SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The tangent function is periodic with a period of π, which implies the identity tan(x + nπ) = tan x for any integer n. In the expression tan(θ - 14π), the value -14π is an integer multiple of the period π, where n = -14. Applying the periodicity property directly, we get tan(θ + (-14)π) = tan θ. Furthermore, since the sine and cosine functions are periodic with 2π, any even multiple of π such as 14π results in sin(θ - 14π) = sin θ and cos(θ - 14π) = cos θ. Thus, the ratio (sin(θ - 14π))/(cos(θ - 14π)) simplifies to (sin θ)/(cos θ), which is tan θ.

Explanation

Given x + 1/x = 2 cos θ, we can use the identity xⁿ + 1/xⁿ = 2 cos(nθ). For n = 3, this gives x³ + 1/x³ = 2 cos(3θ), which is derived from the expansion (x + 1/x)³ - 3(x + 1/x) = 8 cos³ θ - 6 cos θ = 2(4 cos³ θ - 3 cos θ) = 2 cos 3θ.

Explanation

The mean proportion between two numbers a and b is calculated using the formula √(a × b). Given the two expressions a = 6 + √8 and b = 3 - √2, we first simplify a by noting that √8 = √(4 × 2) = 2√2. Thus: a = 6 + 2√2 = 2(3 + √2)

Next, we find the product of a and b: a × b = [2(3 + √2)] × (3 - √2) a × b = 2[(3 + √2)(3 - √2)]

Applying the difference of squares identity (x + y)(x - y) = x² - y², the expression inside the brackets becomes: 3² - (√2)² = 9 - 2 = 7

Substituting this back into the product: a × b = 2 × 7 = 14

The mean proportion is the square root of this result: Mean Proportion = √(14)