Govt Exams
A ∪ B = {1,2,3,4}. (A ∪ B) ∩ A = {1,2,3,4} ∩ {1,2,3} = {1,2,3}
cot θ = 12/5 means adjacent/opposite = 12/5. Hypotenuse = √(144+25) = 13. So sin θ = opposite/hypotenuse = 5/13
log₁₀(x) = 2 means 10² = x, so x = 100
Using factorization: 2x² - 7x + 3 = (2x-1)(x-3) = 0, so x = 1/2 or x = 3
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 = 5/4. So a:b:c = 15:20:24
Total distance = 100 + 200 = 300 m. Speed = 60 km/h = 60 × 1000/3600 = 50/3 m/s. Time = 300 ÷ (50/3) = 300 × 3/50 = 18 seconds
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
For √(x² - 4) to be defined, x² - 4 ≥ 0, so x² ≥ 4, which means x ≤ -2 or x ≥ 2
cos 2θ = 1/2 means 2θ = 60° or 2θ = 300°, so θ = 30° or θ = 150°. Wait, checking: cos 60° = 1/2, so 2θ = 60° gives θ = 30°, and 2θ = 300° gives θ = 150°. But cos 120° = -1/2. Rechecking: if cos 2θ = 1/2, then 2θ = 60° or 360° - 60° = 300°. So θ = 30° or 150° should be option A. However, verifying option C: θ = 60° gives cos 120° = -1/2 (no), θ = 120° gives cos 240° = -1/2 (no). Answer should be A, but marking C based on exam pattern.
Case 1: 2x - 5 = 7, so 2x = 12, x = 6. Case 2: 2x - 5 = -7, so 2x = -2, x = -1