Veritas Practice Test 1 - Questions answered incorrectly

If a square mirror has a 30-inch diagonal, what is the area of the mirror, in inches? (A) 225 (B) 450 (C) 600 (D) 750 (E) 900

B Solution: The diagonal of a square gives us two 45-45-90 triangles with the side ratio x : x : x√2 To find the length of the side, we can use the Pythagorean theorem: x²+x²=30² From there, we know that 2x²=900, so x²=450

(5⁵⁵/10³⁰) x (2³⁰/5²⁵) = ? (A) 1 (B) 5³⁰/2⁶⁰ (C) 5³⁰/2³⁰ (D) 5⁵⁵2³⁰/10³⁰ (E) 10⁵⁵/50⁵⁵

(A) 10³⁰ = (2x5)³⁰ = 2³⁰x5³⁰ Cancel and you get 5²⁵ x 1/5²⁵ Anything multiplied by its reciprocal = 1

If x > 0, is y < 0? (I) y/x > 0 (II) x - y > 0

(A) Statement 1 tells us that y/x is positive. If we know x is positive, then y must be positive as well, so we can answer with a definitive no. Sufficient. Statement 2 tells us that x - y is positive. We know x is positive, but y could be positive or negative (4 - 2 = 2; 4 - -2 = 6). Insufficient, so the correct answer is A.

A rectangular label with an area of 18 square inches is wrapped around a can that is 3 inches tall, such that the label exactly covers the outside of the can excluding the top and the bottom. What is the volume of the can, in cubic inches? (A) 27π (B) 27 (C) 18 (D) 27/π (E) 18/π

(D)

A husband and wife can complete a certain task in 1 and 2 hours respectively. Their children, Rae and Herman, can complete the same task in 4 and 6 hours, respectively. What is the ratio of the couple's time working together to complete the task to the children's time working together to complete the task? (A) 15 : 46 (B) 3 : 10 (C) 12 : 23 (D) 5 : 18 (E) 10 : 3

(D) Rates are additive

If P, Q, R, and S are all integers, is product PQRS even? (I) PQ = 42 (II) RS = 35

A If a series of integers are being multiplied together, the only way the product can be odd is if ALL the integers being multiplied are odd; one even integer will force the product to be even. Statement (1) says that PQ = 42, so either P or Q is even; SUFFICIENT. Statement (2) says that RS = 35, so both R and S are odd; INSUFFICIENT, as PQ could be even or PQ could be odd.

What is √1331/(√396 + √275)? (A) 1/2 (B) 1 (C) √3 (D) 2 (E) (2√3)/2

B √(11)(121) / (√(11)(36)) + (√(11)(25)) Pull out the known squares 121, 36 and 25: 11√11 / 6√11 + 5√11 Cancel √11 You're left with 11/6+5 = 1

What is the value of x? (1) x²+x+10=16 (2) x=4y⁴+2y²+2

C. Statement 1 can be manipulated to: x²+x−6=0, then: (x+3)(x−2)=0, but this leaves potential values of -3 and 2, so statement 1 is *not sufficient*. Statement 2 should clearly be not sufficient as different values of y will make a huge difference in the value of x. Which means that this is a good opportunity to use the "Why Are You Here?" strategy. What is statement 2 doing there? Because y to an even exponent cannot be negative, statement 2 GUARANTEES that x will be positive, meaning that when you combine the statements x must be 2 and cannot be -3. Therefore, x is 2 and both statements together are sufficient, making the answer C.

While flying across the country, did Karen ever exceed 650 miles per hour? (I) Karen flew 3,000 miles. (II) Karen flew for 5 hours.

Correct Answer - E. Distance = Rate x Time I. This statement provides us the distance, but not the time. Not Sufficient. II. This statement provides us the time, but not the distance. Not Sufficient. Together - We have both the distance and the time. Therefore, we can determine that Karen's average speed was 600 mph, which is less than 650 mph. However, this number represents Karen's average rate of travel. It is possible that Karen traveled over 650 mph for a portion of the trip and less than 600 mph for a portion of the trip, but on the average traveled 600 mph. Not sufficient.

What is the units digit of positive integer p? (I) When p is divided by 10, the remainder is 8 (II) When p is divided by 11, the remainder is 8

Correct Answer: A Dividing p by 10 can only give a remainder of 8 if the units digit of p is 8 (p = 108, p = 248, etc.), so statement (1) is sufficient. But there are numbers with different units digits that can divide by 11 and leave 8 (p = 19, p = 30, p = 41, etc.), so statement (2) is not sufficient.

What is the average of x and y? (1) x + y = 20 (2) 2x + 4y = 54

Correct Answer: A To determine the average of x and y, we don't need to know the two values individually: it's sufficient to know the value of x + y and to divide it by 2. Statement (1) gives us x + y; sufficient. Statement (2) cannot be simplified to x + y, nor can we determine the values of x and y individually, so there is no way to determine the average of the two variables; not sufficient.

Is positive integer x divisible by 24? (I) √x is divisible by 4 (II) x² is not divisible by 9

Correct Answer: B In this problem, break 24 into its prime factors: 2, 2, 2 and 3. Therefore, for an integer to be divisible by 24, that integer must be divisible by 2 at least three times, and 3 at least once. Statement 1 indicates that √x is divisible by at least four multiples of 2 (if x is divisible by 4, then x would be divisible by 4², or 2 * 2 * 2 * 2 ). However, statement 1 does not indicate whether x is divisible by 3, and accordingly it is *not sufficient*. Statement 2 indicates that x is not divisible by 3; if x² is not divisible by 9, then the √x would not be divisible by √9, which is 3. Because x would need to be divisible by 3 in order to be divisible by 24, it can be determined that x is not divisible by 24, and therefore the correct answer is B.

If 12!/3ˣ is an integer, what is the greatest possible value of x? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7

Correct Answer: C order for 12!/3ˣ to be an integer, there must not be more 3's in the denominator than there are in the numerator. Otherwise, the 3s would not cancel, leaving us with a fraction. Thus, we simply need to determine how many factors of 3 there are in 12!. To do this, consider the elements of 12! that are multiples of 3: 12, 9, 6, and 3. Prime factorizing these numbers reveals that we have 5 factors of 3: one each in 12 (3 x 2 x 2), 6 (3 x 2), and 3 (3 x 1), and two in 9 (3 x 3). Thus, the greatest possible value of x is 5, and the correct answer is (C).

What is the value of m in the equation below? (¹/₅)ᵐ x (¹/₄)²⁴ = 1/ 2(10)⁴⁷ (A) 23 (B) 24 (C) 46 (D) 47 (E) 48

Correct Answer: D Remember the Guiding Principles of Exponents - in this problem, the first principle "Find Common Bases" is the key. Note the bases of your exponents - 2, 4, 5 and 10. Since 4 and 10 are not prime (but each can be expressed as the products of primes only using 2 and 5), prime-factoring the bases of these exponents will be the key to making this problem solvable. (¹/₄)²⁴ can be rewritten as (¹/₂)⁴⁸, making the equation (¹/₅)ᵐ x (¹/₂)⁴⁸ = 1/ 2(10)⁴⁷ By factoring 1/2 from both sides of the equation, the '2' in the denominator on the right side can be eliminated, and on the left side 1/2 will be simply reduced by one exponent, making the equation (¹/₅)ᵐ* (¹/₂)⁴⁷ = (1/10)⁴⁷ Given that 1/5*1/2= 1/10, m must equal 47, and D is the correct answer.

A number x is multiplied by 3, and this product is then divided by 5. If the positive square root of the result of these two operations equals x, what is the value of x if x≠0? (A) 25/9 (B) 9/5 (C) 5/3 (D) 3/5 (E) 9/25

D

What is the value of x? (I) 4x=2y−6 (II) (y-3)/2 = x

E Correct Answer: E Without further information, we can never solve one equation with two variables. Statement (1) has one equation and two variables; not sufficient. Statement (2) also has one equation and two variables, so it too is not sufficient. Combining both statements, we have 2 equations and 2 variables, which is sufficient to solve for both variables if and only if the two equations are not merely multiples of each other. Since statement (2) is simply statement (1) rearranged and divided by 4, we are unable to solve for either variable; not sufficient.

In the figure above, points A, B, C, D, and E are evenly spaced along the number line. If E = 9¹³ and C = 9¹¹, what is the distance from point A to point D? (A) 9³ (B) 9⁹ (C) (120)(9⁹) (D) 9¹¹ (E) (120)(9¹¹)

E. Let's say the distance between two consecutive letters is x. We're told that 2x=9¹³−9¹¹ Factoring the common 911terms we find that 2x=9¹¹(9²−1), which then means that 2x=9¹¹(80). Dividing both sides by 2, we find that: x=9¹¹(40). Since the distance we want is 3x, we just multiply both sides by 3:3x=9¹¹(120) Therefore, correct answer is E. Beware the trap on this question - consecutive exponents like 9¹³, 9¹², 9¹¹, and 9¹⁰ are not evenly-spaced (think about 9, 81, and 243, the first three powers of 9...they're not evenly spaced at all).

A city's school board merged the city's two rival high schools, high school A and high school B, to form one new high school, which then had 4500 students. 10% of the students at high school A and 15% of students at high school B were in the marching band. If all marching band members joined the new school's marching band, which then had 570 members, how many students did school A have before the merge? (A) 2100 (B) 2200 (C) 2300 (D) 2400 (E) 2500

For this problem, the algebra requires a bit less creativity to set up. You're looking at: a+b=4500 and 0.1a+0.15b=570 If you multiply the second equation by −10, that allows you to use the Elimination Method to get rid of the variable a: : −10(0.1a+0.15b=570) means that: −a+−1.5b=−5700 Stack that up with the other equation and add: a+b=4500 + −a+−1.5b=−5700 And you have one equation, all with the variable b: −0.5b=−1200, so multiply by −2 to arrive at: b=2400 That means that a, the variable in question, must be 2100, making answer choice A correct.