LFM Pure and Mechanics

jee-main 2023 Q87 Normal Vector and Plane Equation View

A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\circ}$, to the $y$-axis at $45^{\circ}$ and to the $z$-axis at an acute angle. If a plane passing through the points $(\sqrt{2}, -1, 1)$ and $(a, b, c)$, is normal to $\vec{v}$, then
(1) $\sqrt{2}a + b + c = 1$
(2) $a + b + \sqrt{2}c = 1$
(3) $a + \sqrt{2}b + c = 1$
(4) $\sqrt{2}a - b + c = 1$

jee-main 2023 Q87 Shortest Distance Between Two Lines View

The shortest distance between the lines $\frac { x - 4 } { 4 } = \frac { y + 2 } { 5 } = \frac { z + 3 } { 3 }$ and $\frac { x - 1 } { 3 } = \frac { y - 3 } { 4 } = \frac { z - 4 } { 2 }$ is
(1) $6 \sqrt { 3 }$
(2) $2 \sqrt { 6 }$
(3) $6 \sqrt { 2 }$
(4) $3 \sqrt { 6 }$

jee-main 2023 Q88 Shortest Distance Between Two Lines View

Consider the lines $L _ { 1 }$ and $L _ { 2 }$ given by $L _ { 1 } : \frac { x - 1 } { 2 } = \frac { y - 3 } { 1 } = \frac { z - 2 } { 2 }$ $L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y - 2 } { 2 } = \frac { z - 3 } { 3 }$
A line $L _ { 3 }$ having direction ratios $1 , - 1 , - 2$, intersects $L _ { 1 }$ and $L _ { 2 }$ at the points $P$ and $Q$ respectively. Then the length of line segment $P Q$ is
(1) $2 \sqrt { 6 }$
(2) $3 \sqrt { 2 }$
(3) $4 \sqrt { 3 }$
(4) 4

jee-main 2023 Q88 Distance from a Point to a Line (Show/Compute) View

Let the co-ordinates of one vertex of $\triangle A B C$ be $A ( 0,2 , \alpha )$ and the other two vertices lie on the line $\frac { \mathrm { x } + \alpha } { 5 } = \frac { \mathrm { y } - 1 } { 2 } = \frac { \mathrm { z } + 4 } { 3 }$. For $\alpha \in \mathbb { Z }$, if the area of $\triangle A B C$ is 21 sq. units and the line segment $B C$ has length $2 \sqrt { 21 }$ units, then $\alpha ^ { 2 }$ is equal to $\_\_\_\_$ .

jee-main 2023 Q88 Normal Vector and Plane Equation View

If a plane passes through the points $(-1, k, 0)$, $(2, k, -1)$, $(1, 1, 2)$ and is parallel to the line $\frac{x-1}{1} = \frac{2y+1}{2} = \frac{z+1}{-1}$, then the value of $\frac{k^{2}+1}{(k-1)(k-2)}$ is
(1) $\frac{17}{5}$
(2) $\frac{5}{17}$
(3) $\frac{6}{13}$
(4) $\frac{13}{6}$

jee-main 2023 Q88 Vector Algebra and Triple Product Computation View

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}| = \sqrt{14}$, $|\vec{b}| = \sqrt{6}$ and $|\vec{a} \times \vec{b}| = \sqrt{48}$. Then $(\vec{a} \cdot \vec{b})^2$ is equal to $\underline{\hspace{1cm}}$.

jee-main 2023 Q88 Normal Vector and Plane Equation View

If the equation of the plane containing the line $x + 2 y + 3 z - 4 = 0 = 2 x + y - z + 5$ and perpendicular to the plane $\vec { r } = ( \hat { i } - \hat { j } ) + \lambda ( \hat { i } + \hat { j } + \hat { k } ) + \mu ( \hat { i } - 2 \hat { j } + 3 \hat { k } )$ is $a x + b y + c z = 4$ then $( a - b + c )$ is equal to
(1) 18
(2) 22
(3) 20
(4) 24

jee-main 2023 Q89 Distance from a Point to a Line (Show/Compute) View

Let a line $L$ pass through the point $P(2, 3, 1)$ and be parallel to the line $x + 3y - 2z - 2 = 0 = x - y + 2z$. If the distance of $L$ from the point $(5, 3, 8)$ is $\alpha$, then $3\alpha^{2}$ is equal to $\_\_\_\_$

jee-main 2023 Q89 Distance from a Point to a Line (Show/Compute) View

Let the line $L: \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-3}{1}$ intersect the plane $2x + y + 3z = 16$ at the point $P$. Let the point $Q$ be the foot of perpendicular from the point $R(1,-1,-3)$ on the line $L$. If $\alpha$ is the area of triangle $PQR$, then $\alpha^2$ is equal to $\underline{\hspace{1cm}}$.

jee-main 2023 Q89 Distance from a Point to a Line (Show/Compute) View

Let $\lambda _ { 1 } , \lambda _ { 2 }$ be the values of $\lambda$ for which the points $\left( \frac { 5 } { 2 } , 1 , \lambda \right)$ and $( - 2 , 0 , 1 )$ are at equal distance from the plane $2 x + 3 y - 6 z + 7 = 0$. If $\lambda _ { 1 } > \lambda _ { 2 }$, then the distance of the point $\left( \lambda _ { 1 } - \lambda _ { 2 } , \lambda _ { 2 } , \lambda _ { 1 } \right)$ from the line $\frac { x - 5 } { 1 } = \frac { y - 1 } { 2 } = \frac { z + 7 } { 2 }$ is $\_\_\_\_$

jee-main 2023 Q90 Dihedral Angle Computation View

Let $\theta$ be the angle between the planes $P_1: \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$ and $P_2: \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$. Let $L$ be the line that meets $P_2$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_2$. If $\alpha$ is the angle between $L$ and $P_2$, then $\tan^2\theta \cdot \cot^2\alpha$ is equal to $\underline{\hspace{1cm}}$.

jee-main 2024 Q77 Section Division and Coordinate Computation View

Consider a $\triangle ABC$ where $A(1,3,2)$, $B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle BAC$ meets the line $BC$ at $D$, then the length of the projection of the vector $\overrightarrow{AD}$ on the vector $\overrightarrow{AC}$ is:
(1) $\frac{37}{2\sqrt{38}}$
(2) $\frac{\sqrt{38}}{2}$
(3) $\frac{39}{2\sqrt{38}}$
(4) $\sqrt{19}$

jee-main 2024 Q77 Section Division and Coordinate Computation View

The position vectors of the vertices $A , B$ and $C$ of a triangle are $2 \hat { i } - 3 \hat { j } + 3 \hat { k } , \quad 2 \hat { i } + 2 \hat { j } + 3 \hat { k }$ and $- \hat { i } + \hat { j } + 3 \hat { k }$ respectively. Let $l$ denotes the length of the angle bisector AD of $\angle \mathrm { BAC }$ where D is on the line segment BC , then $2 l ^ { 2 }$ equals :
(1) 49
(2) 42
(3) 50
(4) 45

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 Q78 Distance from a Point to a Line (Show/Compute) View

Let the position vectors of the vertices $A , B$ and $C$ of a triangle be $2 \hat { i } + 2 \hat { j } + \hat { k } , \hat { i } + 2 \hat { j } + 2 \hat { k }$ and $2 \hat { i } + \hat { j } + 2 \hat { k }$ respectively. Let $l _ { 1 } , l _ { 2 }$ and $l _ { 3 }$ be the lengths of perpendiculars drawn from the ortho centre of the triangle on the sides $A B , B C$ and $C A$ respectively, then $l _ { 1 } ^ { 2 } + l _ { 2 } ^ { 2 } + l _ { 3 } ^ { 2 }$ equals :
(1) $\frac { 1 } { 5 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$

jee-main 2024 Q78 Vector Algebra and Triple Product Computation View

Let $\overrightarrow { \mathrm { a } } = \hat { i } + 2 \hat { j } + 3 \hat { k } , \overrightarrow { \mathrm {~b} } = 2 \hat { i } + 3 \hat { j } - 5 \hat { k }$ and $\overrightarrow { \mathrm { c } } = 3 \hat { i } - \hat { j } + \lambda \hat { k }$ be three vectors. Let $\overrightarrow { \mathrm { r } }$ be a unit vector along $\vec { b } + \vec { c }$. If $\vec { r } \cdot \vec { a } = 3$, then $3 \lambda$ is equal to: (1) 21 (2) 30 (3) 25 (4) 27

jee-main 2024 Q79 MCQ: Distance or Length Optimization on a Line View

Let $P$ and $Q$ be the points on the line $\frac{x+3}{8} = \frac{y-4}{2} = \frac{z+1}{2}$ which are at a distance of 6 units from the point $R(1,2,3)$. If the centroid of the triangle $PQR$ is $(\alpha, \beta, \gamma)$, then $\alpha^2 + \beta^2 + \gamma^2$ is:
(1) 26
(2) 36
(3) 18
(4) 24

jee-main 2024 Q79 Vector Algebra and Triple Product Computation View

Let $\mathrm { P } ( 3,2,3 ) , \mathrm { Q } ( 4,6,2 )$ and $\mathrm { R } ( 7,3,2 )$ be the vertices of $\triangle \mathrm { PQR }$. Then, the angle $\angle \mathrm { QPR }$ is
(1) $\frac { \pi } { 6 }$
(2) $\cos ^ { - 1 } \left( \frac { 7 } { 18 } \right)$
(3) $\cos ^ { - 1 } \left( \frac { 1 } { 18 } \right)$
(4) $\frac { \pi } { 3 }$

jee-main 2024 Q79 MCQ: Point Membership on a Line View

Let $L_1: \vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(\hat{i} - \hat{j} + 2\hat{k})$, $\lambda \in R$, $L_2: \vec{r} = (\hat{j} - \hat{k}) + \mu(3\hat{i} + \hat{j} + p\hat{k})$, $\mu \in R$ and $L_3: \vec{r} = \delta(l\hat{i} + m\hat{j} + n\hat{k})$, $\delta \in R$ be three lines such that $L_1$ is perpendicular to $L_2$ and $L_3$ is perpendicular to both $L_1$ and $L_2$. Then the point which lies on $L_3$ is
(1) $(-1, 7, 4)$
(2) $(-1, -7, 4)$
(3) $(1, 7, -4)$
(4) $(1, -7, 4)$

jee-main 2024 Q79 Distance from a Point to a Line (Show/Compute) View

Let $\mathrm { P } ( \alpha , \beta , \gamma )$ be the image of the point $\mathrm { Q } ( 3 , - 3,1 )$ in the line $\frac { x - 0 } { 1 } = \frac { y - 3 } { 1 } = \frac { z - 1 } { - 1 }$ and R be the point $( 2,5 , - 1 )$. If the area of the triangle $PQR$ is $\lambda$ and $\lambda ^ { 2 } = 14 K$, then $K$ is equal to:
(1) 36
(2) 81
(3) 72
(4) 18

jee-main 2024 Q89 Shortest Distance Between Two Lines View

If the shortest distance between the lines $\frac { x - \lambda } { 3 } = \frac { y - 2 } { - 1 } = \frac { z - 1 } { 1 }$ and $\frac { x + 2 } { - 3 } = \frac { y + 5 } { 2 } = \frac { z - 4 } { 4 }$ is $\frac { 44 } { \sqrt { 30 } }$, then the largest possible value of $| \lambda |$ is equal to $\_\_\_\_$

jee-main 2024 Q90 Distance from a Point to a Line (Show/Compute) View

A line with direction ratio $2,1,2$ meets the lines $\mathrm { x } = \mathrm { y } + 2 = \mathrm { z }$ and $\mathrm { x } + 2 = 2 \mathrm { y } = 2 \mathrm { z }$ respectively at the point P and Q . if the length of the perpendicular from the point $( 1,2,12 )$ to the line PQ is $l$, then $l ^ { 2 }$ is

jee-main 2024 Q90 Multi-Part 3D Geometry Problem View

Let a line passing through the point $(-1, 2, 3)$ intersect the lines $L_1: \frac{x-1}{3} = \frac{y-2}{2} = \frac{z+1}{-2}$ at $M(\alpha, \beta, \gamma)$ and $L_2: \frac{x+2}{-3} = \frac{y-2}{-2} = \frac{z-1}{4}$ at $N(a, b, c)$. Then the value of $\frac{(\alpha + \beta + \gamma)^2}{(a + b + c)^2}$ equals $\underline{\hspace{1cm}}$.

jee-main 2024 Q90 Distance from a Point to a Line (Show/Compute) View

The square of the distance of the image of the point $( 6,1,5 )$ in the line $\frac { x - 1 } { 3 } = \frac { y } { 2 } = \frac { z - 2 } { 4 }$, from the origin is $\_\_\_\_$

jee-main 2025 Q2 Distance from a Point to a Line (Show/Compute) View

Let in a $\triangle ABC$, the length of the side $AC$ be 6, the vertex $B$ be $(1,2,3)$ and the vertices $A, C$ lie on the line $\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}$. Then the area (in sq. units) of $\triangle ABC$ is:
(1) 17
(2) 21
(3) 56
(4) 42

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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