Name: Shape, Space and Measure Prep for Paper 2 Including Pythagoras Trigonometry: SOHCAHTOA Sine Rule Cosine Rule Area using 1-2 ab sin C Transforming Trig Graphs 3D Pythag-Trig Plans and Elevations Area and circumference of circles Circle Theorems Congruency and Similarity Surface Area Spheres, Cones and Pyramids Transformations: Translations Enlargements Density, Mass, Volume Bearings Constructions Loci Vectors
Q1 - Pythagoras. ABCD is a trapezium. AD = 10 cm AB = 9 cm DC = 3 cm Angle ABC = angle BCD = 90 Calculate the length of AC. Give your answer correct to 3 significant figures.... (Total for Question is 5 marks)
Q2 Pythagoras and SOHCAHTOA. The diagram shows a quadrilateral ABCD. drawn Diagram NOT accurately AB = 16 cm. AD = 12 cm. Angle BCD = 40. Angle ADB = angle CBD = 90. Calculate the length of CD. Give your answer correct to 3 significant figures....................... cm (Total for Question is 5 marks) Q3 SOHCAHTOA missing angle.
LMN is a right-angled triangle. MN = 9.6 cm. LM = 6.4 cm. Calculate the size of the angle marked x. Give your answer correct to 1 decimal place. Diagram NOT accurately drawn Q4 Cosine Rule....................... (Total for Question is 3 marks) Calculate the length of PR. Give your answer correct to 3 significant figures...................... cm (Total for Question is 3 marks)
Q5 Sine rule. ABCD is a parallelogram. Diagram NOT accurately drawn AC = 9 cm DC = 11 cm Angle DAC = 100 Calculate the area of the parallelogram. Give your answer correct to 3 significant figures....cm 2 (Total for Question is 5 marks)
Q6 ½ ab sin C & cosine rule. ABC is a triangle. (a) Work out the area of triangle ABC. Give your answer correct to 3 significant figures. (b) Work out the length of the side AB. Give your answer correct to 3 significant figures....................... cm 2 (2)... (3) (Total for Question is 5 marks)
Q7 Trig graphs. A sketch of the curve y = sin x for 0 < x < 360 is shown below. y 2 1 O 90 180 270 360 x 1 2 (a) Using the sketch above, or otherwise, find the equation of each of the following two curves. (i) y 2 1 O 90 180 270 360 x 1 2 (i) Equation y =...
(ii) y 2 1 O 90 180 270 360 x 1 2 (ii) Equation y =... (2) (b) Describe fully the sequence of two transformations that maps the graph of y = sin x onto the graph of y = 3 sin 2x............ (3) (Total 5 marks)
Q8 Trig graphs. Diagram 1 is a sketch of part of the graph of y = sin x. (a) Write down the coordinates of (i) P, (, ) (ii) Q. (, ) (2) Diagram 2 is a sketch of part of the graph of y = 3 cos 2x. (b) Write down the coordinates of (i) R, (, ) (ii) S (, ) (2) (Total 4 marks)
Q9 3D Pythag/Trig. The diagram represents a cuboid ABCDEFGH. AB = 5 cm. BC = 7 cm. AE = 3 cm. (a) Calculate the length of AG. Give your answer correct to 3 significant figures. E 3 cm A 5 cm F B H D 7 cm G C Diagram NOT accurately drawn... cm (2) (b) Calculate the size of the angle between AG and the face ABCD. Give your answer correct to 1 decimal place. (2) (Total 4 marks)
Q10 Plans and Elevations. The diagram shows a solid prism. On the centimetre square grid, draw the side elevation of the solid prism from the direction shown by the arrow. (Total for Question is 2 marks)
Q11 Circumference of a Circle. * Saphia is organising a conference. People at the conference will sit at circular tables. Each table has a diameter of 140 cm. Each person needs 60 cm around the circumference of the table. There are 12 of these tables in the conference room. A total of 90 people will be at the conference. Are there enough tables in the conference room? (Total for question = 4 marks)
Q12 Area of circles. *Mr Weaver's garden is in the shape of a rectangle. In the garden there is a patio in the shape of a rectangle and two ponds in the shape of circles with diameter 3.8 m. The rest of the garden is grass. Diagram NOT accurately drawn Mr Weaver is going to spread fertiliser over all the grass. One box of fertiliser will cover 25 m 2 of grass. How many boxes of fertiliser does Mr Weaver need? You must show your working. (Total for Question is 5 marks)
Q13 Sector Area. OAB is a sector of a circle, centre O. The radius of the circle is 15 cm. The angle of the sector is 30. Calculate the area of sector OAB. Give your answer correct to 3 significant figures...................... cm 2 (Total for Question is 2 marks)
Q14 Circle Theorems. * S and T are points on the circumference of a circle, centre O. PT is a tangent to the circle. SOP is a straight line. Angle OPT = 32 Work out the size of the angle marked x. Give reasons for your answer.... (Total for Question is 5 marks)
Q15 - Circle Theorems. A, B, C and D are points on the circumference of a circle, centre O. AC is a diameter of the circle. AC and BD intersect at E. Angle CAB = 25 Angle DEC = 100 Work out the size of angle DAC. You must show all your working.... (Total for question = 4 marks)
*Q16 Circle Theorems/Congruent Triangles. AOC and BOD are diameters of a circle, centre O. Prove that triangle ABD and triangle DCA are congruent. (Total for Question is 3 marks)
Q17 Similar shapes lengths. Diagram NOT accurately drawn Quadrilaterals ABCD and LMNP are mathematically similar. Angle A = angle L Angle B = angle M Angle C = angle N Angle D = angle P (a) Work out the length of LP. (b) Work out the length of BC....cm (2)...cm (2) (Total for Question is 4 marks)
Q18 Similar shapes volume. A frustrum is made by removing a small cone from a similar large cone. The height of the small cone is 20 cm. The height of the large cone is 40 cm. The diameter of the base of the large cone is 30 cm. Work out the volume of the frustrum. Give your answer correct to 3 significant figures.........................cm 3 (Total for Question is 4 marks)
Q19 Surface area. Here is a cuboid. The cuboid is 6 cm by 1.5 cm by 1.5 cm. Work out the total surface area of the cuboid....................... cm 2 (Total for Question is 3 marks)
Q20 - Volume of Pyramid. The diagram shows a pyramid. BCDE is a square with sides of length 10 cm. The other faces of the pyramid are equilateral triangles with sides of length 10 cm. (a) Calculate the volume of the pyramid. Give your answer correct to 3 significant figures. (b) Find the size of angle DAB......................... cm 3 (4)... (2) (Total for Question is 6 marks)
Q21 Hemisphere and cone volume. The diagram shows a solid made from a hemisphere and a cone. Diagram NOT accurately drawn The radius of the hemisphere is 4 cm. The radius of the base of the cone is 4 cm. Calculate the volume of the solid. Give your answer correct to 3 significant figures....cm 3 (Total for Question is 3 marks)
Q22 Hemisphere and bounds. A clay bowl is in the shape of a hollow hemisphere. 8.2 cm 7.7 cm Diagram NOT accurately drawn The external radius of the bowl is 8.2 cm. The internal radius of the bowl is 7.7 cm. Both measurements are correct to the nearest 0.1 cm. The upper bound for the volume of clay is kπ cm 3. Find the exact value of k. k =.. (Total 4 marks)
Q23 - Transformations. Describe fully the single transformation that maps triangle P onto triangle Q....... (Total for Question is 2 marks)
Q24 - Transformations. Enlarge triangle B by scale factor 3, centre (1, 2). (Total for Question is 3 marks) Q25 - DMV. 3.8 cm 2.5 cm Diagram NOT accurately drawn An ice hockey puck is in the shape of a cylinder with a radius of 3.8 cm, and a thickness of 2.5 cm. It is made out of rubber with a density of 1.5 grams per cm 3. Work out the mass of the ice hockey puck. Give your answer correct to 3 significant figures.... grams (Total 4 marks)
Q26 - Bearings. The diagram shows the positions of two villages, Beckhampton (B) and West Kennett (W). Scale: 4 cm represents 1 km. (a) Work out the real distance, in km, of Beckhampton from West Kennett.... The village, Avebury (A), is on a bearing of 038 from Beckhampton. On the diagram, A is 6 cm from B. (b) On the diagram, mark A with a cross ( ). Label the cross A. (2) (2) (Total for Question is 4 marks) Q27 - Bearings. Work out the bearing of (i) B from P,... (i) P from A, N 63 138 P A Diagram NOT accurately drawn... B (Total 3 marks)
Q28 - Bearings. N Diagram NOT accurately drawn N Hospital Cinema 72 Art gallery The diagram shows the position of each of three buildings in a town. The bearing of the Hospital from the Art gallery is 072. The Cinema is due East of the Hospital. The distance from the Hospital to the Art gallery is equal to the distance from the Hospital to the Cinema. Work out the bearing of the Cinema from the Art gallery. (Total 3 marks)
Q29 SAS accurate drawing. Make an accurate drawing of this triangle. The line AB has been drawn for you. (Total for Question is 2 marks) Q30 SSS accurate drawing. Here is a sketch of a triangle Diagram NOT accurately drawn 3.6 cm 6 cm 7.5 cm
In the space below, use ruler and compasses to construct the triangle accurately. You must show all construction lines. (Total 3 marks) Q31 Angle bisector. A B C Use ruler and compasses to construct the bisector of angle ABC. You must show all construction lines. (Total 2 marks)
Q32 Construct given angle. In the space below, use ruler and compasses to construct an angle of size 30. You must show all construction lines. (Total 3 marks) Q33 Perpendicular bisector. A B Use ruler and compass to construct the perpendicular bisector of the line segment AB. You must show all construction lines. (Total 2 marks)
Q34 - Perpendicular bisector through point on line. Use the ruler and compasses to construct the perpendicular to the line segment AB that passes through the point P. You must show all construction lines. B P A (Total 2 marks) Q35 Perpendicular from external point. A B C On the grid, draw a line from the point C perpendicular to the line AB. (Total 1 marks) Q36. The diagram represents a triangular garden ABC. The scale of the diagram is 1 cm represents 1 m. A tree is to be planted in the garden so that it is nearer to AB than to AC, within 5 m of point A. On the diagram, shade the region where the tree may be planted.
B A C (Total 3 marks) Q37. A B D C ABCD is a rectangle. Shade the set of points inside the rectangle which are both and more than 4 centimetres from the point A more than 1 centimetre from the line DC. (Total 4 marks)
Q38 - Vectors. A B 6a Diagram NOT accurately drawn P O OABC is a parallelogram. 6c C P is the point on AC such that AP = 2 3 AC. OA = 6a. OC = 6c. (a) Find the vector OP. Give your answer in terms of a and c.... (3) The midpoint of CB is M. (b) Prove that OPM is a straight line. (2) (Total 5 marks)
Q39 - Vectors. Diagram NOT accurately drawn OAB is a triangle. = a = b (a) Find in terms of a and b.... (1) P is the point on AB such that AP : PB = 3 : 1 (b) Find in terms of a and b.give your answer in its simplest form.... (3) (Total for Question is 4 marks)
Mark Scheme Shape, Space and Measure Q1. Q2.
Q3.
Q4. Q5.
Q6. 7. (a) (i) y = 1 + sin x 2 B1 for y = 1 + sin x (ii) y = 2sin x B1 for y = 2sin x SC both (i) f(x) + 1, (ii) 2f(x) B1 (b) Stretch parallel to y-axis scale factor 3 3 1 Stretch parallel to x-axis scale factor 2 M1 for stretch A1 for Stretch parallel to y-axis scale factor 3 oe 1 A1 for Stretch parallel to x-axis scale factor 2 oe 1 SC if M0 award BI for sf 3 vertically and sf horizon. 2 [5] 8. (a) (i) (90, 1) 2 B1 cao could be indicated on diagram (ii) (180, 0) B1 cao could be indicated on diagram (b) (i) (45, 0) 2 B1 cao could be indicated on diagram (ii) (90, 3) B1 cao could be indicated on diagram [4] 9. (a) 9.11 2 3 2 + 5 2 + 7 2 = 83 M1 for correct use of 3D Pythagoras formula or 2 correct applications of the 2D formula A1 for 9.11 to 9.12
(b) 19.2 2 Tan GAC = 3 (5 2 + 7 2 ) M1 correct trig expression for angle GAC A1 for 19.2 to 19.3 [4] Q10. Q11. Q12. Q13.
Q14.
Q15. Q16. Q17.
Q18. Q19.
Q20.
Q21. 22. 75.879 1 4 8.25 3 1 4 7.65 3 2 3 2 3 = (374.34375 98.46475) = 75.879 4 B1 for 8.25 or 7.65 seen M1 for expression using r = 8.25 minus same expression using r = 7.65 1 4 3 1 4 3 M1 for R r used 2 3 2 3 A1 cao [4] Q23. Q24.
25. 170 4 Q26. Vol = π 3.8 2 2.5 = π 14.44 2.5 = 45.36 2.5 = 113.411 Mass = 113 1.5 = 170.1165 M1 for π r 2 2.5 where r = is 3.8 or 7.6 A1 if r = 3.8 M1 for 113 1.5 A1 for 169.5 170.3 cao [4] 27. (i) 222 2 360 138 = 222 M1 for 360 138 63 or 159 seen or for 180 138 or 42 seen A1 cao (ii) 243 1 360 (180 63) B1cao for 243 [3] 28. 081 or 81 3 H C A y x = 72 180 162 y 9 2 72 + 9 = 81 M1 for (AHC=) 90 + 72 (=162) accept x marked as 72 and CHS as 90 or symbol ( y )180 "162" M1 dep for (= 9) 2 A1 cao
ALTn Draws line from A parallel to HC M1 for z = w and y + z = 90 72 (= 18) "18" M1 for y (or z) = 2 A1 cao [3] Q29. 30. 3 B1 for base 7.5 cm (± 0.1 cm) M1 for 1 correct arc (± 0.1 cm) A1 for 3 rd vertex within tolerance [3] 31. Bisector 2 M1 pair of relevant arcs on line segments and corresponding relevant pair of intersecting arcs within guidelines A1 bisector line within guidelines [2] 32. Correct construction 3 B1 for 60 construction (with construction arcs seen) M1 for angle bisector construction (intersecting arcs) A1 for 30 within tolerance [3] 33. Correct construction 2 M1 for relevent intersecting arcs aligned vertically A1 for straight line within guidelines [2] 34. Perpendicular from P to intersecting arcs (within tramlines); perpendicular at least 2 cm long 2 M1 relevant pair of arcs crossing within tramlines
A1 SC M1A0 for full construction of a line perpendicular to AB not through P [2] 35. (a) Draws perp. 1 B1 for correctly drawing perp must touch line or cut line AB ± 2mm (b) Sketches a cylinder 1 B1 for sketching cylinder [2] 36. See overlay 3 Bisector of BAC Arc around A Region B3 cao (B2 for either two correct boundaries, no shading/ wrong shading or one correct boundary, one incorrect boundary with valid shading) (B1 for either two incorrect boundaries but one drawn from A and one intersection, with valid shading or one correct boundary) Ignore shading outside the triangle [3] 37. overlay 4 M1 quarter circle drawn centre A inside rectangle (ignore lines outside the rectangle) A1 radius 4 cm ±2mm B1 line drawn 1 cm ±2mm from DC. B1 ft (dep on two loci attempts drawn) region shaded [4] 38. (a) 2a + 4c 3 OP OA AP 2 = OA (6c 6a) 3 = 6a + 4c 4a M1 for OP OA AP or any correct vector journey involving OP 2 1 M1 for AP (6c 6a) oe or CP = ( 6c + 6a) oe or reverse 3 3 vectors A1 for 2a + 4c oe (accept unsimplified)
(b) OM 1. 5OP so OPM is a straight line 2 Q39. Eg OM OC CM = 6c + 3a OM 1. 5OP 1 B1 for OM = 6c + (6a) or PM = 2c + a unsimplified or reverse 2 vectors B1 for a fully correct proof. [5]