Mahajana Mathematics Competition Sample Grade 11 & 12 _____________________________________________________________________ Part A (10 Questions, Each Question Is Worth 2 Marks) 1. 2. In the diagram below, the five smaller rectangles are identical in size and shape. The ratio of DC: AD is (A) 3:2 (B) 3:5 (C) 5:3 (D) 2:3 3. In 2011 Mahajanan Contest Grade 9+10, 30 students wrote a test. If 20 students scored 80, 8 students scored 90, and 2 students scored 100, then the average on this Grade 9+10 test was (A) 90 (B) 84 (C) 82 (D) 86 4. If the expression 256 x 286 x 557 was evaluated, it would end with a string of consecutive zeros. The numbers of consecutive zeros in this string when evaluated are: (A) 10 (B) 12 (C) 8 (D) 14 5. In the diagram, OP = 3, PQ = 4 and OQ = 5. The slope of OP is? (A) 3/4 (B) 4/3 (C) 5/3 (D) 3/5 Mahajana Mathematics Competition Sample Grade 11 & 12 _____________________________________________________________________ 6. If a + b = 12, b + c = 15 and c + d = 19. Then the value of a + d is? (A) 17 (B) 16 (C) 18 (D) 14 7. A square floor is tiled, as partially shown, with a large number of regular hexagonal tiles. The tiles are coloured blue or white. Each blue tile is surrounded by 6 white tiles and each white tile is surrounded by 3 white and 3 blue tiles. Ignoring part tiles, the ratio of the number of blue tiles to the number of white tiles is closest to (A) 1:6 (B) 2:3 (C) 1:2 (D) 1:4 8. The equation of a circle is x2 + y2 = 16. The length of the longest possible cord passing through the points (1, -2) is? (A) 2√7 (B) 2π (C) 8 (D) 16 9. The graph y = 10 (x + 1) (x - 3) intersects the x-axis at two points A and B. The length of AB is? (A) 2 (B) 6 (C) 4 (D) 8 Mahajana Mathematics Competition Sample Grade 11 & 12 _____________________________________________________________________ 10. A rectangular wooden prism is made up of three pieces, each consisting of four cubes of wood glued together. Which of the pieces below has the same shape as the black piece? Part B (10 Questions, Each Question Is Worth 4 Marks) 11. A cube measures 20 cm ×20 cm ×20 cm. Three cuts are made parallel to the faces of the cube as shown creating eight separate solids which are then separated. What is the increase in the total surface area? (A) 300 cm2 (B) 2400 cm2 (C) 1200 cm2 (D) 600 cm2 12. The median of a set of five positive integers is the middle integer when the set is arranged in increasing order. A set of 5 positive integers has all of the following properties: • its median is 9, • its average (mean) is 10 and • the only integer in the list that occurs more than once is 8 The largest possible integer the above set could appear in the list (A) 15 (B) 16 (C) 17 (D) 24 Mahajana Mathematics Competition Sample Grade 11 & 12 _____________________________________________________________________ 13. A regular pentagon is placed in the line PQ as shown and then reflected on line PQ and rotated clockwise 216° about the centre of the pentagon, Its new position is 14. When x2 – kx2- 10kx + 25 is divided by (x - 2), the remainder is 5. The value of k is? (A) 25 (B) 1 (C) (D) 15. In the square ABCD, two pairs of identical isosceles triangles are cut off, leaving rectangle PQRS. The total area cut off is 400 m2. The length of PR, in metres, is A) 40√2 B) 20 C) 20√3 D) 40√3 Mahajana Mathematics Competition Sample Grade 11 & 12 _____________________________________________________________________ 16. If W= 2129 x381 x 5131, Z=2125 x 382 x 5132 X=2127 x381x5131, Y=2126x382 x5131, Then the order from the smallest to largest is (A) w, x, y, z (B) x, w, y, z (C) x, y, z, w (D) z, y, x, w 17. In the diagram below, all triangles are equilateral. If AB = 32, then the total area of all the white triangles is (A) 37√ 3 (B) 108√ 3 (C) 148√ 3 (D) 74/3 18. In the triangular pattern of numbers shown on the right side, every row begins with a 1 and ends with a 2. Each of the numbers, not on the end of a row, is the sum of the two numbers located immediately above and to the right, and immediately above and to the left. For example, in the fourth row the 9 is the sum of the 4 and the 5 in the third row. If this pattern continues, the sum of all of the numbers in the 14th row is A) 12,288 B) 24,576 C) 12 294 D) 18,984 Mahajana Mathematics Competition Sample Grade 11 & 12 _____________________________________________________________________ 19. Two 3-digit numbers, p, q, r and x, y, z, have the following property: None of the value of p, q, r, x, y and z is zero. The sum of p+q+r+x+y+z is (A) 10 (B) 19 (C) 21 (D) 28 20. In the figure below, the four circles have a common centre, and have radii of 1 cm, 2 cm, 3 cm, and 4 cm. The ratio of the area of the shaded regions to the area of the largest circle with radius 4 cm is A) 5: 8 (B) 3: 8 (C) 7: 16 (D) 1: 2 Part C (5 Questions, Each Question Is Worth 6 Marks) 21. In the diagram, the tangents to the two circles intersect at 90° as shown. If the radius of the smaller circle is 2, and the radius of the larger circle is 5, what is the distance between the centers of the two circles? (A) 5 (B) 72 (C) 7√2 (D) 7 Mahajana Mathematics Competition Sample Grade 11 & 12 _____________________________________________________________________ 22. Sunday morning Banu and Shirley have to move 10 large bags and 16 small bags. Look at the chart below which indicates the time that each person takes to move each type of box. BANU 6min SHIRLY 5min LARGE BAG 2min 3min SMALL BAG Banu and Shirley start moving the boxes at 10:00 a.m. The earliest time at which they can be finished moving all of the boxes is (A) 10:41 AM. (B) 10:42 AM. (C) 10:43 AM. (D) 10:44 AM. 23. There are five circles in the diagram below and they are tangent at points B, C, D and E. An ant wants to move from point A to point F entirely on the circumferences of the circles so that no portion of any circumference is covered more than once. The number of possible paths is (A) 2 (B) 8 (C) 16 (D) 32 Mahajana Mathematics Competition Sample Grade 11 & 12 _____________________________________________________________________ 24. A square spiral of numbers is created, as shown, starting with the number 1. If the pattern of the square spiral continues, in what configuration will the numbers 483, 484 and 485 appear? (A) 483 484 485 (B) 483 400399 (C) (D) 25. In the diagram, each of the three identical circles touches the other two. The circumference of each circle is 72. What is the perimeter of the shaded region? (A) 18 (B) 6 (C) 36 (D) 12 Mahajana Mathematics Competition Sample Grade 11 & 12 _____________________________________________________________________ BONUS QUESTION: (10 Marks ) A pattern of regular pentagon family with a fixed top vertex T is shown below in Figure 1, Figure 2 and Figure 3. Figure 1 is a regular pentagon with side length 1 cm. Figure 2 is a regular pentagon of side length 2cm drawn around Figure 1 sothat the two shapes share the fixed top vertex, T, and the sides on either side of T overlap. Figure 3 is a regular pentagon of side length 3cm drawn around Figure 2 so that the two shapes share the fixed top vertex, T, and the sides on either side of T overlap. The tracing length along the sides of Figure 1 is 5cm.The tracing length along the sides of Figure 2 is 13cm. The tracing length along the sides of Figure 3 is 24cm.The pattern continues so that for each n > 1, Figure n is a regular pentagon of side length n cm drawn around the previous Figure so that the two shapes share the fixed top vertex, T, and the sides on either side of T overlap. The tracing length of each Figure is the sum of the lengths of all of the line segments in the respective Figure. a) Determine the tracing length of Figure 4. b) Determine the tracing length of Figure 100 Good Luck!