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Geometry SMC QuestionsAll the following questions were from the last 10 questions (i.e. Q16-25) of Senior Maths Challenge papers from 2005-2012).48723551344295390017064135The diagram shows the ellipse whose equation is x2+y2-xy+x-4y=12. The curve cuts the y-axis at points A and C and cuts the x-axis at points B and D. What is the area of the inscribed quadrilateral ABCD?A 28 B 36 C 42 D 48 E 56The diagram shows a pattern found on a floor tile in the cathedral in Spoleto, Umbria. A circle of radius 1 surrounds four quarter circles, also of radius 1, which enclose a square. The pattern has four axes of symmetry. What is the side length of the square?A 12 B 2-2 C 13 D 12 E 2-1499491094678546374051905The diagram shows two squares, with sides of length 12, inclined at an angle 2α to one another. What is the value of x?A cosα B 1cosα C sinα D 1sinα E tanαIn trapezium PQRS, SR=PQ=25cm and SP is parallel to RQ. All four sides of PQRS are tangent to a circle with centre C. The area of the trapezium is 600cm2. What is the radius of the circle?A 7.5cm B 8cm C 9cm D 10cm E 12cm491871015138404080510-2540A semicircle of radius r is drawn with centre V and diameter UW. The line UW is then extended to the point X, such that UW and WX are of equal length. An arc of the circle with centre X and radius 4r is then drawn so that the line XY is a tangent to the semicircle at Z, as shown. What, in terms of r, is the area of the triangle YVW?A 4r29 B 2r23 C r2 D 4r23 E 2r2The top diagram on the right shows a shape that tiles the plane, as shown in the lower diagram. The tile has nine sides, six of which have length 1. It may be divided into three congruent quadrilaterals as shown. What is the area of the tile?A 1+232 B 433 C 6 D 3+434 E 332496443011963404636770-38100PQRS is a rectangle. The area of triangle QRT is 15 of the area of PQRS, and the area of triangle TSU is 18 of the area of PQRS. What fraction of the area of rectangle PQRS is the area of triangle QTU?A 2740 B 2140 C 12 D 1940 E 236048882301206500The diagram shows a small regular octagram (an eight-sided star) surrounded by eight squares (dark grey) and eight kites (light grey) to make a large regular octagram. Each square has area 1. What is the area of one of the light grey kites?A 2 B 2+1 C 218 D 42-3 E 11443929301468755In the diagram ∠ABE=10°; ∠EBC=70°; ∠ACD=50°; ∠DCB=20°; ∠DEF=α.Which of the following is equal to tan α?A tan10°tan20°tan50° B tan10°tan20°tan70° C tan10°tan50°tan70° D tan20°tan50°tan70° E tan10°tan70°tan50° 48882301332865Three circles and the lines PQ and QR touch as shown. The distance between the centres of the smallest and the biggest circles is 16 times the radius of the smallest circle. What is the size of ∠PQR?A 45° B 60° C 75° D 90° E 135°A solid sculpture consists of a 4×4×4 cube with a 3×3×3 cube sticking out, as shown. Three vertices of the smaller cube lie on edges of the larger cube, the same distance along each.What is the total volume of the sculpture?A 79 B 81 C 82 D 84 E 854964430847725PQRS is a quadrilateral inscribed in a circle of which PR is a diameter. The lengths of PQ, QR and RS are 60, 25 and 52 respectively. What is the length of SP?A 2123 B 281113 C 33 D 36 E 39A solid cube of side 2cm is cut into two triangular prisms by a plane passing through four vertices, as shown. What is the total surface area of these two prisms?A 83+2 B 28+2 C 83+22 D 163+2 E 824095750-91440The diagrams show two different shaded rhombuses, each inside a square with sides of length 6. Each rhombus is formed by joining vertices of the square to midpoints of the sides of the square. What is the difference between the shaded areas?A 4 B 3 C 2 D 1 E 05149850-18097552539901196340The diagram shows a regular hexagon, with sides of length 1, inside a square. Two vertices of the hexagon lie on a diagonal of the square and the other four lie on the edges.What is the area of the square?A 2+3 B 4 C 3+2 D 1+332 E 72The diagram shows two different semicircles inside a square with sides of length 2. The common centre of the semicircles lies on a diagonal of the square.What is the total shaded area?A π B 6π3-22 C π2 D 3π2-2 E 8π22-3Three spheres of radius 1 are placed on a horizontal table and inside a vertical hollow cylinder of height 2 units which is just large enough to surround them. What fraction of the internal volume of the cylinder is occupied by the spheres?A 27+43 B 22+3 C 13 D 32+3 E 67+43A solid cube is divided into two pieces by a single rectangular cut. As a result, the total surface area increases by a fraction fof the surface area of the original cube. What is the greatest possible value of f?A 13 B 34 C 23 D 12 E 135040630824865A point P is chosen at random inside a square QRST. What is the probability that ∠RPQ is acute?A 34 B 2-1 C 12 D π4 E 1-π8A frustum is the solid obtained by slicing a right-circular cone perpendicular to its axis and removing the small cone above the slice. This leaves a shape with two circular faces and a curved surface. The original cone has base radius 6cm and height 8cm, and the curved surface area of the frustum is equal to the area of the two circles. What is the height of the frustum?A 3cm B 4cm C 5cm D 6cm E 7cmM and N are the midpoints of sides GH and FG respectively, of parallelogram EFGH. The area of triangle ENM is 12cm2. What is the area of the parallelogram EFGH?A 20cm2 B 24cm2 C 32cm2 D 48cm2 E more information is required504063011741154972050635A figure in the shape of a cross is made from five 1×1 squares, as shown. The cross is inscribed in a large square whose sides are parallel to the dashed square, formed by four of the vertices of the cross. What is the area of the large outer square?A 9 B 495 C 10 D 818 E 323The shaded square of the lattice shown has area 1. What is the area of the circle through the points X, Y and Z?A 9π2 B 8π C 25π2 D 25π E 50π514731011709404857750-2540The diagram shows four semicircles symmetrically placed between two circles. The shaded circle has area 4 and each semicircle has area 18. What is the area of the outer circle?A 722 B 100 C 98 D 96 E 32333261301040765A pentagon is made by attaching an equilateral triangle to a square with the same edge length. Four such pentagons are placed into the rectangle, as shown.What is the ratio of the length of the rectangle to its width?A 3:1 B 2:1 C 2:1 D 3:2 E 4:34617720979805The two triangles have equal areas and the four marked lengths are equal. What is the value of x?A 30 B 45 C 60 D 75 E more information neededThe largest circle which it is possible to draw inside triangle PQR touches the triangle at S, T, U, as shown in the diagram. The size of ∠STU=55°. What is the size of ∠PQR?A 55° B 60° C 65° D 70° E 75°A triangle is cut from the corner of a rectangle. The resulting pentagon has sides of length 8, 10, 13, 15 and 20 units, though not necessarily in that order. What is the area of the pentagon?A 252.5 B 260 C 270 D 275.5 E 282.54050030-144780In triangle PQR, S and T are the midpoints of PR and PQ respectively; QS is perpendicular to RT; QS=8; RT=12. What is the area of triangle PQR?A 24 B 32 C 48 D 64 E 96The sum of the lengths of the 12 edges of a cuboid is x cm. The distance from one corner of the cuboid to the furthest corner is y cm. What, in cm2, is the total surface area of the cuboid?A x2-2y22 B x2+y2 C x2-4y24 D xy2 E x2-16y216A paperweight is made from a glass cube of side 2 units by first shearing off the eight tetrahedral corners which touch at the midpoints of the edges of the cube. The remaining inner core of the cube is discarded and replaced by a sphere. The eight corner pieces are now stuck onto the sphere so that they have the same positions relative to each other as they did originally. What is the diameter of the sphere?A 8-1 B 8+1 C 136+3 D 433 E 23If α<β, how many different values are there among the following expressions?sinαsinβ sinαcosβ cosαsinβ cosαcosβA 1 B 2 C 3 D 4 E It depends on the value of α48196501141730A trapezium is bounded by four lines, the equations of which are x=0, x=4, 4y=3x+8 and y=k, where k<2.For which values of k is he numerical value of the perimeter of the trapezium equal to the numerical value of the area of the trapezium?A 32 B 1 C 12 D -12 E -148577501664335A toy pool table is 6 feet long and 3 feet wide. It has pockets at each of the four corners P, Q, R and S. When a ball hits a side of the table, it bounces off the side at the same angle as it hit that side. A ball, initially 1 foot to the left of pocket P, is hit from the side SP twards the side PQ as shown.How many feet from P does the ball hit side PQ if it lands in pocket S after two bounces?A 1 B 67 C 34 D 23 E 35 In the diagram, the circle and the two semicircles have radius 1. What is the perimeter of the square?A 6+42 B 2+42+23 C 32+43 D 4+22+26 E 124861560-36830A solid red plastic cube, volume 1cm3, is painted white on its outside. The cube is cut by a plane passing through the mid-points of various edges, as shown.What, in cm2, is the total red area exposed by the cut?A 332 B 2 C 925 D 3 E 33+24 4861560-2540Eight identical octagons are placed edge to edge in a ring in such a way that a symmetrical star shape is formed by the interior edges. If each octagon has sides of length 1, what is the area of the star?A 5+102 B 82 C 9+42 D 16-42 E 8+42A cube exactly fits inside a sphere and another sphere exactly fits inside this cube. What is the ratio of the volume of the smaller sphere to the volume of the larger sphere?A 1:33 B 1:4 C 1:3 D 2:3 E 1:24861560991235It takes two weeks to clean the 3312 panes of glass in the 6000m2 glass roof of the British Museum, a task performance once every two years. Assuming that all the panes are equilateral triangles of the same size, roughly how long is the side of each pane?A 50cm B 1m C 2m D 3m E 4mThe diagram shows four touching circles, each of which also touches the sides of an equilateral triangle with sides of length 3.What is the area of the shaded region?A 11π12 B π C 4+3π6 D 3+3π4 E 37π36 ................
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