UFM Pure

jee-main 2023 Q79 Distance Computation (Point-to-Plane or Line-to-Line) View

Let $N$ be the foot of perpendicular from the point $P ( 1 , - 2 , 3 )$ on the line passing through the points $( 4 , 5 , 8 )$ and $( 1 , - 7 , 5 )$. Then the distance of $N$ from the plane $2x - 2y + z + 5 = 0$ is $\_\_\_\_$.

jee-main 2023 Q79 Distance Computation (Point-to-Plane or Line-to-Line) View

Let $S$ be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac { x - \lambda } { 0 } = \frac { y - 3 } { 4 } = \frac { z + 6 } { 1 }$ and $\frac { x + \lambda } { 3 } = \frac { y } { - 4 } = \frac { z - 6 } { 0 }$ is 13. Then $8 \left| \sum _ { \lambda \in S } \lambda \right|$ is equal to
(1) 306
(2) 304
(3) 308
(4) 302

jee-main 2023 Q86 Distance Computation (Point-to-Plane or Line-to-Line) View

The shortest distance between the lines $x + 1 = 2 y = - 12 z$ and $x = y + 2 = 6 z - 6$ is
(1) 2
(2) 3
(3) $\frac { 5 } { 2 }$
(4) $\frac { 3 } { 2 }$

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 Perpendicular/Orthogonal Projection onto a Plane View

The foot of perpendicular of the point $( 2,0,5 )$ on the line $\frac { x + 1 } { 2 } = \frac { y - 1 } { 5 } = \frac { z + 1 } { - 1 }$ is $( \alpha , \beta , \gamma )$. Then, which of the following is NOT correct?
(1) $\frac { \alpha \beta } { \gamma } = \frac { 4 } { 15 }$
(2) $\frac { \alpha } { \beta } = - 8$
(3) $\frac { \beta } { \gamma } = - 5$
(4) $\frac { \gamma } { \alpha } = \frac { 5 } { 8 }$

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 Coplanarity and Relative Position of Planes View

Let the lines $L _ { 1 } : \frac { x + 5 } { 3 } = \frac { y + 4 } { 1 } = \frac { z - \alpha } { - 2 }$ and $L _ { 2 } : 3 x + 2 y + z - 2 = 0 = x - 3 y + 2 z - 13$ be coplanar. If the point $P ( a , b , c )$ on $L _ { 1 }$ is nearest to the point $Q ( - 4 , - 3,2 )$, then $| a | + | b | + | c |$ is equal to
(1) 12
(2) 14
(3) 8
(4) 10

jee-main 2023 Q88 Distance Computation (Point-to-Plane or Line-to-Line) View

If the shortest distance between the line joining the points $( 1,2,3 )$ and $( 2,3,4 )$, and the line $\frac { x - 1 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z - 2 } { 0 }$ is $\alpha$, then $28 \alpha ^ { 2 }$ is equal to $\_\_\_\_$.

jee-main 2023 Q88 Find Intersection of a Line and a Plane View

The plane $2 x - y + z = 4$ intersects the line segment joining the points $\mathrm { A } ( \mathrm { a } , - 2,4 )$ and $\mathrm { B } ( 2 , \mathrm {~b} , - 3 )$ at the point C in the ratio 2 : 1 and the distance of the point C from the origin is $\sqrt { 5 }$. If $a b < 0$ and P is the point $( \mathrm { a } - \mathrm { b } , \mathrm { b } , 2 \mathrm {~b} - \mathrm { a } )$ then $\mathrm { CP } ^ { 2 }$ is equal to: (1) $\frac { 17 } { 3 }$ (2) $\frac { 16 } { 3 }$ (3) $\frac { 73 } { 3 }$ (4) $\frac { 97 } { 3 }$

jee-main 2023 Q88 Find Cartesian Equation of a Plane View

Let the plane $P : 4 x - y + z = 10$ be rotated by an angle $\frac { \pi } { 2 }$ about its line of intersection with the plane $x + y - z = 4$. If $\alpha$ is the distance of the point $( 2,3 , - 4 )$ from the new position of the plane $P$, then $35 \alpha$ is equal to
(1) 85
(2) 105
(3) 126
(4) 90

jee-main 2023 Q89 Find Cartesian Equation of a Plane View

Let the equation of the plane passing through the line $x - 2 y - z - 5 = 0 = x + y + 3 z - 5$ and parallel to the line $x + y + 2 z - 7 = 0 = 2 x + 3 y + z - 2$ be $a x + b y + c z = 65$. Then the distance of the point $( a , b , c )$ from the plane $2 x + 2 y - z + 16 = 0$ is $\_\_\_\_$.

jee-main 2023 Q89 Find Intersection of a Line and a Plane View

If the lines $\frac { x - 1 } { 1 } = \frac { y - 2 } { 2 } = \frac { z + 3 } { 1 }$ and $\frac { x - a } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 3 } { 1 }$ intersects at the point $P$, then the distance of the point $P$ from the plane $z = a$ is : (1) 16 (2) 28 (3) 10 (4) 22

jee-main 2024 Q78 Perpendicular/Orthogonal Projection onto a Plane View

If the mirror image of the point $P(3,4,9)$ in the line $\frac{x-1}{3} = \frac{y+1}{2} = \frac{z-2}{1}$ is $(\alpha, \beta, \gamma)$, then $14\alpha + \beta + \gamma$ is:
(1) 102
(2) 138
(3) 108
(4) 132

jee-main 2024 Q78 Distance Computation (Point-to-Plane or Line-to-Line) View

If the shortest distance between the lines $\begin{aligned} & L _ { 1 } : \vec { r } = ( 2 + \lambda ) \hat { i } + ( 1 - 3 \lambda ) \hat { j } + ( 3 + 4 \lambda ) \hat { k } , \quad \lambda \in \mathbb { R } \\ & L _ { 2 } : \vec { r } = 2 ( 1 + \mu ) \hat { i } + 3 ( 1 + \mu ) \hat { j } + ( 5 + \mu ) \hat { k } , \quad \mu \in \mathbb { R } \end{aligned}$ is $\frac { m } { \sqrt { n } }$, where $\operatorname { gcd } ( m , n ) = 1$, then the value of $m + n$ equals
(1) 390
(2) 384
(3) 377
(4) 387

jee-main 2024 Q79 Distance Computation (Point-to-Plane or Line-to-Line) View

The shortest distance between the lines $\frac { x - 3 } { 2 } = \frac { y + 15 } { - 7 } = \frac { z - 9 } { 5 }$ and $\frac { x + 1 } { 2 } = \frac { y - 1 } { 1 } = \frac { z - 9 } { - 3 }$ is
(1) $8 \sqrt { 3 }$
(2) $4 \sqrt { 3 }$
(3) $5 \sqrt { 3 }$
(4) $6 \sqrt { 3 }$

jee-main 2024 Q79 Distance Computation (Point-to-Plane or Line-to-Line) View

The distance, of the point $( 7 , - 2,11 )$ from the line $\frac { x - 6 } { 1 } = \frac { y - 4 } { 0 } = \frac { z - 8 } { 3 }$ along the line $\frac { x - 5 } { 2 } = \frac { y - 1 } { - 3 } = \frac { z - 5 } { 6 }$, is:
(1) 12
(2) 14
(3) 18
(4) 21

jee-main 2024 Q79 Find Intersection of a Line and a Plane View

Let $P Q R$ be a triangle with $R ( - 1,4,2 )$. Suppose $M ( 2,1,2 )$ is the mid point of $P Q$. The distance of the centroid of $\triangle P Q R$ from the point of intersection of the line $\frac { x - 2 } { 0 } = \frac { y } { 2 } = \frac { z + 3 } { - 1 }$ and $\frac { x - 1 } { 1 } = \frac { y + 3 } { - 3 } = \frac { z + 1 } { 1 }$ is
(1) 69
(2) 9
(3) $\sqrt { 69 }$
(4) $\sqrt { 99 }$

jee-main 2024 Q79 Perpendicular/Orthogonal Projection onto a Plane View

Let the image of the point $( 1,0,7 )$ in the line $\frac { x } { 1 } = \frac { y - 1 } { 2 } = \frac { z - 2 } { 3 }$ be the point $( \alpha , \beta , \gamma )$. Then which one of the following points lies on the line passing through $( \alpha , \beta , \gamma )$ and making angles $\frac { 2 \pi } { 3 }$ and $\frac { 3 \pi } { 4 }$ with y - axis and z axis respectively and an acute angle with x -axis?
(1) $( 1 , - 2,1 + \sqrt { 2 } )$
(2) $( 1,2,1 - \sqrt { 2 } )$
(3) $( 3,4,3 - 2 \sqrt { 2 } )$
(4) $( 3 , - 4,3 + 2 \sqrt { 2 } )$

jee-main 2024 Q79 Perpendicular/Orthogonal Projection onto a Plane View

Let $( \alpha , \beta , \gamma )$ be the image of the point $( 8,5,7 )$ in the line $\frac { x - 1 } { 2 } = \frac { y + 1 } { 3 } = \frac { z - 2 } { 5 }$. Then $\alpha + \beta + \gamma$ is equal to :
(1) 16
(2) 20
(3) 14
(4) 18

jee-main 2024 Q79 Distance Computation (Point-to-Plane or Line-to-Line) View

If the shortest distance between the lines $\frac { x - \lambda } { 2 } = \frac { y - 4 } { 3 } = \frac { z - 3 } { 4 }$ and $\frac { x - 2 } { 4 } = \frac { y - 4 } { 6 } = \frac { z - 7 } { 8 }$ is $\frac { 13 } { \sqrt { 29 } }$, then a value of $\lambda$ is : (1) - 1 (2) $- \frac { 13 } { 25 }$ (3) $\frac { 13 } { 25 }$ (4) 1

jee-main 2024 Q79 Distance Computation (Point-to-Plane or Line-to-Line) View

Consider the line $L$ passing through the points $( 1,2,3 )$ and $( 2,3,5 )$. The distance of the point $\left( \frac { 11 } { 3 } , \frac { 11 } { 3 } , \frac { 19 } { 3 } \right)$ from the line L along the line $\frac { 3 x - 11 } { 2 } = \frac { 3 y - 11 } { 1 } = \frac { 3 z - 19 } { 2 }$ is equal to
(1) 6
(2) 5
(3) 4
(4) 3

jee-main 2024 Q80 Distance Computation (Point-to-Plane or Line-to-Line) View

If the shortest distance between the lines $\frac { x - 4 } { 1 } = \frac { y + 1 } { 2 } = \frac { z } { - 3 }$ and $\frac { x - \lambda } { 2 } = \frac { y + 1 } { 4 } = \frac { z - 2 } { - 5 }$ is $\frac { 6 } { \sqrt { 5 } }$, then the sum of all possible values of $\lambda$ is:
(1) 5
(2) 8
(3) 7
(4) 10

jee-main 2024 Q80 Distance Computation (Point-to-Plane or Line-to-Line) View

Let P be the point of intersection of the lines $\frac { x - 2 } { 1 } = \frac { y - 4 } { 5 } = \frac { z - 2 } { 1 }$ and $\frac { x - 3 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { 2 }$. Then, the shortest distance of P from the line $4 x = 2 y = z$ is
(1) $\frac { 5 \sqrt { 14 } } { 7 }$
(2) $\frac { 3 \sqrt { 14 } } { 7 }$
(3) $\frac { \sqrt { 14 } } { 7 }$
(4) $\frac { 6 \sqrt { 14 } } { 7 }$

jee-main 2024 Q89 Perpendicular/Orthogonal Projection onto a Plane View

Consider a line L passing through the points $\mathrm { P } ( 1,2,1 )$ and $\mathrm { Q } ( 2,1 , - 1 )$. If the mirror image of the point $\mathrm { A } ( 2,2,2 )$ in the line L is $( \alpha , \beta , \gamma )$, then $\alpha + \beta + 6 \gamma$ is equal to $\_\_\_\_$

UFM Pure

Sequences and series, recurrence and convergence 375

Roots of polynomials 146

Polar coordinates 22

Conic sections 291

Taylor series 183

Hyperbolic functions 10

Integration using inverse trig and hyperbolic functions 6

Integration with Partial Fractions 7

Vectors: Lines & Planes 293

Vectors: Cross Product & Distances 20

First order differential equations (integrating factor) 60

Complex numbers 2 113

Second order differential equations 127

Systems of differential equations 41