The sign │x│ means the absolute value of x (the distance between the number and zero onthe number line). Absolute values are always positive. │-25 │ = 25 8. 230 + 230 + 230 + 230 = A. 8120 B. 830 C. 232 D. 230 E. 226 Correct Answer: C All four terms are identical therefore we have 4 (230). But 4 = 2 2 , and so we can write 2 2 X 2 30 Which is equivalent to 2 32 Jo's collection contains US, Indian and British stamps. If the ratio of US to Indian stamps is 5 to 2 and the ratio of Indian to British stamps is 5 to 1, what is the ratio of US to British stamps? A. 5 : 1 B. 10 : 5 C. 15 : 2 D. 20 : 2 E. 25 : 2 Correct Answer: E Explanation: Indian stamps are common to both ratios. Multiply both ratios by factors such that the Indian stamps are represented by the same number. US : Indian = 5 : 2, and Indian : British = 5 : 1. Multiply the first by 5, and the second by 2. Now US : Indian = 25 : 10, and Indian : British = 10 : 2 Hence the two ratios can be combined and US : British = 25 : 2 A 3 by 4 rectangle is inscribed in circle. What is the circumference of the circle? A. 2.5π B. 3π C. 5π D. 4π E. 10π Correct Answer: C Explanation: Draw the diagram. The diagonal of the rectangle is the diameter of the circle. The diagonal is the hypotenuse of a 3,4,5 triangle, and is therefore, 5. Circumference = π.diameter = 5π n and p are integers greater than 1 5n is the square of a number 75np is the cube of a number. The smallest value for n + p is A. 14 B. 18 C. 20 D. 30 E. 50 Correct Answer: A Explanation: The smallest value for n such that 5n is a square is 5. 75np can now be written as 75 x 5 x p. This gives prime factors.... 3 x 5 x 5 x 5 x p To make the expression a perfect cube, p will have to have factors 3 x 3 , and hence p =9 n + p = 5 + 9 = 14 9- In a certain game of 50 questions, the final score is calculated by subtracting twice the number of wrong answers from the total number of correct answers. If a player attempted all questions and received a final score of 35, how many wrong answers did he give? Correct Answer: 5 Explanation: Let the number of wrong answers = x Right answers = 50 - x Subtracting twice wrong from right will give 35 (50 - x) - 2x = 35; 50 - 3x = 35 15 = 3x; 5 = x 10. What positive value for k would make the following the equations of a pair of parallel lines on the same coordinate axes? y = kx – 2 and ky = 9x – 7 Correct Answer: 3 Explanation: First rearrange the second equation to fit the form y = mx + c, where m = slope We get y = (9/k) x - 7/k Parallel lines have the same slope. From equation 1, slope = k From equation 2, slope = 9/k . Equating the two slopes we get k = 9/k; k² = 9; k = 3 Areas square = a 2 rectangle = ab parallelogram = bh trapezoid = h/2 (b 1 + b 2 ) circle = pi r 2 ellipse = pi r 1 r 2 triangle = (1/2) b h equilateral triangle = (1/4) (3) a 2 triangle given SAS = (1/2) a b sin C triangle given a,b,c = [s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula) regular polygon = (1/2) n sin(360°/n) S 2 when n = # of sides and S = length from center to a corner Volumes cube = a 3 rectangular prism = a b c irregular prism = b h cylinder = b h = r 2 h pyramid = (1/3) b h cone = (1/3) b h = 1/3 r 2 h sphere = (4/3) r 3 ellipsoid = (4/3) pi r 1 r 2 r 3 Surface Areas cube = 6 a 2 prism: (lateral area) = perimeter(b) L (total area) = perimeter(b) L + 2b sphere = 4 r 2 One gallon of fuel mixture contains antifreeze in the ratio of 5 parts fuel to one part antifreeze. To this is added half a gallon of mixture which is one third antifreeze and two thirds fuel. What is the ratio of fuel to antifreeze in the final mixture? (Grid your answer as a fraction: fuel/antifreeze) Correct Answer: 7/2 Explanation: The antifreeze in the initial fuel mixture is 1/6 of one gallon. The added mixture contains 1/3 of half a gallon = 1/6 of a gallon The final mixture contains 1/6 + 1/6 in 1.5 gallon 2/6 antifreeze in total 9/6 gallons Fuel in the mixture is 9/6 - 2/6 = 7/6 Ratio fuel to antifreeze = 7/6 : 2/6 = 7 : 2 Grid in as 7/2 3. Of 60 students in a class 2/3 are girls, and 2/5 of the class are taking music lessons. What is the maximum number of girls that are not taking music lessons? Correct Answer: 36 Explanation: If 2/3 of the class of 60 are girls (=40), 1/3 must be boys (=20) 2/5 of the class (=24) are taking music. The maximum number of girls not taking music will be when all the boys take music. If all 20 boys take music, then only 4 girls need to take it, so 36 girls do not take music lessons. 1. The distance from town A to town B is five miles. C is six miles from B. Which of the following could be the distance from A to C? I 11 II 1 III 7 A. I only B. II only C. I and II only D. II and III only E. I, II, or III. Correct Answer: E Explanation: Do not assume that AB and C are on a straight line. Make a diagram with A and B marked 5 miles apart. Draw a circle centered on B, with radius 6. C could be anywhere on this circle. The minimum distance will be 1, and maximum 11, but anywhere in between is possible. 4. A. 1/5 B. 6/5 C. 6³ D. 64 / 5 E. 64 Correct Answer: E Explanation: 6 5 = 6 4 x 6 (6 4 x 6) - 6 4 = 6 4 (6 - 1) = 6 4 x 5 Now, dividing by 5 will give us 6 4 6. If f(x) = x² – 3, where x is an integer, which of the following could be a value of f(x)? I 6 II 0 III -6 A. I only B. I and II only C. II and III only D. I and III only E. I, II and III Correct Answer: A Explanation: Choice I is correct because f(x) = 6 when x=3 Choice II is incorrect because to make f(x) = 0, x² would have to be 3. But 3 is not the square of an integer. Choice III is incorrect because to make f(x) = -6, x² would have to be –3 but squares cannot be negative. (The minimum value for x² is zero; hence, the minimum value for f(x) = -3) 8. In the above correctly worked addition sum, A,B,C and D represent different digits, and all the digits in the sum are different. What is the sum of A,B,C and D? A. 23 B. 22 C. 18 D. 16 E. 14 Correct Answer: B Explanation: First you must realize that the sum of two 2-digit numbers cannot be more that 198 (99 + 99). Therefore in the given problem D must be 1. Now use trial and error to satisfy the sum 5A + BC = 143 A + C must give 3 in the units place, but neither can be 1 since all the digits have to be different. Therefore A + C = 13. With one to carry over into the tens column, 1 + 5 + B = 14, and B = 8. A + C + B + D = 13 + 8 + 1 = 22 10. Six years ago Anita was P times as old as Ben was. If Anita is now 17 years old, how old is Ben now in terms of P ? A. 11/P + 6 B. P/11 +6 C. 17 - P/6 D. 17/P E. 11.5P Correct Answer: A Explanation: Let Ben’s age now be B Anita’s age now is A. (A - 6) = P(B - 6) But A is 17 and therefore 11 = P(B - 6) 11/P = B-6 (11/P) + 6 = B 9. All the dots in the array are 2 units apart vertically and horizontally. What is the length of the longest line segment that can be drawn joining any two points in the array without passing through any other point ? A. 2 B. 2√2 C. 3 D. √10 E. √20 Correct Answer: E Explanation: The longest line segment that can be drawn without passing through any dots other than those at the beginning and end of the segment, such a line could go from the middle dot in the top row to either the bottom left or right dot. In any case the segment will be the hypotenuse of a right triangle with sides 2 and 4. Using Pythagoras theorem the hypotenuse will be √(2 ² + 4 ² ) = √20 7. n is an integer chosen at random from the set {5, 7, 9, 11 } p is chosen at random from the set {2, 6, 10, 14, 18} What is the probability that n + p = 23 ? A. 0.1 B. 0.2 C. 0.25 D. 0.3 E. 0.4 Correct Answer: A Explanation: Each of the integers in the first set could be combined with any from the second set, giving a total of 4 x 5 = 20 possible pairs. Of these the combinations that could give a sum of 23 are (5 + 18), and (9 + 14) This means that the probability of getting a sum of 23 is 2/20 = 1/10 3. One side of a triangle has length 8 and a second side has length 5. Which of the following could be the area of the triangle? I 24 II 20 III 5 A. I only B. II only C. III only D. II and III only E. I, II and III Correct Answer: D Explanation: The maximum area of the triangle will come when the given sides are placed at right angles. If we take 8 as the base and 5 as the height the area = ½ x 8 x 5 = 20 We can alter the angle between the sides to make it less or more than 90, but this will only reduce the area. (Draw it out for yourself). Hence the area can be anything less than or equal to 20. 9. The fraction x/y is altered by decreasing x by 25 per cent and increasing y by 25 percent. The new fraction is what percent less than the original? Correct Answer: 40 Explanation: Decreasing x by 25 per cent gives 0.75x Increasing y by 25 percent gives 1.25y The new fraction is 0.75x/1.25y = 0.6x/y This is 0.4 less (= 40 percent) 6. If k is a positive integer, what is the smallest value for k to make 60k a perfect square? Correct Answer: 15 Explanation: To find a perfect square we need the prime factors. 60 = 2 x 2 x 3 x 5 To turn it into a perfect square we need to multiply by 3 x 5 = 15 (So that the prime factors are in groups of two = 2 x 2 x 3 x 3 x 5 x 5) 6. Different four-letter passwords can be constructed using the letters A, B, C and D only once. How many such passwords exist if either C or B must be in second position? Correct Answer: 12 Explanation: First imagine C in second position. This gives a choice of 1 out of three for first place, and one out of two for third place. This is equivalent to 6 possible combinations. Now putting B in second place and using the same logic there are 6 more combinations. Total = 12 3. The sum of four consecutive integers is 410. What is the value of the least of these integers? Correct Answer: 101 Explanation: Four consecutive integers can be written n + (n + 1) + (n + 2) + (n + 3) Their sum = 4n + 6 = 410 4n = 404; n = 101 5. A solid cube of side 6 is first painted pink and then cut into smaller cubes of side 2. How many of the smaller cubes have paint on exactly 2 sides? A. 30 B. 24 C. 12 D. 8 E. 6 Correct Answer: C Explanation: When the larger cube is cut into smaller cubes, the corner cubes will have paint on three sides. The cubes in the middle of the faces will have paint on only one side, but the cubes cut from the edges will have paint on two sides. In this case, there will be only one cube one each edge (excluding the corners), and since there are 12 edges, there will be 12 cubes with paint on two sides. (Drawing a diagram will make this much clearer.) (8 on the edges with more than 1 side painted, and 6 in the middle with only 1 side painted and 1 with 0 side painted) 7. In the figure above the square has two sides which are tangent to the circle. If the area of the circle is 4a²π, what is the area of the square? A. 2a² B. 4a C. 4a² D. 16a² E. 64a² Correct Answer: D Explanation: If the area of the circle is 4a 2 π , the radius will be the square root of 4a 2 (i.e. 2a) The diameter will be 4a. The diameter is also the side of the square. Area of the square is (4a) 2 = 16a 2 8. A triangle has a perimeter 13. The two shorter sides have integer lengths equal to x and x + 1. Which of the following could be the length of the other side? A. 2 B. 4 C. 6 D. 8 E. 10 Correct Answer: C Explanation: The measure of the third side of a triangle must lie between the sum and the difference of the other two sides. Using this fact along with the answer choices, we can eliminate the wrong answers. (A) cannot be correct because 2 would not be the longest side. (B) cannot be correct because again it would not be the longest side (C) could be correct because the other two sides would be 3 and 4. (D) and (E) cannot be correct because this third side would be greater than the sum of the other two sides. 6. Line l contains the points (3,1) and (4,4). If line m is a different line, parallel to line l in the same coordinate plane, which of the following could be the equation of line m? A. y = 3x - 8 B. y = 1/3x - 3 C. y = -3x - 8 D. y = 3x + 1 E. y = -8x + 3 Correct Answer: D Explanation: Slope of the line l = (4 – 1) / (4 – 3) = 3/1 = 3 Parallel lines have the same slope. In the equation of a straight line, y = mx + c, m represents the slope, so we are looking for an equation where m = 3 But both answer choices A and D have slope 3. Now we calculate the intercept, c, for line l. Using the coordinate of one of the points onl, we get: 4 = (3 x 4) + c ; 4 – 12 = c; c = -8 Hence, answer A is the equation of line l, and answer D must be the equation of a parallel line. 6. Which of the following could be a solution of the equation │x│ = │4x - 3│ A. -1 B. -0.6 C. 0 D. 0.6 E. 1.5 Correct Answer: D Explanation: This could be tricky unless you realize that the absolute value signs lead to the conclusion that x = 4x – 3 and also –x = 4x – 3 If we solve these equations we get x = 1 and x = 3/5 = 0.6 and so since 1 is not an option, the answer is D. (You could also have substituted the answer choices in the equation.) 10. A wheel has a diameter of x inches and a second wheel has a diameter of y inches. The first wheel covers a distance of d feet in 100 revolutions. How many revolutions does the second wheel make in covering d feet? A. 100xy B. 100y - x C. 100x - y D. 100y / x E. 100x / y Correct Answer: E Explanation: Total distance covered by the first wheel = d, which is the equivalent of 100 x circumference. Circumference = xπ Circumference of the second wheel = yπ Revolutions covered by the second wheel = 100(xπ / yπ) , which simplifies to 100x/y