CET/CAT I MATHEMATICS
Answer: A. 35% of 15% of a number is 0.35 * 0.15 of that number: 0.35 * 0.15 = 0.0525x
35% of 15% of x is equivalent to which of the following? A. 0.0525x B. 0.125x C. 0.25x D. 0.525x
Answer: D. Collinear points are also coplanar. Choice a is not the answer because noncollinear points determine planes, not a single line of collinear points.
Collinear points A. determine a plane. B. are circular. C. are noncoplanar. D. are coplanar.
Answer: D. She runs for 20 minutes and arrived 5 minutes late → She needs to be exactly there in 15 minutes. Using a bike with a speed of 1/3 km per minute → t = d/r → t = 2/1/3 → t = 6 15 minutes - 6 minutes = 9 minutes earlier.
Jenny's house is 2 km away from her school. One day when going to school, Jenny runs for 20 minutes and arrived 5 minutes late. How many minutes earlier would Jenny be if she would use her bicycle at a rate of one-third kilometer per minute? A. 5 minutes B. 6 minutes C. 8 minutes D. 9 minutes
Answer: D. Always assume that in plane geometry a line is a straight line unless otherwise stated. Process of elimination works well with this question: Lines have one dimension, length, and no substance; they are definitely not solid. Lines extend to infinity; they are not finite. Finally, we defined noncollinear as a set of points that "do not line up"; we take our cue from the last part of that statement. Choice c is not our answer.
Lines are always A. solid. B. finite. C. noncollinear. D. straight.
Answer: D. For this expression, use the product property of radicals and combine the factors in the radicand and outside the radical signs. (9√𝑎²𝑏)(3a√𝑏) = 9 x 3a√𝑎²𝑏 𝑥 𝑏 = 27a√𝑎²𝑏² = 27a x ab = 27a²b
Simplify the following radical expression: (9√𝑎²𝑏 )(3𝑎√𝑏) A. 18ab² B. 9ab C. a²b D. 27a²b
Answer: D. Use the distributive property of multiplication. 3(1) − 3(3x) ≥ −3(x) − 3(27) Simplify terms. 3 − 9x ≥ −3x − 81 Add 9x to both sides. 3 − 9x + 9x ≥ 9x − 3x − 81 Combine like-terms. 3 ≥ 6x − 81 Add 81 to both sides of the inequality. 3 + 81 ≥ 6x − 81 + 81 Combine like-terms. 84 ≥ 6x Divide both sides of the inequality by 6. 84/6 ≥ 6𝑥/6 Simplify. 14 ≥ x
Solve the inequality 3(1 − 3x) ≥ −3 (x + 27) A. 3 ≥ x B. 7 C. 14x D. 14 ≥ x
Answer: C. Use an equation to represent the situation, x + x + 2 + x + 4 + x + 6 = 36. Solve for x to find 6, but recognize that this is the smallest integer in the set, the 3rd largest would be 10.
The sum of 4 consecutive even integers is 36, what is the 3rd largest integer in the set? A. 6 B. 8 C. 10 D. 12
Answer: C. The probability of selecting a red marble on the first draw is 10⁄32 because there are 10 red marbles and 32 total marbles. After removing the first red marble there are now 9 red marbles and 31 total marbles left so 9⁄31 chance of selecting the second red marble. To find the probability of both events occurring, we multiply the probabilities to get 9 * 10⁄32 * 31 which reduces to 45⁄496.
There are 32 marbles in a jar: 14 blue, 10 red, 5 green, and 3 yellow. Sally pulls one marble, randomly, from the jar. Without replacing the marble, she pulls another marble. What is the probability that both marbles will be red? A. 41/456 B. 44/476 C. 45/496 D. 40/426
Answer: C. The first term a₁ is 2 and the common difference is equal to 5 - 2 = 8 - 5 = 3 Hence using the formula for the nth term, aₙ = a₁ + (n - 1)d to the term equal to 227, we can write the equation: 227 = 2 + (n - 1)3 Solve the above for n n - 1 = (227 - 2) / 3 = 75 and n = 76 The 76th term is equal to 227
Which term of the arithmetic sequence 2, 5, 8... is equal to 227? A. 74th term B. 75th term C. 76th term D. 77th term
Answer: D The most straightforward method for solving problems of this type is to begin by supplying our own initial value prior to any increase or decrease in profits. Because percents are essentially a division by 100, choosing 100 as our starting value for any percent problem is wise. We are told that for the first year, the percent rose by 12%, so: 100 * 0.12 = 12 So at the start of 2011, the company now has 112, for which there is another increase of 18%, so: 112 * 0.18 = 20.16 The final amount in 2012 = 112 + 20.16 = 132.16 To find the overall percent increase, we find the total increase, then divide this value by the original amount, so: 132.16 − 100 = 32.16 32.16 ÷ 100 = 0.32 = 32%
A company's profits increased by 12% from 2010 to 2011 and by 18% from 2011 to 2012. By what percent did the company's profits increase from 2010 to 2012? A. 37% B. 54% C. 12% D. 32%
Answer: B. Let x = the number of months. The number of months (x), times 12 (pounds per month), plus the starting weight (20), will be equal to 200 pounds. An equation that represents these words would be 12x + 20 = 200. Subtract 20 from both sides of the equation. 12x + 20 − 20 = 200 − 20 Associate like-terms. 12x + (20 − 20) = 200 − 20 Perform numerical operations. 12x + (0) = 180 Divide both sides of the equation by 12. 12𝑥 = 180 12 12 x = 15 The farmer would have to wait 15 months before selling his hog.
A farmer is raising a hog that weighed 20 lbs. when he bought it. He expects it to gain 12 pounds per month. He will sell it when it weighs 200 lbs. How many months will it be before he will sell the animal? A. 14 B. 15 C. 24 D. 25
Answer: -15 If you think of distance above sea level as a positive number, then you must think of going below sea level as a negative number. Going up is in the positive direction, while going down is in the negative direction. Give all the descending distances a negative sign and the ascending distances a positive sign. The resulting numerical expression would be as follows: (−80) + (+25) + (−12) + (+52) Because addition is commutative, you can associate like-signed numbers: (−80 + −12) + (+25 + +52) Evaluate the numerical expression in each parentheses: [−80 + −12 = −92] [ +25 + +52 = +77] Substitute the values into the numerical expression: (−92) + (+77) Signs different? Subtract the value of the numbers and give the result the sign of the higher value number. [92 − 77 = 15] The diver took his rest stop at −15 feet.
A scuba diver descends 80 feet, rises 25 feet, descends 12 feet, and then rises 52 feet where he will do a safety stop for five minutes before surfacing. At what depth did he do his safety stop?
Answer: A. Recall that the angles of a triangle sum to 180 degrees: 3x + 10 + −2x + 40 + x + 40 = 180 2x + 90 = 180 2x = 90 x = 45
A triangle is composed of angles represented as 3x + 10, −2x + 40, and x + 40. What is the value of x? A. 45 B. 50 C. 55 D. 60
Answer: D Substitute the values for the variables into the expression. ( 1/2 ) {( (6)/2 − 3 ) − 4(3)} Evaluate the expression in the innermost parentheses. [( (6)/2 − 3) = 6/2 − 3] PEMDAS: Division before subtraction. Substitute the result into the numerical expression. [ 6/2 − 3 = 3 − 3 = 0] ( 1/2 ){(0) − 4(3)} Evaluate the expression inside the parentheses. [{0 − 4(3)} = 0 − 4 · 3] PEMDAS: Multiply before subtraction. [0 − 4 · 3 = 0 − 12] Change subtraction to addition and the sign of the term that follows. [0 − 12 = 0 + −12 = −12] Substitute the result into the numerical expression ( 1/2 ){−12} = 1/2 · −12 Are signs different? Multiply numbers and give the result a negative sign. [ 1 /2 · 12 = 6] 1/2 · −12 = −6 The simplified value of the expression is as follows: y{(𝑥/2 − 3) − 4a} = -6
Evaluate the algebraic expression 𝑦{( 𝑥/2 − 3) − 4𝑎} when a = 3 x = 6 y = 1 2 A. 𝑦 {( 𝑥/2 − 3) − 4𝑎} = −5 B. 𝑦 {( 𝑥/2 − 3) − 4𝑎} = −7 C. 𝑦 {( 𝑥/2 − 3) − 4𝑎} = 5 D. 𝑦 {( 𝑥/2 − 3) − 4𝑎} = −6
Answer: B. Substitute x by -2 in f(x) as follows f(-2) = 4(-2)ᶟ - 4(-2)² + 10 = 4(-8) - 4(4) + 10 = - 32 - 16 + 10 = - 38
If f(x) = 4xᶟ - 4x² + 10, then f(-2) = A. 26 B. -38 C. 10 D. 38
Answer: B. We solve this problem by replacing every x in h(x) with 2x − 3 and evaluating the expression: h(2x − 3) = 3(2x − 3) + 4 = 6x − 9 + 4 = 6x - 5
If h(x) = 3x + 4, what is h(2x − 3)? A. -6X + 5 B. 6X - 5 C. 5X - 2 D. 5X + 2
Answer: A. This expression can be factored using the trinomial method. The factors of v⁴ are (v²)(v²), and the factors of 48 are (1)(48) or (2)(24) or (3)(16) or (4)(12) or (6)(8). Only the product of a positive and a negative numerical term will result in −48. The only factors of 48 that can be added or subtracted in any way to equal 13 are 3 and 16. Use 3 and 16 and a positive and negative sign in the terms of the trinomial factors. Check your answer using FOIL. (v² + 3)(v² − 16) = v⁴ − 16v² + 3v2 − 48 = v⁴ − 13v - 48 You may notice that one of the two factors of the trinomial expression can itself be factored in. The second term is the difference between two perfect squares. Factor (v² − 16) using the form for factoring the difference of two perfect squares. (v + 4)(v − 4) = v2 − 4v + 4v − 16 = v² - 16 This now makes the complete factorization of v⁴ − 13v² − 48 = (v² + 3)(v + 4)(v − 4).
Factor the following polynomial: v⁴ − 13v² - 48 A. v⁴ − 13v² − 48 = (v² + 3)(v + 4)(v − 4) B. v⁴ + 13v² − 48 = (v² + 3)(v + 4)(v − 4) C. (v + 4) = v² − 4v + 4v − 16 = v² - 16 D. (v − 4) = v² − 4v + 4v − 16 = v² - 16
Answer: D. The first term a₁ = 9 and d = 2 (the difference between any two consecutive odd integers). Hence the sum Sₙ of the n terms may be written as follows Sₙ = (n/2)[2*a₁ + (n - 1)d] = 15,960 With a₁ = 9 and d=2, the above equation in n may be written as follows n₂ + 8 n - 15860 = 0 Solve the above for n n = 122 and n = -130 The solution to the problem is that 122 consecutive odd numbers must be added in order to obtain a sum of 15,860
How many consecutive odd integers of an arithmetic sequence, starting from 9, must be added in order to obtain a sum of 15,860? A. 119 B. 120 C. 121 D. 122
Answer: C. The number of ways of arranging n objects in a round table is (n - 1)! Ways. For the five students the number of arrangements is (5 - 1)! = 4! = 24
How many different ways can five students be seated in a round table? A. 5 B. 10 C. 24 D. 25
Answer: B. P = 2L + 2W L = (P - 2W) ÷ 2 = [(16x + 8y) - 2 (5x - 2y)] ÷ 2 = [16x + 8y - 10x + 4y] ÷ 2 = [6x + 12y] ÷ 2 L = 3x + 6y
If 16x + 8y represents the perimeter of a rectangle, and 5x - 2y represents its width, then its length is represented by the expression A. 3x + 2y B. 3x + 6y C. 11x + 6y D. 21x + 6y
Answer: D. Translate the given expressions into equations: x * 0.6 = 12 x = 20 20 * 1.65 = 33
If 60% of a number is 12, what is 165% of the same number? A. 30 B. 31 C. 32 D. 33
Answer: C. Begin by rounding the number to the nearest hundredth: 89.88. Now add the tenths place, 8, and the hundredths, 8, to get 16
If the number 89.8756 is rounded to the nearest hundredth, what will be the sum of the tenths and hundredths place of the resulting number? A. 12 B. 14 C. 16 D. 20
Answer: B. Solve this problem by setting up a proportion. We are told the ratio of milk to juice is 13:x and that there are 39 milk and 18 juice so → 13/x = 39/18, cross multiply and solve for x to get 6.
If the ratio of milk cartons to juice boxes is 13:x and there are 39 milk cartons and 18 juice boxes, what is the value of x? A. 4 B. 6 C. 8 D. 12
Answer: C. Multiply the exponents of each factor inside the parentheses by the exponent outside the parentheses. 3²x²y¹⁰ - 11x²y²4²y⁸ Use the commutative property of multiplication. 3²x²y¹⁰ - 11 × 4²x²y²y⁸ When similar factors, or bases, are multiplied, add the exponents of the variables. 3²x²y¹⁰ - 11 × 16x²y¹⁰ Evaluate numerical factors. 9x²y¹⁰ - 176x²y¹⁰ Combine like terms. −167x²y¹⁰
Simplify the equation (3xy⁵)² - 11x²y²(4y⁴)² A. 176x²y¹⁰ B. -176x²y¹⁰ C. -167x²y¹⁰ D. 167x²y¹⁰
Answer: C. The trigonometric ratios sine and cosine never equal or exceed 1.000 because the hypotenuse, the longest side of a right triangle, is always their denominator. The trigonometric ratio Tangent can equal and exceed the value 1.000 because the hypotenuse is never its denominator.
Which trigonometric function can equal or be greater than 1.000? A. Sine B. Cosine C. Tangent D. none of the above
Answer: A. Method 1: Quick Method: The number of girls participating is 105. It was stated in the problem that there are twenty-five more girls than the number of boys that participate. This means that the number of boys participating is twenty-five less than the number of girls participating. Thus, we can simply subtract 25 from the number of girls that participate (which is 105) to obtain the number of boys that participate: 105-25=80 Hence, the number of boys that participate is 80. Method 2: Algebraic Method: Let b be the number of boys that participate. Since there are twenty-five more girls than the number of boys, we can express the number of girls that participate as follows: b + 25 Since the number of girls participating is 105, we can equate b + 25 to 105: b+25=105 We will now solve for b in the equation to find the number of boys that participate. Subtracting both sides of the equation by 25: b=80 Hence, the number of boys that participate is 80.
Twenty-five more girls than the number of boys participate in interscholastic sports at a local high school. If the number of girls participating is 105, how many boys participate? A. 80 B. 81 C. 82 D. 83
Answer: D. If a triangle has side lengths a, b, and c, the sum of the lengths of any 2 sides must be larger than the length of the 3rd side. So in this case, 5 + 6 = 11 must be larger than side length c. From the answer choices, 12 is the only length greater than 11, so it cannot be the length of the third side.
Two side lengths of a triangle are 5 and 6, which of the following CANNOT be the length of the third side? A. 3 B. 6 C. 9 D. 12
Answer: D. Since in this case the number of scores is even, the median is the average of the two middlemost scores. median = 50 + 51 2 = 101 2 = 50.5
What is the median of the first 100 non-negative integers? A. 49 B. 49.5 C. 50 D. 50.5
Answer: B. Recall that slope-intercept form is y = mx + b where m is the slope and b is the y intercept. Solve for y: 8x − 2y = −6 2y = 8x + 6 Divide everything by 2: y = 4x + 3
What is the slope-intercept form of (13x − 5x) + 12 − 2y = 6? A. y = 3x - 9 B. y = 4x + 3 C. y = 6x + 2 D. y = -2x + 5
Answer: D. A line, a line segment, and a ray are sets of points. How many points make a set? An infinite number. Since a limit cannot be put on infinity, not one of the answer choices has more than the other.
Which choice below has the most points? A. a line B. a line segment C. a ray D. No determination can be made.
Answer: C. A. The sum of two odd integers. 15 + 15 = 30, 30 ÷ 10 = 3 (odd integer) B. The product of two prime numbers. 2 x 5 = 10, 10 ÷ 10 = 1 (odd integer) C. The product of two odd integers. 3 x 5 = 15, 15 ÷ 10 = 1.5 (not an integer) D. The sum of three consecutive integers. 9 + 10 + 11 = 30, 30 ÷ 10 = 3 (odd integer)
Which of the following CANNOT yield an odd integer when divided by 10? A. The sum of two odd integers. B. The product of two prime numbers. C. The product of two odd integers. D. The sum of three consecutive integers.
Answer: B. 16 - x² ≥ 0 x² - 16 ≤ 0 → (x - 4)(x + 4) ≤ 0 → -4 ≤ x ≤ 4
Which of the following describes the values of x for which 16 - x² ≥ 0? A. x ≤ - 4 or x ≥ 4 B. -4 ≤ x ≤ 4 C. 0 ≤ x ≤ 4 D. x ≥ 0
Answer: C. Choice A is not an acute triangle because it has one right angle. In choice B, the sum of interior angle measures exceeds 180°. Choice D suffers the reverse problem; its sum does not make 180°. Though choice C describes an equilateral triangle; it also describes an isosceles triangle.
Which of the following sets of interior angle measures would describe an acute isosceles triangle? A. 90°, 45°, 45° B. 80°, 60°, 60° C. 60°, 60°, 60° D. 60°, 50°, 50°
Answer: D. A square is a rhombus and a rectangle. Therefore, some rhombuses are rectangles.
Which of the following statements is true? A. The diagonals of a rhombus are congruent. B. All rectangles are similar. C. All rectangles are rhombuses. D. Some rhombuses are rectangles.
Answer: A A probability is always greater than or equal to 0 and less than or equal to 1, hence only A above cannot represent probabilities.
Which of these numbers cannot be a probability? A. -0.00001 B. 0.5 C. 0 D. 1