If [x] denotes the greatest integer less than or equal to x, what is the value of [x]?
(1) When x^{2} is divided by 9, the quotient is zero.
(2) 2x2+9x+10=0
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find: [x] = ?
Step 3: Analyze Statement 1 independently
(1) When x^{2} is divided by 9, the quotient is zero
3 < x <3
If x lies between 3 and 2, then [x] = 3
If x lies between 2 and 1, then [x] = 2 and so on.
Thus, Statement 1 is not sufficient to find a unique value of [x]
Step 4: Analyze Statement 2 independently
Step 5: Analyze Both Statements Together (if needed)
From Statement 1: 3 < x < 3
From Statement 2: x = 2 or x = 2.5
Corresponding values of [x] = 2 or 3
Both values of x from Statement 2 conform to the inequality from Statement 1
So, even after combining the two statements, we still get two values of x, and correspondingly, two values of [x]
Thus, the two statements together are not sufficient to find a unique answer to the question.
Answer: Option E
For the positive integers k, m and n, k(m)n means that the remainder when m is divided by n is k. If k(13766)9 and p(137)k, where p is a positive integer, then p is equal to.
Given:
To find: p = ?
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
If f(z) = 2^{z+1}, then which of the following can be the value of x for which f(1)∗f(2x)=f(x−1)∗{f(4)+f(−1)}−f(1)?
Given:
f(z) = 2^{z+1}
To find: The value (out of the 5 given options) of x for whichf(1)∗f(2x)=f(x−1)∗{f(4)+f(−1)}−f(1)
Approach:
Working Out:
Looking at the answer choices, we see that only the value in Option A corresponds to one of the roots we found. So, the correct answer is Option A
For a positive integer x, the function f(x) is defined as f(x)=2x^{2}+3x+4
. Is the value of the function f(x) an even number?
Step 1 & Step 2: Understanding the Question statement and Drawing Inferences
Step 3: Analyze statement 1 independently
Statement 1:
Step 4: Analyze statement 2 independently
Statement 2:
Step 5: Analyze the two statements together
For all positive integers x, when x is even and when x is odd. What is the value of positive integer x?
(1) The highest power of 2 that divides &(4x) completely is 36
(2) The highest power of 2 that divides (&4)*x completely is 8
Steps 1 & 2: Understand Question and Draw Inferences
Given:
Step 3: Analyze Statement 1 independently
(1) The highest power of 2 that divides &(4x) completely is 36
Thus, Statement 1 is sufficient to determine a unique value of x.
Step 4: Analyze Statement 2 independently
(2) The highest power of 2 that divides (&4)*x completely is 8
Thus, Statement 2 is not sufficient to determine a unique value of x.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 3, this step is not required
Answer: Option A
The function PF is defined as PF(a) = k, where k is the number of prime factors of positive integer a. If PF(x) = PF(2x)=PF(3x) = 2 and PF(y) = PF(5y) = PF(7y) = 2, where x and y are positive integers, what is the value of PF(least common multiple of x and y)?
Given
To Find: PF(LCM(x, y)?
Approach
Working Out
Answer: C
For any integers x and y, min(x, y) and max(x, y) denote the minimum and the maximum of x and y, respectively. For example, min(2, 1) = 1 and max(2,1) = 2. If a, b, c and d are distinct positive integers, is max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))) ?
(1) b, c and d are factors of a
(2) a – 2d = b + c
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Is max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))) ?
Step 3: Analyze Statement 1 independently
(1) b, c and d are factors of a
So, max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))).
Sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) a – 2d = b + c
So, max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))).
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
The function f(x) is defined as where x is not equal to 1. If a is not equal to 1 or 0, which of the following expressions must be true for all values of a?
Given:
To find: Which of the 3 expressions is/are true for all values of a?
Approach:
Working Out:
Evaluating Expression III
Looking at the answer choices, we see that the correct answer is Option B
If % denotes one of the arithmetic operations, addition, subtraction, multiplication or division, and if z is an integer, what is the value of 2z%z?
(1) 3%(2z)=−(z%1)
(2) (3z)%(z−1)=5
Steps 1 & 2: Understand Question and Draw Inferences
Given: Integer z
To find: 2z%z =
So, in order to find the value of 2z%z, we need to find out what % is and (if % is not division, then) also what z is.
Step 3: Analyze Statement 1 independently
(1) 3%(2z) =  (z%1)
As is clear from the table, we get 2 possible values of 2z%z from Statement 1.
So, Statement 1 is not sufficient.
Step 4: Analyze Statement 2 independently
(2) (3z)%(z  1) = 5
This statement leads us to a unique value of the required expression.
So, this statement alone is sufficient.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
For all positive numbers a and b, the function {a, b} = k, where k is the multiple of b nearest to a.. If x and y are positive integers such that x/y= 15.25 and {xy,y} = 16, what is the value of y?
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Value of y
As we’ve inferred above, the possible values of y are {4, 8, 16}
Step 3: Analyze Statement 1 independently
Since it’s given that {xy,y} = 16, using Statement 1 we can write: {x/y,2y} = 16. So, 16 is a multiple of 2y.
Since, we do not have a unique value of y, the statement is insufficient to answer.
Step 4: Analyze Statement 2 independently
Since it’s given that {xy,y} = 16, using Statement 1 we can write: {x/y,y^{2}} = 16. So, 16 is a multiple of y^{2}.
As we have a unique value of y, statement is sufficient to answer
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step4, this step is not required.
Answer: B
The distance travelled (D) by a car in miles when it uses more than 1 liter of fuel is given by D= x +kv, where v is the liters of fuel used, and x, and k are constants. If the car uses 4 liters of fuel to travel 53 miles and 8 liters of fuel to travel 101 miles, how much fuel in litres will the car require to travel 149 miles?
Given
To Find: Liters of fuel used to travel 149 miles?
Approach
Working Out
(2) – (1), we have
4k = 48, i.e. k =12.
Substituting the value of k in (1), we have x = 5
Putting the values of x and k in the equation 149 = x+ kv, we have
149 = 5 +12v, i.e. v = 12 liters
Hence, it will require 12 liters to travel a distance of 149 miles.
Answer: C
If [z] denotes the least integer greater than or equal to z and [z^{2}] = 2, which of the following could be the value of [z]?
I. 2
II. 1
III. 2
Given:
To find: Can [z] be {2, 1, 2}?
Approach:
Working Out:
Thus, we see that [z] is either equal to 1 or equal to 2. Out of the 3 given values, only value I (which is 2) is therefore possible.
Looking at the answer choices, we see that the correct answer is Option A
For distinct positive integers x and y, where x < y, the function FP(x, y) returns the smallest prime number between x and y, exclusive, or the text string ‘NULL’ if no such number is found. If FP(a, b) +FP(c, d) = FP(e, f), where a, b, c, d, e and f are distinct positive integers, what is the value of c^{a} ?
(1) FP(g, h) = a, where g and h are distinct positive integers
(2) c is less than the minimum possible value of the function FP(x,y).
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Value of c
Step 3: Analyze Statement 1 independently
(1) FP(g, h) = a, where g and h are distinct positive integers
If c = 1, c^{a} = 1
Sufficient to answer
Step 4: Analyze Statement 2 independently
(2) c is less than the minimum possible value of the function FP(x,y).
Hence c^{a} = 1
Sufficient to answer
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
If the operation a* is defined by what is the value of
Given:
Approach:
Working Out:
For any nonzero numbers p and q, What is the value of p$q?
Steps 1 & 2: Understand Question and Draw Inferences
Given:
In order to find the required value, we need to know the value of
Step 3: Analyze Statement 1 independently
=> q= p^{2 }( Since p^{2} is always positive, we can remove the modulus sign)
Since we do not know the value of p, we cannot find a unique value of
Statement 1 is not sufficient.
Step 4: Analyze Statement 2 independently
Thus, Statement 2 is not sufficient to find a unique value of the ratio
Step 5: Analyze Both Statements Together (if needed)
Thus, the two statements together are sufficient to obtain a unique value of
Answer: Option C
A function f(x) is defined as f(x) = x + x^{2} for an integer x. If a is an integer, is f(a) a prime number?
1. f(a)+f(a+1) is a prime number
2. The greatest common divisor of f(3) & f(4) is f(a)*2.
Step 1 & Step 2: Understanding the Question statement and Drawing Inferences
Given Info:
To find:
Subtracting 2 from both sides of the equation, we get
Step 3: Analyze statement 1 independently
Statement 1:
Step 4: Analyze statement 2 independently
Statement 2:
⇒ f(a) = 2
Step 5: Analyze the two statements together
Hence the correct answer is option B Only statement 2 is sufficient to arrive at a unique answer.
The function Ro rounds a positive decimal number to its nearest integer starting from its rightmost digit. What is the value of Ro(64.4ab5), where a and b are single digit positive integers?
(1) Ro(36.4a7) = 37
(2) Ro(87.b6) = 88
Steps 1 & 2: Understand Question and Draw Inferences
To Find: value of Ro(64.4ab5)
Step 3: Analyze Statement 1 independently
(1)Ro(36.4a7) = 37
If a = 4, we need to know the value of b to find the value of Ro(64.4a(b+1))
If a > 4, Ro(64.4a(b+1)) = 65
As we do not have a unique answer, the statement is insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) Ro(87.b6) = 88
Need to know the value of a. Insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
(1) From Statement 1, a ≥ 4
(2) From Statement 2, b ≥ 4
Sufficient to answer.
Answer: C
The operation @ is defined for all distinct and nonzero a and b by
Given:
To find: x = ?
Approach:
Looking at the answer choices, we see that the correct answer is Option D
The symbols @ and # each denote one of the four arithmetic operations: addition, subtraction, multiplication and division. If the symbols denote different arithmetic operations such thatand a@b#c = c#b@a, where a, b and c are distinct positive integers, what is the value of 4@5#4?
Given
To Find: value of 4@5#4?
Approach
Working Out
Thus, we have a unique value of 4@5#4 = 24
Answer: D
A function f(x) is defined as f(x)=3x^{2}−20x+c, where c is a constant. Also given f(1) = 16. What is the value of f(c) + f(c) ?
Given Info:
To Find:
⇒f(c)+f(−c)=6c^{2}+2c
Approach:
Working out:
⇒ f(1)=3(12)−20(1)+c
⇒ f(1)=c−17
⇒ c−17=−16
⇒ c=1
⇒ f(c)+f(−c)=6(1)^{2}+2(1)
⇒ f(c)+f(−c)=8
Answer
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