**Each of the data sufficiency problems consists of a question followed by two statements, namely (A) and (B). Here, you must decide whether the given data to solve the question is sufficient or not.**

**The correct answer to each question will be one of the following:**

a. if statement (A) alone is sufficient to answer the question but statement (B) alone is not sufficient

b. if statement (B) alone is sufficient to answer the question but statement (A) alone is not sufficient

c. if the two statements (A) and (B) taken together, are sufficient to answer the question, but neither statement alone is sufficient

d. if each statement alone is sufficient to answer the question

**1. Identify whether the following equation is true or false.**

**(2)x * (2)y = (2)x+y**

**A. x is a whole number**

**B. y is a whole number**

c. if statement (A) alone is sufficient to answer the question but statement (B) alone is not sufficient

d. if statement (B) alone is sufficient to answer the question but statement (A) alone is not sufficient

e. if the two statements (A) and (B) taken together, are sufficient to answer the question, but neither statement alone is sufficient

f. if each statement alone is sufficient to answer the question

g. if the two statements taken together are still not sufficient to answer the question

Correct answer: e

Explanation:

It is given that both x and y are whole numbers; therefore 2x and 2y will also belong to the set of whole numbers.

From data (A), we get x is a whole number. Suppose x = 4, then after substituting this value in the given equation, we get 24 * 2y = 24 + y

Hence only data (A) is not sufficient to answer the question.

Now, from data (B), we get y is whole number. Suppose y = 2, then after substituting this value in the given equation, we get 24 * 22 = 24 + 2 = 26 = 64

Hence only data (A) is also not sufficient to answer the question.

Hence option e is the correct answer.

**2. Find out whether the value of ‘m’ for the equation m5 – 48 = (-16) is divisible by 2 or not?**

**A. ‘m’ is a positive integer**

**B. ‘m’ is an even number**

a. if statement (A) alone is sufficient to answer the question but statement (B) alone is not sufficient

b. if statement (B) alone is sufficient to answer the question but statement (A) alone is not sufficient

c. if the two statements (A) and (B) taken together, are sufficient to answer the question, but neither statement alone is sufficient

d. if each statement alone is sufficient to answer the question

e. if the two statements taken together are still not sufficient to answer the question

Correct answer: d

Explanation:

According to data in (A), m is a positive integer. Using only this data we can find out the value of ‘m’ as follows:

m5 – 48 = (-16) → m5 = (-16) + 48 → m5 = 32 → m= 321/5 → m = 2. Therefore, ‘m’ is divisible by

2.

Hence, we can find whether the value of ‘m’ is divisible by 2 or not using only the data (A).

Similarly, it is possible to find out the correct answer by using only the data in (B) i.e. ‘m’ is an even number.

Hence, the correct answer is option d

**3. Find out the value of ‘x’?**

**A. 6x – 4y = 4**

**B. y = 5**

a. if statement (A) alone is sufficient to answer the question but statement (B) alone is not sufficient

b. if statement (B) alone is sufficient to answer the question but statement (A) alone is not sufficient

c. if the two statements (A) and (B) taken together, are sufficient to answer the question, but neither statement alone is sufficient

d. if each statement alone is sufficient to answer the question

e. if the two statements taken together are still not sufficient to answer the question

Correct answer: c

Explanation:

By using the data in (A), we get, 6x – 4y = 4 → 6x = 4 + 4y → x = (4 + 4y) / 6

Hence, we are not able to find the value of ‘x’ with the help only the data in (A).

Now, according to the data in (B), y = 5.

After substituting the value of y in the equation given in data (A), we get,

6x – 4(5) = 4 → 6x = 4 + 20 → 6x = 24 → x = 24/6 → x = 4.

Therefore, we are able to find the value of ‘x’ using both the data (A) as well as (B), but not using any of the statements alone.

Hence, the correct answer is option c.

**4. Find out whether the number ‘x’ is completely divisible by 8 or not?**

**A. x is an odd number and belongs to the set of integers**

**B. x = 4567**

d. if each statement alone is sufficient to answer the question

e. if the two statements taken together are still not sufficient to answer the question

Correct answer: b

Explanation:

According to the data in (A), x is an odd number and belongs to the set of integers. Using this data it is not possible to find out the answer for the question.

Hence, data (A) alone is not sufficient to find the answer to the question.

Now, according to the data in (B), x = 4567

The divisibility rule for the integer 8 states that, a number is completely divisible by 8 if the last three digits are zero or the number formed by the last three digits is divisible by 8 completely.

According to this rule, the last three numbers are not equal to 0. Hence, we will check whether the number formed by the last three digits i.e. ‘567’ is divisible by 8 or not.

The number ‘567’ is not divisible by 8. Hence, it the number ‘4567’ i.e. ‘x’ is not divisible by 8 completely.

Hence, it is proved that, we can answer the question using the data in (B) alone, but not the data in (A).

Hence, the correct answer is option b.

**5. Find out the value of the divisor ‘d’, where d is a positive integer.**

**A. ‘d’ when divided by 7 gives a remainder of 3**

**B. The quotient is a prime number between 3 and 7**

d. if each statement alone is sufficient to answer the question

e. if the two statements taken together are still not sufficient to answer the question

Correct answer: c

Explanation:

According to the rule of divisibility,

Dividend = divisor * quotient + remainder--------(I)

Now, according to the data in (A), 7 is the divisor and 3 is the remainder,

After substituting these values in the equation (I), we get,

Dividend (d) = 7 * quotient + 3

Hence, it is not possible to find the value of the dividend using only the data in (A).

Now, according to the data in (B), quotient is a prime number between 3 and 7.

Hence, we can conclude that the quotient is 5 as it is the only prime number that lies between 3 and 7.

Hence, we can write equation (I) as follows,

Dividend (d) = 7 * 5 + 3 → d = 35 + 3 → d = 38.

Thus we are able to answer the question if we use both the data in (A) and (B) together.

Hence, the correct answer is option c.

**6. Find the value of the variable 'x'; if 24 = 4x.**

**A. x belongs to the set of integers**

**B. x is an even number**

a. (A) ALONE is sufficient to answer the question but (B) alone is not sufficient

b. (B) ALONE is sufficient to answer the question but (A) alone is not sufficient

c. Both (A) and (B) TAKEN TOGETHER, are sufficient to answer the question, but NEITHER (A) nor (B) ALONE is sufficient

d. Both (A) and (B) ALONE are sufficient to answer the question

e. Both (A) and (B) TAKEN TOGETHER are still NOT sufficient to answer the question

Correct answer: d

Explanation:

According to the data in statement A, 'x' belongs to the set of integers. Means that 'x' can attain either positive or negative values.

Now, using this data, let us try to solve the question.

It is given that 24 = 4x

We have to find the value of 'x' which is an integer.

Now, 2 * 2 * 2 * 2 = 16

And 4 * 4 = 16

Hence, the value of 'x' = 2

Thus, with the help of this data alone we can find the exact answer to the question.

Now, according to the data in statement B, 'x' is an even number.

With the help of this data also we can find out the answer to the question by starting with the smallest even number and substituting it in place of 'x' as follows.

The smallest even number = 2.

Now, substituting x = 2 in 4x we get the answer as 16.

Also, 24 = 16, hence, the value of x is 2.

Since, we are able to answer the question using each of the data alone hence the correct answer is option d.

**7. Find out whether 'a' is a positive or a negative integer?**

**A . a + b = 10**

**B. a - b = 5**

a. (A) ALONE is sufficient to answer the question but (B) alone is not sufficient

c. (B) ALONE is sufficient to answer the question but (A) alone is not sufficient

d. Both (A) and (B) TAKEN TOGETHER, are sufficient to answer the question, but NEITHER (A) nor (B) ALONE is sufficient

e. Both (A) and (B) ALONE are sufficient to answer the question

f. Both (A) and (B) TAKEN TOGETHER are still NOT sufficient to answer the question

Correct answer: c

Explanation:

From the data in statement A, a + b = 10.

Using this data alone, it is not possible to say whether the value of 'a' is positive or negative.

Hence, option 'a' and option 'd' cannot be the right answer.

Now, from the data in statement B, a – b = 5.

Using this data alone also it is not possible to say whether the value of 'a' is positive or negative.

Hence, option 'b' and option 'd' cannot be the right answer.

Now, using both the data together as follows,

a + b = 10 …………i

a – b = 5 …………..ii

Adding equations I and ii, we get,

2a = 15

∴ a = 7.5

This, proves that using both the data A and B together, we can say that the value of 'a' is a positive integer.

Hence, the correct answer is option c.

**8. Find out whether y = 7 is true or false?**

**A. (x – 6) (y – 7) = 0**

**B. y – 7 = 1**

a. (A) ALONE is sufficient to answer the question but (B) alone is not sufficient

c. (B) ALONE is sufficient to answer the question but (A) alone is not sufficient

d. Both (A) and (B) TAKEN TOGETHER, are sufficient to answer the question, but NEITHER (A) nor (B) ALONE is sufficient

e. Both (A) and (B) ALONE are sufficient to answer the question

f. Both (A) and (B) TAKEN TOGETHER are still NOT sufficient to answer the question

Correct answer: b

Explanation:

As per the data in statement (A), we can say that, (x - 6) = 0 or (y - 7) = 0 i.e. either x = 6 or y = 7.

But, there is no proof that y = 7 is always true. Hence data (A) alone is not sufficient to answer the question.

Hence, option 'a' cannot be the answer.

Now, as per the data in statement (B), y – 7 = 1 i.e. y = 1 + 7 → y = 8.

Thus, we are able to answer the question with the help of only the data in (B).

Hence, the correct answer is option b.

**9. What is the value of q for the following?**

**A . Sin q = 1**

**B. Cos q is positive**

a. (A) ALONE is sufficient to answer the question but (B) alone is not sufficient

b. (B) ALONE is sufficient to answer the question but (A) alone is not sufficient

c. Both (A) and (B) TAKEN TOGETHER, are sufficient to answer the question, but NEITHER (A) nor (B) ALONE is sufficient

d. Both (A) and (B) ALONE are sufficient to answer the question

e. Both (A) and (B) TAKEN TOGETHER are still NOT sufficient to answer the question

Correct answer: a

Explanation:

As per the data given in (A), the value of sin q = 1.

∴ q = sin-1 (1)

∴ q = 90 degree

Thus, we are able to find out the answer of the question, using only the data in statement (A).

Now, let us see if we can find out the answer using the data in statement (B).

As per the data in (B), Cos Θ is positive.

But, the value of Cos Θ is positive for more than one values of Θ

Hence, we cannot determine the answer to the question using only the data in (B).

Hence, the correct answer is option a.

**10. Calculate the time required to cover a distance of 10 km.**

**A. Speed is positive**

**B. Speed = 5 km / hr**

a. (A) ALONE is sufficient to answer the question but (B) alone is not sufficient

b. (B) ALONE is sufficient to answer the question but (A) alone is not sufficient

c. Both (A) and (B) TAKEN TOGETHER, are sufficient to answer the question, but NEITHER (A) nor (B) ALONE is sufficient

d. Both (A) and (B) ALONE are sufficient to answer the question

e. Both (A) and (B) TAKEN TOGETHER are still NOT sufficient to answer the question

Correct answer: b

Explanation:

According to the data in (A), speed is a positive factor. But, there is no data about the exact value of speed.

Thus, it is not possible to calculate the time factor using only the data in (A).

Now, as per the data in statement (B), speed = 5 km / hr

According to the standard mathematical formula for calculating distance, time and speed,

Time = distance / speed

After substituting the values of distance and speed in this formula, we can calculate the time factor as,

Time = 10 km / 5 km per hour

∴ Time = 2 hours.

Thus, we are able to answer the question using the data in (B) alone.

Hence, the correct answer is option b.

**11. Which is the smallest number that is divisible by both 14 as well as 21?**

**A. The result is a two digit number**

**B. The sum of the digits is equal to 12**

a. (A) ALONE is sufficient to answer the question but (B) alone is not sufficient

b. (B) ALONE is sufficient to answer the question but (A) alone is not sufficient

d. Both (A) and (B) ALONE are sufficient to answer the question

e. Both (A) and (B) TAKEN TOGETHER are still NOT sufficient to answer the question

Correct answer: c

Explanation:

According to the data in statement (A), the smallest number that is divisible by both 14 as well as 21 is a two digit number.

But, it is not possible to find out the correct answer using only (A), because there are so many two digit numbers.

Hence, option (a) and option (d) cannot be the right answer.

Using only data in (B), we end up with so many numbers whose digits sum up to 12. Hence, it is not possible to find out the correct answer using only (B) as well.

Thus, option (b) also cannot be the right answer.

Now, as per the data in both statement (A) and statement (B), the sum of the digits of the resultant number is equal to 12.

Now, we can find out the numbers whose digits sum upto 12 as follows.

39, 48, 84 and 93

Of all these numbers we can easily find out the number that is divisible by both 14 and 21 as 84.

Hence, we are able to find out the answer using both (A) and (B), therefore the correct answer is option c.

**12. Find out the product of 'a' and 'b' i.e. ab.**

**A. a + b = 3**

**B. a2 + b2 = 8**

a. (A) ALONE is sufficient to answer the question but (B) alone is not sufficient

b. (B) ALONE is sufficient to answer the question but (A) alone is not sufficient

d. Both (A) and (B) ALONE are sufficient to answer the question

e. Both (A) and (B) TAKEN TOGETHER are still NOT sufficient to answer the question

Correct answer: c

Explanation:

As per the data in statement (A), a + b = 3.

Using this data alone, it is not possible to find out the product of 'a' and 'b'.

Hence, option 'a' and option 'd' cannot be the right answer.

Now, as per the data in statement (B), a2 + b2 = 8.

Using this data alone also it is not possible to find out the product of 'a' and 'b'.

Hence, option 'b' and option 'd' cannot be the right answer.

Now, using both the data (A) and (B) together let us try to solve the question as follows

According to the standard mathematical (a + b)2 = a2 + 2ab + b2

Now, substituting the values in this formula we get,

(3)2 = 8 + 2ab → 9 – 8 = 2ab → 1 = 2ab → ab = 1 / 2

Thus, we are able to find the answer using both (A) and (B) together, but neither of them alone.

Hence, the correct answer is option c.

**13. Find out the length of the side AB of the following triangle.**

**A. Triangle ABC is a right angled triangle**

**B. Length of side AC = 5 cm**

a. (A) ALONE is sufficient to answer the question but (B) alone is not sufficient

b. (B) ALONE is sufficient to answer the question but (A) alone is not sufficient

d. Both (A) and (B) ALONE are sufficient to answer the question

e. Both (A) and (B) TAKEN TOGETHER are still NOT sufficient to answer the question

Correct answer: e

Explanation:

As per the data in statement (A), triangle ABC is a right angled triangle.

From this data we can only say that the side AC is the hypotenuse, but, we cannot find the length of the side AB.

Hence, option 'a' and option 'd' cannot be the right answer.

Now, as per the data (B), length of side AC i.e. the hypotenuse = 5 cm.

But, from this data also we cannot find the length of side AB as well. Hence, option 'b' also is not the right answer.

Now, using both data (A) and (B), together

According to the Pythagoras theorem, AC2 = AB2 + BC2

But, we know the length of only side AC. Thus, we cannot determine the length of side AC.

Hence, option 'c' is not the right answer.

Hence, the correct answer is option e.

**14. What is the area of the square?**

**A. Side of a square = 6 cm**

**B. Perimeter of a square = 24 cm**

a. (A) ALONE is sufficient to answer the question but (B) alone is not sufficient

b. (B) ALONE is sufficient to answer the question but (A) alone is not sufficient

d. Both (A) and (B) ALONE are sufficient to answer the question

e. Both (A) and (B) TAKEN TOGETHER are still NOT sufficient to answer the question

Correct answer: d

Explanation:

As per the data given in statement (A), the length of the side of a square = 6 cm.

∴ We can calculate the area of the square as follows,

Area of square = side * side = 6 * 6 = 36 square cm.

Hence, option 'c' and option 'e' cannot be the right answers.

Now, as per the data in statement (B), the perimeter of a square = 24 cm.

Perimeter = sum of all the sides

Since a square has 4 sides, thus perimeter of a square = 4 * side

∴ The length of each side of the square = 24 / 4 = 6 cm.

Hence, we can calculate the area of a square as side * side = 46 * 6 = 36 square cm.

This proves that, we can find the answer using both the data (A) as well as (B) alone.

Hence, the correct answer is option d.

**15. Calculate the distance to be covered if**

**A. Time = 2 hours**

**B. Speed = 3 km / hr**

a. (A) ALONE is sufficient to answer the question but (B) alone is not sufficient

b. (B) ALONE is sufficient to answer the question but (A) alone is not sufficient

d. Both (A) and (B) ALONE are sufficient to answer the question

e. Both (A) and (B) TAKEN TOGETHER are still NOT sufficient to answer the question

Correct answer: c

Explanation:

As per the data in (A), time = 2 hours.

But, according to the standard mathematical formula for calculating distance, time and speed,

Distance = Speed * Time

As we know only the time factor, it is not possible to calculate the distance using only the data in (A).

Hence, option 'a' and option 'd' cannot be the right answers.

Now, as per the data in statement (B), speed = 3 km / hr

According to the standard mathematical formula for calculating distance, time and speed,

Distance = Time * Speed

As we know only the speed factor, it is not possible to calculate the distance using only the data in (B).

Hence, option 'b' cannot be the right answer.

Now, using both the data in (A) and (B) together

After substituting the values of time and speed in this formula, we can calculate the distance covered as,

Distance = 2 hours * 3 km / hr

∴ Distance = 6 km.

Thus, we are able to answer the question using data (A) as well as (B), but neither of them alone.

Hence, the correct answer is option c.

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