LFM Pure and Mechanics

jee-main 2017 Q88 Distance from a Point to a Line (Show/Compute) View

The distance of the point $(1, 3, -7)$ from the plane passing through the point $(1, -1, -1)$, having normal perpendicular to both the lines $\dfrac{x-1}{1} = \dfrac{y+2}{-2} = \dfrac{z-4}{3}$ and $\dfrac{x-2}{2} = \dfrac{y+1}{-1} = \dfrac{z+7}{-1}$, is:
(1) $\dfrac{20}{\sqrt{74}}$
(2) $\dfrac{10}{\sqrt{83}}$
(3) $\dfrac{5}{\sqrt{83}}$
(4) $\dfrac{10}{\sqrt{74}}$

jee-main 2018 Q89 Distance from a Point to a Line (Show/Compute) View

The length of the projection of the line segment joining the points $( 5 , - 1,4 )$ and $( 4 , - 1,3 )$ on the plane, $x + y + z = 7$ is
(1) $\sqrt { \frac { 2 } { 3 } }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$

jee-main 2019 Q87 Vector Algebra and Triple Product Computation View

Let $\vec{a} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}},\, \vec{b} = b_1\hat{\mathrm{i}} + b_2\hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}}$ and $\vec{c} = 5\hat{\mathrm{i}} + \hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}}$ be three vectors such that the projection vector of $\vec{b}$ on $\vec{a}$ is $|\vec{a}|$. If $\vec{a} + \vec{b}$ is perpendicular to $\vec{c}$, then $|\vec{b}|$ is equal to:
(1) $\sqrt{22}$
(2) $\sqrt{32}$
(3) 6
(4) 4

jee-main 2019 Q88 Distance from a Point to a Line (Show/Compute) View

The vertices $B$ and $C$ of a $\triangle A B C$ lie on the line, $\frac { x + 2 } { 3 } = \frac { y - 1 } { 0 } = \frac { z } { 4 }$ such that $B C = 5$ units. Then the area (in sq. units) of this triangle, given the point $A ( 1 , - 1,2 )$, is
(1) 6
(2) $2 \sqrt { 34 }$
(3) $\sqrt { 34 }$
(4) $5 \sqrt { 17 }$

jee-main 2019 Q89 Section Division and Coordinate Computation View

If a point $R ( 4 , y , z )$ lies on the line segment joining the points $P ( 2 , - 3 , 4 )$ and $Q ( 8 , 0 , 10 )$, then the distance of $R$ from the origin is
(1) $2 \sqrt { 21 }$
(2) $\sqrt { 53 }$
(3) 6
(4) $2 \sqrt { 14 }$

jee-main 2020 Q68 Vector Algebra and Triple Product Computation View

A vector $\vec { a } = \alpha \hat { i } + 2 \hat { j } + \beta \hat { k }$ $(\alpha, \beta \in R)$ lies in the plane of the vectors, $\vec { b } = \hat { i } + \hat { j }$ and $\vec { c } = \hat { i } - \hat { j } + 4 \hat { k }$. If $\vec { a }$ bisects the angle between $\vec { b }$ and $\vec { c }$, then
(1) $\vec { a } \cdot \hat { i } + 3 = 0$
(2) $\vec { a } \cdot \hat { i } + 1 = 0$
(3) $\vec { a } \cdot \widehat { k } + 2 = 0$
(4) $\vec { a } \cdot \widehat { k } + 4 = 0$

jee-main 2020 Q69 MCQ: Relationship Between Two Lines View

The lines $\vec { r } = ( \hat { i } - \hat { j } ) + l ( 2 \hat { i } + \widehat { k } )$ and $\vec { r } = ( 2 \hat { i } - \hat { j } ) + m ( \hat { i } + \hat { j } - \widehat { k } )$
(1) Do not intersect for any values of $l$ and $m$
(2) Intersect for all values of $l$ and $m$
(3) Intersect when $l = 2$ and $m = \frac { 1 } { 2 }$
(4) Intersect when $l = 1$ and $m = 2$

jee-main 2021 Q77 Distance from a Point to a Line (Show/Compute) View

If for $a > 0$, the feet of perpendiculars from the points $A ( a , - 2 a , 3 )$ and $B ( 0,4,5 )$ on the plane $l x + m y + n z = 0$ are points $C ( 0 , - a , - 1 )$ and $D$ respectively, then the length of line segment $C D$ is equal to :
(1) $\sqrt { 31 }$
(2) $\sqrt { 41 }$
(3) $\sqrt { 55 }$
(4) $\sqrt { 66 }$

jee-main 2021 Q78 Vector Algebra and Triple Product Computation View

Let $\vec { a } = \hat { i } + \hat { j } + \hat { k }$ and $\vec { b } = \hat { j } - \hat { k }$. If $\vec { c }$ is a vector such that $\vec { a } \times \vec { c } = \vec { b }$ and $\vec { a } \cdot \vec { c } = 3$, then $\vec { a } \cdot ( \vec { b } \times \vec { c } )$ is equal to

jee-main 2021 Q78 Parametric Representation of a Line View

The equation of the line through the point $( 0,1,2 )$ and perpendicular to the line $\frac { x - 1 } { 2 } = \frac { y + 1 } { 3 } = \frac { z - 1 } { - 2 }$ is :
(1) $\frac { x } { 3 } = \frac { y - 1 } { - 4 } = \frac { z - 2 } { 3 }$
(2) $\frac { x } { 3 } = \frac { y - 1 } { 4 } = \frac { z - 2 } { 3 }$
(3) $\frac { x } { - 3 } = \frac { y - 1 } { 4 } = \frac { z - 2 } { 3 }$
(4) $\frac { x } { 3 } = \frac { y - 1 } { 4 } = \frac { z - 2 } { - 3 }$

jee-main 2021 Q78 Normal Vector and Plane Equation View

A plane passes through the points $A ( 1,2,3 ) , B ( 2,3,1 )$ and $C ( 2,4,2 )$. If $O$ is the origin and $P$ is $( 2 , - 1,1 )$, then the projection of $\overrightarrow { O P }$ on this plane is of length:
(1) $\sqrt { \frac { 2 } { 5 } }$
(2) $\sqrt { \frac { 2 } { 7 } }$
(3) $\sqrt { \frac { 2 } { 3 } }$
(4) $\sqrt { \frac { 2 } { 11 } }$

jee-main 2021 Q78 Shortest Distance Between Two Lines View

Let the position vectors of two points $P$ and $Q$ be $3 \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \widehat { \mathrm { k } }$ and $\hat { \mathrm { i } } + 2 \hat { \mathrm { j } } - 4 \widehat { \mathrm { k } }$, respectively. Let $R$ and $S$ be two points such that the direction ratios of lines $P R$ and $Q S$ are $( 4 , - 1,2 )$ and $( - 2,1 , - 2 )$, respectively. Let lines $P R$ and $Q S$ intersect at $T$. If the vector $\overrightarrow { T A }$ is perpendicular to both $\overrightarrow { P R }$ and $\overrightarrow { Q S }$ and the length of vector $\overrightarrow { T A }$ is $\sqrt { 5 }$ units, then the modulus of a position vector of $A$ is :
(1) $\sqrt { 482 }$
(2) $\sqrt { 171 }$
(3) $\sqrt { 5 }$
(4) $\sqrt { 227 }$

jee-main 2021 Q79 Line-Plane Intersection View

Let $P$ be a plane $l x + m y + n z = 0$ containing the line, $\frac { 1 - x } { 1 } = \frac { y + 4 } { 2 } = \frac { z + 2 } { 3 }$. If plane $P$ divides the line segment $A B$ joining points $A ( - 3 , - 6,1 )$ and $B ( 2,4 , - 3 )$ in ratio $k : 1$ then the value of $k$ is equal to :
(1) 1.5
(2) 3
(3) 2
(4) 4

jee-main 2021 Q79 MCQ: Relationship Between Two Lines View

The angle between the straight lines, whose direction cosines $l , m , n$ are given by the equations $2 l + 2 m - n = 0$ and $m n + n l + \operatorname { lm } = 0$, is: (1) $\frac { \pi } { 3 }$ (2) $\frac { \pi } { 2 }$ (3) $\cos ^ { - 1 } \left( \frac { 8 } { 9 } \right)$ (4) $\pi - \cos ^ { - 1 } \left( \frac { 4 } { 9 } \right)$

jee-main 2021 Q89 Line-Plane Intersection View

The square of the distance of the point of intersection of the line $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z + 1 } { 6 }$ and the plane $2 x - y + z = 6$ from the point $( - 1 , - 1,2 )$ is

jee-main 2021 Q89 Distance from a Point to a Line (Show/Compute) View

Let $S$ be the mirror image of the point $Q ( 1,3,4 )$ with respect to the plane $2 x - y + z + 3 = 0$ and let $R ( 3,5 , \gamma )$ be a point of this plane. Then the square of the length of the line segment $S R$ is

jee-main 2021 Q90 Vector Algebra and Triple Product Computation View

Let $\vec { a } = \hat { i } + 2 \hat { j } - \widehat { k } , \vec { b } = \hat { i } - \hat { j }$ and $\vec { c } = \hat { i } - \hat { j } - \hat { k }$ be three given vectors. If $\vec { r }$ is a vector such that $\vec { r } \times \vec { a } = \vec { c } \times \vec { a }$ and $\vec { r } \cdot \vec { b } = 0$, then $\vec { r } \cdot \vec { a }$ is equal to

jee-main 2021 Q90 Shortest Distance Between Two Lines View

A line $l$ passing through origin is perpendicular to the lines $l _ { 1 } : \vec { r } = ( 3 + t ) \hat { \mathrm { i } } + ( - 1 + 2 t ) \hat { \mathrm { j } } + ( 4 + 2 t ) \hat { \mathrm { k } }$ $l _ { 2 } : \vec { r } = ( 3 + 2 s ) \hat { \mathrm { i } } + ( 3 + 2 s ) \hat { \mathrm { j } } + ( 2 + s ) \hat { \mathrm { k } }$ If the co-ordinates of the point in the first octant on $l _ { 2 }$ at a distance of $\sqrt { 17 }$ from the point of intersection of $l$ and $l _ { 1 }$ are $( a , b , c )$, then $18 ( a + b + c )$ is equal to $\underline{\hspace{1cm}}$.

jee-main 2022 Q77 Normal Vector and Plane Equation View

If $(2, 3, 9)$, $(5, 2, 1)$, $(1, \lambda, 8)$ and $(\lambda, 2, 3)$ are coplanar, then the product of all possible values of $\lambda$ is
(1) $\frac{21}{2}$
(2) $\frac{59}{8}$
(3) $\frac{57}{8}$
(4) $\frac{95}{8}$

jee-main 2022 Q77 Distance from a Point to a Line (Show/Compute) View

If the length of the perpendicular drawn from the point $P ( a , 4,2 ) , a > 0$ on the line $\frac { x + 1 } { 2 } = \frac { y - 3 } { 3 } = \frac { z - 1 } { - 1 }$ is $2 \sqrt { 6 }$ units and $Q \left( \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } \right)$ is the image of the point $P$ in this line, then $a + \sum _ { i = 1 } ^ { 3 } \alpha _ { i }$ is equal to
(1) 7
(2) 8
(3) 12
(4) 14

jee-main 2022 Q78 Shortest Distance Between Two Lines View

If the shortest distance between the lines $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { \lambda }$ and $\frac { x - 2 } { 1 } = \frac { y - 4 } { 4 } = \frac { z - 5 } { 5 }$ is $\frac { 1 } { \sqrt { 3 } }$, then the sum of all possible values of $\lambda$ is:
(1) 16
(2) 6
(3) 12
(4) 15

jee-main 2022 Q78 Normal Vector and Plane Equation View

A plane $E$ is perpendicular to the two planes $2x - 2y + z = 0$ and $x - y + 2z = 4$, and passes through the point $P(1, -1, 1)$. If the distance of the plane $E$ from the point $Q(a, a, 2)$ is $3\sqrt{2}$, then $(PQ)^2$ is equal to
(1) 9
(2) 12
(3) 21
(4) 33

jee-main 2022 Q78 MCQ: Relationship Between Two Lines View

If the two lines $l _ { 1 } : \frac { x - 2 } { 3 } = \frac { y + 1 } { - 2 } , z = 2$ and $l _ { 2 } : \frac { x - 1 } { 1 } = \frac { 2 y + 3 } { \alpha } = \frac { z + 5 } { 2 }$ are perpendicular, then an angle between the lines $l _ { 2 }$ and $l _ { 3 } : \frac { 1 - x } { 3 } = \frac { 2 y - 1 } { - 4 } = \frac { z } { 4 }$ is
(1) $\cos ^ { - 1 } \left( \frac { 29 } { 4 } \right)$
(2) $\sec ^ { - 1 } \left( \frac { 29 } { 4 } \right)$
(3) $\cos ^ { - 1 } \left( \frac { 2 } { 29 } \right)$
(4) $\cos ^ { - 1 } \left( \frac { 2 } { \sqrt { 29 } } \right)$

jee-main 2022 Q78 Distance from a Point to a Line (Show/Compute) View

The length of the perpendicular from the point $( 1 , - 2,5 )$ on the line passing through $( 1,2,4 )$ and parallel to the line $x + y - z = 0 = x - 2 y + 3 z - 5$ is:
(1) $\sqrt { \frac { 21 } { 2 } }$
(2) $\sqrt { \frac { 9 } { 2 } }$
(3) $\sqrt { \frac { 73 } { 2 } }$
(4) 1

jee-main 2022 Q79 Dihedral Angle Computation View

Let the points on the plane $P$ be equidistant from the points $( - 4,2,1 )$ and $( 2 , - 2,3 )$. Then the acute angle between the plane $P$ and the plane $2 x + y + 3 z = 1$ is
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { 5 \pi } { 12 }$

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40