LFM Pure and Mechanics

jee-main 2016 Q87 Coplanarity and Relative Position of Planes View

The number of distinct real values of $\lambda$, for which the lines $\frac { x - 1 } { 1 } = \frac { y - 2 } { 2 } = \frac { z + 3 } { \lambda ^ { 2 } }$ and $\frac { x - 3 } { 1 } = \frac { y - 2 } { \lambda ^ { 2 } } = \frac { z - 1 } { 2 }$, are coplanar is
(1) 2
(2) 4
(3) 3
(4) 1

jee-main 2016 Q88 Section Division and Coordinate Computation View

$ABC$ is a triangle in a plane with vertices $A ( 2,3,5 ) , B ( - 1,3,2 )$ and $C ( \lambda , 5 , \mu )$. If the median through $A$ is equally inclined to the coordinate axes, then the value of $\left( \lambda ^ { 3 } + \mu ^ { 3 } + 5 \right)$ is
(1) 1130
(2) 1348
(3) 1077
(4) 676

jee-main 2016 Q89 Section Division and Coordinate Computation View

Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors $A , B , C$ and $P$ are $\vec{a}, \vec{b}, \vec{c}$ and $\frac { \vec { a } + \vec { b } + \vec { c } } { 4 }$ respectively, then the position vector of the orthocentre of this triangle, is :
(1) $- \left( \frac { \vec { a } + \vec { b } + \vec { c } } { 2 } \right)$
(2) $\vec { a } + \vec { b } + \vec { c }$
(3) $\frac { ( \vec { a } + \vec { b } + \vec { c } ) } { 2 }$
(4) $\overrightarrow { 0 }$

jee-main 2017 Q86 Perpendicular/Orthogonal Projection onto a Plane View

If the image of the point $P ( 1 , - 2 , 3 )$ in the plane $2 x + 3 y - 4 z + 22 = 0$ measured parallel to the line $\frac { x } { 1 } = \frac { y } { 4 } = \frac { z } { 5 }$ is $Q$, then $P Q$ is equal to:
(1) $3 \sqrt { 5 }$
(2) $2 \sqrt { 42 }$
(3) $\sqrt { 42 }$
(4) $6 \sqrt { 5 }$

jee-main 2017 Q86 Area Computation Using Vectors View

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8 \hat { \mathrm { i } } - 6 \hat { \mathrm { j } }$ and $3 \hat { \mathrm { i } } + 4 \hat { \mathrm { j } } - 12 \widehat { \mathrm { k } }$, is:
(1) 20
(2) 65
(3) 52
(4) 26

jee-main 2017 Q87 Vector Algebra and Triple Product Computation View

Let $\vec { u } = \hat { i } + \hat { j }$, $\vec { v } = \hat { i } - \hat { j }$ and $\vec { w } = \hat { i } + 2 \hat { j } + 3 \hat { k }$. If $\hat { n }$ is a unit vector such that $\vec { u } \cdot \hat { n } = 0$ and $\vec { v } \cdot \hat { n } = 0$, then $| \vec { w } \cdot \hat { n } |$ is equal to:
(1) 0
(2) 1
(3) 2
(4) 3

jee-main 2018 Q87 Perpendicularity or Parallel Condition View

Let $\vec { u }$ be a vector coplanar with the vectors $\vec { a } = 2 \hat { i } + 3 \hat { j } - \widehat { k }$ and $\vec { b } = \hat { j } + \widehat { k }$. If $\vec { u }$ is perpendicular to $\vec { a }$ and $\vec { u } \cdot \vec { b } = 24$, then $| \vec { u } | ^ { 2 }$ is equal to:
(1) 84
(2) 336
(3) 315
(4) 256

jee-main 2018 Q87 Dot Product Computation View

If $\vec { a } , \vec { b } , \vec { c }$ are unit vectors such that $\vec { a } + 2 \vec { b } + 2 \vec { c } = \overrightarrow { 0 }$, then $| \vec { a } \times \vec { c } |$ is equal to :
(1) $\frac { 1 } { 4 }$
(2) $\frac { 15 } { 16 }$
(3) $\frac { \sqrt { 15 } } { 4 }$
(4) $\frac { \sqrt { 15 } } { 16 }$

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 Q86 Vector Algebra and Triple Product Computation View

Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three unit vectors, out of which vectors $\vec { b }$ and $\vec { c }$ are non-parallel. If $\alpha$ and $\beta$ are the angles which vector $\vec { a }$ makes with vectors $\vec { b }$ and $\vec { c }$ respectively and $\vec { a } \times ( \vec { b } \times \vec { c } ) = \frac { 1 } { 2 } \vec { b }$, then $| \alpha - \beta |$ is equal to :
(1) $90 ^ { \circ }$
(2) $60 ^ { \circ }$
(3) $45 ^ { \circ }$
(4) $30 ^ { \circ }$

jee-main 2019 Q86 Vector Algebra and Triple Product Computation View

Let $y = y(x)$ be the solution of the differential equation, $\left(x^2 + 1\right)^2 \frac{dy}{dx} + 2x\left(x^2 + 1\right)y = 1$ such that $y(0) = 0$. If $\sqrt{a}\, y(1) = \frac{\pi}{32}$, then the value of $a$ is
(1) $\frac{1}{16}$
(2) $\frac{1}{2}$
(3) $\frac{1}{4}$
(4) $1$

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 Q87 Perpendicularity or Parallel Condition View

Let $\vec { a } = 2 \hat { i } + \lambda _ { 1 } \hat { j } + 3 \hat { k } , \vec { b } = 4 \hat { i } + \left( 3 - \lambda _ { 2 } \right) \hat { j } + 6 \hat { k }$ and $\vec { c } = 3 \hat { i } + 6 \hat { j } + \left( \lambda _ { 3 } - 1 \right) \hat { k }$ be three vectors such that $\vec { b } = 2 \vec { a }$ and $\vec { a }$ is perpendicular to $\vec { c }$. Then a possible value of $\left( \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \right)$ is
(1) $\left( - \frac { 1 } { 2 } , 4,0 \right)$
(2) $( 1,5,1 )$
(3) $\left( \frac { 1 } { 2 } , 4 , - 2 \right)$
(4) $( 1,3,1 )$

jee-main 2019 Q87 Optimization of a Vector Expression View

Let $\vec { a } = 3 \hat { i } + 2 \hat { j } + x \hat { k }$ and $\vec { b } = \hat { i } - \hat { j } + \hat { k }$, for some real $x$. Then the condition for $| \vec { a } \times \vec { b } | = r$ to follow
(1) $0 < r \leq \sqrt { \frac { 3 } { 2 } }$
(2) $r \geq 5 \sqrt { \frac { 3 } { 2 } }$
(3) $\sqrt { \frac { 3 } { 2 } } < r \leq 3 \sqrt { \frac { 3 } { 2 } }$
(4) $3 \sqrt { \frac { 3 } { 2 } } < r < 5 \sqrt { \frac { 3 } { 2 } }$

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

Let $\vec { \alpha } = 3 \hat { i } + \hat { j }$ and $\vec { \beta } = 2 \hat { i } - \hat { j } + 3 \hat { k }$. If $\vec { \beta } = \overrightarrow { \beta _ { 1 } } - \overrightarrow { \beta _ { 2 } }$, where $\overrightarrow { \beta _ { 1 } }$ is parallel to $\vec { \alpha }$ and $\overrightarrow { \beta _ { 2 } }$ is perpendicular to $\vec { \alpha }$, then $\overrightarrow { \beta _ { 1 } } \times \overrightarrow { \beta _ { 2 } }$ is equal to:
(1) $\frac { 1 } { 2 } ( - 3 \hat { i } + 9 \hat { j } + 5 \widehat { k } )$
(2) $3 \hat { i } - 9 \hat { j } - 5 \widehat { k }$
(3) $- 3 \hat { i } + 9 \hat { j } + 5 \widehat { k }$
(4) $\frac { 1 } { 2 } ( 3 \hat { i } - 9 \hat { j } + 5 \hat { k } )$

jee-main 2019 Q87 Angle or Cosine Between Vectors View

If a unit vector $\vec { a }$ makes angles $\frac { \pi } { 3 }$ with $\hat { i } , \frac { \pi } { 4 }$ with $\hat { j }$ and $\theta \in ( 0 , \pi )$ with $\widehat { k }$, then a value of $\theta$ is:
(1) $\frac { 5 \pi } { 6 }$
(2) $\frac { 5 \pi } { 12 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { 2 \pi } { 3 }$

jee-main 2019 Q88 Parallelism Between Line and Plane or Constraint on Parameters View

Let $A$ be a point on the line $\vec { r } = ( 1 - 3 \mu ) \hat { i } + ( \mu - 1 ) \hat { j } + ( 2 + 5 \mu ) \hat { k }$ and $B ( 3,2,6 )$ be a point in the space. Then the value of $\mu$ for which the vector $\overrightarrow { A B }$ is parallel to the plane $x - 4 y + 3 z = 1$ is
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $- \frac { 1 } { 4 }$
(4) $\frac { 1 } { 8 }$

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 Find Cartesian Equation of a Plane View

The plane passing through the point $( 4 , - 1,2 )$ and parallel to the lines $\frac { x + 2 } { 3 } = \frac { y - 2 } { - 1 } = \frac { z + 1 } { 2 }$ and $\frac { x - 2 } { 1 } = \frac { y - 3 } { 2 } = \frac { z - 4 } { 3 }$ also passes through the point
(1) $( 1,1 , - 1 )$
(2) $( - 1 , - 1 , - 1 )$
(3) $( - 1 , - 1,1 )$
(4) $( 1,1,1 )$

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 2019 Q89 Distance Computation (Point-to-Plane or Line-to-Line) View

Let $P$ be the plane, which contains the line of intersection of the planes, $x + y + z - 6 = 0$ and $2 x + 3 y + z + 5 = 0$ and it is perpendicular to the $x y$-plane. Then the distance of the point $( 0,0,256 )$ from $P$ is equal to:
(1) $205 \sqrt { 5 }$ units
(2) $\frac { 17 } { \sqrt { 5 } }$ units
(3) $\frac { 11 } { \sqrt { 5 } }$ units
(4) $63 \sqrt { 5 }$ units

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 Q68 Perpendicular/Orthogonal Projection onto a Plane View

The foot of the perpendicular drawn from the point $( 4,2,3 )$ to the line joining the points $( 1 , - 2,3 )$ and $( 1,1,0 )$ lies on the plane
(1) $2 x + y - z = 1$
(2) $x - y - 2 z = 1$
(3) $x - 2 y + z = 1$
(4) $x + 2 y - z = 1$

jee-main 2020 Q69 Distance Computation (Point-to-Plane or Line-to-Line) View

The distance of the point $( 1 , - 2,3 )$ from the plane $x - y + z = 5$ measured parallel to the line $\frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { - 6 }$ is:
(1) $\frac { 7 } { 5 }$
(2) 1
(3) $\frac { 1 } { 7 }$
(4) 7

jee-main 2020 Q70 Perpendicular/Orthogonal Projection onto a Plane View

If $( a , b , c )$ is the image of the point $( 1,2 , - 3 )$ in the line, $\frac { x + 1 } { 2 } = \frac { y - 3 } { - 2 } = \frac { z } { - 1 }$, then $a + b + c$ is equal to:
(1) 2
(2) - 1
(3) 3
(4) 1

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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