UPSC Prelims 2024 · CSAT
Question 39: Basic Numeracy
Let p, q, r and s be distinct positive integers. Let p, q be odd and r, s be even. Consider the following statements :
- 1.(p − r)^2 (qs) is even.
- 2.(q − s) q^2 s is even.
- 3.(q + r)^2 (p + s) is odd.
Which of the statements given above are correct?
AnswerUPSC’s answer is (d).
p and q are odd; r and s are even. Statement 1: qs is odd × even = even, so the product is even. Statement 2: s is even and is a factor of q²s, so the product is even whatever (q − s) is. Statement 3: q + r is odd + even = odd, so its square is odd; p + s is odd + even = odd; odd × odd = odd. All three statements are correct.
- 1. qs is even, so the whole product is even.
- 2. s is even, so q²s is even and the product is even.
- 3. Both (q + r)² and (p + s) are odd, so the product is odd.
How you could have got this
- If you were sure statement 1 is correct, (b) goes: it leaves 1 out.
What you needed to know: qs is even, so the whole product is even.
- Then, if you were sure statement 2 is correct, (c) goes: it leaves 2 out.
What you needed to know: s is even, so q²s is even and the product is even.
- Then, if you were sure statement 3 is correct, (a) goes: it leaves 3 out.
What you needed to know: Both (q + r)² and (p + s) are odd, so the product is odd.
Only (d) is left. This one needs every statement.
Which options go is worked out from UPSC’s key and the options, not by a model.
Why each option is right or wrong
- (a) Wrong. It leaves out statement 3, which is correct.
- (b) Wrong. It leaves out statement 1, which is correct.
- (c) Wrong. It leaves out statement 2, which is correct.
- (d) Right. All three statements hold.
Answer: UPSC’s key. Explanation: written by UPSC Grade, not by UPSC.
Source: UPSC, Civil Services (Preliminary) Examination 2024, General Studies Paper II (CSAT), Series A. The paper on upsc.gov.in.