Statement I :
Two forces (P⃗+Q⃗) and (P⃗-Q⃗) where P⃗⊥ Q⃗, when act at an angle θ₁ to each other, the magnitude of their resultant is √3(P²+Q²), when they act at an angle θ₂, the magnitude of their resultant becomes √2(P²+Q²). This is possible only when θ₁<θ₂.
Statement II :
In the situation given above.
θ₁ = 60° and θ₂ = 90°
In the light of the above statements, choose the most appropriate answer from the options given below :-

Solution

Physics sir · 4 steps

Since P is perpendicular to Q, both forces have the same size, so the resultant depends only on the angle.

1. Equal magnitudes. Because P⃗⊥Q⃗, the dot product P⃗·Q⃗ = 0. So |P⃗+Q⃗|² = |P⃗-Q⃗|² = P²+Q². Call this common magnitude A, with A² = P²+Q².

   R² = A² + A² + 2A²cos θ = 2A²(1+cos θ)

2. Find both angles. For R² = 3A²: 1+cos θ₁ = 3/2, so cos θ₁ = 1/2 and θ₁ = 60°. For R² = 2A²: 1+cos θ₂ = 1, so cos θ₂ = 0 and θ₂ = 90°. Statement II is true.
3. Check Statement I. As θ goes from 0° to 180°, cos θ falls, so R falls. A larger resultant needs a smaller angle, so θ₁ < θ₂ is the only possibility. Statement I is true.
4. So. Both statements are true, which is option (b).

The idea to keep. For two vectors of equal size A at angle θ, R = 2A cos θ/2. The resultant shrinks as the angle grows, so a bigger resultant always means a smaller angle.

Answer from the NTA final answer key. Solution by PrepWiser.