of freedom. Of the 400 people Tony interviewed, 220 were male and 180 were female. 80 of the people had chosen engineering as a profession. alculate the expected number of female engineers. Tony used a 5 % level of significance for his test and obtained a p-value of correct to 3 significant figures. (d) State Tony s conclusion to the test. Give a reason for this conclusion. (Total 6 marks) 19) For his Mathematical Studies project, Marty set out to discover if stress was related to the amount of time that students spent travelling to or from school. The results of one of his surveys are shown in the table below. Travel time (t mins) Number of students high stress moderate stress low stress t t t He used a χ 2 test at the 5 level of significance to find out if there was any relationship between student stress and travel time. (i) Write down the null and alternative hypotheses for this test.

10 (ii) (iii) (iv) Write down the table of expected values. Give values to the nearest integer. Show that there are 4 degrees of freedom. alculate the χ 2 statistic for this data. The χ 2 critical value for 4 degrees of freedom at the 5 level of significance is (v) What conclusion can Marty draw from this test? Give a reason for your answer. Marty asked some of his classmates to rate their level of stress out of 10, with 10 being very high. He also asked them to measure the number of minutes it took them to get from home to school. random selection of his results is listed below. (i) Travel time (x) Stress rating (y) Write down the value of the (linear) coefficient of correlation for this information. (ii) (iii) (iv) Explain what a positive value for the coefficient of correlation indicates. Write down the linear regression equation of y on x in the form y = ax + b Use your equation in part (iii) to determine the stress rating for a student who takes three quarters of an hour to travel to school. (v) an your answer in part (iv) be considered reliable? Give a reason for your answer. (Total 18 marks) 20) marine biologist records as a frequency distribution the lengths (L), measured to the nearest centimetre, of 100 mackerel. The results are given in the table below. Length of mackerel (L cm) Number of mackerel 27 < L < L < L < L < L < L < L < L onstruct a cumulative frequency table for the data in the table. Draw a cumulative frequency curve. Hint: Plot your cumulative frequencies at the top of each interval.

11 Use the cumulative frequency curve to find an estimate, to the nearest cm for (i) (ii) the median length of mackerel; the interquartile range of mackerel length. (Total 9 marks)