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Angle between two forces from a condition on their resultant

This type of question, asked 3 times: 2021 Paper 23, 4 Apr 2024, 9 Apr 2024. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 23

    4
    Statement I: Two forces (P→+Q→)(\overrightarrow{P}+\overrightarrow{Q}) and (P→−Q→)(\overrightarrow{P}-\overrightarrow{Q}) where P→⊥Q→\overrightarrow{P}\bot\overrightarrow{Q}, when act at an angle θ1\theta_{1} to each other, the magnitude of their resultant is 3(P2+Q2)\sqrt{3\left( P^{2}+Q^{2} \right)}, when they act at an angle θ2\theta_{2}, the magnitude of their resultant becomes 2(P2+Q2)\sqrt{2\left( P^{2}+Q^{2} \right)}. This is possible only when θ1<θ2\theta_{1}<\theta_{2}. Statement II: In the situation given above. θ1=60∘\theta_{1}=60^{\circ} and θ2=90∘\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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  2. Question 2

    4 Apr 2024

    29
    Two forces F→1{\overrightarrow{F}}_{1} and F→2{\overrightarrow{F}}_{2} are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between F→1{\overrightarrow{F}}_{1} and F→2{\overrightarrow{F}}_{2} is cos⁡−1(1n)\cos^{- 1}\left( \frac{1}{n} \right). The value of ∣n∣|n| is___

  3. Question 3

    9 Apr 2024

    117
    The resultant of two vectors A→\overrightarrow{A} and B→\overrightarrow{B} is perpendicular to A→\overrightarrow{A} and its magnitude is half that of B→\overrightarrow{B}. The angle between vectors A→\overrightarrow{A} and B→\overrightarrow{B} Is..........