Considering the alarming importance of CSAT in UPSC CSE Prelims exam and with enormous requests we received recently, InsightsIAS has started Daily CSAT Test to ensure students practice CSAT Questions on a daily basis. Regular Practice would help one overcome the fear of CSAT too.We are naming this initiative as Insta– DART – Daily Aptitude and Reasoning Test. We hope you will be able to use DART to hit bull’s eye in CSAT paper and comfortably score 100+ even in the most difficult question paper that UPSC can give you in CSP-2021. Your peace of mind after every step of this exam is very important for us.
Looking forward to your enthusiastic participation (both in sending us questions and solving them on daily basis on this portal).
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Question 1 of 5
1. Question
Consider the following statements for the five-digit number AB0AB, where AB stands for the 10th prime number and the middle digit is zero.
Statement 1: AB0AB is divisible by 101.
Statement 2: Number of factors of AB0AB is 12.Which of the statements given above is/are correct?
Correct
Answer: (a) 1 only
Explanation:
The 10th prime number is 29. So AB0AB = 29029.Now,
29029 ÷ 101 = 287So AB0AB is divisible by 101. Hence Statement 1 is correct.
Further,
287 = 7 × 41So,
29029 = 101 × 7 × 41These are three distinct prime factors.
Therefore, number of factors = (1 + 1)(1 + 1)(1 + 1) = 8
So Statement 2 is incorrect.
Hence, the correct answer is (a) 1 only.
Incorrect
Answer: (a) 1 only
Explanation:
The 10th prime number is 29. So AB0AB = 29029.Now,
29029 ÷ 101 = 287So AB0AB is divisible by 101. Hence Statement 1 is correct.
Further,
287 = 7 × 41So,
29029 = 101 × 7 × 41These are three distinct prime factors.
Therefore, number of factors = (1 + 1)(1 + 1)(1 + 1) = 8
So Statement 2 is incorrect.
Hence, the correct answer is (a) 1 only.
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Question 2 of 5
2. Question
If x and y are non-negative real numbers such that 2x + y = 10, then the average of the maximum and minimum possible values of (x + y) is?
Correct
Answer: (b) 7
Explanation:
From 2x + y = 10 → y = 10 − 2xSo, x + y = x + (10 − 2x) = 10 − x
For maximum: x = 0 → y = 10 → x + y = 10
For minimum: y = 0 → 2x = 10 → x = 5 → x + y = 5
Average = (10 + 5) / 2 = 7
Incorrect
Answer: (b) 7
Explanation:
From 2x + y = 10 → y = 10 − 2xSo, x + y = x + (10 − 2x) = 10 − x
For maximum: x = 0 → y = 10 → x + y = 10
For minimum: y = 0 → 2x = 10 → x = 5 → x + y = 5
Average = (10 + 5) / 2 = 7
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Question 3 of 5
3. Question
If , then which of the following statements must be true?
I. At least one of x or y is zero.
II. Both x and y must be zero.
III. If x is non-zero, then y must be zero.Select the correct answer using the code given below.
Correct
Answer: (b)
Explanation:
If product xy = 0, then at least one of the numbers must be zero → Statement I is correct.Both need not be zero → Statement II is incorrect.
If x is non-zero, then y must be zero → Statement III is correct.
Hence, option (b) is correct.
Incorrect
Answer: (b)
Explanation:
If product xy = 0, then at least one of the numbers must be zero → Statement I is correct.Both need not be zero → Statement II is incorrect.
If x is non-zero, then y must be zero → Statement III is correct.
Hence, option (b) is correct.
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Question 4 of 5
4. Question
The price (p) of a commodity is increased by x% and then decreased by x%. If the final price is q, then what is the relation between p and q?
Correct
Answer: (a)
Explanation:
After increase and decrease:
q = p × (100 + x)(100 − x) / 100²q = p(100² − x²)/100²
⇒ q = p(10⁴ − x²)/10⁴
⇒ p(10⁴ − x²) = q × 10⁴
Incorrect
Answer: (a)
Explanation:
After increase and decrease:
q = p × (100 + x)(100 − x) / 100²q = p(100² − x²)/100²
⇒ q = p(10⁴ − x²)/10⁴
⇒ p(10⁴ − x²) = q × 10⁴
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Question 5 of 5
5. Question
Consider the following statements:
I. There exists a natural number which when increased by 50% becomes a prime number having same number of factors.
II. There exists a natural number which when increased by 100% becomes a prime number having same number of factors.Which of the statements given above is/are correct?
Correct
Answer: (a)
Explanation:
Statement I:
Take number = 2
After 50% increase → 3
Both 2 and 3 are prime → both have 2 factors
So Statement I is correctStatement II:
Take number = 2
After 100% increase → 4
Factors change from 2 to 3 → not sameTry number = 3 → becomes 6 → not prime
So Statement II is incorrect
Incorrect
Answer: (a)
Explanation:
Statement I:
Take number = 2
After 50% increase → 3
Both 2 and 3 are prime → both have 2 factors
So Statement I is correctStatement II:
Take number = 2
After 100% increase → 4
Factors change from 2 to 3 → not sameTry number = 3 → becomes 6 → not prime
So Statement II is incorrect
