LFM Pure and Mechanics

jee-main 2022 Q90 Dot Product Computation View

Let $\vec { a } , \vec { b } , \vec { c }$ be three non-coplanar vectors such that $\vec { a } \times \vec { b } = \overrightarrow { 4 c } , \vec { b } \times \vec { c } = 9 \vec { a }$ and $\vec { c } \times \vec { a } = \alpha \vec { b } , \alpha > 0$. If $| \vec { a } | + | \vec { b } | + | \vec { c } | = 36$, then $\alpha$ is equal to $\_\_\_\_$ .

jee-main 2023 Q65 Vector Algebra and Triple Product Computation View

Let $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$, $\vec{b} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{c} = \hat{i} + \hat{j} - \hat{k}$. A vector in the plane of $\vec{a}$ and $\vec{b}$ whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$ is
(1) $4\hat{i} - \hat{j} + 4\hat{k}$
(2) $3\hat{i} + \hat{j} - 3\hat{k}$
(3) $2\hat{i} + \hat{j} - 2\hat{k}$
(4) $4\hat{i} + \hat{j} - 4\hat{k}$

jee-main 2023 Q71 Line-Plane Intersection View

Let the system of linear equations $- x + 2 y - 9 z = 7$ $- x + 3 y + 7 z = 9$ $- 2 x + y + 5 z = 8$ $- 3 x + y + 13 z = \lambda$ has a unique solution $x = \alpha , y = \beta , z = \gamma$. Then the distance of the point $( \alpha , \beta , \gamma )$ from the plane $2 x - 2 y + z = \lambda$ is
(1) 11
(2) 7
(3) 9
(4) 13

jee-main 2023 Q76 Vector Algebra and Triple Product Computation View

Let $S$ be the set of all $( \lambda , \mu )$ for which the vectors $\lambda \hat { i } - \hat { j } + \widehat { k } , \hat { i } + 2 \hat { j } + \mu \widehat { k }$ and $3 \hat { i } - 4 \hat { j } + 5 \widehat { k }$, where $\lambda - \mu = 5$, are coplanar, then $\sum _ { ( \lambda , \mu ) \in S } 80 \left( \lambda ^ { 2 } + \mu ^ { 2 } \right)$ is equal to
(1) 2210
(2) 2130
(3) 2290
(4) 2370

jee-main 2023 Q77 Section Division and Coordinate Computation View

Let two vertices of a triangle $ABC$ be $(2, 4, 6)$ and $(0, -2, -5)$, and its centroid be $(2, 1, -1)$. If the image of the third vertex in the plane $x + 2y + 4z = 11$ is $(\alpha, \beta, \gamma)$, then $\alpha\beta + \beta\gamma + \gamma\alpha$ is equal to
(1) 70
(2) 76
(3) 74
(4) 72

jee-main 2023 Q78 Line-Plane Intersection View

Let the image of the point $P(2, -1, 3)$ in the plane $x + 2y - z = 0$ be $Q$. Then the distance of the plane $3x + 2y + z + 29 = 0$ from the point $Q$ is
(1) $\frac{22\sqrt{2}}{7}$
(2) $\frac{24\sqrt{2}}{7}$
(3) $2\sqrt{14}$
(4) $3\sqrt{14}$

jee-main 2023 Q78 Vector Algebra and Triple Product Computation View

Let $\vec{a} = 2\hat{i} + \hat{j} + \hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a} + \vec{b} + \vec{c}| = |\vec{a} + \vec{b} - \vec{c}|$ and $\vec{b} \cdot \vec{c} = 0$. Consider the following two statements: $A$: $|\vec{a} + \lambda\vec{c}| \geq |\vec{a}|$ for all $\lambda \in \mathbb{R}$. $B$: $\vec{a}$ and $\vec{c}$ are always parallel.
(1) only (B) is correct
(2) neither (A) nor (B) is correct
(3) only (A) is correct
(4) both (A) and (B) are correct.

jee-main 2023 Q79 Shortest Distance Between Two Lines View

Let the shortest distance between the lines $L: \frac{x-5}{-2} = \frac{y-\lambda}{0} = \frac{z+\lambda}{1}$, $\lambda \geq 0$ and $L_1: x+1 = y-1 = 4-z$ be $2\sqrt{6}$. If $(\alpha, \beta, \gamma)$ lies on $L$, then which of the following is NOT possible?
(1) $\alpha + 2\gamma = 24$
(2) $2\alpha + \gamma = 7$
(3) $2\alpha - \gamma = 9$
(4) $\alpha - 2\gamma = 19$

jee-main 2023 Q79 Line-Plane Intersection View

Let $P$ be the point of intersection of the line $\frac{x+3}{3} = \frac{y+2}{1} = \frac{1-z}{2}$ and the plane $x + y + z = 2$. If the distance of the point $P$ from the plane $3x - 4y + 12z = 32$ is $q$, then $q$ and $2q$ are the roots of the equation
(1) $x^2 - 18x - 72 = 0$
(2) $x^2 - 18x + 72 = 0$
(3) $x^2 + 18x + 72 = 0$
(4) $x^2 + 18x - 72 = 0$

jee-main 2023 Q84 Vector Algebra and Triple Product Computation View

If the four points, whose position vectors are $3 \hat { i } - 4 \hat { j } + 2 \widehat { k } , \hat { i } + 2 \hat { j } - \widehat { k } , - 2 \hat { i } - \hat { j } + 3 \widehat { k }$ and $5 \hat { i } - 2 \alpha \hat { j } + 4 \widehat { k }$ are coplanar, then $\alpha$ is equal to
(1) $\frac { 73 } { 17 }$
(2) $- \frac { 107 } { 17 }$
(3) $- \frac { 73 } { 17 }$
(4) $\frac { 107 } { 17 }$

jee-main 2023 Q84 Geometric Property Identification via Vectors View

Let $a , b , c$ be three distinct real numbers, none equal to one. If the vectors $a \hat { i } + \hat { j } + \widehat { k } , \hat { i } + b \hat { j } + \widehat { k }$ and $\hat { i } + \hat { j } + c \hat { k }$ are coplanar, then $\frac { 1 } { 1 - a } + \frac { 1 } { 1 - b } + \frac { 1 } { 1 - c }$ is equal to
(1) 2
(2) - 1
(3) - 2
(4) 1

jee-main 2023 Q85 Vector Algebra and Triple Product Computation View

Let $\vec { a } = - \hat { i } - \hat { j } + \hat { k } , \vec { a } \cdot \vec { b } = 1$ and $\vec { a } \times \vec { b } = \hat { i } - \hat { j }$. Then $\vec { a } - 6 \vec { b }$ is equal to
(1) $3 ( \hat { i } - \hat { j } - \widehat { k } )$
(2) $3 ( \hat { i } + \hat { j } + \hat { k } )$
(3) $3 ( \hat { i } - \hat { j } + \widehat { k } )$
(4) $3 ( \hat { i } + \hat { j } - \widehat { k } )$

jee-main 2023 Q85 Vector Algebra and Triple Product Computation View

If the vectors $\vec { a } = \lambda \hat { i } + \mu \hat { j } + 4 \widehat { k } , \vec { b } = - 2 \hat { i } + 4 \hat { j } - 2 \widehat { k }$ and $\vec { c } = 2 \hat { i } + 3 \hat { j } + \widehat { k }$ are coplanar and the projection of $\vec { a }$ on the vector $\vec { b }$ is $\sqrt { 54 }$ units, then the sum of all possible values of $\lambda + \mu$ is equal to
(1) 0
(2) 6
(3) 24
(4) 18

jee-main 2023 Q85 Dot Product Computation View

If $\vec { a } = \hat { i } + 2 \widehat { k } , \vec { b } = \hat { i } + \hat { j } + \widehat { k } , \vec { c } = 7 \hat { i } - 3 \hat { j } + 4 \widehat { k } , \vec { r } \times \vec { b } + \vec { b } \times \vec { c } = \overrightarrow { 0 }$ and $\vec { r } \cdot \vec { a } = 0$ then $\vec { r } \cdot \vec { c }$ is equal to: (1) 34 (2) 12 (3) 36 (4) 30

jee-main 2023 Q85 Section Division and Coordinate Computation View

If the points with position vectors $\alpha \hat { \mathrm { i } } + 10 \hat { \mathrm { j } } + 13 \hat { \mathrm { k } } , 6 \hat { \mathrm { i } } + 11 \hat { \mathrm { j } } + 11 \hat { \mathrm { k } } , \frac { 9 } { 2 } \hat { \mathrm { i } } + \beta \hat { \mathrm { j } } - 8 \hat { \mathrm { k } }$ are collinear, then $( 19 \alpha - 6 \beta ) ^ { 2 }$ is equal to
(1) 36
(2) 25
(3) 49
(4) 16

jee-main 2023 Q85 Magnitude of Vector Expression View

Let $\lambda \in \mathbb { Z } , \vec { a } = \lambda \hat { i } + \hat { j } - \widehat { k }$ and $\vec { b } = 3 \hat { i } - \hat { j } + 2 \widehat { k }$. Let $\vec { c }$ be a vector such that $( \vec { a } + \vec { b } + \vec { c } ) \times \vec { c } = \overrightarrow { 0 } , \vec { a } \cdot \vec { c } = - 17$ and $\vec { b } \cdot \vec { c } = - 20$. Then $| \vec { c } \times ( \lambda \hat { i } + \hat { j } + \hat { k } ) | ^ { 2 }$ is equal to
(1) 46
(2) 53
(3) 62
(4) 49

jee-main 2023 Q86 Dot Product Computation View

The vector $\vec { a } = - \hat { i } + 2 \hat { j } + \hat { k }$ is rotated through a right angle, passing through the $y$-axis in its way and the resulting vector is $\vec { b }$. Then the projection of $3 \vec { a } + \sqrt { 2 } \vec { b }$ on $\vec { c } = 5 \hat { i } + 4 \hat { j } + 3 \hat { k }$ is
(1) $3 \sqrt { 2 }$
(2) 1
(3) $\sqrt { 6 }$
(4) $2 \sqrt { 3 }$

jee-main 2023 Q86 Vector Algebra and Triple Product Computation View

Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A , B , C$ and $D$ be $\overrightarrow { \mathrm { a } } - \overrightarrow { \mathrm { b } } + \overrightarrow { \mathrm { c } } , \lambda \overrightarrow { \mathrm { a } } - 3 \overrightarrow { \mathrm {~b} } + 4 \overrightarrow { \mathrm { c } } , - \vec { a } + 2 \vec { b } - 3 \vec { c }$ and $2 \vec { a } - 4 \vec { b } + 6 \vec { c }$ respectively. If $\overrightarrow { A B } , \overrightarrow { A C }$ and $\overrightarrow { A D }$ are coplanar, then $\lambda$ is :

jee-main 2023 Q86 Dot Product Computation View

Let $\vec { a } = 4 \hat { i } + 3 \hat { j }$ and $\vec { b } = 3 \hat { i } - 4 \hat { j } + 5 \hat { k }$ and $\overrightarrow { \mathrm { c } }$ is a vector such that $\vec { c } \cdot ( \vec { a } \times \vec { b } ) + 25 = 0 , \vec { c } \cdot ( \hat { i } + \hat { j } + \hat { k } ) = 4$ and projection of $\vec { c }$ on $\overrightarrow { \mathrm { a } }$ is 1 , then the projection of $\vec { c }$ on $\vec { b }$ equals: (1) $\frac { 5 } { \sqrt { 2 } }$ (2) $\frac { 1 } { 5 }$ (3) $\frac { 1 } { \sqrt { 2 } }$ (4) $\frac { 3 } { \sqrt { 2 } }$

jee-main 2023 Q86 Vector Algebra and Triple Product Computation View

Let $\vec { a } = 6 \hat { i } + 9 \hat { j } + 12 \hat { k } , \vec { b } = \alpha \hat { i } + 11 \hat { j } - 2 \hat { k }$ and $\vec { c }$ be vectors such that $\vec { a } \times \vec { c } = \vec { a } \times \vec { b }$. If $\vec { a } \cdot \vec { c } = - 12$, and $\vec { c } \cdot ( \hat { i } - 2 \hat { j } + \hat { k } ) = 5$ then $\vec { c } \cdot ( \hat { i } + \hat { j } + \hat { k } )$ is equal to $\_\_\_\_$

jee-main 2023 Q86 Perpendicular/Orthogonal Projection onto a Plane View

Let the plane $x + 3 y - 2 z + 6 = 0$ meet the co-ordinate axes at the points $A , B , C$. If the orthocenter of the triangle $A B C$ is $\left( \alpha , \beta , \frac { 6 } { 7 } \right)$, then $98 ( \alpha + \beta ) ^ { 2 }$ is equal to $\_\_\_\_$ .

jee-main 2023 Q87 Distance from a Point to a Line (Show/Compute) View

The distance of the point $P ( 4,6 , - 2 )$ from the line passing through the point $( - 3,2,3 )$ and parallel to a line with direction ratios $3,3 , - 1$ is equal to:
(1) 3
(2) $\sqrt { 6 }$
(3) $2 \sqrt { 3 }$
(4) $\sqrt { 14 }$

jee-main 2023 Q87 Find Cartesian Equation of a Plane View

Let the equation of the plane P containing the line $x + 10 = \frac { 8 - y } { 2 } = z$ be $a x + b y + 3 z = 2 ( a + b )$ and the distance of the plane P from the point $( 1,27,7 )$ be $c$. Then $\mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } + \mathrm { c } ^ { 2 }$ is equal to

jee-main 2023 Q87 Distance Computation (Point-to-Plane or Line-to-Line) View

Shortest distance between the lines $\frac { x - 1 } { 2 } = \frac { y + 8 } { - 7 } = \frac { z - 4 } { 5 }$ and $\frac { x - 1 } { 2 } = \frac { y - 2 } { 1 } = \frac { z - 6 } { - 3 }$ is (1) $2 \sqrt { 3 }$ (2) $4 \sqrt { 3 }$ (3) $3 \sqrt { 3 }$ (4) $5 \sqrt { 3 }$

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) $3 \sqrt { 6 }$

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LFM Pure and Mechanics

Indices and Surds 96

Exponential Functions 197

Laws of Logarithms 242

Exponential Equations & Modelling 28

Arithmetic Sequences and Series 363

Geometric Sequences and Series 159

Differentiation from First Principles 45

Tangents, normals and gradients 140

Stationary points and optimisation 505

Applied differentiation 76

Indefinite & Definite Integrals 502

Areas Between Curves 71

Areas by integration 221

Volumes of Revolution 68

Numerical integration 49

Vectors Introduction & 2D 211

Vectors 3D & Lines 524

Travel graphs 14

SUVAT in 2D & Gravity 31

Constant acceleration (SUVAT) 78

Variable acceleration (1D) 55

Variable acceleration (vectors) 32

Forces, equilibrium and resultants 24

Newton's laws and connected particles 21

Pulley systems 6

Motion on a slope 6

Friction 8

Momentum and Collisions 34

Moments 28

Parametric curves and Cartesian conversion 9

Parametric differentiation 13

Parametric integration 6

Projectiles 49