Q1·CSAT · Prelims 2026
Algebra — Multi-Variable Inequalities
NumericalAlgebra — Linear InequalitiesStatement-based● Hard
Question
For 1/3<x<y<2, which of the following statements is/are always correct? I. x+1/x<y+1/y II. √(1+y^2){y}<√(1+x^2){x} Select the answer using the code given below.
Options
a
I only
bAnswer
II only
c
Both I and II
d
Neither I nor II
Explanation
Let us analyze the mathematical functions within the defined domain 1/3 < x < y < 2.
Statement I Evaluation: Consider the function f(t) = t + 1/t. Its derivative is f'(t) = 1 - 1/t². For t > 1, f'(t) > 0 (increasing), but for t < 1, f'(t) < 0 (decreasing). Since our domain straddles across 1 (e.g., if x = 0.5 and y = 1, then 0.5 + 2 = 2.5 and 1 + 1 = 2, which gives 2.5 > 2, breaking the inequality), Statement I is not always correct.
Statement II Evaluation: Consider the function g(t) = √(1+t²){t} = √(1/t² + 1). As t increases in the positive real domain, 1/t² strictly decreases, which means g(t) strictly decreases. Since x < y, it must be that g(x) > g(y), which can be rewritten as g(y) < g(x). This matches the statement exactly, so Statement II is always correct.
Monotonicity analysis shows that a strictly decreasing function preserves inverted inequality bounds across its entire domain.
Answer: (b).
Question details
Year
2026
Paper
CSAT
Question
Q1
Section
Quantitative Aptitude
Sub-topic
Algebra — Linear Inequalities
Type
Statement-based
Difficulty
Hard
Source hint
Standard inequalities — algebraic property substitution
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