NDA / CDS / AFCAT - MCQ Practice Questions

tan 75° = tan(45° + 30°) = (tan 45° + tan 30°)/(1 - tan 45° tan 30°) = (1 + 1/√3)/(1 - 1/√3) = (√3 + 1)/(√3 - 1) × (√3 + 1)/(√3 + 1) = (3 + 2√3 + 1)/(3 - 1) = (4 + 2√3)/2 = 2 + √3

Total distance = 100 + 200 = 300 m. Speed = 60 km/h = 60 × $\frac{1000}{3600}$ = $\frac{50}{3}$ m/s. Time = 300 ÷ ($\frac{50}{3}$) = 300 × $\frac{3}{50}$ = 18 seconds

3x + 7 > 16 implies 3x > 9, so x > 3

We know sec²θ - tan²θ = 1. So (sec θ - tan θ)(sec θ + tan θ) = 1. Given sec θ - tan θ = $\frac{1}{5}$, we get ($\frac{1}{5}$)(sec θ + tan θ) = 1, so sec θ + tan θ = 5

# Sum of Interior Angles of a Hexagon

The sum of interior angles in any polygon depends on the number of sides it has, following a fixed mathematical formula.

Step 1: Identify the Number of Sides

A hexagon is a polygon with 6 sides. This is the key information we need to apply the formula.

Step 2: Apply the Interior Angles Formula

The sum of interior angles of any polygon is given by the formula: Sum = (n - 2) × 180°, where n is the number of sides.

The sum of interior angles of a hexagon is 720°.

Answer: (A) 720°

2^5 = 32, therefore x = 5

Circumference = 2πr = 2 × ($\frac{22}{7}$) × 7 = 44 cm

(5³)^($\frac{2}{3}$) = 5^(3 × $\frac{2}{3}$) = 5² = 25

From a:b = 3:4, let a=3k, b=4k. From b:c = 5:6, let b=5m, c=6m. Since b=4k=5m, k/m = $\frac{5}{4}$. So a:b:c = 15:20:24

Using factorization: 2x² - 7x + 3 = (2x-1)(x-3) = 0, so x = $\frac{1}{2}$ or x = 3

log₁₀(x) = 2 means 10² = x, so x = 100

SI = (P × R × T)/100 = (5000 × 8 × 2)/100 = 800

cot θ = $\frac{12}{5}$ means adjacent/opposite = $\frac{12}{5}$. Hypotenuse = √(144+25) = 13. So sin θ = opposite/hypotenuse = $\frac{5}{13}$

Area of parallelogram = base × height = 15 × 8 = 120 cm²

A ∪ B = {1,2,3,4}. (A ∪ B) ∩ A = {1,2,3,4} ∩ {1,2,3} = {1,2,3}

If roots are α and β, equation is x² - (α+β)x + αβ = 0. Here (2-3)x + (2)(-3) = 0 gives x² + x - 6 = 0

# Solution: Exponential Equations with Same Base

When multiplying powers with the same base, add the exponents together.

Step 1: Apply the Law of Exponents

Using the rule $a^m\times a^n=a^{m+n}$, we combine the left side:

Step 2: Express 243 as a Power of 3

Convert the right side to the same base by finding what power of 3 equals 243:

Step 3: Equate the Exponents

Now our equation becomes:

Since the bases are equal, the exponents must be equal:

Step 4: Solve for x

Verification: $3^2\times 3^3=9\times 27=243$ ✓

The answer is (A) 2

Speed = $\frac{120}{3}$ = 40 km/h = 40 × ($\frac{5}{18}$) = 11.11 m/s

tan A = 1 means A = 45°. sin 45° = cos 45° = 1/√2. So 2 × (1/√2) × (1/√2) = $\frac{2}{2}$ = 1

Arranging in order: {1,2,3,5,7,8,9}. Median is the middle value = 5