SSC CGL 2024 Tier-I · 9 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 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 then the value of is:

Answer

Why

Correct — D. Rule: every term of the given ratio has the same degree in sinθ and cosθ, so dividing through by cosθ turns the whole thing into tanθ.

(7tanθ + 4)/(9tanθ − 2) = 5/4
Cross-multiply: 28tanθ + 16 = 45tanθ − 10
26 = 17tanθ → tanθ = 26/17
tan²θ = 676/289

Now the expression, multiplying top and bottom by 289:
(676 + 5×289)/(676 − 5×289) = (676 + 1445)/(676 − 1445)
= 2121/(−769) = −2121/769 → option (d)

The sign is negative because tan²θ ≈ 2.34 is smaller than 5.

Why the others are wrong

(a)
−769/2121 is the answer turned upside down. It is what you get by evaluating (tan²θ − 5) over (tan²θ + 5); read the fraction in the order the question prints it.
(b)
769/2121 is wrong twice over — inverted and positive. 769 belongs in the denominator, and it arrives negative because tan²θ ≈ 2.34 falls short of 5.
(c)
The magnitude is right and the sign is not. tan²θ − 5 = 676/289 − 5 is a negative number, so the fraction as a whole is negative.

Concept

An expression is homogeneous when every term of the numerator and every term of the denominator carries the same power of sinθ and cosθ combined. Dividing it through by cosθ replaces each sinθ/cosθ with tanθ and the angle itself never has to be found.

That matters here because tanθ = 26/17 is not a standard angle — there is no θ to look up, and none is needed.

The last decision is the sign. Once tan²θ is known, whether tan²θ − 5 comes out positive or negative settles it.

The question is set as two images. The condition reads (7sinθ + 4cosθ)/(9sinθ − 2cosθ) = 5/4, and the expression asked for is (tan²θ + 5)/(tan²θ − 5).

All four options are images of the same two numbers — 2121/769 and 769/2121 — each with and without a minus sign, so sign and orientation are the only things separating them.

Key facts

Dividing a homogeneous sin-cos ratio by cosθ replaces every sinθ/cosθ with tanθ.
Cross-multiplying 4(7sinθ + 4cosθ) = 5(9sinθ − 2cosθ) reduces to 26cosθ = 17sinθ.
So tanθ = 26/17 and tan²θ = 676/289 ≈ 2.34, which is less than 5.
Multiplying (tan²θ + 5)/(tan²θ − 5) top and bottom by 289 gives 2121/(−769).

Study next

Homogeneous trigonometric ratios and when to divide by cosθ rather than cos²θRecovering sinθ and cosθ from tanθ using a right triangle with legs 26 and 17Identities linking tan²θ and sec²θ, which convert the same expression a second way

Common traps

Mis-collecting after cross-multiplication: 28sinθ + 16cosθ = 45sinθ − 10cosθ gives 26cosθ = 17sinθ.
Dropping the minus sign and marking 2121/769.
Chasing the value of θ, which is neither a standard angle nor needed at any point.

SSC gives one linear sin-cos ratio and asks for a second expression built out of tanθ. In this item the four options are the same two numbers in all four sign-and-inversion combinations, so the sign check is worth as many marks as the algebra.

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