SSC CGL 2024 Tier-I · 26 September 2024, Shift 3 · Quantitative Aptitude · Question 14 of 100

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 10 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 cot θ = √11, then the value of (cosec²θ − sec²θ) ⁄ (cosec²θ + sec²θ) is:

Answer

Why

Correct — B. Express both terms through cot θ, whose square you are handed.

cosec²θ = 1 + cot²θ = 1 + 11 = 12
tan²θ = 1⁄11, so sec²θ = 1 + tan²θ = 12⁄11

Numerator: 12 − 12⁄11 = (132 − 12)⁄11 = 120⁄11
Denominator: 12 + 12⁄11 = (132 + 12)⁄11 = 144⁄11

The 11s cancel: 120⁄144 = 5⁄6 → option (b).

Why the others are wrong

(a)
6⁄7 is the value at cot²θ = 13, since (13 − 1)⁄(13 + 1) = 6⁄7. The paper gives cot θ = √11, so cot²θ is 11.
(c)
3⁄5 belongs to cot θ = 2: (4 − 1)⁄(4 + 1) = 3⁄5. Squaring √11 gives 11, not 4.
(d)
4⁄5 is what cot θ = 3 gives: (9 − 1)⁄(9 + 1) = 4⁄5. It is the answer you land on if √11 is rounded to 3 before squaring.

Concept

Both squared reciprocal ratios are one more than a square: cosec²θ = 1 + cot²θ and sec²θ = 1 + tan²θ.

Given cot θ, the first is immediate. The second follows because tan θ is the reciprocal of cot θ, so tan²θ = 1⁄11 here.

Both terms then carry 11 in the denominator, the 11s cancel in the ratio, and the answer is a clean fraction. No angle is ever found, and √11 is never turned into a decimal.

The whole expression reduces to (cot²θ − 1)⁄(cot²θ + 1) — divide numerator and denominator by cosec²θ and use sec²θ⁄cosec²θ = tan²θ.

With cot²θ = 11 that is (11 − 1)⁄(11 + 1) = 10⁄12 = 5⁄6, in one line. The same form is what tests each wrong option below.

Key facts

cosec²A = 1 + cot²A and sec²A = 1 + tan²A.
tan A and cot A are reciprocals, so tan²A = 1⁄cot²A.
(cosec²A − sec²A)⁄(cosec²A + sec²A) simplifies to (cot²A − 1)⁄(cot²A + 1).

Study next

Reciprocal and Pythagorean identities used as one toolkitSimplifying a ratio of trigonometric expressions before substituting a valueBuilding a right triangle from cot θ = √11 to read every ratio off its sides

Common traps

Trying to find θ from cot θ = √11 — it is not a standard angle and is not needed.
Writing sec²θ = 1 + cot²θ, swapping which identity carries which ratio.
Cancelling the 11s before both fractions have been put over a common denominator.

SSC gives one ratio and asks for a symmetric expression built from the others, so the route is identity substitution rather than angle-finding. A related identity simplification is asked on 25 Sep 2024, 09:00, Quant Q.1.

Related PYQs

No directly related past PYQ was found.