Question 1

What is the area of a rectangle?
Statement I: The length of the rectangle is 12 cm.
Statement II: The perimeter of the rectangle is 40 cm.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

To find the area of a rectangle, we need both length AND width. Statement I gives only length (12 cm). Statement II gives perimeter (40 cm), from which we can derive: 2(l + w) = 40, so l + w = 20. If l = 12, then w = 8. Now we can calculate area = 12 × 8 = 96 cm². Therefore, both statements together are sufficient.

Question 2

What is the value of y?
Statement I: 3y + 5 = 20
Statement II: y is a natural number less than 10.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Statement I: 3y + 5 = 20 ⟹ 3y = 15 ⟹ y = 5. This uniquely determines y. Statement II alone only narrows y to {1, 2, 3, 4, 5, 6, 7, 8, 9}, which is insufficient. Statement I alone is sufficient to find y = 5.

Question 3

What is the speed of a car?
Statement I: The car travels 120 km in 2 hours.
Statement II: The car's speed is greater than 50 km/h.
(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Statement I: Speed = Distance ÷ Time = 120 ÷ 2 = 60 km/h. This uniquely determines the speed. Statement II: Speed > 50 km/h only provides a lower bound, allowing many possible speeds. Statement I alone is sufficient.

Question 4

What is the area of a rectangle?
Statement I: The length of the rectangle is 12 cm.
Statement II: The perimeter of the rectangle is 40 cm.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Both statements together are not sufficient

Accepted Answer

To find the area of a rectangle, we need both length AND width (or length and perimeter). Statement I gives only length (12 cm), which is insufficient alone. Statement II gives perimeter (40 cm), which is also insufficient alone because perimeter = 2(l + w), so knowing perimeter doesn't uniquely determine l and w. However, combining both: From Statement I, length = 12 cm. From Statement II, perimeter = 40 cm, so 2(12 + w) = 40, which gives 12 + w = 20, thus w = 8 cm. Now we can calculate area = 12 × 8 = 96 cm². Therefore, both statements together are sufficient.

Question 5

Question: What is the value of x?
Statement I: x² = 49
Statement II: x is a positive integer

Which of the following is correct?

Accepted Answer

Statement I alone: x² = 49 gives x = +7 or x = –7. Two values, so not sufficient alone.
Statement II alone: x is a positive integer — no specific value, so not sufficient alone.
Combining both: x² = 49 AND x is positive → x = 7 (unique value). So both statements together are sufficient.

Question 6

How many students are in the class?
Statement I: There are 8 more boys than girls.
Statement II: The total number of students is 3 times the number of girls.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Let g = girls, b = boys. Statement I: b = g + 8 (one equation, two unknowns—insufficient). Statement II: b + g = 3g → b = 2g (one equation, two unknowns—insufficient). Together: b = g + 8 and b = 2g → g + 8 = 2g → g = 8, b = 16. Total = 24 students. Both statements together are sufficient.

Question 7

Is x a positive number?
Statement I: x² = 16
Statement II: x > 0
(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Statement I: x² = 16 means x = 4 or x = −4. This does NOT confirm x is positive. Statement II: x > 0 directly states x is positive. Therefore, Statement II alone is sufficient to answer the question.

Question 8

What is the value of a + b?
Statement I: a − b = 5.
Statement II: a² − b² = 45.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Statement I: a − b = 5 (one equation, two unknowns—insufficient). Statement II: a² − b² = 45. Using the identity a² − b² = (a+b)(a−b), we get (a+b)(a−b) = 45. From Statement I, a − b = 5, so (a+b)(5) = 45 → a+b = 9. Both statements together are sufficient.

Question 9

What is the area of a rectangle?
Statement I: The length of the rectangle is 12 cm.
Statement II: The perimeter of the rectangle is 40 cm.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

To find the area of a rectangle, we need both length AND breadth (Area = length × breadth).

Statement I: Length = 12 cm. Without breadth, we cannot find area. INSUFFICIENT.

Statement II: Perimeter = 40 cm. Using Perimeter = 2(L + B), we get L + B = 20. This gives us a relationship but infinite combinations exist (e.g., 12×8, 10×10, 15×5, etc.). INSUFFICIENT.

Both statements together: From Statement I, L = 12. From Statement II, L + B = 20, so B = 8. Area = 12 × 8 = 96 cm². SUFFICIENT.

Answer: (C) Both statements together are sufficient.

Question 10

What is the value of y?
Statement I: 3y + 5 = 20
Statement II: y is a whole number less than 10.
(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Statement I: 3y + 5 = 20 → 3y = 15 → y = 5. This uniquely determines y. Statement II: y is a whole number less than 10 gives multiple possible values (0, 1, 2, ..., 9). Statement I alone is sufficient.

Question 11

What is the average of three numbers?
Statement I: The sum of the three numbers is 45.
Statement II: The three numbers are consecutive integers.
(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Statement I: Sum = 45 → Average = 45 ÷ 3 = 15. This directly gives the average without needing to know the individual numbers. Statement II: Consecutive integers tell us the pattern but not the actual values. Statement I alone is sufficient.

Question 12

Is x > y?
Statement I: x = 2a and y = a, where a > 0.
Statement II: x + y = 30 and x − y = 10.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Either statement alone is sufficient

Accepted Answer

Statement I: x = 2a, y = a, where a > 0. Since a > 0, we have x = 2a > a = y. So x > y is TRUE. Statement II: x + y = 30 and x − y = 10. Solving: 2x = 40 → x = 20; y = 10. So x > y is TRUE. Either statement alone is sufficient to answer the question.

Question 13

What is the cost price of an item?
Statement I: The selling price is Rs. 120 and profit is 20%.
Statement II: The profit is Rs. 20.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Statement I: Selling Price = 120, Profit% = 20. Using Profit% = (SP − CP) / CP × 100, we get 20 = (120 − CP) / CP × 100 → 0.2 × CP = 120 − CP → 1.2 × CP = 120 → CP = 100. Statement I alone is sufficient. Statement II: Profit = Rs. 20, but without knowing either SP or CP, we cannot determine CP uniquely. Therefore, Statement I alone is sufficient.

Question 14

What is the average of three numbers?
Statement I: The sum of the three numbers is 45.
Statement II: The three numbers are in the ratio 2:3:4.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Average = Sum / Count. Statement I: Sum = 45. Average = 45 / 3 = 15. Statement I alone is sufficient to find the average, regardless of what the individual numbers are. Statement II gives only the ratio, which does not determine the sum or average. Therefore, Statement I alone is sufficient.

Question 15

What is the value of n?
Statement I: n is a positive integer less than 10.
Statement II: n² = 49.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Statement I: n is a positive integer less than 10. This gives us n ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9}, but does not uniquely determine n. Statement II: n² = 49 means n = ±7. Since we need a unique value and Statement II gives two possible values (7 or −7), Statement II alone is insufficient. However, combining both: n is positive (from I) and n² = 49 (from II) → n = 7. Both statements together are sufficient.

Question 16

What is the cost price of an item?
Statement I: The selling price is ₹120 and profit is 20%.
Statement II: The profit amount is ₹20.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Statement I: Selling Price = ₹120, Profit% = 20%. Using SP = CP(1 + Profit%/100), we get 120 = CP(1.2) ⟹ CP = ₹100. This is sufficient. Statement II: Profit = ₹20 alone doesn't determine CP (CP could be ₹100, ₹200, etc.). Statement I alone is sufficient.

Question 17

Is x a positive integer?
Statement I: x² = 16
Statement II: x > 0

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

Statement I: x² = 16 means x = 4 or x = -4 (x could be negative, so not sufficient to confirm x is positive). Statement II: x > 0 tells us x is positive but doesn't confirm it's an integer. Together: x² = 16 AND x > 0 means x = 4 (positive integer). Both statements together are sufficient.

Question 18

What is the speed of a car?
Statement I: The car travels 240 km in 4 hours.
Statement II: The car covers 60 km in 1 hour.

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Either statement alone is sufficient

Accepted Answer

Statement I: Speed = Distance/Time = 240/4 = 60 km/h. This is sufficient. Statement II: Speed = 60 km in 1 hour = 60 km/h. This is also sufficient. Both statements independently give the same answer, so either statement alone is sufficient.

Question 19

A company has employees in three departments: Sales, Marketing, and Operations. Statement I: The ratio of employees in Sales to Marketing is 3:2. Statement II: The total number of employees in Sales and Marketing combined is 100. What is the total number of employees in all three departments?
(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are necessary
(D) Neither statement alone nor together is sufficient

Accepted Answer

Let Sales = S, Marketing = M, Operations = O. Statement I: S:M = 3:2, so S = 3k, M = 2k for some k. This alone does not determine total employees. Statement II: S + M = 100. Combined: 3k + 2k = 100 → 5k = 100 → k = 20. Thus S = 60, M = 40. However, this gives only S + M = 100, not the total of all three departments. The number of Operations employees (O) is never specified. Therefore, even both statements together are insufficient to find the total of all three departments.

Question 20

Question: What is the value of the two-digit number?
Statement I: The sum of the digits of the number is 9.
Statement II: The number is divisible by 9 and is greater than 50.

Which of the following is correct?

Accepted Answer

Statement I alone: Sum of digits = 9 gives multiple possibilities: 18, 27, 36, 45, 54, 63, 72, 81, 90. Not sufficient.
Statement II alone: Two-digit multiples of 9 greater than 50: 54, 63, 72, 81, 90. Multiple answers, not sufficient.
Both together: Multiples of 9 greater than 50 where digit sum = 9: 54 (5+4=9), 63 (6+3=9), 72 (7+2=9), 81 (8+1=9), 90 (9+0=9). All multiples of 9 have digit sum 9, so still multiple answers. Both statements together are NOT sufficient.

CDS Data Sufficiency — Maths

Data Sufficiency — Maths— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Data Sufficiency — Maths — Practice Questions

60-Second Revision — Data Sufficiency — Maths