Statement I: Two forces (P + Q) and (P - Q) where P ⊥ Q, when act at an angle θ 1 to each other, the magnitude of their resultant is √(3(P² + Q²)), when they act at an angle θ 2, the magnitude of their resultant becomes √(2(P² + Q²)). This is possible only when θ₁ < θ₂. Statement II: In the situation given above. θ 1 = 60 ° and θ 2 = 90 ° In the light of the above statements, choose the most appropriate answer from the options given below:- (a) Statement I is false but Statement II is true (b) Both Statement I and Statement II are true (c) Statement I is true but Statement II is false (d) Both Statement I and Statement II are false.

Answer: (b). Since P ⊥ Q: |P + Q| = |P - Q| = √(P² + Q²) = S. Two equal forces S at angle θ: R² = 2S²(1 + cos θ). R² = 3S² ⇒ cos θ₁ = 1/2, θ₁ = 60°; R² = 2S² ⇒ cos θ₂ = 0, θ₂ = 90°. So θ₁ < θ₂ and both statements are true: answer (b).

Statement I: Two forces (P + Q) and (P - Q) where P ⊥ Q, when act at… | JEE Main 2021 (31 Aug, Shift 2)

Q21JEE Main 2021 · 31 AUG · SHIFT 2Difficult

Addition & Subtraction of Vectors

Statement I:

Two forces $\left(\vec{P} + \vec{Q}\right)$ and $\left(\vec{P} - \vec{Q}\right)$ where $\vec{P} \perp \vec{Q}$ , when act at an angle $\theta$ 1 to each other, the magnitude of their resultant is $\sqrt{3(P^2 + Q^2)}$ , when they act at an angle $\theta$ 2, the magnitude of their resultant becomes $\sqrt{2(P^2 + Q^2)}$ . This is possible only when $\theta_1 < \theta_2$ .

Statement II:

In the situation given above.

$\theta$ 1 = 60 ${}^\circ$ and $\theta$ 2 = 90 ${}^\circ$

In the light of the above statements, choose the most appropriate answer from the options given below :-

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Answer: (b)

Since $\vec{P} \perp \vec{Q}$ : $|\vec{P} + \vec{Q}| = |\vec{P} - \vec{Q}| = \sqrt{P^2 + Q^2} = S$ .
Two equal forces $S$ at angle $\theta$ : $R^2 = 2S^2(1 + \cos\theta)$ .
$R^2 = 3S^2 \Rightarrow \cos\theta_1 = \frac{1}{2},\ \theta_1 = 60^\circ$ ; $R^2 = 2S^2 \Rightarrow \cos\theta_2 = 0,\ \theta_2 = 90^\circ$ .
So $\theta_1 < \theta_2$ and both statements are true: answer (b).

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Addition & Subtraction of Vectors question from JEE Main 2021 (31 Aug, Shift 2)

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