JEE Advanced Physics Vectors PYQ | Previous Year Questions with Solutions

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Q 1 :

Three vectors $\vec{P}, \vec{Q}$ and $\vec{R}$ are shown in the figure. Let S be any point on the vector $\vec{R}$ . The distance between the points P and S is $b | \vec{R} |$ . The general relation among vectors $\vec{P}, \vec{Q}$ and $\vec{S}$ is [2017]

[IMAGE 18]

$\vec{S} = (1 - b) \vec{P} + b \vec{Q}$
$\vec{S} = (b - 1) \vec{P} + b \vec{Q}$
$\vec{S} = (1 - b^{2}) \vec{P} + b \vec{Q}$
$\vec{S} = (1 - b) \vec{P} + b^{2} \vec{Q}$

(1)

$Here \vec{P} + b \vec{R} = \vec{S} \Rightarrow \vec{R} = \frac{\vec{S} - \vec{P}}{b}$

$Also \vec{R} = \vec{Q} - \vec{P}$

$∴$ $\frac{\vec{S} - \vec{P}}{b} = \vec{Q} - \vec{P} \Rightarrow \vec{S} - \vec{P} = b \vec{Q} - b \vec{P}$

$∴$ $\vec{S} = b \vec{Q} + (1 - b) \vec{P}$

Q 2 :

Two vectors $\vec{A}$ and $\vec{B}$ are defined as $\vec{A} = a \hat{i}$ and $\vec{B} = a (\cos ω t \hat{i} + \sin ω t \hat{j})$ , where $a$ is a constant and $ω = \frac{π}{6} {rad s}^{- 1}$ . If $| \vec{A} + \vec{B} |$ $= \sqrt{3}$ $| \vec{A} - \vec{B} |$ at time $t = τ$ for the first time, the value of $τ$ , in seconds, is ______. [2018]

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(2)

$Given:$ $| \vec{A} + \vec{B} |$ $= \sqrt{3}$ $| \vec{A} - \vec{B} |$

$∴$ $| a \hat{i} + a \cos ω t \hat{i} + a \sin ω t \hat{j} |$ $= \sqrt{3}$ $| a \hat{i} - a \cos ω t \hat{i} - a \sin ω t \hat{j} |$

$\Rightarrow$ $| (1 + \cos ω t) \hat{i} + \sin ω t \hat{j} |$ $= \sqrt{3}$ $| (1 - \cos ω t) \hat{i} - \sin ω t \hat{j} |$

$\sqrt{2 + 2 \cos ω t} = \sqrt{3} \sqrt{2 - 2 \cos ω t}$

$∴$ $1 + \cos ω t = 3 (1 - \cos ω t)$

$\Rightarrow 4 \cos ω t = 2$ $∴ \cos ω t = \frac{1}{2}$ $or, ω t = \frac{π}{3}$

$∴$ $\frac{π}{6} \times τ = \frac{π}{3}$ $∴ τ = 2.00 seconds$

Topic Question Set

Q 1 :

Three vectors $\vec{P}, \vec{Q}$ and $\vec{R}$ are shown in the figure. Let S be any point on the vector $\vec{R}$ . The distance between the points P and S is $b | \vec{R} |$ . The general relation among vectors $\vec{P}, \vec{Q}$ and $\vec{S}$ is [2017]

[IMAGE 18]

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Topic Question Set

Q 1 : Three vectors P→,Q→ and R→ are shown in the figure. Let S be any point on the vector R→. The distance between the points P and S is b|R→|. The general relation among vectors P→,Q→ and S→ is [2017] [IMAGE 18]

Q 2 : Two vectors A→ and B→ are defined as A→=ai^ and B→=a(cosωt i^+sinωt j^), where a is a constant and ω=π6 rad s-1. If |A→+B→|=3|A→-B→| at time t=τ for the first time, the value of τ, in seconds, is ______. [2018]

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Q 1 :

Three vectors $\vec{P}, \vec{Q}$ and $\vec{R}$ are shown in the figure. Let S be any point on the vector $\vec{R}$ . The distance between the points P and S is $b | \vec{R} |$ . The general relation among vectors $\vec{P}, \vec{Q}$ and $\vec{S}$ is [2017]

[IMAGE 18]