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All questions of Functions for GMAT Exam

If f(x) = -2x + 8 & f(p) = 16, find the value of p?
  • a)
    -12
  • b)
    -8
  • c)
    -4
  • d)
    4
  • e)
    12
Correct answer is option 'C'. Can you explain this answer?

Palak Saha answered
Given, f(x) = -2x + 8 and f(p) = 16

To find: The value of p

Solution:

Substitute f(p) = 16 in the equation f(x) = -2x + 8

f(p) = -2p + 8 = 16

-2p = 16 - 8

-2p = 8

Divide both sides by -2

p = 8/-2

p = -4

Therefore, the value of p is -4.

Hence, the correct option is (c) -4.
Clear all your doubts with EduRev

If g(x) = -2x2 + 8 and g (-q) = -24, which of the following could be the value of q?
  • a)
    -4
  • b)
    -2
  • c)
    -1
  • d)
    1
  • e)
    2
Correct answer is option 'A'. Can you explain this answer?

Parth Singh answered
Given:
- g(x) = -2x^2 + 8
- g(-q) = -24

To find:
- Possible values of q

Solution:

Substitute -q in place of x in g(x) to get g(-q)

g(-q) = -2(-q)^2 + 8
g(-q) = -2q^2 + 8

Given that g(-q) = -24, we can set up the equation:

-2q^2 + 8 = -24

Simplifying, we get:

-2q^2 = -32

Dividing by -2, we get:

q^2 = 16

Taking the square root of both sides, we get:

q = ±4

Therefore, the possible values of q are -4 and 4.

Option A (-4) is the correct answer.

A polynomial function P(x) is defined as,
P(x) = 4x3 – 2x2
If P (z -2) =0 & z ≠ 2, find the value of z?
  • a)
    -3/2
  • b)
    +1/2
  • c)
    1
  • d)
    +5/2
  • e)
    +7/2
Correct answer is option 'D'. Can you explain this answer?

Siddharth Pillai answered
Solution:

Given, P(x) = 4x3 - 2x2

Let z-2 = k

P(z-2) = P(k) = 4k3 - 2k2

We know that P(k) = 0

Therefore, 4k3 - 2k2 = 0

2k2(2k - 1) = 0

k = 0 or k = 1/2

Now, z-2 = k

So, z = k+2

For k = 0, z = 2

For k = 1/2, z = 5/2

Therefore, the possible values of z are 2 and 5/2, but since z ≤ 2, the only possible value of z is 5/2.

Hence, the correct answer is option (D) 5/2.

$x = 6x +4 and £x = 8x – 2
Find the value of x for which $x = £x?
  • a)
    -3
  • b)
    -2
  • c)
    1
  • d)
    2
  • e)
    3
Correct answer is option 'E'. Can you explain this answer?

Bhavana Kulkarni answered
Solution:

Given, $x = 6x + 4$ and $x = 8x - 2$

Simplifying the above equations, we get

$5x = -4$ and $7x = 2$

Solving for x, we get

$x = -\frac{4}{5}$ and $x = \frac{2}{7}$

Since both values of x are not equal, we cannot find the value of x for which $x = x$ from the given equations.

Therefore, the answer is none of the given options.

Find the minimum value of the function fix) = log2 (x2 - 2x + 5).
  • a)
    -4
  • b)
    2
  • c)
    4
  • d)
    -2
Correct answer is option 'B'. Can you explain this answer?

Alok Verma answered
Method to Solve :

y= x2 – 2x + 5
Step 1  : Differentiate with respect to x
Step 2 : Equate to 0
Step 3 : Find the value of x
dy/dx=2x-2 =0 implies x=1
Hence f(1)= 12 – 2 + 5= 4
Thus minimum value of the argument of the log is 4.
So minimum value of the function is log 4 (base 2) =2

Read the instructions below and solve.
f(x) = f(x – 2) – f(x – 1), x is a natural number
f(1) = 0, f(2) = 1
The value of f(8) is
  • a)
    0
  • b)
    13
  • c)
    –5
  • d)
    –9
Correct answer is option 'B'. Can you explain this answer?

Alok Verma answered
f(1) = 0, f(2) = 1,
f(3) = f(1) – f(2) = –1
f(4) = f(2) – f(3) = 2
f(5) = f(3) – f(4) = –3
f(6) = f(4) – f(5) = 5
f(7) = f(5) – f(6) = –8
f(8) = f(6) – f(7) = 13
f(9) = f(7) – f(8) = –21
13

If 0 < a <1 and 0 < b < 1 and if a < b, which of the following expressions will have the highestvalue?
  • a)
    (a M b)
  • b)
    (a D b)
  • c)
    (a P b)
  • d)
    Cannot be determined
Correct answer is option 'D'. Can you explain this answer?

Yash Patel answered
Again (a + b) or a/b can both be greater than each other depending on the values we take for a and b.
E.g. for a = 0.9 and b = 0.91, a + b > a/b.
For a = 0.1 and b = 0.11, a + b < a/b

Find the maximum value of the function 1/(x2 – 3x + 2).
  • a)
    11/4
  • b)
    1/4
  • c)
    0
  • d)
    None of these
Correct answer is option 'D'. Can you explain this answer?

Aarav Sharma answered
To find the maximum value of the function 1/(x^2 - 3x + 2), we need to determine the critical points of the function and then evaluate the function at those critical points.

1. Finding the critical points:
To find the critical points, we need to find the values of x where the derivative of the function is equal to zero or undefined. The derivative of the given function is given by:
f'(x) = (-2x + 3)/(x^2 - 3x + 2)^2

Setting f'(x) equal to zero and solving for x:
(-2x + 3)/(x^2 - 3x + 2)^2 = 0

Since the numerator can never be equal to zero, the only way for the fraction to be zero is if the denominator is equal to zero. Thus, we need to solve the equation:
x^2 - 3x + 2 = 0

Factoring this quadratic equation, we get:
(x - 1)(x - 2) = 0

Setting each factor equal to zero, we find two critical points:
x - 1 = 0 => x = 1
x - 2 = 0 => x = 2

2. Evaluating the function at the critical points:
We substitute the critical points into the original function to find the corresponding y-values:
f(1) = 1/(1^2 - 3(1) + 2) = 1/0 (undefined)
f(2) = 1/(2^2 - 3(2) + 2) = 1/0 (undefined)

Since the function is undefined at both critical points, there is no maximum value for the given function.

Therefore, the correct answer is option 'D' (None of these).

Read the instructions below and solve.
f(x) = f(x – 2) – f(x – 1), x is a natural number
f(1) = 0, f(2) = 1
What will be the domain of the definition of the function f(x) = 8–xC 5–x for positive values of x?
  • a)
    {1, 2, 3}
  • b)
    {1, 2, 3, 4}
  • c)
    {1, 2, 3, 4, 5}
  • d)
    {1, 2, 3, 4, 5, 6, 7, 8}
Correct answer is option 'C'. Can you explain this answer?

Nikita Singh answered
f(1) = 0, f(2) = 1,
f(3) = f(1) – f(2) = –1
f(4) = f(2) – f(3) = 2
f(5) = f(3) – f(4) = –3
f(6) = f(4) – f(5) = 5
f(7) = f(5) – f(6) = –8
f(8) = f(6) – f(7) = 13
f(9) = f(7) – f(8) = –21
For any nCr, n should be positive and r ≥ 0.
Thus, for positive x, 5 – x ≥ 0
fi x = 1, 2, 3, 4, 5.

Ajesh saves Rs 50,000 every year and deposits the money in a bank at compound interest of 10%(compunded annually).What would be his total saving at the end of the 5th year?
  • a)
    Rs 3,05,255
  • b)
    Rs 1,05,255
  • c)
    Rs 8,05,255
  • d)
    Rs 4,05,255
Correct answer is option 'A'. Can you explain this answer?

Rajeev Kumar answered
 
At the end of the 1st year, he will get Rs 50000, it will give him interest for 4 years compounded annually
Hence at the end of 5 years, this amount will become 50000(1.1)4
Similarly, the amount deposited in the 2nd year will give interest for 3 years. Hence it will become 50000(1.1)3
Similarly, we can calculate for the remaining years.
The total saving at the end of the 5th year would be a GP, given by
Net saving = 50000(1.1)4 +  50000(1.1)3  ..... 50000
Thus net saving = =  Rs 3,05,255

If f(x) = 3x + 6, then what is the value of f (2) + f(7)? 
  • a)
    f(8)
  • b)
    f(9)
  • c)
    f(10)
  • d)
    f(11)
  • e)
    f(12)
Correct answer is option 'D'. Can you explain this answer?

Arjun Iyer answered
Solution:

Given: f(x) = 3x - 6

We need to find f(2) + f(7)

Substituting x = 2 and x = 7 in the given equation, we get:

f(2) = 3(2) - 6 = 0

f(7) = 3(7) - 6 = 15

Therefore, f(2) + f(7) = 0 + 15 = 15

Hence, the correct answer is option D.

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
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?

Sounak Iyer answered
Problem Analysis:
We are given the expressions max(a, max(b, min(c, d))) and max(d, max(a, min(b, c))). We need to determine if these two expressions are equal.

Statement 1: b, c, and d are factors of a.
If b, c, and d are factors of a, it means that a is divisible by b, c, and d. In other words, a should be the multiple of b, c, and d.

Statement 2: a - 2d = b and ca
From this statement, we can deduce that b = a - 2d and c = a - 2d. This implies that b and c are both positive integers.

Combined Analysis:
From Statement 1, we know that a is divisible by b, c, and d. Therefore, a is a multiple of b, c, and d.

From Statement 2, we know that b = a - 2d and c = a - 2d. Substituting these values in the expression max(a, max(b, min(c, d))) gives us:
max(a, max(a - 2d, min(a - 2d, d)))

We can simplify this expression as follows:
1. If a > a - 2d, then max(a - 2d, min(a - 2d, d)) = max(a - 2d, d) = a - 2d
2. If a < a="" -="" 2d,="" then="" max(a="" -="" 2d,="" min(a="" -="" 2d,="" d))="max(a" -="" 2d,="" a="" -="" 2d)="a" -="" />
3. If a = a - 2d, then max(a - 2d, min(a - 2d, d)) = max(a - 2d, d) = a - 2d

Therefore, in all cases, max(a, max(b, min(c, d))) = a - 2d.

Similarly, substituting the values in the expression max(d, max(a, min(b, c))) gives us:
max(d, max(a, min(a - 2d, d)))

We can simplify this expression as follows:
1. If d > a, then max(a, min(a - 2d, d)) = max(a, d) = d
2. If d < a,="" then="" max(a,="" min(a="" -="" 2d,="" d))="max(a," a="" -="" 2d)="" />
3. If d = a, then max(a, min(a - 2d, d)) = max(a, d) = a

Therefore, in all cases, max(d, max(a, min(b, c))) = a - 2d.

Since both expressions simplify to a - 2d, we can conclude that max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))).

Therefore, both statements together are sufficient to answer the question.

Answer: (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

If f(x) = 3x2 – 5x + 9 and g(x) = 4x – 5, then find the value of g( f(x)) at x = 3. 
  • a)
    7
  • b)
    51
  • c)
    56
  • d)
    79
  • e)
    121
Correct answer is option 'D'. Can you explain this answer?

Kiran Nambiar answered
The information provided in the question is:
f(x) = 3x2 – 5x + 9
g(x) = 4x – 5
We have to find out the value of g( f(x)) at x = 3.  
f(x) = 3x2 – 5x + 9 
f(3) = 3*(3)2 – 5*3 + 9 
        = 27 – 15 + 9 
        = 21 
 
g(x) = 4x – 5
g(f(x)) = 4f(x) -5
g( f(3)) = 4f(3) – 5
= 4*21 – 5
               = 84 – 5
               = 79   
 
Answer: Option (D)  

For which of the following functions is f(ab) = f(a) * f(b)?
  1. f(x) = x2
  2. f (x) = √x
  3. f (x) = 2x
  • a)
    I only
  • b)
    I, II and III
  • c)
    III only
  • d)
    I and II
Correct answer is 'D'. Can you explain this answer?

Srestha Basu answered
Ex

The function f(x) = ex satisfies the property f(ab) = f(a) * f(b) because:

f(ab) = eab = ea * eb = f(a) * f(b)

Using the laws of exponents, we can see that eab = ea * eb, which means that f(ab) = f(a) * f(b) for this function.

If f(x) = x + x2, is f(a+1) – f(a) divisible by 4, where a is an integer > 0
(1) f(a) is divisible by 4
(2)  (-1)a < (-1)a+1
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'B'. Can you explain this answer?

Disha Mehta answered
Steps 1 & 2: Understand Question and Draw Inferences
The question wants us to know whether f(a+1) – f(a) is divisible by 4. Let’s simplify the expression given to us.
f(a+1) – f(a) = (a+1) + (a+1)2 – (a + a2)
Simplifying we get f(a+1) – f(a) = (a + 1 –a) + ((a+1)2 -a2))
  • 1 + (a + 1 –a)(a+1+a) = 2 + 2a = 2(1+a)           … (using a2 – b2 = (a-b)(a+b))
  • From the statement above we can conclude that the given expression is always divisible by 2.
Hence, for f(a+1) – f(a) to be divisible by 4, (1+a) must be divisible by 2, which means that a must be odd.
Step 3: Analyze Statement 1
Statement 1 says that f(a) is divisible by 4.
f(a) = a(1+a)
a(1+a) is the product of two consecutive integers. Therefore, one term out of a and 1+a will be even and the other will be odd. The product of these two terms will be even and will always be divisible by 2.
But, we are given that a(1+a) is divisible by 4 also.
This can happen only if
a)      a is divisible by 4 or
b)      1+a is divisible by 4 or
c)       Both a and 1+a are divisible by 2
Case c) is ruled out since a and 1+a are consecutive terms. Therefore, they cannot be both even.
If a is divisible by 4, then a is even.
If 1+a is divisible by 4, then a is odd.
Thus, we cannot determine with confidence whether a is odd or not.
Since Statement 1 does not give us a unique answer, it is not sufficient.
Step 4: Analyze Statement 2
Statement 2 says that (-1)a < (-1)a+1
This is only possible if a is odd, implying that a+1 is even. 
Thus, a is indeed odd.
Since statement 2 gives us a unique answer, it is sufficient to arrive at the conclusion.
 Step 5: Analyze Both Statements Together (if needed)
Since statement 2 gave us a unique answer, this step is not needed.
Correct Answer: B

The function {x} is defined as the lowest odd integer greater than x. What is the value of {x}?
(1) -3.1 < x < -2.5
(2) x2 < 9
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'C'. Can you explain this answer?

Sreemoyee Nambiar answered
Steps 1 & 2: Understand Question and Draw Inferences
Let’s understand the meaning of {x}. If x = 2, then the odd integers greater than 2 are 3, 5, etc. The lowest among them is 3. Therefore {2} = 3.
Step 3: Analyze Statement 1
If x is between -3.1 and -3, then {x} = -3. If x is between -3 and -2.5, then {x} = -1.
INSUFFICIENT.
Step 4: Analyze Statement 2
x2 < 9 only for values of x between -3 and 3.
If x = is between -3 and – 1, {x} = -1, if x is between -1 and 1, {x} = 1 and if x is between 1 and 3, {x} = 3.
 Therefore, {x} can be -1, 1 or 3.
INSUFFICIENT.
Step 5: Analyze Both Statements Together (if needed)
When we combine statements (1) and (2),
We know that
-3.1 < x < -2.5
AND
-3 < x < 3
The common solution is: -3 < x < -2.5
For this set of values, {x} = -1.
SUFFICIENT.
 Answer: Option (C)

If x ≠ 0, for which of following functions is  , for all values of x?
  • a)
  • b)
  • c)
  • d)
  • e)
Correct answer is 'D'. Can you explain this answer?

KS Coaching Center answered
Step 1: Question statement and Inferences
We have to pick the only option for which  , Since the option will have to be true for all the values of x, substituting numbers is not the best approach. We shall try to replace x with 1/x in each option.
Step 2 & 3: Finding required values and finding the final answer
Let’s check each option one by one.
 
A. .
Not equal to f(x)
 B.  
Not equal to f(x)
C. .  Not equal to f(x)
D.  Not equal to f(x)
In the exam, you need not evaluate option (E) – since we already know that Option D is the answer- but here for practice, let’s evaluate it.
Not equal to f(x).

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.
  • a)
    1
  • b)
    2
  • c)
    5
  • d)
    7
  • e)
    8
Correct answer is option 'B'. Can you explain this answer?

Mrinalini Dasgupta answered
To solve this problem, we need to understand the properties of the operation k(m)n and use them to find the value of p in p(137)k.

Understanding the operation k(m)n:
- In the operation k(m)n, the remainder when m is divided by n is k.
- This means that k is the difference between m and the largest multiple of n that is less than or equal to m.

Solving k(13766)9:
- We are given that k(13766)9.
- This means that the remainder when 13766 is divided by 9 is k.
- To find k, we need to find the largest multiple of 9 that is less than or equal to 13766.
- The largest multiple of 9 less than or equal to 13766 is 13761 (9 * 1529).
- Therefore, k = 13766 - 13761 = 5.

Finding the value of p in p(137)k:
- We are given that p(137)k.
- This means that the remainder when 137 is divided by k is p.
- To find p, we need to find the largest multiple of k that is less than or equal to 137.
- Since k = 5, we need to find the largest multiple of 5 that is less than or equal to 137.
- The largest multiple of 5 less than or equal to 137 is 135 (5 * 27).
- Therefore, p = 137 - 135 = 2.

Conclusion:
- The value of p in p(137)k is 2.
- Therefore, the correct answer is option B.

The number of real-valued solutions of the equation 2x+ 2-x = 2 - (x - 2)2 is:
  • a)
    1
  • b)
    2
  • c)
    infinite
  • d)
    0
Correct answer is option 'D'. Can you explain this answer?

Anshul Ghoshal answered
Understanding the Equation
To solve the equation 2x + 2 - x = 2 - (x - 2)², we first simplify both sides.
Left Side Simplification
- The left side is: 2x + 2 - x = x + 2.
Right Side Simplification
- The right side expands as follows:
- (x - 2)² = x² - 4x + 4
- Therefore, 2 - (x - 2)² = 2 - (x² - 4x + 4) = -x² + 4x - 2.
So, the equation simplifies to:
x + 2 = -x² + 4x - 2.
Rearranging the Equation
Now, we bring all terms to one side:
x² - 3x + 4 = 0.
Discriminant Calculation
Next, we check the discriminant (D) of this quadratic equation:
- D = b² - 4ac
- Here, a = 1, b = -3, c = 4.
- D = (-3)² - 4(1)(4) = 9 - 16 = -7.
Conclusion on Real Solutions
Since the discriminant is negative (D < 0),="" the="" quadratic="" equation="" has="" no="" real="" solutions.="" thus,="" the="" number="" of="" real-valued="" solutions="" for="" the="" given="" equation="" is:="" />Final Answer
- 0 (option D).

If [z] denotes the least integer greater than or equal to z and [z2] = 2, which of the following could be the value of [z]?
I. 2
II. 1
III. -2
  • a)
    I only
  • b)
    II only
  • c)
    III only
  • d)
    I and II only
  • e)
    II and III only
Correct answer is option 'A'. Can you explain this answer?

Arjun Iyer answered
Given:
  • The function [z]
  • [z2] = 2
To find: Can [z] be {2, 1, -2}?
Approach:
  1. To find which values of [z] are possible and which are not, we need to first know the range of possible values of z. Once we know what z can be, we’ll be able to find what [z] can be.
  2. We’ll get an idea of the possible values of z from the fact that [z2] = 2
Working Out:
  • [z2] = 2
    • This means, the least integer that is greater than or equal to z2 is 2.
  • So, the value of z2 must be equal to 2 or must lie between 1 and 2 (because if z2 is less than 1 then [z2] would be 1 or less and not 2)
    • We can write: 1 < z2 ≤ 2
  • The above inequality contains 2 inequalities: z2 > 1 AND z2 ≤ 2
    • So now, we’ll solve these inequalities one by one, and then find the values of z that satisfy both these inequalities
    • Solving z2 > 1
  • This means, z < - 1 or z > 1 . . . (1)
  • The gray zones in the above figure represent the overlap zones, that is, those values of z that satisfy both the inequalities
  • So, either -√2 ≤ z < -1 or 1 < z ≤ √2
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

If v* = (1/v), find the value of
v* + (v*)* + ((v*)*)* if v = 1/2.
  • a)
    3
  • b)
    7/2
  • c)
    4
  • d)
    9/2
  • e)
    5
Correct answer is option 'D'. Can you explain this answer?

Abhishek Choudhury answered
Step 1: Question statement and Inferences
A function has been defined called v* such that v* = 1v
 
Step 2 & 3: Finding required values and calculating the final answer
 v = 1/2
=> v*=1v=2
 
=> (v*)* = (2)* = 1/2
=> ((v*)*)* = (1/2)* = 2
=> v* + (v*)* + ((v*)*)* = 2 + 1/2 + 2 = 9/2.
 
Answer: Option (D)

A function f(x) is defined as f(x)=3x2−20x+c, where c is a constant. Also given f(1) =  -16. What is the value of f(c) + f(-c) ?
  • a)
    6
  • b)
    8
  • c)
    10
  • d)
    12
  • e)
    30
Correct answer is option 'B'. Can you explain this answer?

Sankar Desai answered
Given Information:
- Function f(x) = 3x^2 + 20x + c
- f(1) = -16

Calculating the value of c:
Given f(1) = -16
Substitute x = 1 into the function:
3(1)^2 + 20(1) + c = -16
3 + 20 + c = -16
23 + c = -16
c = -16 - 23
c = -39

Calculating f(c) and f(-c):
Substitute x = c into the function:
f(c) = 3c^2 + 20c + c
f(c) = 3(-39)^2 + 20(-39) - 39
f(c) = 3(1521) - 780 - 39
f(c) = 4563 - 780 - 39
f(c) = 3744 - 39
f(c) = 3705
Substitute x = -c into the function:
f(-c) = 3(-c)^2 + 20(-c) - 39
f(-c) = 3(39)^2 - 20(39) - 39
f(-c) = 3(1521) - 780 - 39
f(-c) = 4563 - 780 - 39
f(-c) = 3744 - 39
f(-c) = 3705

Calculating f(c) + f(-c):
f(c) + f(-c) = 3705 + 3705
f(c) + f(-c) = 7410
Therefore, the value of f(c) + f(-c) is 7410, which corresponds to option B.

The function SCI is defined as SCI(x, y) = z, where z is the sum of y consecutive positive integers starting from positive integer x. If a and n are positive integers, is SCI(a,n) divisible by n?
(1) 3n +2a  is not divisible by 2
(2) 3a +2n  is divisible by 2
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'A'. Can you explain this answer?

Kiran Nambiar answered
Steps 1 & 2: Understand Question and Draw Inferences
  • SCI(x, y) = z, where z = x+ x+1 +x+2…….x+y-1 and x, y, z are integers > 0
  • a, n are integers > 0
     
To Find: Is SCI(a, n) = nb, where quotient b is an integer?
  • SCI(a, n) = a+ a+1….a+n -1
  • The terms a, a + 1, a + 2…. a + n -1 are in arithmetic sequence with common difference  = 1
  • SCI(a, n) = Sum of an arithmetic sequence = 
  • For SCI  to be divisible by n,   should be an integer.
    • That is  should be an integer
    • That is,  should be an integer, i.e. n -1 should be divisible by 2.
  • For n -1 to be divisible by 2, n-1 = even, i.e. n = odd.
  • Thus, if n is odd SCI(a, n) is divisible by n, else it is not. Hence, we need to find the even-odd nature of n.
     
    Step 3: Analyze Statement 1 independently
    (1) 3n +2a  is not divisible by 2
  • As 3n + 2a is not divisible by 2, 3n + 2a is odd
  • 3n +2a = odd
    • 3n + even = odd
    • 3n = odd
    • n = odd
  • Since we know the even-odd nature of n, the statement is sufficient to answer.
     
    Step 4: Analyze Statement 2 independently
    (2) 3a +2n  is divisible by 2
  • As 3a + 2n is divisible by 2, 3a + 2n is even
  • 3a + 2n = even
    • 3a + even = even
    • 3a = even
  • Does not tell us anything about the even-odd nature of n.
    Insufficient to answer.
     
    Step 5: Analyze Both Statements Together (if needed)
    As we have a unique answer from step-3, this step is not required.
    Answer: A

Which one of the following is not a document related to fulfill the customs formalities
  • a)
    Letter of insurance
  • b)
    Shipping bill
  • c)
    Export licence
  • d)
    Proforma invoice
Correct answer is option 'D'. Can you explain this answer?

Notes Wala answered
Document Related to Fulfilling Customs Formalities:


There are several documents that are typically required to fulfill customs formalities. However, one of the following is not a document related to fulfilling customs formalities:


A: Letter of insurance
- The letter of insurance is not directly related to customs formalities but rather pertains to insurance coverage for the shipment.
B: Shipping bill
- The shipping bill is a document that contains details about the exported goods and is required by the customs authorities to process the shipment.
C: Export license
- An export license is a document issued by the appropriate government authority that grants permission to export specific goods. It is an essential document for customs clearance.
D: Proforma invoice
- The proforma invoice is a preliminary invoice that provides a detailed description of the goods to be exported, including their value and other relevant information. It is a crucial document for customs purposes.
Answer: D
- The proforma invoice is indeed a document related to fulfilling customs formalities. The correct answer is D, as stated in the question.

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 ca ?
(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).
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?

Pallavi Sharma answered
Steps 1 & 2: Understand Question and Draw Inferences
  • For positive integers x and y where x < y, FP(x, y) returns the smallest prime number between x and y, exclusive
  • FP(a, b) +FP(c, d) = FP(e, f)
    • FP(a, b), FP(c, d) and FP(e, f) each return a prime number
    • So, the above equation conveys that the Sum of two prime numbers is equal to another prime number
    • As all prime numbers except 2 are odd, FP(e, f) will be odd and either of FP(a, b) or FP(c, d) will be even i.e. 2
      • For example, if both FP(a,b) and FP(c, d) are odd, then their sum i.e. FP(e, f) will be even, i.e. 2, which is not possible as there are no prime numbers less than 2
      • So, the only case possible is that either of FP(a, b) or FP(c, d) is 2 and hence the sum of FP(a, b) and FP(c, d), which is FP (e, f) is odd
    • If FP(a, b) = 2
      • a = 1, ca = c
    • If FP(c, d) = 2
      • If c = 1, ca = 1
To Find: Value of c
 
Step 3: Analyze Statement 1 independently
(1) FP(g, h) = a, where g and h are distinct positive integers
  • Since we know that FP(g, h) will return a prime number, we can infer that a is a prime number
  • As a is a prime number, a > 1. So, FP(a, b) ≠ 2
  • Hence FP(c, d) = 2
    • For this to be possible, c = 1
If c = 1, ca = 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).
  • Minimum possible value of the function FP(x, y) = smallest prime number = 2
  • So, c < 2,
  • As c is given to be a positive integer and now we know that c < 2, the only possible value of c = 1
Hence ca = 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

For any non-zero numbers p and q,  What is the value of p$q?
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'C'. Can you explain this answer?

Nitya Kumar answered
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= p2  ( 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
  • However, q may be positive or negative
  • This equation conveys that p is positive. So, |p| = p
  • =13(if q is positive) or − 13 (if q is negative)
Thus, Statement 2 is not sufficient to find a unique value of the ratio 
Step 5: Analyze Both Statements Together (if needed)
  • From Statement 1: q = p2
     
  • From Statement 2: =13 (if q is positive ) or − 13 ( if q is negative)
  • Since q is the square of a non-zero number, q must be positive
  • Combining both statements: =13 (since q is positive ) 
Thus, the two statements together are sufficient to obtain a unique value of 
Answer: Option C

 If [x] denotes the greatest integer less than or equal to x, what is the value of [x]?
(1) When x2 is divided by 9, the quotient is zero.
(2) 2x2+9x+10=0
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'E'. Can you explain this answer?

Anirban Das answered
Steps 1 & 2: Understand Question and Draw Inferences
Given:
  • If x is an integer, then [x] = x
  • If x is a decimal of the form a.bc… where a, b, c etc. are the units, tenths, hundredths . . . digits respectively of x, then [x] = a
    • Example, if x = 2.45, then the greatest integer that is less than x is 2
  • If x is a decimal of the form –a.bc . . . , where here a, b, c etc. are the units, tenths, hundredths . . . digits respectively of x, then [x] = -a – 1
    • Example, if x = -2.45, then the greatest integer that is less than x is -3 (that is, -2 – 1)
To find: [x] = ?
  • To answer this question, we need to know the value of x
  
Step 3: Analyze Statement 1 independently
(1) When x2 is divided by 9, the quotient is zero
  • x2=9∗0+r , where remainder r is less than 9
  • x2=r, where r is less than 9
  • This means, x2 < 9
  • That is, x2 - 32 < 0
  • (x + 3)(x - 3) < 0
  • From wavy line method, we know that the above inequality will be satisfied for: 
-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
 

The operation @ is defined for all distinct and nonzero a and b by 
  • a)
    9
  • b)
    15/2
  • c)
    15/8
  • d)
    1
  • e)
    -15/8
Correct answer is option 'D'. Can you explain this answer?

Pranab Dasgupta answered
Given: 
To find: x = ?
Approach:
  1. We’ll first evaluate the expression for 3@x
  2. Then, we’ll evaluate the expression for 2@(3@x), and finally, will equate the resulting expression with 4. We’ll thus get an equation in x, upon solving which, we’ll get the value of x.
  • Since we’re given that 2@(3@x) = 4, we can write:
Looking at the answer choices, we see that the correct answer is Option D

A function is defined as f (n) = the number of factors of n. If f (p*q*r) = 8, where p, q and r are prime numbers, what is the value of p?
(1) p + q + r is an even number    
(2) q < p < r   
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'E'. Can you explain this answer?

Kirti Roy answered
Steps 1 & 2: Understand Question and Draw Inferences
We are given that:
 f(n) = the number of factors of n and 
f(p*q*r) = 8
This means,
Number of factors of pqr = 8
We have learnt that in order to find the total number of factors of any number X, we should write it in terms of its prime factors:
X = P1m * P2n * P3r . . . where P1, P2, P3 . . .  are the prime factors of X and m, n, r . . . are non-negative integers
The total number of prime factors of X = (m+1)(n+1)(r+1) . . .
Here, it is given that:
Total number of factors of pqr = 8
Now, 8 = 23 = (1+1)(1+1)(1+1)
Thus, total number of factors of pqr = (1+1)(1+1)(1+1)
Comparing this with the above formula, we can deduce that p, q and r are distinct prime numbers.
[Note: We can deduce that p, q and r are distinct prime numbers because if say, p was equal to r, in that case, we could write:
pqr = p2q
The total number of factors in this case would be = (2+1)(1+1) = 3*2 = 6.
Since we are given the total number of factors is not 6, we can say confidently that no two prime numbers are equal to one another]
We know that except 2, all the prime numbers are odd. 2 is the only even prime number.
Step 3: Analyze Statement 1
Statement (1) says:
p + q + r = even number.
 Now, the sum of three integers can be an even number only if:
  1. Either all the integers are even numbers.                     (Think 2 + 4 + 6 = 12)
  2. Or one of them is even and two are odd numbers.     (Think 2 + 3 + 5 = 10)
 
Since p, q, and r are distinct prime numbers, all three of them can’t be even because there is only one even prime number.
 So, the only possible condition is that one of p, q, and r is even number and the other two are odd numbers.
 So, this statement tells us that the value of one of the numbers p, q, and r is 2. However, it does not tell us what the value of p is. 
To conclude
One of p, q, or r is equal to 2.
So, statement (1) alone is not sufficient to answer the question “what is the value of p”.
Step 4: Analyze Statement 2
Statement (2) tells us that q < p < r. So, we know that the prime number p is greater than q and smaller than r. 
However, since there are infinite number of prime numbers possible, we still cannot determine the value of p.
To conclude
q < p < r  
Thus, statement (2) alone is not sufficient to answer the question “what is the value of p”. 
 
Step 5: Analyze Both Statements Together (if needed)
Since the individual analysis of the choices does not tell us the value of p, let’s analyse both the statements together.
From statement (1): One of p, q, or r is equal to 2.
From statement (2): q < p < r
Now, since 2 is the smallest prime number, the value of q is 2. So, p is a prime number greater than 2, but we still don’t know the value of p.
So, analyzing both the statements together is not enough to determine the value of p.  
Answer: Option (E)

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?
  • a)
    I only
  • b)
    II only
  • c)
    III only
  • d)
    I, II and III
  • e)
    None of the above
Correct answer is option 'B'. Can you explain this answer?

Nandita Chauhan answered
Given:
  • Function f(x) is defined as  
  • To find: Which of the 3 expressions is/are true for all values of a?
    Approach:
  • Since the three expressions contain f(a – 1), f(a) and f(1), we’ll first find the expressions for these 3 functions
  • Then, we’ll evaluate the given expressions one by one.
  • Working Out:
  • Finding the expressions for f(a – 1), f(a) and f(1)
​ 
  • Evaluating Expression I
  • f(a−1)=f(a)–f(1)
  • We will assume that f(a-1) = f(a) – f(1) is true and then see if it leads to a real value of a or not.
  • Equating the above expression with the expression for f(a-1):
  • For no real value of a can the square of a+1/2  be negative
  • Therefore, no real value of a satisfies the equation given in Expression I.
  • So, Expression I is not true.
  • Since Left hand side of Expression II has the same value as the right hand side, this expression will hold true for all values of a.
Evaluating Expression III
  • f(a−1)=f(a)∗f(1)
  • The value of the left hand side of Expression III = 1/a
  • The value of the right hand side of Expression III = 1/2(a+1)
  •  
  • Equating the 2 values, we get:
  • Thus, Expression III is true for only 1 value of a (-2) – not for other values.
    • So, Expression III is not a must be true expression.
  • Getting to the answer
    • Thus, we have seen that out of the 3 expressions, only Expression II is a must be true expression.
Looking at the answer choices, we see that the correct answer is Option B
 

If # and $ are mathematical operations such that p # q = (p+ q)2 and p $ q = (p – q)2, what is the value of  (2 # 2) + (3 $ 3) – (5 # 5) + (6 $ 6)? 
  • a)
    -84
  • b)
    -48
  • c)
    48
  • d)
    84
  • e)
    96
Correct answer is option 'A'. Can you explain this answer?

Pranav Das answered
We are given that
p # q = (p+ q)2 and
p $ q = (p – q)2
 
To calculate the value of the whole expression, we need to calculate the values of the individual expressions:
(2 # 2) = (2 + 2)2 = 42 = 16
(3 $ 3)  = (3 – 3)2 = 0
(5 # 5) = (5 + 5)2 = 100
(6 $ 6) = (6 – 6)2 = 0
So,
(2 # 2) + (3 $ 3) – (5 # 5) + (6 $ 6) = 16 – 0 – 100 – 0 = -84  
Answer: Option (A)

A function f(x) is defined as f(x) = x + x2  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.
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'B'. Can you explain this answer?

Pallavi Sharma answered
Step 1 & Step 2: Understanding the Question statement and Drawing Inferences
Given Info:
  • A function f(x) is defined as f(x) = x + x2 , where x is an integer.
  • So function f(x) is given as a quadratic function in x and can be written as
    f(x) = x(x+1)
To find:
  • Is f(a) a prime number?
  • So f(a) = a(a+1)
  • Now f(a) is the product of two consecutive numbers.
  • Product of any two consecutive number will always have one even number and one odd number and thus the product will always be an even number. →
    EvenNumber*OddNumber=EvenNumber
  • , and the only even number which is a prime number is 2.
  • So, f(a) will be prime only when f(a) = 2
  • So we have to find the values of a, for which f(a)=2, to determine whether f(a) is a prime number or not.
  • If f(a) = 2
Subtracting 2 from both sides of the equation, we get
  • So this gives either a=-2 or a=1
  • So for a = -2 or a=1 the above expression of f(a) will be a prime number otherwise it will be not. So from the given statements we will determine the value of a and then conclude whether f(a) is a prime number or not.
Step 3: Analyze statement 1 independently
Statement 1:
  • f(a)+f(a+1) is a prime number
  • We have, f(a)=a(a+1) (Already established above)
  • The above expression will be prime when (a+1)2=1 or f(a) + f(a+1) = 2 . This is so because multiplication of (a+1)2 with 2 will always result in even number and the only even number which is prime is 2.
  • Subtracting 1 from both sides of the equation, we get
  • This means either a=0 or a=-2
  • Now f(a) will be prime in case when a=-2 but it will not be prime when a=0 (As already established above)
  • Hence statement 1 is not sufficient to answer the question
 
Step 4: Analyze statement 2 independently
Statement 2:
  • The greatest common divisor of f(3) & f(4) is f(a)*2.
  • Now, we know  f(x)=x(x+1)
  • So,f(3)=3(3+1)=12
  • And,f(4)=4(4+1)=20
    • Now 12 can be expressed in prime factors as → 12 = 3*22
    • And 20 can be expresses in prime factors as → 20 = 5*22
    • The greatest common factor of 12 and 20 is 22.
    • So the greatest common divisor of f(3) and f(4) will be equal to 4.
  • Now we are given this greatest common divisor is equal to f(a) * 2
  • So we have, f(a) * 2 = 4
⇒ f(a) = 2
  • Now since f(a) = 2, we will have f(a) is a prime number, because 2 is a prime number.
  • Hence statement 2 is sufficient to arrive at a unique answer.
     
Step 5: Analyze the two statements together
  • Since from statement 2 we are able to arrive at a unique answer (as shown in step 4) combining and analysing both statements together is not required.
Hence the correct answer is option B- Only statement 2 is sufficient to arrive at a unique answer.

For a positive integer x, the function f(x) is defined as f(x)=2x2+3x+4
. Is the value of the function f(x) an even number?
  1. x is a multiple of 3
  2. x is a prime number
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'C'. Can you explain this answer?

Nitya Kumar answered
Step 1 & Step 2: Understanding the Question statement and Drawing Inferences
  • The function f(x) is defined as f(x)=2x2+3x+4, where x ≥ 1
  • Now in function f(x), we have three terms being added
  • 1st part: 2x2
    • Now any number multiplied by 2 will always result in an even number.
    • So 2x2 will always be even
  • 2nd part: 3x
    • 3x will be even, when x is even.
    • This is so because, 3 is an odd number and if x → even number, it will result in od dnumber * even number=even number
    • 3x will be odd, when x is odd.
    • This is so because, 3 is an odd number and if x → odd number, it will result in odd number*odd number=odd number
  • 3rd part: 4
    • 4 will always be even
  • Now, 2x2+4=evennumber+evennumber=evennumber
  • So, for f(x) to be even → 3x should be even
  • 3x would be even in cases when x is even (Already established above)
  • So we have to ultimately find the even-odd nature of x to find whether f(x) is even or not.
 
Step 3: Analyze statement 1 independently
Statement 1:
  • x is a multiple of 3.
  • So x can be even (Example: 6) or it can be odd (Example :3)
  • So, we will not be able to determine the nature of x whether it is even or odd,
  • Hence won’t be able to tell whether function f(x) will give an even value or not.
  • Hence statement 1 is not sufficient to answer the question
 
Step 4: Analyze statement 2 independently
Statement 2:
  • x is a prime number.
  • So x can be an even prime number ‘2’ or it can be odd prime number like 3, 5,7.
  • So, we will not be able to determine the nature of x whether it is even or odd
  • Hence won’t be able to tell whether function f(x) will give an even value or not.
  • Hence statement 2 is not sufficient to answer the question.
     
Step 5: Analyze the two statements together
  • From statement 1, we have x is a multiple of 3
  • From statement 2, we have x is a prime number and thus can be {2,3,5,7,….}
  • Only one number 3’ will satisfy both the statements because the number ‘3’ is a multiple of 3 and is also a prime number. All other numbers except ‘3’ will be multiple of 3 and hence will not be prime.
  • Now when x=3, f(x) will be odd, hence its value will not be an even number.
  • So by combining both statements we are able to arrive at a unique answer.
  • Hence the correct answer is option C- Both statements together are necessary to arrive at a unique answer.

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)?
  • a)
    2
  • b)
    3
  • c)
    4
  • d)
    5
  • e)
    Can't be determined
Correct answer is option 'C'. Can you explain this answer?

Pallavi Sharma answered
Given
  • PF(a) = k, where k is the number of prime factors of positive integer a
  • PF(x) = PF(2x)=PF(3x) = 2, where x is an integer > 0
  • PF(y) = PF(5y) = PF(7y) = 2, where y is an integer > 0
To Find: PF(LCM(x, y)?
 
Approach
  1. PF(LCM(x, y) = Number of prime factors of LCM(x, y)
    1. LCM(x, y) will have the same prime factors as x and y
    2. As PF(x) = 2 and PF(y) = 2, LCM(x, y) can have:
      1.  a maximum of 4 prime factors if the prime factors of x and y are distinct
      2. a minimum of 2 prime factors if the prime factors of x and y are same
         
  2. For finding the prime factors of LCM(x, y), we need to know the prime factors of x and y
     
  3. Finding prime factors of x
    1. As PF(x) = PF(2x) = 2, i.e. both x and 2x have 2 prime factors
    2. As PF(x) = PF(3x) = 2, i.e. both x and 3x also have 2 prime factors
    3. We will use the above 2 relations to find the prime factors of x
       
  4. Finding prime factors of y
    1. As PF(y) = PF(5y) = 2, i.e. both y and 5y have 2 prime factors
    2. As PF(y) = PF(7y) = 2, i.e. both y and 7y have 2 prime factors
    3. We will use the above 2 relations to find the prime factors of y
       
  5. Once we find the prime factors of x and y, we can calculate the prime factors of LCM(x, y)

Working Out
  1. Finding prime factors of x
    1. As PF(x) = PF(2x) = 2, i.e. both x and 2x have 2 prime factors  and 2 is one of the prime factors of x
    2. As As PF(x) = PF(3x) = 2, i.e. both x and 3x also have 2 prime factors and 3 is one of the prime factors of x
    3. Thus, x has 2 and 3 as its prime factors.
  2. Finding prime factors of y
    1. As PF(y) = PF(5y) = 2, i.e. both y and 5y have 2 prime factors and 5 is one of the prime factors of y.
    2. As PF(y) = PF(7y) = 2, i.e. both y and 7y have 2 prime factors and 7 is one of the prime factors of y.
    3. Thus, y has 5 and 7 as its prime factors.
  3. As x and y do not have any common prime factor, LCM(x, y) will have {2, 3, 5, 7} as its prime factors.
  4. Thus PF(LCM(x, y)) = 4
 
Answer: C

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?
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'B'. Can you explain this answer?

Devansh Chawla answered
Steps 1 & 2: Understand Question and Draw Inferences
  • Positive integers x and y such that {x/y,y} = 16
  • As 16 is a multiple of y, possible values of y = factors of 16 = {1, 2, 4, 8, 16}
 
  • As x is an integer, y/4 has to be an integer
  • Out of {1, 2, 4, 8, 16}, y/4  is an integer for y = {4, 8, 16]
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.
  • In Steps 1 and 2 above, we’ve inferred that y = {4, 8 or 16}. So, 2y = {8, 16 or 32}. But 16 is not a multiple of 32. So, 2y ≠ 32.
  • Therefore, 2y = 8 or 16
  • Possible values of y = {4 or 8}
 
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,y2} = 16. So, 16 is a multiple of y2.
  • In Steps 1 and 2 above, we’ve inferred that y = {4, 8 or 16}. So, y2 = {16, 64 or 1024}. But 16 is not a multiple of 64 or 1024. Hence y2 = 16
  • Hence y = 4 (since y is given to be a positive integer, y cannot be equal to -4)
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 step-4, this step is not required.
 
Answer: B

If  and  then for positive integers (a, b) the maximum value of  
 is
  • a)
    less than 1
  • b)
    0
  • c)
    -1
  • d)
    Occurs at a = b
  • e)
    cannot be estimated
Correct answer is option 'A'. Can you explain this answer?

Pranav Das answered
To find the maximum value, let’s first create a consolidated expression:
Simplifying we get,
The maximum value of the above expressions happens when a is much larger than b. Let’s say that a is 100 times larger than b, and b = 1 (the smallest allowed value for a positive integer);
hence a = 100.
The first term on right hand side is very close to 1, while the second time is very close to 0. Hence, we can say that the maximum value of the expression is close to but less than 1.
As the difference between a and b increases even further, the first term on the Right Hand Side of (1) will get still closer to 1, and the second term will approach zero, but the sum will still be lesser than 1.
Therefore, the correct answer choice is A

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
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'C'. Can you explain this answer?

Pallavi Sharma answered
Steps 1 & 2: Understand Question and Draw Inferences
  • Function Ro rounds a positive decimal number to its nearest integer starting from its rightmost digit
  • a and b are single digit positive integers
  • 0 < a, b < 10
To Find: value of Ro(64.4ab5)
  • As the function Ro rounds off the number starting from its rightmost digit, in the first step Ro(64.4ab5) = Ro(64.4a(b+1)). This is because the last digit is 5 and hence 64. 4ab5 will be rounded up to 64.4a(b+1).
  • Now, the final value of Ro(64.4a(b+1)) = 64 or 65. Let’s evaluate the possible cases:
    • Ro(64.4a(b+1)) = 64. This will be possible when Ro(64.4a) = Ro(64.4). So, for all the cases where a < 4, Ro(64. 4a) = Ro(64.4) = 64. Please note that if a < 4, the value of b is immaterial, because even if b ≥ 5, the maximum value of a = 4, in which case Ro(64.4a) = Ro(64.4) = 64
  • Ro64.4a(b+1)) = 65. This will be possible if Ro(64.4a) = Ro(64.5). This can happen in following ways:
    • If a ≥ 5 and b is any value OR
    • a =4 and b=>4, then Ro(64.4a(b+1)) = Ro(64.4(a+1)) = Ro(64.45) = Ro(64.5) = 65
Step 3: Analyze Statement 1 independently
(1)Ro(36.4a7) = 37
  • Since the function Ro rounds off starting from the rightmost digit, the first digit that will be rounded off will be a. The rounding of a will depend on the value to the right of it, i.e. 7. So, we have Ro(36.4a7) = Ro(36.4(a+1))
  • For Ro(36.4(a+1)) = 37, Ro(36.4(a+1)) = Ro(36.5), i.e. 4 should be rounded up to 5. This will be possible when a+1 ≥ 5, i.e. a ≥ 4
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
  • Since the function Ro rounds off starting from the rightmost digit, the first digit that will be rounded off will be b. The rounding of b will depend on the value to the right of it, i.e. 6. So, we have Ro(87.b6) = Ro(87.(b+1)) = 88
  • So, b+1 ≥ 5, i.e. b ≥ 4
  • Using the value of b in Ro(64.4a(b+1)) = Ro(64. 4(a+1))
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
  • Ro(64.4ab5) = Ro(64.4a(b+1))
  • As b ≥ 4, b +1 ≥5. So,  Ro(64.4a(b+1)) = Ro(64.4(a+1))
  • Since a ≥ 4, a +1 ≥5. So, Ro(64.4(a+1)) = Ro(64.5) = 65
Sufficient to answer.
 
Answer: C

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
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'B'. Can you explain this answer?

Snehal Banerjee answered
Steps 1 & 2: Understand Question and Draw Inferences
Given: Integer z
  • % operation denotes:
    • either +
    • or –
    • or ×
    • or ÷
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)
  • Since we don’t know what operation % is, let’s consider all the 4 possibilities and see which of these satisfy the constraint that z is an integer:
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
  • Since we don’t know what operation % is, let’s consider all the 4 possibilities and see which of these satisfy the constraint that z is an integer:
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 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
 
 
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are neede
Correct answer is option 'A'. Can you explain this answer?

Maya Choudhury answered
Steps 1 & 2: Understand Question and Draw Inferences
Given:
  • Integer x > 0
Step 3: Analyze Statement 1 independently
(1) The highest power of 2 that divides &(4x) completely is 36
  • This means, &(4x) contains 236 but not any higher power of 2
  • 4x is definitely an even integer.
  • In this case, the power of 2 will only come from the term 24x+2
  • Since the highest power of 2 in &(4x) is 36, we can write:
    • 4x + 2 = 36
  • That is, 4x = 34
  • This doesn’t give an integral value of x. So, this case is rejected.
  • Case 2: x is even
    • In this case, x contains at least 21 but may also contain higher powers of 2.
    • So, we will have to consider these possibilities:
  • So, the only possible value of x = 22 = 4
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
  • Statement 2 tells us that the highest power of 2 that divides 26x is 8
    • 26 is a constant. The power of 2 in this number is 6
  • So, from Statement 2, we infer that the highest power of 2 contained in x is 2
    • So, x is divisible by 22 (that is, 4) but not be 23, 24 etc.
    • Thus, x must be of the form 4 * odd number
  • So, x = 4(2k+1) where k is a positive integer
  • Values such as 4*1, 4*3, 4*5 etc. will all satisfy Statement 2
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
 

If f(z) = 2z+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)?
  • a)
    -3
  • b)
    -2
  • c)
    -1/8
  • d)
    1/8
  • e)
    4
Correct answer is option 'A'. Can you explain this answer?

Pallavi Sharma answered
Given:
f(z) = 2z+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:
  1. Our first step will be to use the definition of f(z) to find the value of each function term in the given equation (for example, the value of f(1), f(2x) etc.)
  2. Then, by substituting these values in the equation, we’ll get an equation in terms of x.
  3. Upon solving this equation, we’ll get the roots of the equation. These are the values of x that satisfy the given equation.
Working Out:
  • Finding the values of all function terms in the equation
  • Solving the equation in terms of x
  • Getting to the answer
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

If the operation a* is defined by  what is the value of 
  • a)
  • b)
    -212
  • c)
    2-12
  • d)
  • e)
    212
Correct answer is option 'E'. Can you explain this answer?

Snehal Banerjee answered
Given:
  • Operation a* is defined by 
  • To find:  
Approach:
  1. We’ll apply the definition of operation a* on the given expression to calculate the value that the expression results in.
    • Note that the given expression applies the operation a* twice: first in the power of 2 within the brackets and then to the whole expression inside the brackets
    • So, we will first find the value of and use it to find the value of  
    • Next, we will find the value of 
​Working Out:
  • Finding the value of
  • Looking at the answer choices, we see that the correct answer is Option E

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?
  • a)
    -24
  • b)
    -16
  • c)
    16
  • d)
    24
  • e)
    Can't be determined
Correct answer is option 'D'. Can you explain this answer?

Nandita Chauhan answered
Given
  • @ = { + , - , *, /}
  • # = { + , - , *, /}
    • @ ≠ #
  • a@b#c = c#b@a, where a, b , c are distinct integers > 0
 
To Find: value of 4@5#4?
  • We need to find the operations that @ and # denote.
Approach
  1. To find the operations that @ and # denote, we will analyse the relation a@b#c = c#b@a.
  2. We will calculate the values of 4@5#4 for all the possible operations denoted by @ and #.
Working Out
  1. As a@(b#c) = (c#b)@a, we can say that @ = {*, +}
  2. Similarly, as (a@b) # c = c#(b@a), we can say that # = {+, *}
  3. Now, let’s evaluate all the possible cases for the operations denoted by @ and #.
    • {@, #} = {*, +} or {*, *} or {+, *} or {+, +}.
    • Since @ and # denote different operations, {*,*} and {+, +} are not the possible cases.
  4. Evaluating the possible cases, we have:
    1. If @ = * and # = +
      1. So, 4@5#4 = 4*5 + 4 = 24
    2. If @ = + and # = *
      1. So, 4@5#4 = 4+5*4 = 24
Thus, we have a unique value of 4@5#4 = 24
 
Answer: D

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