LFM Pure and Mechanics

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jee-main 2021 Q79 Magnitude of Vector Expression View
Let $\vec { a }$ and $\vec { b }$ be two vectors such that $| 2 \vec { a } + 3 \vec { b } | = | 3 \vec { a } + \vec { b } |$ and the angle between $\vec { a }$ and $\vec { b }$ is $60 ^ { \circ }$. If $\frac { 1 } { 8 } \vec { a }$ is a unit vector, then $| \vec { b } |$ is equal to :
(1) 8
(2) 4
(3) 6
(4) 5
jee-main 2021 Q90 Angle or Cosine Between Vectors View
For $p > 0$, a vector $\vec { v } _ { 2 } = 2 \hat { i } + ( p + 1 ) \hat { j }$ is obtained by rotating the vector $\vec { v } _ { 1 } = \sqrt { 3 } p \hat { i } + \hat { j }$ by an angle $\theta$ about origin in counter clockwise direction. If $\tan \theta = \frac { ( \alpha \sqrt { 3 } - 2 ) } { ( 4 \sqrt { 3 } + 3 ) }$, then the value of $\alpha$ is equal to $\underline{\hspace{1cm}}$.
jee-main 2022 Q1 Angle or Cosine Between Vectors View
Two vectors $\vec { A }$ and $\vec { B }$ have equal magnitudes. If magnitude of $\vec { A } + \vec { B }$ is equal to two times the magnitude of $\vec { A } - \vec { B }$, then the angle between $\vec { A }$ and $\vec { B }$ will be
(1) $\cos ^ { - 1 } \left( \frac { 3 } { 5 } \right)$
(2) $\cos ^ { - 1 } \left( \frac { 1 } { 3 } \right)$
(3) $\sin ^ { - 1 } \left( \frac { 1 } { 3 } \right)$
(4) $\sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$
jee-main 2022 Q6 Magnitude of Vector Expression View
Two bodies of mass 1 kg and 3 kg have position vectors $\hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ and $- 3 \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :
(1) $\hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$
(2) $- 3 \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$
(3) $- 2 \hat { \mathrm { i } } + 2 \widehat { \mathrm { k } }$
(4) $- 2 \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \widehat { \mathrm { k } }$
jee-main 2022 Q21 Dot Product Computation View
If $\vec { A } = 2 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } - \hat { \mathrm { k } }$ m and $\vec { B } = \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ m. The magnitude of component of vector $\vec { A }$ along vector $\vec { B }$ will be $\_\_\_\_$ m.
jee-main 2022 Q21 Perpendicularity or Parallel Condition View
If the projection of $2\hat{i} + 4\hat{j} - 2\hat{k}$ on $\hat{i} + 2\hat{j} + \alpha\hat{k}$ is zero. Then, the value of $\alpha$ will be
jee-main 2022 Q77 True/False or Multiple-Statement Verification View
Let $a$ and $b$ be two unit vectors such that $| ( a + b ) + 2 ( a \times b ) | = 2$. If $\theta \in ( 0 , \pi )$ is the angle between $\hat { \mathrm { a } }$ and $\widehat { \mathrm { b } }$, then among the statements: $( S 1 ) : 2 | \widehat { a } \times \hat { b } | = | \widehat { a } - \hat { b } |$ $( S 2 )$ : The projection of $\widehat { a }$ on $( \widehat { a } + \widehat { b } )$ is $\frac { 1 } { 2 }$
(1) Only $( S 1 )$ is true.
(2) Only $( S 2 )$ is true.
(3) Both $( S 1 )$ and $( S 2 )$ are true.
(4) Both $( S 1 )$ and $( S 2 )$ are false.
jee-main 2022 Q77 Angle or Cosine Between Vectors View
Let $\vec { a }$ and $\vec { b }$ be the vectors along the diagonal of a parallelogram having area $2 \sqrt { 2 }$. Let the angle between $\vec { a }$ and $\vec { b }$ be acute. $| \vec { a } | = 1$ and $| \vec { a } \cdot \vec { b } | = | \vec { a } \times \vec { b } |$. If $\vec { c } = 2 \sqrt { 2 } ( \vec { a } \times \vec { b } ) - 2 \vec { b }$, then an angle between $\vec { b }$ and $\vec { c }$ is
(1) $\frac { - \pi } { 4 }$
(2) $\frac { 5 \pi } { 6 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { 3 \pi } { 4 }$
jee-main 2022 Q77 True/False or Multiple-Statement Verification View
Let $ABC$ be a triangle such that $\overrightarrow { BC } = \vec { a }$, $\overrightarrow { CA } = \vec { b }$, $\overrightarrow { AB } = \vec { c }$, $|\vec{a}| = 6\sqrt{2}$, $|\vec{b}| = 2\sqrt{3}$ and $\vec{b} \cdot \vec{c} = 12$. Consider the statements: S1: $|\vec{a} \times \vec{b} + \vec{c} \times \vec{b}| - |\vec{c}| = 6(2\sqrt{2} - 1)$ S2: $\angle ABC = \cos^{-1}\sqrt{\frac{2}{3}}$. Then
(1) both $S1$ and $S2$ are true
(2) only $S1$ is true
(3) only $S2$ is true
(4) both $S1$ and $S2$ are false
jee-main 2022 Q78 Expressing a Vector as a Linear Combination View
Let $\vec { a } = \hat { i } + \hat { j } + 2 \widehat { k } , \vec { b } = 2 \hat { i } - 3 \hat { j } + \widehat { k }$ and $\vec { c } = \hat { i } - \hat { j } + \widehat { k }$ be the three given vectors. Let $\vec { v }$ be a vector in the plane of $\vec { a }$ and $\vec { b }$ whose projection on $\vec { c }$ is $\frac { 2 } { \sqrt { 3 } }$. If $\vec { v } \cdot \hat { j } = 7$, then $\vec { v } \cdot ( \hat { i } + \hat { k } )$ is equal to
(1) 6
(2) 7
(3) 8
(4) 9
jee-main 2023 Q1 Magnitude of Vector Expression View
When vector $\vec { A } = 2 \hat { i } + 3 \hat { j } + 2 \widehat { k }$ is subtracted from vector $\vec { B }$, it gives a vector equal to $2 \hat { j }$. Then the magnitude of vector $\vec { B }$ will be:
(1) $\sqrt { 5 }$
(2) 3
(3) $\sqrt { 6 }$
(4) $\sqrt { 33 }$
jee-main 2023 Q21 Perpendicularity or Parallel Condition View
Vectors $a \hat { i } + b \hat { j } + \hat { k }$ and $2 \hat { i } - 3 \hat { j } + 4 \hat { k }$ are perpendicular to each other when $3 a + 2 b = 7$, the ratio of $a$ to $b$ is $\frac { x } { 2 }$. The value of $x$ is $\_\_\_\_$ .
jee-main 2023 Q75 Expressing a Vector as a Linear Combination View
An arc $PQ$ of a circle subtends a right angle at its centre $O$. The mid point of the arc $PQ$ is $R$. If $\overrightarrow{OP} = \vec{u}$, $\overrightarrow{OR} = \vec{v}$ and $\overrightarrow{OQ} = \alpha\vec{u} + \beta\vec{v}$, then $\alpha$, $\beta^2$ are the roots of the equation
(1) $x^2 + x - 2 = 0$
(2) $x^2 - x - 2 = 0$
(3) $3x^2 - 2x - 1 = 0$
(4) $3x^2 + 2x - 1 = 0$
jee-main 2023 Q77 Expressing a Vector as a Linear Combination View
Let $ABCD$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and $( \overrightarrow { AB } - \overrightarrow { BC } ) + ( \overrightarrow { AD } - \overrightarrow { DC } ) = k \overrightarrow { FE }$, then $k$ is equal to
(1) 4
(2) $- 2$
(3) 2
(4) $- 4$
jee-main 2023 Q89 Optimization of a Vector Expression View
Let $\vec{v} = \alpha\hat{i} + 2\hat{j} - 3\hat{k}$, $\vec{w} = 2\alpha\hat{i} + \hat{j} - \hat{k}$, and $\vec{u}$ be a vector such that $|\vec{u}| = \alpha > 0$. If the minimum value of the scalar triple product $[\vec{u}\, \vec{v}\, \vec{w}]$ is $-\alpha\sqrt{3401}$, and $|\vec{u} \cdot \hat{i}|^2 = \frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m + n$ is equal to $\_\_\_\_$.
jee-main 2024 Q21 Perpendicularity or Parallel Condition View
For three vectors $\vec { A } = ( - x \hat { i } - 6 \hat { j } - 2 \hat { k } ) , \vec { B } = ( - \hat { i } + 4 \hat { j } + 3 \hat { k } )$ and $\vec { C } = ( - 8 \hat { i } - \hat { j } + 3 \hat { k } )$, if $\vec { A } \cdot ( \vec { B } \times \vec { C } ) = 0$, then value of $x$ is $\_\_\_\_$
jee-main 2024 Q21 Perpendicularity or Parallel Condition View
If $\vec { a }$ and $\vec { b }$ makes an angle $\cos ^ { - 1 } \left( \frac { 5 } { 9 } \right)$ with each other, then $| \vec { a } + \vec { b } | = \sqrt { 2 } | \vec { a } - \vec { b } |$ for $| \vec { a } | = n | \vec { b } |$ The integer value of n is $\_\_\_\_$
jee-main 2024 Q77 Area Computation Using Vectors View
Let $\overrightarrow { \mathrm { OA } } = \overrightarrow { \mathrm { a } } , \overrightarrow { \mathrm { OB } } = 12 \overrightarrow { \mathrm { a } } + 4 \overrightarrow { \mathrm {~b} }$ and $\overrightarrow { \mathrm { OC } } = \overrightarrow { \mathrm { b } }$, where O is the origin. If $S$ is the parallelogram with adjacent sides OA and OC, then $\frac { \text { area of the quadrilateral } \mathrm { OABC } } { \text { area of } \mathrm { S } }$ is equal to
(1) 6
(2) 10
(3) 7
(4) 8
jee-main 2024 Q78 Vector Algebra and Triple Product Computation View
Let a unit vector $\widehat { u } = x \hat { i } + y \hat { j } + z \widehat { k }$ make angles $\frac { \pi } { 2 } , \frac { \pi } { 3 }$ and $\frac { 2 \pi } { 3 }$ with the vectors $\frac { 1 } { \sqrt { 2 } } \hat { i } + \frac { 1 } { \sqrt { 2 } } \widehat { k } , \frac { 1 } { \sqrt { 2 } } \hat { j } + \frac { 1 } { \sqrt { 2 } } \widehat { k }$ and $\frac { 1 } { \sqrt { 2 } } \hat { i } + \frac { 1 } { \sqrt { 2 } } \hat { j }$ respectively. If $\vec { v } = \frac { 1 } { \sqrt { 2 } } \hat { i } + \frac { 1 } { \sqrt { 2 } } \hat { j } + \frac { 1 } { \sqrt { 2 } } \hat { k }$, then $| \hat { u } - \vec { v } | ^ { 2 }$ is equal to
(1) $\frac { 11 } { 2 }$
(2) $\frac { 5 } { 2 }$
(3) 9
(4) 7
jee-main 2024 Q89 Angle or Cosine Between Vectors View
The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + 2 \widehat { \mathrm { k } }$ and $\alpha \hat { \mathrm { i } } + 2 \alpha \hat { \mathrm { j } } - 2 \widehat { \mathrm { k } }$ is acute, is $\_\_\_\_$.
jee-main 2025 Q1 Angle or Cosine Between Vectors View
Q1. The angle between vector $\vec { Q }$ and the resultant of $( 2 \vec { Q } + 2 \vec { P } )$ and $( 2 \vec { Q } - 2 \vec { P } )$ is :
(1) $\tan ^ { - 1 } \frac { ( 2 \vec { Q } - 2 \vec { P } ) } { 2 \vec { Q } + 2 \vec { P } }$
(2) $0 ^ { \circ }$
(3) $\tan ^ { - 1 } ( \mathrm { P } / \mathrm { Q } )$
(4) $\tan ^ { - 1 } ( 2 Q / P )$
jee-main 2025 Q2 Magnitude of Vector Expression View
If the components of $\overrightarrow { \mathrm { a } } = \alpha \hat { i } + \beta \hat { j } + \gamma \hat { k }$ along and perpendicular to $\overrightarrow { \mathrm { b } } = 3 \hat { i } + \hat { j } - \hat { k }$ respectively, are $\frac { 16 } { 11 } ( 3 \hat { i } + \hat { j } - \hat { k } )$ and $\frac { 1 } { 11 } ( - 4 \hat { i } - 5 \hat { j } - 17 \hat { k } )$, then $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$ is equal to :
(1) 26
(2) 18
(3) 23
(4) 16
jee-main 2025 Q2 Vector Word Problem / Physical Application View
Q2. A cyclist starts from the point $P$ of a circular ground of radius 2 km and travels along its circumference to the [Figure] point S . The displacement of a cyclist is:
(1) $\sqrt { 8 } \mathrm {~km}$
(2) 8 km
(3) 6 km
(4) 4 km
jee-main 2025 Q3 Point-to-Line Distance Computation View
Let $\mathrm { A } , \mathrm { B } , \mathrm { C }$ be three points in $xy$-plane, whose position vector are given by $\sqrt { 3 } \hat { i } + \hat { j } , \hat { i } + \sqrt { 3 } \hat { j }$ and $\mathrm { a } \hat { i } + ( 1 - \mathrm { a } ) \hat { j }$ respectively with respect to the origin O. If the distance of the point C from the line bisecting the angle between the vectors $\overrightarrow { \mathrm { OA } }$ and $\overrightarrow { \mathrm { OB } }$ is $\frac { 9 } { \sqrt { 2 } }$, then the sum of all the possible values of $a$ is :
(1) 2
(2) $9/2$
(3) 1
(4) 0
jee-main 2025 Q8 Section Ratios and Intersection via Vectors View
Let the point A divide the line segment joining the points $P ( - 1 , - 1,2 )$ and $Q ( 5,5,10 )$ internally in the ratio $\mathrm { r } : 1 ( \mathrm { r } > 0 )$. If O is the origin and $( \overrightarrow { \mathrm { OQ } } \cdot \overrightarrow { \mathrm { OA } } ) - \frac { 1 } { 5 } | \overrightarrow { \mathrm { OP } } \times \overrightarrow { \mathrm { OA } } | ^ { 2 } = 10$, then the value of r is :
(1) $\sqrt { 7 }$
(2) 14
(3) 3
(4) 7

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