SSC CGL 2024 Tier-I · 19 September 2024, Shift 2 · Quantitative Aptitude · Question 21 of 99

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.

The maximum value of (2 sin θ + 3 cos θ) is:

Answer

Why

Correct — D. The stem is printed as an image and reads: The maximum value of (2 sin θ + 3 cos θ) is. The options are images as well, and option (d) shows √13.

Rule: a sin θ + b cos θ never leaves the band from −√(a² + b²) to +√(a² + b²).

Here a = 2 and b = 3.
a² + b² = 4 + 9 = 13
Maximum = √(a² + b²) = √13 ≈ 3.61 → option (d)

Why the others are wrong

(a)
Option (a) shows 9, which is only b². Even the crude ceiling 2 + 3 = 5, which would need sin θ and cos θ both to equal 1 at once, is well under 9.
(b)
Option (b) shows √17, and 17 is not 2² + 3². It is 1² + 4², the answer to a stem with coefficients 1 and 4.
(c)
Option (c) shows √11 ≈ 3.32, but at θ = 45° the expression already equals 5/√2 ≈ 3.54. A maximum cannot sit below a value the expression actually reaches.

Concept

Any expression a sin θ + b cos θ can be rewritten as R sin(θ + φ), with R = √(a² + b²). A sine never leaves −1 to 1, so the expression is confined to the band −R to +R and touches each end at some angle.

The two coefficients settle everything. You need neither φ nor the angle at which the peak occurs, unless the question asks for it.

A constant shifts the band without changing its width: c + a sin θ + b cos θ runs from c − R to c + R. Check for such a constant before reaching for the formula.

Key facts

a sin θ + b cos θ equals R sin(θ + φ) with R = √(a² + b²).
Its maximum is +√(a² + b²) and its minimum is −√(a² + b²).
For a = 2 and b = 3 the amplitude is √13, about 3.61.
At θ = 45° the expression here is 5/√2, about 3.54, just short of its peak.

Study next

Writing a sin θ + b cos θ as R sin(θ + φ) and locating φMaximum and minimum of a sin²θ + b cos²θ, where the answer is simply the larger and smaller coefficientRange of an expression with a constant added, such as 5 + 3 sin θ − 4 cos θ

Common traps

Adding the coefficients to 5, which would need sin θ and cos θ to be 1 at the same angle.
Answering 13 and forgetting the square root.
Taking the minimum as 0 rather than −√13.

SSC also puts trigonometric identity work in this same paper: Quant Q.3 simplifies 1 − sin²32° + 1/(1 + tan²58°), and Quant Q.12 evaluates (1 − sin 2t)/(1 + sin 2t) × (cos t + sin t)/(cos t − sin t).

A range question like this one is different in kind — nothing has to be simplified, because the coefficients alone decide the answer.

Related PYQs

No directly related past PYQ was found.