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All questions of Algebra for Computer Science Engineering (CSE) Exam

What is the number of integral solutions of the equation 2x2 – 3x – 2 = 0?
  • a)
    0
  • b)
    1
  • c)
    2
  • d)
    3
  • e)
    4
Correct answer is option 'B'. Can you explain this answer?

Sandeep Mehra answered
Solution:
Given equation is 2x2 + 3x + 2 = 0
We can solve this quadratic equation using the quadratic formula, which states that for an equation of the form ax2 + bx + c = 0, the solutions (also called roots or zeros) are given by:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

Applying this formula to our equation, we get:

x = (-3 ± sqrt(3^2 - 4(2)(2))) / 2(2)
x = (-3 ± sqrt(1)) / 4
x = (-3 ± 1) / 4

Therefore, the solutions are x = -1/2 and x = -1.5
But we have to find the number of integral solutions, i.e., solutions that are integers. In this case, there is only one such solution: x = -1
Therefore, the answer is option B (1).
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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.

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.

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.

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.

 If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is       
  • a)
    13
  • b)
    9       
  • c)
    21
  • d)
    17
Correct answer is option 'C'. Can you explain this answer?

Kirti Roy answered
Let 3 consecutive terms A.P is a –d, a , a + d. and the sum is 51
so, (a –d) + a + (a + d) = 51
⇒ 3a –d + d = 51
⇒ 3a = 51
⇒ a = 17
The product of first and third terms is 273
So  it stand for ( a –d) (a + d) = 273
⇒ a2 –d2 = 273
⇒ 172 –d 2 = 273
⇒ 289 –d 2 = 273
⇒  d 2 = 289 –273
⇒ d 2 = 16
⇒ d = 4
Hence the 3rd terms ( a+d ) is 21

What is the sum of all values of x that satisfies the equation:
  • a)
    20
  • b)
    24
  • c)
    28
  • d)
    32
  • e)
    36
Correct answer is option 'C'. Can you explain this answer?

Yash Patel answered
Correct Answer :- c
Explanation : By picking from the options, we can check which value satisfies the equation 
For x = 28
√168 = √143+1
12.9 = (11.9+1)
12.9 = 12.9

If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?
  • a)
    29
  • b)
    34
  • c)
    81
  • d)
    86
  • e)
    91
Correct answer is option 'A'. Can you explain this answer?

Lavanya Menon answered
Given:
  • Sum of the first 5 terms of an arithmetic sequence = 120
  • Sum of the next 5 terms of the same arithmetic sequence = 245
  • Let the first term of this arithmetic sequence be x1 and let the common difference be d.
To Find:
  • 4th term of the arithmetic sequence.
  • So the 4th term of the sequence will become x1+3d
  • So we need to find the value of x1 and d or the value of x1+3d to find the 4th term of the sequence.
 
Approach:
  • We know that the sum of first n terms of the Arithmetic Sequence is given as
  •  where n is the number of terms in the arithmetic sequence.
  • Using the formula above for the sum of first 5 terms of the sequence, we will get an equation in terms of  and common difference d, as we are given the sum of first 5 terms of the sequence.
  • We are also given the sum of next 5 terms of the sequence. So, we will be able to calculate the sum of first 10 terms of the sequence.
    → Sum of first 10 terms of sequence = Sum of first 5 terms + sum of next 5 terms.
  • Using the formula above for the sum of first 10 terms of sequence, we will get another equation in terms of x1 and common difference d.
  • Using these two equations in x1 and d, we will be able to calculate the value of x1 and d.
  • Knowing the values of x1 and d, we will be able to calculate the fourth term of the sequence, which is equal to x1+3d
Working out:
 
  • Sum of first 5 terms of the arithmetic sequence = 120
  • Putting this in formula of sum of first n terms, where n=5 and z=120, we get
  • Sum of the next 5 terms of the sequence = 245
     
  • Sum of the first 10 terms of the sequence = Sum of the first five terms + Sum of the next five terms.
    • Sum of the first 10 terms of the sequence = 120+245 = 365
       
  • Now, using the formula of the sum of first n terms of an arithmetic sequence, we get
  • Solving Equations 1 and 2.
    • Multiplying ‘equation 1’ by 2, we have 10x1+20d  =240 ...(Equation 3)
Now that we have values of x1  and d. The value of 4th term of the sequence will be
⇒ x1+3(d)=14+3(5)=29
Answer:
  • The value of 4th term of the sequence is 29.
  • Hence the correct answer is option A
Alternate method 
  • Let the first term be 'a' and common dfference between any two cosecutive terms be 'd'
Therefore, 
  • 1st term = a
  • 5th term = a + 4d
  • 6th term = a + 5d
  • 10th term = a + 9d
  • Average of first five terms of an arithemetic sequence  = (First term + Last term)/2 = (a + a +4d) / 2 = a + 2d 
  • Sum of first five terms = Average of first five terms * 5 = (a + 2d) * 5 = 120
  • a + 2d = 120/5 = 24 ---------------- Eq(1)
  • Average of next five terms of the arithemetic sequence  = (First term + Last term)/2 = (a+ 5d + a +9d) / 2 = a + 7d 
  • Sum of five terms = Average of five terms * 5 = (a + 7d) * 5 = 245
  • a + 7d = 245/5 = 49---------------- Eq(2)
Solving Eq(1) and (2) we get
  • d = 5
  • 4th term =
  • a + 3d = (a+2d)+ d = 24 + 5 = 29
Correct Answer: Option A

$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.

List A consists of 10 distinct integers arranged in ascending order. Is the difference between the sixth term and the fifth term of list A greater than 5?
(1) The difference between any two integers in list A is a multiple of 5.
(2) The median of the list is an integer.
  • 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?

Rhea Gupta answered
Given:         
  1. List A = {a1, a2, a3, a4, a5, a6, a7, a8, a9, a10}
    1. consists of all integers,
    2. where a10 > a9> a8…> a2> a1
To Find:  Is a6 – a5 > 5?
Step 4: Analyse Statement 2 independently
The median of the list is an integer.
  • Median =?
    • List contains 10 elements
    • a5 & a6 are integers at the centre of the list (in ascending order)
    • Median of the list = a5+a62
    •  = m, where m is an integer
  • Rearranging, we get a5+ a6 = 2m = even
  • If sum of a6 & a5 is even, then
    • Difference of a6 & a5 is also even
  • No information about values of the terms , so cannot tell if (a6 – a5) > 5
  • Hence, statement 2 is insufficient to answer the question.
Step 5: Analyse Both Statements Together (if needed)
  • From statement 1, a6 – a5 = {5, 10, 15}
  • From statement 2, a6 – a5 = even
  • Combining the two,
    • a6 – a5 = even multiples of 5 = {10, 20, 30…}
    • a6 – a5 > 5
    • So, we can answer the question.
  • Hence statement 1 and statement 2 together are sufficient to arrive at a definite answer.
Answer: C
Step 3: Analyse Statement 1 independently
The difference between any two integers in list A is a multiple of 5.
  • Difference between any two terms of A can be = {5,10,15…}
    • a6 – a5 = {5, 10, 15…}
    • Since the difference can also = 5, we cannot be certain that a6 – a5 > 5.
  • Hence, Statement 1 is insufficient to answer the question.

Ted and Robin start from the same point at 7 AM and drive in opposite directions. Ted doubles his speed after every 90 minutes whereas Robin reduces her speed by half after every 120 minutes. If Ted starts driving at a speed of 10 kilometers/hour and Robin starts driving at a speed of 120 kilometers/hour, how far in kilometers will they be from one another at 1 PM?
  • a)
    195
  • b)
    485
  • c)
    525
  • d)
    645
  • e)
    675
Correct answer is option 'D'. Can you explain this answer?

Srestha Basu answered
Given
  • Ted starts at 7 AM
    • Ted’s initial speed = 10 kilometers/hour
    • Ted doubles his speed after every 1.5 hours
 
  • Robin starts at 7 AM from the same point as Ted and travels in opposite directions
    • Robin’s initial speed = 120 kilometers/hour
    • Robin reduces her speed by half after every 2 hours
To Find: Distance between Ted and Robin at 1 PM
  • Find the distance travelled by Ted in 6 hours
  • Find the distance travelled by Robin in 6 hours
 
Approach
Distance = Speed * Time
1. We need to find the distance travelled by Ted in 6 hours As Ted doubles his speed after every 1.5 hours, he will travel at 6/1.5=4
  1.  different speeds during the period
  2. We know his initial speed and we know that he doubles his speed after every interval of 1.5 hours. So, we can calculate his speed for each interval
 
2.  We need to find the distance travelled by Robin in 6 hours
  1. As Robin reduces her speed after every 2 hours, she will travel at 6/2=3
  1.  different speeds during the period
  2. We know her initial speed and we know that she reduces her speed by half after every interval of 2 hours. So, we can calculate her speed for each interval.
 
Working Out
1.  Ted 
2.  Robin
Distance between Ted and Robin at 1 PM = 225 + 420 = 645 kilometers
Answer: D

If one of the roots of the quadratic equation x2 + bx + 98 = 0 is the average (arithmetic mean) of the  roots of the equation x2 + 28x – 588 = 0, what is the other root of the equation x2 + bx + 98 = 0?
  • a)
    -7
  • b)
    −5/2
  • c)
    5/2
  • d)
    7
  • e)
    21
Correct answer is option 'A'. Can you explain this answer?

Pranab Dasgupta answered
Given
To Find: value of n?
Approach
  1. We know that the product of roots m and n is equal to c/a
  1. So, we can write m*n = 98, i.e. n = 98/m
  2. So, for finding n, we need to find m
  • We are given that m = (p+q)/2
  1. We know that sum of roots of the quadratic equation is −b/a
  2. We will use this relation to find the value of p + q and hence the value of m
Hence, the other root of the equation x2 + bx + 98 = 0 is -7
Answer: A

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?

Snehal Banerjee answered
Steps 1 & 2: Understand Question and Draw Inferences
  • min(x, y) → minimum of x and y, where x and y are integers
  • max(x, y) → maximum of x and y, where x and y are integers
  • a, b, c, d are distinct integers > 0
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, we can write a = bx = cy = dz, where x, y and z are distinct integers > 1
    • For example, since 2, 3 and 6 are factors of 36, we can write: 36 = 2x  =3y = 6z (in this example, x, y and z will be equal to 18, 12 and 6 respectively)
  • Hence, a > {b, c, d}
    • Like, in the above example, 36 > {2, 3, 6}
  • For solving the expression, we need to start from the innermost bracket. Since we do not know which of c or d is smaller, max(a, max(b, min(c, d))) = max(a, max(b, (c or d)))
  • Now, we do not know, which of b, c or d is the greatest,  max(a, max(b, (c or d))) = max (a, (b, c or d))
  • However, we do know that a > b, c, d. So, we can write  max (a, (b, c or d)) = a
  • Similarly, we do not know which b or c is smaller, so, max(d, max(a, min(b, c))) = max (d, max (a, (b or c)))
  • However, we do know that a > b , c. So, we can write  max (d, max (a, (b or c)))= max(d, a)
  • Also, as a > d, we can write max (d, a) = 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
  • Rearranging the terms: a = b + c + 2d. So, a > b, c and d
  • We know that a > b, c, d and for this case we have already evaluated in statement-1 that max(a, max(b, min(c, d))) = max(d, max(a, min(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

Given the equation x2 + bx + c = 0, where b and c are constants, what is the value of c?
(1)   The sum of the roots of the equation is zero.
(2)   The sum of the squares of the roots of the equation is equal to 18.
  • 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?

Kiran Nambiar answered
Solution:

Given equation is x2 bx c = 0, where b and c are constants

To find: value of c

Statement 1: The sum of the roots of the equation is zero.

Let the roots of the equation be α and β. Then, we know that:

α + β = −b
αβ = c

The sum of the roots is zero, so:

α + β = 0
−b = 0
b = 0

From the equation, we can conclude that c = αβ = 0. This statement alone is sufficient to answer the question.

Statement 2: The sum of the squares of the roots of the equation is equal to 18.

Let the roots of the equation be α and β. Then, we know that:

α + β = −b
αβ = c

We are given that:

α2 + β2 = 18

Squaring the equation α + β = −b, we get:

α2 + 2αβ + β2 = b2

Substituting the values of αβ and b2, we get:

α2 + 2c + β2 = b2

α2 + β2 = b2 − 2c

Substituting the given value of α2 + β2, we get:

18 = b2 − 2c

We can solve for c in terms of b:

c = (b2 − 18)/2

However, we do not know the value of b, so we cannot determine the value of c. This statement alone is not sufficient to answer the question.

Together, statements 1 and 2 give us:

b = 0
α2 + β2 = b2 − 2c = 0 − 2c = −2c

Substituting the given value of α2 + β2, we get:

18 = −2c

c = −9

Therefore, both statements together are sufficient to answer the question. The answer is (C).

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)  

If x2+4x+p=13  , where p is a constant, what is the product of the roots of this quadratic equation?
(1) -2 is one of the roots of the quadratic equation
(2) x2+4x+p=13  has equal roots
  • 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?

Devansh Shah answered
Statement (1): -2 is one of the roots of the quadratic equation
Let's assume the quadratic equation is in the form ax^2 + bx + c = 0.
From the given information, we know that one of the roots is -2. This means that when x = -2, the quadratic equation is satisfied. Plugging this value into the equation, we get:
(-2)^2 + 4(-2) + p = 13
4 - 8 + p = 13
-4 + p = 13
p = 17

Statement (2): x^2 + 4x + p = 13 has equal roots
This means that the discriminant of the quadratic equation is zero. The discriminant is given by b^2 - 4ac.
In this case, a = 1, b = 4, and c = p - 13.
So, the discriminant is:
(4)^2 - 4(1)(p - 13) = 0
16 - 4p + 52 = 0
-4p + 68 = 0
-4p = -68
p = 17

Combined solution:
From both statements, we have determined that p = 17. Therefore, we can find the roots of the quadratic equation:
x^2 + 4x + 17 = 0
Using the quadratic formula, the roots can be calculated as follows:
x = (-b ± √(b^2 - 4ac)) / (2a)
x = (-4 ± √(4^2 - 4(1)(17))) / (2(1))
x = (-4 ± √(16 - 68)) / 2
x = (-4 ± √(-52)) / 2
Since the discriminant is negative, the roots are imaginary and the product of the roots will also be imaginary. Therefore, the product of the roots cannot be determined based on the given information.

Answer:
Neither statement alone nor both statements together are sufficient to answer the question asked.

A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following could be the value of the ordered set (a, b, c)?
I. (-1, 4, -3)
II. (1, 4, 3)
III. (3, -10√3, 9) 
  • a)
    I Only
  • b)
    II Only
  • c)
    III Only
  • d)
    I and II Only
  • e)
    I, II and III Only
Correct answer is option 'D'. Can you explain this answer?

Nitya Kumar answered
Given
To Find: Values of (a, b, c)
Approach
  • For finding the values of (a, b, c), we would first need to find the value of x1 , x2
    • We will use the relation    
      • to find out the values of x1 , x2
      • Also, we will keep in mind the constraint that x1 , x2 are integers
    • Now, we know that  
  • We will use the above relation to find out the possible values of (a, b, c)
As x1 , x2 are integers, the possible cases for (x1 , x2) is either  (3,1) or (-3,-1)
  •  Using (1), (2) and (3), the values of (a, b, c) can be of the form ( a, 4a, 3a) or (a, -4a, 3a)
  • Among the options,
    • Option-I (-1, 4, -3) is of the form (a, -4a, 3a) and
    • Option- II (1, 4, 3) is of the form (a, 4a, 3a)
Hence, options I and II can be the value of ordered set (a, b, c).
Answer: D

The sequence a1, a2,…an is such that an = an-1 +n*d for all n > 1, where d is a positive integer. If a3 = 20 and a5 = 47, what is the value of a7?
  • a)
    53
  • b)
    65
  • c)
    75
  • d)
    80
  • e)
    86
Correct answer is option 'E'. Can you explain this answer?

Kavya Chakraborty answered
Given
  • A sequence a1, a2,…an
  • an = an-1 +n*d for all n > 1, where d is an integer > 0
  • a3 = 20
  • a5 = 47
To Find: a7?
Approach
  1. As an = an-1 +n*d, we can express a7 in terms of a1 and d
    1. So, we need to find the value of a1 and d.
  2. As we are given the values of a3 and a5, we will express them in terms of a1 and d to get 2 equations in a1 and d.
  3. We will then solve these two equations to find out the value of a1 and d.
Working Out
Solving (1) and (2), we have a1 = 5 and d = 3
Answer: E

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 p and q are the roots of the quadratic equation ax2 + bx + c = 0 where a ≠ 0, what are the roots of the equation ayx2 + byx + cy = 0 where 0 < y ≤ 1?
  • a)
    p and q
  • b)
    py and qy
  • c)
    p/yand q/y
  • d)
    y and -y
  • e)
    Cannot be determined
Correct answer is option 'A'. Can you explain this answer?

Rajdeep Nair answered
Given
  • ax2+bx+c=0
  •  has two roots p and q a ≠ 0, 0 < y ≤ 1
To Find 
  • Roots of ayx2+byx+cy=0
 
Approach
  1. To find the roots of the equation ayx2+byx+cy=0 , we will do the following steps:
    1. Take y common and the get the equation in terms of ax2+bx+c=0
    2. As we know the roots of ax2+bx+c=0, we will use this knowledge to find the roots of the equation ayx2+byx+cy=0
  •  
    Working Out
    Solving the equation ayx2+byx+cy=0
 
  • ayx2+byx+cy=0
  • y(ax2+bx+c)=0
So we have either y = 0 or ax2+bx+c=0
Since we are given that y > 0, y ≠ 0.
Hence ax2+bx+c=0
.As the roots of the equation ax2+bx+c=0  are p and q, the roots of the equation ayx2+byx+cy=0  will also be p and q.
Answer: A

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?

Snehal Banerjee answered
Given Info:
  • Function f(x) is given as f(x)=3x2−20x+c
  • , where c is a constant
  • The above function is a quadratic function in x
  • Also given f(1)=−16
 
To Find:
  • Value of f(c)+f(−c)
⇒f(c)+f(−c)=6c2+2c
Approach:
 
Working out:
  • Now, f(c)=3c2−20c+c
  • (Putting value of c in the given function → f(x)=3x2−20x+c
  • And, f(−c)=3c2+20c+c
  • Adding both functions, we get
    ⇒f(c)+f(−c)=3c2−20c+c+3c2+20c+c
    • Now in order to calculate the above value, we need to determine the value of c.
    • To determine the value of c, we will work on the quadratic function in x. We know the value of function at x=1 as’ -16’ as given in the question, so we will calculate the value of c by putting the value of f(x) and the value of x.
    • After knowing the value of c from the given value of the function at x=1, we will find the value of f(c)+f(−c)
    • f(x)=3x2−20x+c
    • Given f(1)=−16
⇒ f(1)=3(12)−20(1)+c
⇒ f(1)=c−17
  • Now f(1)=−16
⇒ c−17=−16
⇒ c=1
  • Now we have already established above, f(c)+f(−c)=6c2+2c
  • Putting value of c from above, we get
⇒ f(c)+f(−c)=6(1)2+2(1)
⇒ f(c)+f(−c)=8
Answer
  • So the value of f(c)+f(−c) is 8
  • Hence answer option B is correct.

Mike took 5 mock tests before appearing for the GMAT. In each mock test he scored 10 points more than the previous mock test. If he scored 760 on the GMAT and his average score for the mocks was 720, what is the difference between his last mock score and his GMAT score?
  • a)
    10
  • b)
    20
  • c)
    30
  • d)
    40
  • e)
    50
Correct answer is option 'B'. Can you explain this answer?

Rhea Gupta answered
Given
Mike took 5 mock tests
  • Let his score in the 1st mock be x.
  • So, his scores in the other mocks = x+ 10, x+20, x+30, x + 40
  • Mike’s score on GMAT = 760
  • Average score of mocks = 720
To Find: 
  • 760 – (x +40)
 
Working Out
Average score of mocks = (x + x + 10 + … x + 40) /5 = (5x + 100)/5 = x + 20
(The other way to think about this is, as Mike’s scores in the mocks are in arithmetic sequence, average will be the middle term)
  • Mike’s average score in the mocks = x + 20 = 720
  • x = 700
Therefore
  • x + 40 = 740
  • 760 – (x+40) = 760 – 740 = 20
 
Correct Answer: Option B

If twice the reciprocal of x is equal to the sum of x and 1, how many values of x are possible?
  • a)
    None
  • b)
    One
  • c)
    Two
  • d)
    Three
  • e)
    An infinite number
Correct answer is option 'C'. Can you explain this answer?

Prashanth Chawla answered
Problem Analysis:
We are given that twice the reciprocal of x is equal to the sum of x and 1. Let's break down the statement:
- The reciprocal of x is 1/x.
- Twice the reciprocal of x is 2/x.
- The sum of x and 1 is x + 1.

So, we can write the given equation as 2/x = x + 1.

Solving the Equation:
To solve the equation, we can multiply both sides by x to eliminate the fraction:
2 = x(x + 1).

Expanding the equation:
2 = x^2 + x.

Rearranging the equation to standard quadratic form:
x^2 + x - 2 = 0.

Now, we need to solve this quadratic equation to find the possible values of x.

Factoring the Quadratic Equation:
To factor the quadratic equation, we need to find two numbers whose product is -2 and whose sum is 1.

The numbers that satisfy these conditions are 2 and -1:
x^2 + 2x - x - 2 = 0.
x(x + 2) - 1(x + 2) = 0.
(x - 1)(x + 2) = 0.

Setting each factor equal to zero:
x - 1 = 0 or x + 2 = 0.

Solving for x:
x = 1 or x = -2.

Conclusion:
We have found two values of x that satisfy the given equation: x = 1 and x = -2. Therefore, the correct answer is option C: Two.

Given the three quadratic equations above, which pair of equations has at least one common root?
  • a)
    I and II
  • b)
    II and III
  • c)
    I and III
  • d)
    I, II and III
  • e)
    None of the above
Correct answer is option 'E'. Can you explain this answer?

Nandita Chauhan answered
Given
  • Three quadratic equations:
    • x2 – 16x + 55 = 0
    • 2x2 – 4x - 70 = 0
    • x (x +7) = 44
To Find: Which pair of equations has at least one common root?
Approach
  1. As we need to find the common roots among the equation, we will solve each quadratic equation to find their roots and then find the pair of equations that have at least one common root.
Working Out
So, the roots of Equations I, II and III are (5, 11), (-5, 7) and (-11, 4) respectively.
Hence, none of the equations have even one root in common.
Answer: E

r and s are the roots of the quadratic equation ax2 + bx + c = 0 where a ≠ 0 & s >0, such that r is 50 percent greater than s. If the product of the roots of the equation is 150, what is the sum of the roots of the equation?
  • a)
    -25
  • b)
    -15
  • c)
    15
  • d)
    25
  • e)
    Cannot be determined
Correct answer is option 'D'. Can you explain this answer?

Hridoy Gupta answered
To solve this problem, let's start by assigning variables to the roots of the quadratic equation. Let's say that r is one root and s is the other root.

Given that r is 50% greater than s, we can express this relationship as:

r = 1.5s

We are also given that the product of the roots is 150. This means that:

rs = 150

Now, let's express the quadratic equation in terms of the roots:

ax^2 + bx + c = 0

Since r and s are the roots, we can write the equation as:

a(x - r)(x - s) = 0

Expanding this equation, we get:

ax^2 - (ar + as)x + rs = 0

Now, let's substitute the values of r and s into the equation:

ax^2 - (1.5as + as)x + 150 = 0

Simplifying further, we get:

ax^2 - 2.5asx + 150 = 0

Since the quadratic equation is in the form ax^2 + bx + c = 0, we can equate the coefficients to find the sum of the roots:

b/a = -2.5as/a

b = -2.5as

Since the sum of the roots is equal to -b/a, we can substitute the value of b:

Sum of roots = -(-2.5as)/a

Sum of roots = 2.5s

Now, we can substitute the value of rs = 150 into the equation rs = 150:

s(1.5s) = 150

1.5s^2 = 150

Dividing both sides by 1.5, we get:

s^2 = 100

Taking the square root of both sides, we get:

s = 10

Now, substituting the value of s into the equation for the sum of the roots:

Sum of roots = 2.5s

Sum of roots = 2.5(10)

Sum of roots = 25

Therefore, the sum of the roots of the quadratic equation is 25. So, the correct answer is option D) 25.

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

What is the value of s+t if the equation 3x2 + tx + s has 2 and 3 as its roots?
  • a)
    1
  • b)
    2
  • c)
    3
  • d)
    4
  • e)
    5
Correct answer is option 'C'. Can you explain this answer?

Palak Yadav answered
Solution:
Given that the equation 3x² - tx + s has 2 and 3 as its roots.
Let's use the sum and product of roots formula to find the value of s and t.
Sum of roots = -b/a = 2 + 3 = 5
Product of roots = c/a = 2 * 3 = 6
Substituting the values in the equation, we get:
3x² - (sum of roots)x + (product of roots) = 0
3x² - 5x + 6 = 0
Comparing this with the given equation, we can see that t = 5 and s = 6.
Therefore, the value of s t is s * t = 5 * 6 = 30.
But the question asks for the value of s t, given that the correct options are integers from 1 to 5. Therefore, we need to find a value of t such that s t is an integer between 1 and 5.
We know that s = 6. Therefore, to get an integer value for s t, we need to choose t such that t is a factor of 6 and an integer between 1 and 5.
The factors of 6 are 1, 2, 3, and 6. Out of these, only 3 is an integer between 1 and 5.
Therefore, t = 5 and s = 6, which gives s t = 6 * 3 = 18.
Hence, the correct answer is option C.

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?

Snehal Banerjee answered
Given:
  • For positive integers k, m and n, k(m)n implies k is the remainder when m is divided by n
  • k(13766)9 → k is the remainder when 13766 is divided by 9
  • p(137)k → p is the remainder when 137 is divided by k
  • p is a positive integer
To find: p = ?
Approach:
  1. To find the value of p, we need to evaluate the expression p(137)k
    • However, this expression also contains k
    • So, we’ll be able to find the value of p only when we first know the value of k
  2. We can find the value of k by evaluating the expression k(13766)9
 
Working Out:
  • Finding the value of k
    • k(13766)9 means that the remainder when 13766 is divided by 9 is k.
    • Sum of digits of 13766 = 1+3+7+6+6 = 23
    • So, remainder when 13766 is divided with 9 = Remainder that 23 leaves with 9 = 5
    • Thus, k = 5
  • Finding the value of p
    • p(137)k means that the remainder when 137 is divided by k is p
      • We’ve inferred above that k = 5
      • So, the above expression means that when 137 is divided by 5, the remainder is p
    • Now, 137 leaves a remainder of 2 with 5
    • So, p= 2
 
Looking at the answer choices, we see that the correct answer is Option B

The nth term of an increasing sequence S is given by Sn = Sn-1 + Sn-2 for n > 2 and the nth term of a sequence S’ is given by S’n = S’n-1 - S’n-2 for n > 2. If S5 = S’5, what is the average (arithmetic mean) of S2 and S’2?
(1) The difference between the fourth term and the second term of sequence S is 14.
(2) The sum of the fourth term and the second term of sequence S’ is 14.
  • 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?

Aarav Sharma answered
Steps 1 & 2: Understand Question and Draw Inferences
  • Increasing sequence S whose nth term is represented as  Sn = Sn-1 + Sn-2 for n > 2
  • For a sequence S’, the nth term of which is represented as S’n = S’n-1 - S’n-2 for n > 2
  • S5 = S’5
​
Thus we need to find the value of S3 to find the average of S2 andS′2
 
Step 3: Analyze Statement 1 independently
(1) The difference between the fourth term and the second term of sequence S is 14
  • S4−S2=14
  • Substituting the expression of S4=S3+S2
  • (S3+S2)−S2=14,i.e.S3=14
As we know the value of S3, the statement is sufficient to answer.
 
Step 4: Analyze Statement 2 independently
(2) The sum of the fourth term and the second term of sequence S’ is 14.
  • S′2+S′4=14
  • Substituting the expression of S′4=S′3−S′2
  • S′2+(S′3−S′2)=14
  • S′3=14
Does not tell us anything about the value of S3, the statement is 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

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)

In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?
(1)    3 and 4 are roots of the equation.
(2)   The product of the roots of the equation is 12.
  • 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?

Ameya Yadav answered
Understanding the Problem
To find the value of \( b \) in the equation \( ax^2 + bx + c = 0 \), we can use the relationships between the roots of the quadratic equation. The roots of the equation can be denoted as \( r_1 \) and \( r_2 \).
Key Relationships
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)
Evaluating Statement (1)
- Statement (1): 3 and 4 are roots of the equation.
- From this, we can calculate:
- Sum of the roots: \( 3 + 4 = 7 \)
- Therefore, \( -\frac{b}{a} = 7 \) implies \( b = -7a \).
- However, without knowing the value of \( a \), we cannot determine a unique value for \( b \). Thus, statement (1) alone is insufficient.
Evaluating Statement (2)
- Statement (2): The product of the roots of the equation is 12.
- This gives us: \( r_1 \cdot r_2 = 12 \) implies \( \frac{c}{a} = 12 \).
- We cannot determine \( b \) as we still lack information about the sum of the roots. Thus, statement (2) alone is also insufficient.
Combining Statements (1) and (2)
- Combining both statements:
- From statement (1), we have \( b = -7a \).
- From statement (2), we know the product of the roots is 12, but we still lack the actual values of \( a \) and \( c \).
- Therefore, even together, the statements do not provide enough information to uniquely determine \( b \).
Conclusion
Thus, the correct answer is option E: Statements (1) and (2) together are NOT sufficient.

Edward invested five-ninths of his money at an annual rate of 2r% compounded semi-annually, and the remaining money at an annual rate of r% compounded annually. If after one year, Edward’s money had grown by one-thirds, the value of r is equal to which of the following?
  • a)
    10%
  • b)
    15%
  • c)
    20%
  • d)
    25%
  • e)
    33%
Correct answer is option 'C'. Can you explain this answer?

Tanishq Choudhury answered
Given: Let the total money be y
  • First Investment:
    • ​
      • Rate of interest = 2r% p.a. compounded every 6 months = r% per 6-months
    • Time = 1 year
  • Second Investment:
    • ​
      • Rate of interest = r% p.a.
      • Time = 1 year
  • Total Interest earned in 1 year 
To find:  The value of r
Approach
1.  Total interest earned in 1 year = (Interest earned from 1st investment) + (Interest earned from 2nd investment)
i.  In the given time frame of 1 year, the 1st investment will pay interest twice (since this investment pays interest every 6 months). So, the formula for compound interest will be applicable for the 1st investment
ii.  The 2nd investment pays interest after 1 year. Since the given time frame is also 1 year, this investment will yield simple interest
2.  The only unknown in the above equation will be r. So, using this equation, we can find the value of r
 
Working Out
  • Amount of the first investment after 1 year 
  • Compound Interest earned from 1st investment 
 
  1. Calculating the interest earned from the 2nd investment:
    1. ​Simple interest earned from 2nd investment =   
    2. So, Total interest earned in 1 year = 
      2. Multiplying both sides of the above equation with 9/y :
  • Rejecting the negative value since the money is given to have grown.
​
Looking at the answer choices, we see that Option C is correct
(Note: You could also have solved this question by framing the first equation in terms of the amount that each investment grows to, as under:
(Total Amount after 1 year) = (Amount that the 1st investment grows to) + (Amount that the 2nd investment grows to) 
This equation leads to a similar calculation and the same result as in the solution above /End of Note)

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?

Pallavi Choudhury answered
**Statement (1) alone**: f(a) is divisible by 4.

In order to determine if f(a+1) - f(a) is divisible by 4, we need to understand the function f(x) = x * x^2.

Let's substitute a into the function:
f(a) = a * a^2 = a^3

So, statement (1) tells us that a^3 is divisible by 4.

To determine if f(a+1) - f(a) is divisible by 4, we can simplify the expression:

f(a+1) - f(a) = (a+1) * (a+1)^2 - a * a^2
= (a+1) * (a+1) * (a+1) - a^3
= (a+1)^3 - a^3

To check if (a+1)^3 - a^3 is divisible by 4, we can factorize it using the binomial theorem:

(a+1)^3 - a^3 = a^3 + 3*a^2 + 3*a + 1 - a^3
= 3*a^2 + 3*a + 1

Now, if we substitute a = 0 into the expression, we get:

3*(0)^2 + 3*(0) + 1 = 1

Since 1 is not divisible by 4, we can conclude that statement (1) alone is not sufficient to determine if f(a+1) - f(a) is divisible by 4.

**Statement (2) alone**: (-1)^a * (-1)^(a+1)

Statement (2) can be simplified as (-1)^a * (-1)^(a+1) = (-1)^a * (-1)^a * (-1) = (-1)^(2a) * (-1)
= 1 * (-1) = -1

This means that (-1)^a * (-1)^(a+1) is always equal to -1, regardless of the value of a. Since -1 is not divisible by 4, we can conclude that statement (2) alone is not sufficient to determine if f(a+1) - f(a) is divisible by 4.

**Statements (1) and (2) together**:

By combining statements (1) and (2), we know that a^3 is divisible by 4 and (-1)^a * (-1)^(a+1) is equal to -1.

However, even with this information, we still cannot determine if f(a+1) - f(a) is divisible by 4.

For example, let's consider two cases:

Case 1: a = 2
In this case, a^3 = 2^3 = 8, which is divisible by 4. Also, (-1)^a * (-1)^(a+1) = (-1)^2 * (-1)^3 = 1 * (-1) = -1.
If we substitute a = 2 into the expression (a+1)^3 - a^3, we get:
(2+1)^3 - 2^3 = 3^3 - 8 = 27 - 8 = 19, which is not divisible by 4.

Case 2: a

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?

Rajeev Kumar 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).

If the sum of the first 30 positive odd integers is k, what is the sum of first 30 non-negative even integers?
  • a)
    k-29
  • b)
    k-30
  • c)
    k
  • d)
    k+29
  • e)
    k+30
Correct answer is option 'B'. Can you explain this answer?

Moumita Sen answered
Given
  • 1 + 3 + 5………..30*2 -1 = k.
    • Let’s call this sequence O
 
To Find: 0 + 2+ 4………30 *2 -2 = ?
  • Let’s call this sequence E
Approach
  1. To express the sum of sequence E in terms of k, we need to express the terms of sequence E in terms of sequence O
    1. Now, we see that we can write 0 = 1 – 1
    2. Similarly, we can write 2 = 3 -1
    3. Continuing the same pattern, we can write 58 = 59 -1
    4. Observe that 1, 3…..59 are terms of sequence O. So, using the above process we have captured all the terms of sequence O in sequence E
  2. We will use the above logic to represent the sum of sequence E in terms of k
Working Out
  1. 0 + 2+ 4……58 = (1-1) + (3-1) +…….(59- 1) = 1+ 3+ 5…….59 – (1 + 1 + 1……….30 times)
  2. 0 + 2 + 4 …….+ 58 = k – 30
 
Answer: B

240 students are to be arranged in n rows in the assembly hall of a school. If  the first row is the closest to the stage and each subsequent row has 10 more students than the row ahead of it, what is the value of n?
(1) There are 45 students in the 4th row from the stage.
(2) The number of students in the nth row is 10 less than 5 times the number of students in the first row.
  • 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?

Rhea Gupta answered
Steps 1 & 2: Understand Question and Draw Inferences
  • As the number of students in a row is 10 more than the number of students in the previous row, the number of students in the rows form an arithmetic sequence with common difference, d = 10
  • Number of rows = n
  • Let the number of students in the first row be a1
 and that in the last row be an
 (average of the 1st and the nth term is the average of all the terms of the sequence)
To Find: Unique value of n
  • Need to find the value of a1  or express a1  in terms of n to find the value of n.
Step 3: Analyze Statement 1 independently
(1) There are 45 students in the 4th row from the stage.
Substituting the value of a1 in (1), we have
As n cannot be negative, n = 6.
Sufficient to answer.
 
Step 4: Analyze Statement 2 independently
(2) The number of students in the nth row is 10 less than 5 times the number of students in the first row.
​
Putting the value of in (1), we have
As n has to be an integer, n = 6.
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

In the sequence S, the difference between any two consecutive terms is equal. If the sum of the fourth term and the fifth term of the sequence is equal to the seventh term of the sequence, what is the value of the second term of the sequence?
  • a)
    -4
  • b)
    0
  • c)
    4
  • d)
    8
  • e)
    Can't be determined
Correct answer is option 'B'. Can you explain this answer?

Sharmila Singh answered
Given
  • In a sequence S, difference between any two terms is equal
    • S is an arithmetic sequence
    • Let the first term of the sequence be a and the common difference be d
  • 4th   term + 5th  term = 7th  term
    • 4th  term = a + (4-1)*d = a+ 3d
    • 5th  term = a + (5-1)*d = a +4d
    • 7th term = a + (7-1)d = a + 6d
To Find: a + d = ?
Approach
  1. To find the value of a + d, we need to either find the values of a and d or a + d
  2. We are given a relation
    1. (a + 3d) + (a + 4d) = a + 6d
    2. As we are given only one equation between a and d, we will be unable to find the values of both a and b. So, we will try to see if we can find the value of a + d
 
Working Out
  1. (a + 3d) + (a+4d) = a+ 6d
    1. 2a + 7d = a + 6d
    2. a + d = 0
Hence, the value of the 2nd term of the sequence = a + d = 0
Answer: B

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 is an integer and   what is the value of x?
  • a)
    -3
  • b)
    -1
  • c)
    0
  • d)
    2
  • e)
    3
Correct answer is option 'E'. Can you explain this answer?

Devika Yadav answered
Given equation is
And, we need to find the value of x, given that x is an integer.
Step 2 & 3: Finding required values and calculating the final answer
Let us rewriting the last term of (I) to get x-1 in the denominator:
Let us now bring the terms with the same denominator together:
  • (x + 1)2 = {2(x – 1)}2
  • [(x+1)2-{2(x-1)}2 ]=0 ………….. (II)
  • Observe that this is of the form  a2 – b2 = (a+b)(a-b)
     
    So let us say:
    a = x + 1             
    b = 2(x – 1) = 2x - 2
     
    Therefore:
    a + b = x + 1 + 2x – 2 = 3x -1
    a – b = x + 1 - 2x + 2 = -x + 3
     
    Therefore we can rewrite (II) as:
    (3x-1)(3-x)=0
  • Putting each factor to zero:
Since it is given that x is an integer,
Thus, x = 3 is the required answer.
 
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 x and y are non-zero numbers, what is the value of 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 'E'. Can you explain this answer?

Manasa Gupta answered
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
Not sufficient to get a unique value of the ratio x/y.
Step 4: Analyze Statement 2 independently
Multiplying both sides of the inequality with  will not impact the sign of inequality:
From the Wavy Line Method:
Not sufficient to determine the exact value.
 
Step 5: Analyze Both Statements Together (if needed)
From Statement 1:
From Statement 2:
Both the values of   obtained from Statement 1 is less than −1/5
Therefore, both these values satisfy Statement 2
So, we are still not able to determine a unique value of the ratio x/y
Answer: Option E

if   and x is not a positive integer, x = ?
  • a)
    0
  • b)
    √2
  • c)
    √3
  • d)
    2√2
  • e)
    4√2
Correct answer is option 'A'. Can you explain this answer?

Devika Yadav answered
Step 1: Question statement and Inferences
Given that x is not a positive integer and we need to find the value of x from the given equation.
Step 2 & 3: Finding required values and calculating the final answer
Given equation is
Squaring on both sides
​Squaring on both sides
8x  =x2
  • x2 -8x=0
  • x(x -8)=0
 Putting each factor to zero, we get:
x = 0 or x = 8
However, since it is already given that x is not a positive integer, x≠8.
Therefore x = 0.
 Answer: Option (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.

If x + y = -10 and xy = 16, which of the following statements must be true?
(I) x – y = -6
(II) x2 + y2 = 68
(III) (x-y)2 leaves a remainder of 1 when divided by 7
  • a)
    I only
  • b)
    II only
  • c)
    III only
  • d)
    I and II
  • e)
    II and III
Correct answer is option 'E'. Can you explain this answer?

Pranav Das answered
Step 1: Question statement and Inferences
It is given that
 x+y=−10 ....(i)
xy=16  ....(ii)
We need to find which of the given statements must be true.
Step 2 & 3: Finding required values and calculating the final answer
Squaring (i) on both sides, we get:
(x+y)2=100
 
x2+y2+2xy=100
....(iii)
We know that
(x−y)2=x2+y2−2xy
(x−y)2=x2+y2+2xy−4xy
(x−y)2=(x+y)2−4xy
Using (ii) and (iii) we get:
(x−y)2=100−64
(x−y)2=36
⇒(x−y)=±6
Therefore (I) is not a must be true statement.
Now let us look at (II).
(iii) – 2*(ii) gives us:
(x2+y2+2xy)−2xy=100−32
⇒x2+y2=68
Therefore statement II is true.
We already observed above that
(x−y)2=36
We know that
36=35+136=7∗5+1
Therefore
(x−y)2
 leaves a remainder of 1 when divided by 7.
Answer: Option (E)

Steven and Stuart took a job in different companies at the same time. Steven’s salary increased by a fixed amount at the end of every year and Stuart’s salary increased by a fixed percentage at the end of every year. If the increase in the salary of Steven at the end of the third year was equal to the increase in the salary of Stuart at the end of the second year, what was the difference in the salaries of Steven and Stuart when they took the job?
(1) Steven’s salary after 2 years was 20% more than the salary at which he took the job
(2) The increase in the salary of Stuart at the end of the second year was 11% of the salary at which he took the job.
  • 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?

Kavya Chakraborty answered
Steps 1 & 2: Understand Question and Draw Inferences
  • Let Steven’s initial salary be S1
 and Stuart’s initial salary be S1
  •  
  • Let Steven’s salary increase by x every year and Steven’s salary increase by y% every year.
  • Steven’s salary increase in the 3rd year = x
  • Stuart’s salary increase at the end of the 2nd year  = y% of (Stuart’s salary in the 2nd year)
  • Steven’s salary increase in the 3rd year = Stuart’s salary increase in the 2nd year
To Find: S1−S2
Step 3: Analyze Statement 1 independently
(1) Steven’s salary after 2 years was 20% more than the salary at which he took the job
  • Steven’s salary after 2 years =S1+2x
However, we do not know anything about the values of S2, x and y.
Insufficient to answer.
 
Step 4: Analyze Statement 2 independently
(2) The increase in the salary of Stuart at the end of the second year was 11% of the salary at which he took the job.
  • Increase in Stuart’s salary at the end of the 2nd year = 
Insufficient to answer as it does give us the value of  S1,S2orx
Step 5: Analyze Both Statements Together (if needed)
Need to know the value of x to answer the question.
Insufficient to answer.
Answer: E

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

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?
  • a)
    10
  • b)
    11
  • c)
    12
  • d)
    13
  • e)
    14
Correct answer is option 'C'. Can you explain this answer?

Nitya Kumar answered
Given
  • D = x + kv, where
    • D is the distance travelled and v is the liter of fuel used
    • x and k are constants
  • Case-I:
    • 4 liters of fuel used to travel 53 miles
  • Case-II:
    • 8 liters of fuel used to travel 101 miles

To Find: Liters of fuel used to travel 149 miles?
  • So, in the context of the equation 149 = x +kv
  • We need to find the value of v

Approach
  1. To find the liters of fuel used to travel 149 miles, we will need to find the values of x and k. Since there are 2 unknowns, we will need 2 equations.
  2. Case-I: Using 4 liters of fuel to travel 53 miles
    1. 53 = x +4k……(1)
  3. Case-II: Using 8 liters of fuel to travel 101 miles
    1. 101 = x + 8k….(2)
  4. As we have 2 equations and 2 unknowns, we can find the value of x and k.
 
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

An increasing sequence consists of 4 negative integers and 6 positive integers. Is the sum of the sequence positive?
(1) The difference between any two consecutive negative integers is 5 and the difference between any two consecutive positive integers is 2
(2) The first term of the sequence is -16
  • 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?

Kirti Roy answered
Steps 1 & 2: Understand Question and Draw Inferences
  • Let the sequence be   where 
To Find: Is 
Step 3: Analyze Statement 1 independently
(1) The difference between any two consecutive negative integers is 5 and the difference between any two consecutive positive integers is 2
Need to know the value of a1 and a5  to know if the sum of the sequence is greater than 0.
Insufficient to answer.
 
Step 4: Analyze Statement 2 independently
(2) The first term of the sequence is -16
  • a1=−16
  • We do not know the difference between consecutive terms
Hence,
  • Statement 2 alone is not sufficient
Step 5: Analyze Both Statements Together (if needed)
From Statement 1 we got
From Statement 2 we got
  • a1 = -16
Combining both we get:
Now, if we notice carefully a5   is a positive integer,
  • a5 ≥ 1
Therefore
  • 6a5≥6
Hence
  • 6a5−4 is always greater than zero.
Combining both statements was sufficient to answer the question
Correct Answer: C

The area of a rectangle is 28 square centimeter. What is the perimeter of the rectangle?
(1) If the length of the rectangle is increased by 10 centimeter and the breadth is decreased by 5 centimeter, the perimeter of the rectangle is eight times the original length of the rectangle.
(2) If the length of the rectangle is increased by 350% and the breadth is increased to 350% of the original breadth, the perimeter of the rectangle is 63 centimeters more than the original perimeter of the rectangle.
  • 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?

Prashanth Chawla answered
Steps 1 & 2: Understand Question and Draw Inferences
Given:
Let the length and breadth of rectangle be L and B respectively
  • LB = 28…..............................................(1)
To find:  The value of 2(L+B)
  • To find perimeter, we need to know the value of L and B.
  
Step 3: Analyze Statement 1 independently
(1) If the length of the rectangle is increased by 10 centimeter and the breadth is decreased by 5 centimeter, the perimeter of the rectangle is eight times the original length of the rectangle.
  • New L = L  +10
  • New B = B - 5
So,
  • New Perimeter = 2(L+10 + B-5)
  • 2(L+10 + B-5) = 8L
  • 2L + 2B + 10 = 8L
  • 6L – 2B – 10 = 0
  • 3L – B – 5 = 0
  • B = 3L - 5 . . . . . . . . . . . . . . .. . . . .(2)
 
Substituting (2) in (1):
  • 2 values of L (roots) will be obtained from this quadratic equation
 
Comparing this with the standard quadratic form : ax2 + bx + c = 0, we get : a = 1 ; b = -5/3 ; c = -28/3
  • Product of these 2 values  = (c/a)  −283
    • Since the product is negative, one root is positive and the other is negative
  • The negative root will be rejected
    • L, being the length of a rectangle, cannot be negative
  • Thus, a unique value of L is obtained
    • Using Equation (2), a unique value of B is also obtained
Hence, Statement 1 alone is sufficient.
Step 4: Analyze Statement 2 independently
  • If the length of the rectangle is increased by 350% and the breadth is increased to 350% of the original breadth, the perimeter of the rectangle is 63 centimeters more than the original perimeter of the rectangle
  • New L = 4.5L
  • New B = 3.5B
So,
Substituting (3) in  (1):
  • 2 values of L (roots) will be obtained from this quadratic equation
Comparing this with the standard quadratic form : ax2 + bx + c = 0, we get :
a = 1 ; b = - 9 ; c = 20
  • Product of these 2 values  = (c/a) =  20
    • Since the product is positive, the two roots are either both positive or both negative
  • Sum of roots = (-b/a) =  -(-9) = 9
    • Since the sum is positive, it means both roots are positive
  • Thus, St. 2 leads to 2 values of L
    • From Equation (3), 2 values of B will be obtained
    • So, 2 values of Perimeter will be obtained.
St. 2 is not sufficient to obtain a unique value of the perimeter.
Step 5: Analyze Both Statements Together (if needed)
Since we get a unique answer in Step 3, this step is not required
Answer: Option A

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