SSC CGL 2024 Tier-I · 17 September 2024, Shift 3 · Quantitative Aptitude · Question 21 of 98

Text as printed on a candidate's response sheet for this shift; answer per SSC's tentative key as marked on that sheet. Solution last reviewed 6 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. The questions and options here are reproduced as printed on a candidate's own response sheet for this shift — the Commission's text, unchanged. Each answer is SSC's tentative key as marked on that sheet; SSC allowed challenges to it and never publishes a final key. The solutions are written by UnlockIAS.

If cosec x = 5⁄4, then the value of (5 sin²x + 8 cot²x) ⁄ (27 tan²x − 125 cos²x) is:

Answer

Why

Correct — C. From cosec x = 5⁄4, sin x = 4⁄5, so sin²x = 16⁄25 and cos²x = 1 − 16⁄25 = 9⁄25 — the 3-4-5 triangle.

Every term in the expression is squared, so the sign of cos x never matters.
tan²x = sin²x ⁄ cos²x = (16⁄25) ⁄ (9⁄25) = 16⁄9
cot²x = 9⁄16

Numerator: 5 sin²x + 8 cot²x = 5(16⁄25) + 8(9⁄16) = 16⁄5 + 9⁄2 = 77⁄10
Denominator: 27 tan²x − 125 cos²x = 27(16⁄9) − 125(9⁄25) = 48 − 45 = 3

Value = (77⁄10) ÷ 3 = 77⁄30, the fraction printed as option (c).

Why the others are wrong

(a)
Option (a) shows 99⁄47, about 2.11 against the true 2.57. It is the right size but the wrong parts: this working produces a numerator of 77⁄10 and a denominator of 3, and no step in it yields a 47.
(b)
Option (b) shows 55⁄87, a value below 1. The denominator collapses to just 3 while the numerator is 7.7, so the answer has to exceed 2 — anything under 1 is out before the arithmetic finishes.
(d)
Option (d) shows 33⁄91, about 0.36. It fails the same size check: 77⁄10 divided by 3 cannot come out smaller than one.

Concept

One trigonometric ratio fixes all the others. cosec x = 5⁄4 gives sin x = 4⁄5, and sin²x + cos²x = 1 then supplies cos²x = 9⁄25 — the 3-4-5 right triangle in disguise.

Because the expression is built entirely from squared ratios, you never need the sign of cos x and never need the angle itself. Convert each term to a fraction and the rest is arithmetic.

Look for the cancellation before multiplying out. 27 × 16⁄9 = 48 and 125 × 9⁄25 = 45 are both whole numbers, a sign the setter intends the denominator to collapse to something small.

cosec x = 5⁄4 is greater than 1, as every cosecant must be, so the given value is consistent. It does not fix the quadrant, since sin x is positive in both the first and the second — but with only even powers in the expression, either quadrant returns the same value.

The four options reach the candidate as images rather than as text, so each is a printed fraction rather than a typed one.

Key facts

cosec x = 5⁄4 gives sin x = 4⁄5 and sin²x = 16⁄25.
sin²x + cos²x = 1 then gives cos²x = 9⁄25, tan²x = 16⁄9 and cot²x = 9⁄16.
The numerator 5 sin²x + 8 cot²x comes to 16⁄5 + 9⁄2 = 77⁄10.
The denominator 27 tan²x − 125 cos²x collapses to 48 − 45 = 3.

Study next

Evaluating an expression in sec and tan from a single given ratioWhy even powers of a ratio make the quadrant irrelevantThe 3-4-5 and 5-12-13 triples as shortcuts for sin, cos and tan

Common traps

Reading cosec x = 5⁄4 as sin x = 5⁄4, a value no sine can take
Writing cot²x as 16⁄9 instead of 9⁄16, the reciprocal slipping the wrong way
Adding 16⁄5 and 9⁄2 without a common denominator and getting 25⁄7

SSC gives one ratio and asks for a composite expression, so the marks lie in converting once and reusing the pieces. The same given-one-value, evaluate-another shape appears in algebra at Quant Q.3 of this paper, where x + 1⁄x = 65⁄8 has to become x − 1⁄x.

Related PYQs

No directly related past PYQ was found.