SSC CGL 2024 Tier-I · 11 September 2024, Shift 3 · Quantitative Aptitude · Question 24 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 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.

Consider a triangle PQR, right angled at R, in which PQ = 29 units, QR = 21 units and ∠PQR = θ. Find the value of cos²θ − sin²θ.

Answer

Why

Correct — B. The image gives triangle PQR right-angled at R, with PQ = 29, QR = 21 and ∠PQR = θ, and asks for cos²θ − sin²θ.

The right angle is at R, so PQ = 29 is the hypotenuse
PR = √(29² − 21²) = √(841 − 441) = √400 = 20
θ sits at Q, so cos θ = 21⁄29 and sin θ = 20⁄29

cos²θ − sin²θ = 441⁄841 − 400⁄841 = 41⁄841, the fraction printed as option (b).

Why the others are wrong

(a)
Option (a) shows the whole number 1, which is cos²θ + sin²θ. The stem prints a minus sign, and 441⁄841 − 400⁄841 leaves 41⁄841.
(c)
Option (c) shows 21⁄841 — the side 21 left unsquared over 29². The squaring has to happen on top as well, giving cos²θ = 441⁄841.
(d)
Option (d) shows 20⁄841, the third side PR = 20 over 29² without being squared. sin²θ is 400⁄841, and 441 − 400 = 41.

Concept

Two things have to be settled before a single ratio is written: which side is the hypotenuse, and which vertex carries θ.

The right angle is at R, so the side facing it — PQ = 29 — is the hypotenuse. θ sits at Q, which makes QR = 21 adjacent to θ and PR opposite it.

Pythagoras supplies the missing side, 20, and 20-21-29 is a triple worth recognising on sight. Then cos θ = 21⁄29 and sin θ = 20⁄29, and both squares share the denominator 841.

cos²θ − sin²θ is cos 2θ, but nothing here needs that identity. The two squares already share the denominator 841, so the subtraction is done on the numerators alone.

Key facts

The hypotenuse is the side opposite the right angle, which here makes PQ = 29 the hypotenuse.
20, 21 and 29 form a Pythagorean triple, since 400 + 441 = 841.
cos²θ + sin²θ is 1 for every θ, whereas cos²θ − sin²θ equals cos 2θ and varies with θ.

Study next

Fixing which side is adjacent and which opposite from the vertex carrying the anglePythagorean triples worth recognising on sight, such as 20-21-29 and 8-15-17The double-angle identity cos 2θ = cos²θ − sin²θ

Common traps

Taking QR = 21 as the hypotenuse because the right angle's position at R is skimmed.
Reaching for cos²θ + sin²θ = 1 by reflex when the stem prints a minus sign.
Leaving the sides unsquared, which produces 21⁄841 or 20⁄841 instead of a difference of squares.

Right-triangle ratios also run at 25 Sep 2024, 09:00, Quant Q.1, where (cosec θ − sin θ)(sec θ − cos θ)(tan θ + cot θ) has to be reduced, and at 25 Sep 2024, 16:00, Quant Q.14, where sec θ + tan θ = x is turned into sin θ.

Quant Q.7 here converts sin 74° + tan 74° into ratios of angles under 45°.

Related PYQs

No directly related past PYQ was found.