EXTENDED MATHEMATICS CLASSIFIEDS GEOMETRY. Compiled & Edited By. Dr. Eltayeb Abdul Rhman. First Edition 2011

Transcription

1 EXTENDED MTHEMTIS LSSIFIEDS GEOMETRY ompiled & Edited y Dr. Eltayeb bdul Rhman First Edition 2011

2 4 7 O y 50 z T is a tangent at to the circle, centre O. ngle O = 50. Find the value of (a) y, t T nswer(a) y = [1] (b) z, nswer(b) z = [1] (c) t. nswer(c) t = [1] 8 Seismic shock waves travel at speed v through rock of density d. v varies inversely as the square root of d. v = 3 when d = Find v when d = nswer v = [3] ULES /21/O/N/11

3 10 16 y (9,7) (1,3) O (3,0) x The co-ordinates of, and are shown on the diagram, which is not to scale. (a) Find the length of the line. nswer(a) = [3] (b) Find the equation of the line. nswer(b) [3] ULES /21/O/N/11

4 cm X 3.84 cm D 9.40 cm,, and D lie on a circle. and D intersect at X. (a) Give a reason why angle X is equal to angle DX. nswer(a) [1] (b) = 4.40 cm, D = 9.40 cm and X = 3.84 cm. (i) alculate the length of X. (ii) The area of triangle X is 5.41 cm 2. alculate the area of triangle DX. nswer(b)(i) X = cm [2] nswer(b)(ii) cm 2 [2] ULES /23/O/N/11

5 6 3 (a) D E D is a quadrilateral with angle D = 40. is extended to E and angle E = 30. = D and D =. (i) alculate angle D. (ii) Give a reason why D is not parallel to E. nswer(a)(i) ngle D = [3] nswer(a)(ii) [1] (b) regular polygon has n sides. Each exterior angle is Find the value of n. 5n degrees. 2 nswer(b) n = [3] ULES /43/O/N/11

6 7 (c) O 25 The diagram shows a circle centre O., and are points on the circumference. O is parallel to. ngle O = 25. alculate angle O. nswer(c)ngle O = [3] ULES /43/O/N/11

7 1 Javed says that his eyes will blink times in 79 years. 2 (a) Write in standard form. nswer (a)... [1] (b) One year is approximately minutes. alculate, correct to the nearest whole number, the average number of times his eyes will blink per minute. nswer (b)... [1] 2 Luis and Hans both have their birthdays on January 1st. In 2002 Luis is 13 and Hans is 17 years old. (a) Which is the next year after 2002 when both their ages will be prime numbers? nswer (a)... [1] (b) In which year was Hans twice as old as Luis? nswer (b)... [1] 3 D Diagram 1 Diagram 2 (a) In Diagram 1, shade the area which represents. [1] (b) Describe in set notation the shaded area in Diagram 2. nswer (b)... [1] 0580/2, 0581/2 Jun02

8 3 4 y D x E D is a parallelogram and E is a straight line. ngle DE = 54 and angle D = 20. Find x and y. nswer x =... nswer y =... [2] 5 alculate the length of the straight line joining the points ( 1, 4) and (5, 4). nswer... [2] 15 (a) (i) omplete quadrilateral D so that the dotted line is the only line of symmetry. [1] (ii) Write down the special name for quadrilateral D. nswer (a)(ii)... [1] D (b) F (i) omplete quadrilateral EFGH so that the dotted line is one of two lines of symmetry. [1] E (ii) Write down the order of rotational symmetry for quadrilateral EFGH. H nswer (b)(ii)... [1] 0580/2, 0581/2 Jun02

9 8 17 4r r Two circles have radii r cm and 4r cm. Find, in terms of π and r. (a) the area of the circle with radius 4r cm, nswer (a)... cm 2 [1] (b) the area of the shaded ring, nswer (b)... cm 2 [1] (c) the total length of the inner and outer edges of the shaded ring. nswer (c)... cm [2] 0580/2, 0581/2 Jun02

10 4 4 b W c d D G 42 a e X sphere, centre, rests on horizontal ground at and touches a vertical wall at D. straight plank of wood, GW, touches the sphere at, rests on the ground at G and against the wall at W. The wall and the ground meet at X. ngle WGX = 42. (a) Find the values of a, b, c, d and e marked on the diagram. [5] (b) Write down one word which completes the following sentence. ngle G is 21 because triangle G and triangle G are. [1] (c) The radius of the sphere is 54 cm. (i) alculate the distance G. Show all your working. [3] (ii) Show that GX = 195 cm correct to the nearest centimetre. [1] (iii) alculate the length of the plank GW. [3] (iv) Find the distance W. [1] 0580/4, 0581/4 Jun02

11 8 (a) sector of a circle, radius 6 cm, has an angle of alculate 6cm (i) the area of the sector, [2] (ii) the arc length of the sector. [2] (b) 20 6cm 5cm whole cheese is a cylinder, radius 6 cm and height 5 cm. The diagram shows a slice of this cheese with sector angle 20. alculate (i) the volume of the slice of cheese, [2] (ii) the total surface area of the slice of cheese. [4] (c) The radius, r, and height, h, of cylindrical cheeses vary but the volume remains constant. (i) Which one of the following statements,, or D is true? : h is proportional to r. : h is proportional to r 2. : h is inversely proportional to r. D: h is inversely proportional to r 2. [2] (ii) What happens to the height h of the cylindrical cheese when the volume remains constant but the radius is doubled? [2] 0580/4, 0581/4 Jun02

12 4 9 y x 1 2 (a) Find the gradient of the line. (b) alculate the angle that makes with the x-axis. nswer (a)... [1] nswer (b)... [2] 0580/2, 0581/2 Jun 2003

13 5 12 q 2p O 3p 5q D,,, D and E lie on a circle, centre O. O is a diameter. Find the value of (a) p, E (b) q. nswer (a) p #... [2] nswer (b) q #... [2] 0580/2, 0581/2 Jun 2003

14 8 9 (a) Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 1 shows a triangle with its base divided in the ratio 1 : 3. Diagram 2 shows a parallelogram with its base divided in the ratio 1 : 3. Diagram 3 shows a kite with a diagonal divided in the ratio 1 : 3. Diagram 4 shows two congruent triangles and a trapezium each of height 1 unit. For each of the four diagrams, write down the percentage of the total area which is shaded. [7] (b) 80 O 80 O 2 1 Diagram 5 Diagram 6 Diagram 7 Diagram 5 shows a semicircle, centre O. Diagram 6 shows two circles with radii 1 unit and 5 units. Diagram 7 shows two sectors, centre O, with radii 2 units and 3 units. For each of diagrams 5, 6 and 7, write down the fraction of the total area which is shaded. [6] 0580/4, 0581/4 Jun/03

15 15 6 The area of triangle PQ is 99 cm 2 and the area of triangle is 11 cm 2. is parallel to PQ and the length of PQ is 12 cm. alculate the length of. P Q nswer = cm [3] ULES /2, 0581/2 Jun/04

16 4 D 11.1 cm E 37 o 70 o 9.5 cm D is a cyclic quadrilateral. = 9.5 cm, = 11.1 cm, angle = 70 o and angle D = 37 o. (a) alculate the length of. [4] (b) Explain why angle D = 110 o. [1] (c) alculate the length of D. [4] (d) point E lies on the circle such that triangle E is isosceles, with E = E. (i) Write down the size of angle E. [1] (ii) alculate the area of triangle E. [3] 0580/4, 0581/4 Jun/04 15 The points (6,2) and (8,5) lie on a straight line. (a) Work out the gradient of this line. 7 nswer (a) [1] (b) Work out the equation of the line, giving your answer in the form y = mx + c. nswer (b) [2] 0580/02, 0581/02 Jun 05

17 cm O 40 o, and are points on a circle, centre O. ngle O = 40 o. (a) (i) Write down the size of angle. (ii) Find the size of angle O. nswer (a)(i) ngle = [1] nswer (a)(ii) ngle O = [1] (b) The radius of the circle is 5 cm. (i) alculate the length of the minor arc. nswer (b)(i) cm[2] (ii) alculate the area of the minor sector O. nswer (b)(ii) cm 2 [2] ULES /02, 0581/02 Jun 05

18 7 18 w o 22 o E x o y o z o 60 o D D is a diameter of the circle DE. ngle = 22 and angle D = 60. and ED are parallel lines. Find the values of w, x, y and z. nswer w= x= y= z= [4] ULES /02, 0581/02 Jun 06

19 10 22 (a) p cm D 4 cm q cm 5 cm 4 cm E 8 cm In the diagram triangles E and D are similar. E is parallel to D. = 5 cm, = 4 cm, E = 4 cm, E = 8 cm, D = p cm and DE = q cm. Work out the values of p and q. nswer(a) p = q = [4] (b) spherical balloon of radius 3 metres has a volume of 36π cubic metres. It is further inflated until its radius is 12 m. alculate its new volume, leaving your answer in terms of π. nswer(b) m 3 [2] ULES /02, 0581/02 Jun 06

20 o The diagram shows three touching circles. is the centre of a circle of radius x centimetres. and are the centres of circles of radius 3.8 centimetres. ngle = 70. Find the value of x. nswer x = [3] ULES /02/J/07

21 8 19 R S 64 w Q O x P y T P, Q, R and S lie on a circle, centre O. TP and TQ are tangents to the circle. PR is a diameter and angle PSQ = 64. (a) Work out the values of w and x. nswer(a) w = [1] x = [1] (b) Showing all your working, find the value of y. nswer(b) y = [2] ULES /02/J/07

22 cm The largest possible circle is drawn inside a semicircle, as shown in the diagram. The distance is 12 centimetres. (a) Find the shaded area. (b) Find the perimeter of the shaded area. nswer(a) cm 2 [4] nswer(b) cm [2] ULES /02/J/07

23 4 3 (a) cm cm X 20.1 cm D,, and D lie on a circle. and D intersect at X. ngle X = 55 and angle X = 92. X = 26.8 cm, X = 40.3 cm and X = 20.1 cm. (i) alculate the area of triangle X. You must show your working. [2] (ii) alculate the length of. You must show your working. [3] (iii) Write down the size of angle D. Give a reason for your answer. [2] (iv) Find the size of angle D. [1] (v) Write down the geometrical word which completes the statement Triangle X is to triangle DX. [1] (vi) alculate the length of XD. You must show your working. [2] ULES /04/M/J/07

24 6 5 E F Q D G O H 12 cm P circle, centre O, touches all the sides of the regular octagon DEFGH shaded in the diagram. The sides of the octagon are of length 12 cm. and GH are extended to meet at P. HG and EF are extended to meet at Q. (a) (i) Show that angle H is 135. [2] (ii) Show that angle PH is 90. [1] (b) alculate (i) the length of PH, [2] (ii) the length of PQ, [2] (iii) the area of triangle PH, [2] (iv) the area of the octagon. [3] (c) alculate (i) the radius of the circle, [2] (ii) the area of the circle as a percentage of the area of the octagon. [3] ULES /04/M/J/08

25 8 7 (a) S D z 40 T O 130 x y,, and D lie on a circle, centre O. ST is the tangent at and is parallel to O. ngle O = 130, and angle T = 40. ngle O = x, angle O = y and angle D = z. (i) Write down the geometrical word which completes the following statement. D is a quadrilateral. [1] (ii) Find the values of x, y and z. [3] (iii) Write down the value of angle OT. [1] (iv) Find the value of the reflex angle O. [1] ULES /04/M/J/08

26 9 (b) P 7 cm Q X S 10 cm R P, Q, R and S lie on a circle. PQ = 7 cm and SR = 10 cm. PR and QS intersect at X. The area of triangle SRX = 20 cm 2. (i) Write down the geometrical word which completes the following statement. Triangle PQX is to triangle SRX. [1] (ii) alculate the area of triangle PQX. [2] (iii) alculate the length of the perpendicular height from X to RS. [2] ULES /04/M/J/08

27 17 E O 38 D is the diameter of a circle, centre O., D and E lie on the circle. E is parallel to and perpendicular to OD. ngle DO is 38. Work out (a) angle O, (b) angle O, (c) angle EDO. nswer(a) ngle O = [1] nswer(b) ngle O = [1] nswer(c) ngle EDO = [2] ULES /21/M/J/10

28 8 15 y 4 y = 4 0 x 2x + y = 8 3x + y = 18 (a) The line y = 4 meets the line 2x + y = 8 at the point. Find the co-ordinates of. nswer(a) (, ) [1] (b) The line 3x + y = 18 meets the x axis at the point. Find the co-ordinates of. nswer(b) (, ) [1] (c) (i) Find the co-ordinates of the mid-point M of the line joining to. (ii) Find the equation of the line through M parallel to 3x + y = 18. nswer(c)(i) M (, ) [1] nswer(c)(ii) [2] ULES /22/M/J/10

29 10 17 Q R P W O 8 m S T V U The diagram shows the junction of four paths. In the junction there is a circular area covered in grass. This circle has centre O and radius 8 m. (a) alculate the area of grass. nswer(a) m 2 [2] (b) Q 12 m P 45 O The arc PQ and the other three identical arcs, RS, TU and VW are each part of a circle, centre O, radius 12m. The angle POQ is 45. The arcs PQ, RS, TU, VW and the circumference of the circle in part(a) are painted white. alculate the total length painted white. nswer(b) m [4] ULES /22/M/J/10

30 16 9 (a) 9 cm x y X 3 cm 6 cm D E The lines and DE are parallel. D and intersect at X. = 9 cm, D = 6 cm and DX = 3 cm. (i) omplete the following statement. Triangle X is to triangle DX. [1] (ii) alculate the length of X. (iii) The area of triangle DX is 6 cm 2. alculate the area of triangle X. nswer(a)(ii) X = cm [2] (iv) ngle X = x and angle X = y. Find angle X and angle XDE in terms of x and/or y. nswer(a)(iii) cm 2 [2] nswer(a)(iv) ngle X = ngle XDE = [2] ULES /42/M/J/10

31 17 (b) R 35 Q O P 42 S P, Q, R and S lie on a circle, centre O. ngle OPS = 42 and angle PRQ = 35. alculate (i) angle POS, (ii) angle PRS, (iii) angle SPQ, (iv) angle PSQ. nswer(b)(i) ngle POS = [1] nswer(b)(ii) ngle PRS = [1] nswer(b)(iii) ngle SPQ = [1] nswer(b)(iv) ngle PSQ = [1] (c) The interior angle of a regular polygon is 8 times as large as the exterior angle. alculate the number of sides of the polygon. nswer(c) [3] ULES /42/M/J/10

32 3 2 3 cm 5 cm 11 cm 8 cm D In the quadrilateral D, = 3 cm, D = 11 cm and D = 8 cm. The diagonal = 5 cm and angle = 90. alculate (a) the length of, nswer(a) = cm [2] (b) angle D, nswer(b) ngle D = [4] (c) the area of the quadrilateral D. nswer(c) cm 2 [3] ULES /43/M/J/10

33 8 5 (a) 3 cm 4 cm Q P 3.6 cm The diagram shows two triangles and PQ. ngle PQ = angle and angle QP = angle. = 4 cm, = 3.6 cm and Q = 3 cm. (i) omplete the following statement. Triangle is to triangle PQ. [1] (ii) alculate the length of PQ. (iii) The area of triangle is 5.6 cm 2. alculate the area of triangle PQ. nswer(a)(ii) PQ = cm [2] nswer(a)(iii) cm 2 [2] ULES /43/M/J/10

34 9 (b) R H S O U M 61 T N R, H, S, T and U lie on a circle, centre O. HT is a diameter and MN is a tangent to the circle at T. ngle RTM = 61. Find (i) angle RTH, (ii) angle RHT, (iii) angle RST, (iv) angle RUT. nswer(b)(i) ngle RTH = [1] nswer(b)(ii) ngle RHT = [1] nswer(b)(iii) ngle RST = [1] nswer(b)(iv) ngle RUT = [1] (c) DEF is a hexagon. The interior angle is 4 greater than interior angle. The interior angle is 4 greater than interior angle, and so on, with each of the next interior angles 4 greater than the previous one. (i) y how many degrees is interior angle F greater than interior angle? (ii) alculate interior angle. nswer(c)(i) [1] nswer(c)(ii) [3] ULES /43/M/J/10 [Turn over

35 4 9 2x 5x x D is parallel to D. alculate the value of x. nswer x = [3] ULES /11/M/J/11

36 9 18 O 24 T, and are points on a circle, centre O. T is a tangent to the circle at and OT is a straight line. is a diameter and angle OT = 24. alculate (a) angle OT, nswer(a) ngle OT = [2] (b) angle O, nswer(b) ngle O = [1] (c) angle O. nswer(c) ngle O = [1] ULES /11/M/J/11

37 ULES /12/M/J/11 2 x 52 straight line intersects two parallel lines as shown in the diagram. Find the value of x. nswer x = [1] 9 2x 5x x D is parallel to D. alculate the value of x. nswer x = [3]

38 8 17 (a) Points, and lie on the circumference of the circle shown above. When angle is 90 write down a statement about the line. nswer(a) [1] (b) O 54 D O is the centre of a circle and the line is a tangent to the circle at. D is a point on the circumference and angle OD = 54. alculate angle D. nswer(b) ngle D = [3] ULES /12/M/J/11

39 8 16 k cm The diagram shows a square of side k cm. The circle inside the square touches all four sides of the square. (a) The shaded area is cm 2. Show that 4 = 4k 2 πk 2. nswer (a) [2] (b) Make k the subject of the formula 4 = 4k 2 πk 2. nswer(b) k = [3] ULES /21/M/J/11

40 9 17 O 24 T, and are points on a circle, centre O. T is a tangent to the circle at and OT is a straight line. is a diameter and angle OT = 24. alculate (a) angle OT, nswer(a) ngle OT = [2] (b) angle, nswer(b) ngle = [1] (c) angle T. nswer(c) ngle T = [2] ULES /21/M/J/11

41 7 14 y x The diagram shows the straight line which passes through the points (0, 1) and (3, 13). Find the equation of the straight line. nswer [3] 15 cylinder has a height of 12 cm and a volume of 920 cm 3. alculate the radius of the base of the cylinder. nswer cm [3] ULES /23/M/J/11

42 10 20 V U W g O 70 h X e f T The diagram shows a circle, centre O. VT is a diameter and T is a tangent to the circle at T. U, V, W and X lie on the circle and angle VOU = 70. alculate the value of (a) e, nswer(a) e = [1] (b) f, nswer(b) f = [1] (c) g, nswer(c) g = [1] (d) h. nswer(d) h = [1] ULES /23/M/J/11

43 y x (a) (i) Find the gradient of the line. (ii) Write down the equation of the line in the form y = mx + c. nswer(a)(i) [2] nswer(a)(ii) y = [2] ULES /31/M/J/11

44 5 (a) The table below shows how many sides different polygons have. omplete the table. 7 Name of polygon Number of sides 3 Quadrilateral 4 5 Hexagon 6 Heptagon 7 8 Nonagon 9 [3] (b) Two sides, and, of a regular nonagon are shown in the diagram below. x (i) Work out the value of x, the exterior angle. nswer(b)(i) x = [2] (ii) Find the value of angle, the interior angle of a regular nonagon. nswer(b)(ii) ngle = [1] ULES /32/M/J/11 [Turn over

45 10 6 (a) p 140 D The diagram shows a triangle with extended to D. = and angle D = 140. Find the value of p. nswer(a) p = [2] (b) 72 q Find the value of q. nswer(b) q = [2] (c) x Find the value of x. nswer(c) x = [1] ULES /33/M/J/11

46 11 (d) 22 In triangle, angle = 90 and angle = 22. alculate angle. nswer(d) ngle = [1] (e) X 10 cm P 8 cm Q Y 10 cm Z In triangle XYZ, P is a point on XY and Q is a point on XZ. PQ is parallel to YZ. (i) omplete the statement. Triangle XPQ is to triangle XYZ. [1] (ii) PQ = 8 cm, XQ = 10 cm and YZ = 10 cm. alculate the length of XZ. nswer(e)(ii) XZ = cm [2] ULES /33/M/J/11

47 3 (ii) North km = = 6 km. Junior students follow a similar path but they only walk 4 km North from, then 4 km on a bearing 110 before returning to. Senior students walk a total of 18.9 km. alculate the distance walked by junior students. nswer(b)(ii) km [3] (c) The total amount, $1380, raised in 2010 was 8% less than the total amount raised in alculate the total amount raised in nswer(c) $ [3] ULES /41/M/J/11

48 6 4 8 cm 6 cm O 9 cm The circle, centre O, passes through the points, and. In the triangle, = 8 cm, = 9 cm and = 6 cm. (a) alculate angle and show that it rounds to 78.6, correct to 1 decimal place. nswer(a) [4] (b) M is the midpoint of. (i) Find angle OM. nswer(b)(i) ngle OM = [1] ULES /43/M/J/11

49 17 (b) F 2x (x + 15) G H J 75 EFG is a triangle. HJ is parallel to FG. ngle FEG = 75. ngle EFG = 2x and angle FGE = (x + 15). (i) Find the value of x. E nswer(b)(i) x = [2] (ii) Find angle HJG. nswer(b)(ii) ngle HJG = [1] ULES /43/M/J/11

50 5 12 F E D In the hexagon DEF, is parallel to ED and D is parallel to EF. ngle DEF = 109 and angle EF = 95. ngle F is equal to angle. Find the size of (a) angle ED, nswer (a) ngle ED =... [1] (b) angle F. nswer (b) ngle F =... [2] 0580/2/O/N/02

51 7 14 S P Q X 80 R PQRS is a cyclic quadrilateral. The diagonals PR and QS intersect at X. ngle SPR = 21, angle PRS = 80 and angle PXQ = 33. alculate (a) angle PQS, nswer (a) ngle PQS =... [1] (b) angle QPR, nswer (b) ngle QPR =... [1] (c) angle PSQ. nswer (c) ngle PSQ =... [1] 15 Solve the simultaneous equations 4x + 5y = 0, 8x 15y = 5. nswer x =... y =... [4] 0580/2/O/N/02

52 11 22 P R 8cm O K 3cm M 8cm 5.5cm L Q In the circle, centre O, the chords KL and PQ are each of length 8 cm. M is the mid-point of KL and R is the mid-point of PQ. OM = 3 cm. (a) alculate the length of OK. nswer (a) OK =...cm [2] (b) RM has a length of 5.5 cm. alculate angle ROM. nswer (b) ngle ROM =... [3] 0580/2/O/N/02

53 2 D 10cm 6cm 8cm 5cm E 5cm F The diagram shows a sketch of the net of a solid tetrahedron (triangular prism). The right-angled triangle is its base. = 8 cm, = 6 cm and = 10 cm. F = E = 5 cm. (a) (i) Show that E = 61 cm. [1] (ii) Write down the length of D. [1] (iii) Explain why D = 89 cm. [2] (b) alculate the size of angle D. [4] (c) alculate the area of triangle D. [3] (d) Find the total surface area of the solid. [3] (e) alculate the volume of the solid. 1 [The volume of a tetrahedron is (area of the base) perpendicular height.] [3] /4,0581/4/O/N02

54 7 17 D X D is a cyclic quadrilateral. D is parallel to. The diagonals D and meet at X. ngle # 62 and angle D # 20. alculate (a) angle D, nswer (a) ngle D #... [1] (b) angle D, nswer (b) ngle D #... [1] (c) angle D, nswer (c) ngle D #... [1] (d) angle X, nswer (d) ngle X #... [1] (e) angle D. nswer (e) ngle D #... [1] 0580/02/0581/02/O/N/03

55 m 35 m The diagram shows an athletics track with six lanes. The distance around the inside of the inner lane is 400 metres. The radius of each semicircular section of the inside of the inner lane is 35 metres. (a) alculate the total length of the two straight sections at the inside of the inner lane. nswer(a) m [3] (b) Each lane is one metre wide. alculate the difference in the distances around the outside of the outer lane and the inside of the inner lane. nswer(b) m [2] ULES /02/O/N/04

56 8 10 Quadrilaterals P and Q each have diagonals which are unequal, intersect at right angles. P has two lines of symmetry. Q has one line of symmetry. (a) (i) Sketch quadrilateral P. Write down its geometrical name. [2] (ii) Sketch quadrilateral Q. Write down its geometrical name. [2] (b) In quadrilateral P, an angle between one diagonal and a side is x. Write down, in terms of x, the four angles of quadrilateral P. [2] (c) The diagonals of quadrilateral Q have lengths 20 cm and 12 cm. alculate the area of quadrilateral Q. [2] (d) Quadrilateral P has the same area as quadrilateral Q. The lengths of the diagonals and sides of quadrilateral P are all integer values. Find the length of a side of quadrilateral P. [3] ULES /04/O/N/04

57 6 15 D 48 o E O 126 o F,, and D lie on a circle centre O. is a diameter of the circle. D, E and F are parallel lines. ngle E = 48 and angle F = 126. Find (a) angle DE, (b) angle E, nswer(a) ngle DE = [1] (c) angle E. nswer(b) ngle E = [1] nswer(c) ngle E = [1] ULES /02, 0581/02 Nov 2005

58 7 17 D E F DE is a regular pentagon. DEF is a straight line. alculate (a) angle EF, nswer(a) ngle EF = [2] (b) angle DE. nswer(b) ngle DE = [1] 18 Simplify (a) 27 x , nswer(a) [2] (b) x nswer(b) [2] ULES /02, 0581/02 Nov 2005

59 8 19 y 5 l 0 (a) alculate the gradient of the line l. 10 x nswer(a) [2] (b) Write down the equation of the line l. nswer(b) [2] ULES /02, 0581/02 Nov 2005

60 10 21 E O D,, and D lie on a circle, centre O, radius 8 cm. and D are tangents to a circle, centre O, radius 4 cm. D is a rectangle. (a) alculate the distance E. nswer(a) E = cm [2] (b) alculate the shaded area. nswer(b) cm 2 [3] ULES /02, 0581/02 Nov 2005

61 6 12 P 38º S 38º T Q R In the diagram PT and QR are parallel. TP and TR are tangents to the circle PQRS. ngle PTR = angle RPQ = 38. (a) What is the special name of triangle TPR. Give a reason for your answer. nswer(a) name reason [1] (b) alculate (i) angle PQR, (ii) angle PSR. nswer(b)(i) ngle PQR = [1] nswer(b)(ii)ngle PSR = [1] 13 statue two metres high has a volume of five cubic metres. similar model of the statue has a height of four centimetres. (a) alculate the volume of the model statue in cubic centimetres. (b) Write your answer to part (a) in cubic metres. nswer(a) cm 3 [2] nswer(b) m 3 [1] ULES /02/N/06

62 x cm 5x cm D 17x cm 12x cm yº D is a trapezium. (a) Find the area of the trapezium in terms of x and simplify your answer. nswer(a) cm 2 [2] (b) ngle D = y. alculate the value of y. nswer(b) y = [2] ULES /02/N/06

63 11 21 y 2x + 3y = 17 4x y = 6 (4,3) x (1, 2) In the diagram, the line has equation 2x + 3y = 17 and the line has equation 4x y = 6. The lines and intersect at (1, 2). The lines and intersect at (4, 3). (a) Use algebra to find the coordinates of the point. nswer(a) [3] (b) Find the equation of the line. nswer(b) [3] ULES /02/N/06

64 8 18 y l x The line l passes through the points (10, 0) and (0, 8) as shown in the diagram. (a) Find the gradient of the line as a fraction in its simplest form. nswer(a) [1] (b) Write down the equation of the line parallel to l which passes through the origin. nswer(b) [1] (c) Find the equation of the line parallel to l which passes through the point (3, 1). nswer(c) y = [2] ULES /02/O/N/07

65 º O 50º 123º D T The points,, and D lie on a circle centre O. ngle O = 90, angle OD = 50 and angle D = 123. The line DT is a tangent to the circle at D. Find (a) angle OD, nswer(a) ngle OD = [1] (b) angle TD, nswer(b) ngle TD = [1] (c) angle, nswer(c) ngle = [1] (d) reflex angle O. nswer(d) ngle O = [1] ULES /02/O/N/07

66 9 8 D y x E z O DE is a pentagon. circle, centre O, passes through the points,, D and E. ngle E = 36, angle = 78 and is parallel to D. (a) Find the values of x, y and z, giving a reason for each. [6] (b) Explain why ED is not parallel to. [1] (c) Find the value of angle EO. [1] (d) =. Find the value of angle. [1] ULES /04/O/N/08

67 8 18 y y = 2x + 4 y = mx + c 6 units x The line y = mx + c is parallel to the line y = 2x + 4. The distance is 6 units. Find the value of m and the value of c. nswer m = and c = [4] ULES /21/O/N/09

68 D O E F Points, and lie on a circle, centre O, with diameter. D, OE and F are parallel lines. ngle D = 68. alculate (a) angle O, nswer(a) ngle O = [2] (b) angle E. nswer(b) ngle E = [2] ULES /21/O/N/09

69 3 4 D O 50 O is the centre of the circle. D is the tangent to the circle at and D is the tangent to the circle at. O is a straight line. ngle O = 50. alculate (a) angle O, (b) angle DO. nswer(a) ngle O = [1] nswer(b) ngle DO = [1] 5 J 20 m 50 m G R JGR is a right-angled triangle. JR = 50m and JG = 20m. alculate angle JRG. nswer ngle JRG = [2] 6 Write (a) in standard form, nswer(a) [1] (b) correct to 2 significant figures. nswer(b) [1] ULES /22/O/N/10

70 E 140 D The pentagon has three angles which are each 140. The other two interior angles are equal. alculate the size of one of these angles. nswer [3] ULES /22/O/N/10

71 6 13 The diagram shows a circle of radius 5cm in a square of side 18cm. alculate the shaded area. nswer cm 2 [3] 14 Draw, accurately, the locus of all the points outside the triangle which are 3 centimetres away from the triangle. [3] ULES /22/O/N/10

72 4 9 8 cm Q P 10 cm 12 cm P and Q are straight lines. PQ is parallel to. P = 8 cm, PQ = 10 cm and = 12 cm. alculate the length of. nswer = cm [2] ULES /23/O/N/10

73 D O The points,, and D lie on the circumference of the circle, centre O. ngle D = 30, angle D = 50 and angle O = 86. (a) Give the reason why angle D = 50. nswer(a) [1] (b) Find (i) angle D, nswer(b)(i) ngle D = [1] (ii) angle D, nswer(b)(ii) ngle D = [1] (iii) angle OD. nswer(b)(iii) ngle OD = [2] ULES /23/O/N/10

74 10 6 (a) P 19.5 cm 16.5 cm 11 cm Q R The diagram shows a toy boat. = 16.5 cm, = 19.5 cm and PR = 11 cm. Triangles and PQR are similar. (i) alculate PQ. (ii) alculate. nswer(a)(i) PQ = cm [2] (iii) alculate angle. nswer(a)(ii) = cm [3] nswer(a)(iii) ngle = [2] ULES /41/O/N/10

75 (iv) The toy boat is mathematically similar to a real boat. The length of the real boat is 32 times the length of the toy boat. The fuel tank in the toy boat holds 0.02 litres of diesel. alculate how many litres of diesel the fuel tank of the real boat holds. 11 nswer(a)(iv) litres [2] (b) E 105 m F m 70 D G The diagram shows a field DEFG, in the shape of a quadrilateral, with a footpath along the diagonal DF. DF = 105 m and FG = 67 m. ngle EDF = 70U, angle EFD = 32U and angle DFG = 143U. (i) alculate DG. (ii) alculate EF. nswer(b)(i) DG = m [4] nswer(b)(ii) EF = m [4] ULES /41/O/N/10

76 12 7 (a) D x w y 62 O z,, and D are points on the circumference of a circle centre O. is a diameter. D = and angle D = 62U. Work out the values of w, x, y and z. Give a reason for each of your answers. w = because [2] x = because [2] y = because [2] z = because [2] ULES /41/O/N/10

77 3 2 R 4 km 7 km Q 4.5 km S P The diagram shows five straight roads. PQ = 4.5 km, QR = 4 km and PR = 7 km. ngle RPS = 40 and angle PSR = 85. (a) alculate angle PQR and show that it rounds to nswer(a) [4] (b) alculate the length of the road RS and show that it rounds to 4.52 km. nswer(b) [3] (c) alculate the area of the quadrilateral PQRS. [Use the value of for angle PQR and the value of 4.52 km for RS.] nswer(c) km 2 [5] ULES /43/O/N/10

78 t t In the pentagon the two angles labelled t are equal. alculate the value of t. nswer t =... [3] 0580/2 W00

79 EXTENDED MTHEMTIS LSSIFIEDS GEOMETRY ompiled & Edited y Dr. Eltayeb bdul Rhman First Edition 2011