UFM Pure

jee-main 2020 Q69 Perpendicular/Orthogonal Projection onto a Plane View

Let $P$ be a plane passing through the points $(2,1,0)$, $(4,1,1)$ and $(5,0,1)$ and $R$ be any point $(2,1,6)$. Then the image of $R$ in the plane $P$ is
(1) $(6,5,2)$
(2) $(6,5,-2)$
(3) $(4,3,2)$
(4) $(3,4,-2)$

jee-main 2020 Q69 Find Cartesian Equation of a Plane View

The plane passing through the points $(1, 2, 1)$, $(2, 1, 2)$ and parallel to the line, $2x = 3y, z = 1$ also passes through the point
(1) $(0, 6, -2)$
(2) $(-2, 0, 1)$
(3) $(0, -6, 2)$
(4) $(2, 0, -1)$

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

jee-main 2020 Q70 Coplanarity and Relative Position of Planes View

If for some $\alpha \in \mathrm{R}$, the lines $L_1: \frac{x+1}{2} = \frac{y-2}{-1} = \frac{z-1}{1}$ and $L_2: \frac{x+2}{\alpha} = \frac{y+1}{5-\alpha} = \frac{z+1}{1}$ are coplanar, then the line $L_2$ passes through the point:
(1) $(10, 2, 2)$
(2) $(2, -10, -2)$
(3) $(10, -2, -2)$
(4) $(-2, 10, 2)$

jee-main 2021 Q77 Dihedral Angle or Angle Between Planes/Lines View

Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l + m - n = 0$ and $l ^ { 2 } + m ^ { 2 } - n ^ { 2 } = 0$. Then the value of $\sin ^ { 4 } \alpha + \cos ^ { 4 } \alpha$ is :
(1) $\frac { 5 } { 8 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 3 } { 4 }$

jee-main 2021 Q78 Coplanarity and Relative Position of Planes View

The lines $x = ay - 1 = z - 2$ and $x = 3y - 2 = bz - 2 , ( ab \neq 0 )$ are coplanar, if:
(1) $b = 1 , a \in R - \{ 0 \}$
(2) $a = 1 , b \in R - \{ 0 \}$
(3) $a = 2 , b = 2$
(4) $a = 2 , b = 3$

jee-main 2021 Q78 Find Cartesian Equation of a Plane View

The equation of the plane passing through the line of intersection of the planes $\vec { r } \cdot ( \hat { i } + \hat { j } + \widehat { k } ) = 1$ and $\vec { r } \cdot ( 2 \hat { i } + 3 \hat { j } - \hat { k } ) + 4 = 0$ and parallel to the $x$-axis, is (1) $\vec { r } \cdot ( \hat { i } + 3 \widehat { k } ) + 6 = 0$ (2) $\vec { r } \cdot ( \hat { i } - 3 \widehat { k } ) + 6 = 0$ (3) $\vec { r } \cdot ( \hat { j } - 3 \widehat { k } ) - 6 = 0$ (4) $\vec { r } \cdot ( \hat { j } - 3 \widehat { k } ) + 6 = 0$

jee-main 2021 Q79 Find Cartesian Equation of a Plane View

If the equation of plane passing through the mirror image of a point $( 2,3,1 )$ with respect to line $\frac { x + 1 } { 2 } = \frac { y - 3 } { 1 } = \frac { z + 2 } { - 1 }$ and containing the line $\frac { x - 2 } { 3 } = \frac { 1 - y } { 2 } = \frac { z + 1 } { 1 }$ is $\alpha x + \beta y + \gamma z = 24$ then $\alpha + \beta + \gamma$ is equal to:
(1) 20
(2) 19
(3) 18
(4) 21

jee-main 2021 Q79 Perpendicular/Orthogonal Projection onto a Plane View

Consider the line $L$ given by the equation $\frac { x - 3 } { 2 } = \frac { y - 1 } { 1 } = \frac { z - 2 } { 1 }$. Let $Q$ be the mirror image of the point $( 2,3 , - 1 )$ with respect to $L$. Let a plane $P$ be such that it passes through $Q$, and the line $L$ is perpendicular to $P$. Then which of the following points is on the plane $P$?
(1) $( - 1,1,2 )$
(2) $( 1,1,1 )$
(3) $( 1,1,2 )$
(4) $( 1,2,2 )$

jee-main 2021 Q80 Find Cartesian Equation of a Plane View

Let the equation of the plane, that passes through the point $( 1,4 , - 3 )$ and contains the line of intersection of the planes $3 x - 2 y + 4 z - 7 = 0$ and $x + 5 y - 2 z + 9 = 0$, be $\alpha x + \beta y + \gamma z + 3 = 0$, then $\alpha + \beta + \gamma$ is equal to :
(1) $- 15$
(2) 15
(3) $- 23$
(4) 23

jee-main 2021 Q84 Parallelism Between Line and Plane or Constraint on Parameters 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

jee-main 2022 Q78 Distance Computation (Point-to-Plane or Line-to-Line) View

Let the foot of the perpendicular from the point $( 1,2,4 )$ on the line $\frac { x + 2 } { 4 } = \frac { y - 1 } { 2 } = \frac { z + 1 } { 3 }$ be $P$. Then the distance of $P$ from the plane $3 x + 4 y + 12 z + 23 = 0$ is
(1) $\frac { 50 } { 13 }$
(2) $\frac { 63 } { 13 }$
(3) $\frac { 65 } { 13 }$
(4) 4

jee-main 2022 Q78 Distance Computation (Point-to-Plane or Line-to-Line) View

Let $P$ be the plane containing the straight line $\frac { x - 3 } { 9 } = \frac { y + 4 } { - 1 } = \frac { z - 7 } { - 5 }$ and perpendicular to the plane containing the straight lines $\frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 5 }$ and $\frac { x } { 3 } = \frac { y } { 7 } = \frac { z } { 8 }$. If $d$ is the distance of $P$ from the point $(2,-5,11)$, then $d ^ { 2 }$ is equal to
(1) $\frac { 147 } { 2 }$
(2) 96
(3) $\frac { 32 } { 3 }$
(4) 54

jee-main 2022 Q78 Dihedral Angle or Angle Between Planes/Lines View

If the line of intersection of the planes $a x + b y = 3$ and $a x + b y + c z = 0 , a > 0$ makes an angle $30 ^ { \circ }$ with the plane $y - z + 2 = 0$, then the direction cosines of the line are
(1) $\frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0$
(2) $\frac { 1 } { \sqrt { 2 } } , \frac { - 1 } { \sqrt { 2 } } , 0$
(3) $\frac { 1 } { \sqrt { 5 } } , - \frac { 2 } { \sqrt { 5 } } , 0$
(4) $\frac { 1 } { 2 } , - \frac { \sqrt { 3 } } { 2 } , 0$

jee-main 2022 Q78 Coplanarity and Relative Position of Planes View

Let the lines $\frac { x - 1 } { \lambda } = \frac { y - 2 } { 1 } = \frac { z - 3 } { 2 }$ and $\frac { x + 26 } { - 2 } = \frac { y + 18 } { 3 } = \frac { z + 28 } { \lambda }$ be coplanar and $P$ be the plane containing these two lines. Then which of the following points does NOT lie on $P$?
(1) $( 0 , - 2 , - 2 )$
(2) $( - 5,0 , - 1 )$
(3) $( 3 , - 1,0 )$
(4) $( 0,4,5 )$

jee-main 2022 Q79 Distance Computation (Point-to-Plane or Line-to-Line) View

The shortest distance between the lines $\frac { x - 3 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 1 } { - 1 }$ and $\frac { x + 3 } { 2 } = \frac { y - 6 } { 1 } = \frac { z - 5 } { 3 }$ is
(1) $\frac { 18 } { \sqrt { 5 } }$
(2) $\frac { 22 } { 3 \sqrt { 5 } }$
(3) $\frac { 46 } { 3 \sqrt { 5 } }$
(4) $6 \sqrt { 3 }$

jee-main 2022 Q79 Find Cartesian Equation of a Plane View

If the plane $P$ passes through the intersection of two mutually perpendicular planes $2 x + k y - 5 z = 1$ and $3 k x - k y + z = 5 , k < 3$ and intercepts a unit length on positive $x$-axis, then the intercept made by the plane $P$ on the $y$-axis is
(1) $\frac { 1 } { 11 }$
(2) $\frac { 5 } { 11 }$
(3) 6
(4) 7

jee-main 2022 Q79 Volume of Pyramid/Tetrahedron Using Planes and Lines View

A plane $P$ is parallel to two lines whose direction ratios are $- 2,1 , - 3$ and $- 1,2 , - 2$ and it contains the point $( 2,2 , - 2 )$. Let $P$ intersect the co-ordinate axes at the points $A , B , C$ making the intercepts $\alpha , \beta , \gamma$. If $V$ is the volume of the tetrahedron $O A B C$, where $O$ is the origin and $p = \alpha + \beta + \gamma$, then the ordered pair $( V , p )$ is equal to
(1) $( 48 , - 13 )$
(2) $( 24 , - 13 )$
(3) $( 48,11 )$
(4) $( 24 , - 5 )$

jee-main 2022 Q90 Dihedral Angle or Angle Between Planes/Lines View

The line of shortest distance between the lines $\frac { x - 2 } { 0 } = \frac { y - 1 } { 1 } = \frac { z } { 1 }$ and $\frac { x - 3 } { 2 } = \frac { y - 5 } { 2 } = \frac { z - 1 } { 1 }$ makes an angle of $\sin ^ { - 1 } \sqrt { \frac { 2 } { 27 } }$ with the plane $P : ax - y - z = 0$, $a > 0$. If the image of the point $(1,1,-5)$ in the plane $P$ is $(\alpha, \beta, \gamma)$, then $\alpha + \beta - \gamma$ is equal to $\_\_\_\_$.

jee-main 2022 Q90 Find Intersection of a Line and a Plane View

Let the line $\frac { x - 3 } { 7 } = \frac { y - 2 } { - 1 } = \frac { z - 3 } { - 4 }$ intersect the plane containing the lines $\frac { x - 4 } { 1 } = \frac { y + 1 } { - 2 } = \frac { z } { 1 }$ and $4 a x - y + 5 z - 7 a = 0 = 2 x - 5 y - z - 3 , a \in \mathbb { R }$ at the point $P ( \alpha , \beta , \gamma )$. Then the value of $\alpha + \beta + \gamma$ equals $\_\_\_\_$ .

jee-main 2023 Q67 Coplanarity and Relative Position of Planes View

Let $P$ be the plane passing through the intersection of the planes $\vec{r}\cdot(\hat{i}+3\hat{j}-\hat{k}) = 5$ and $\vec{r}\cdot(2\hat{i}-\hat{j}+\hat{k}) = 3$, and the point $(2, 1, -2)$. Let the position vectors of the points $X$ and $Y$ be $\hat{i} - 2\hat{j} + 4\hat{k}$ and $5\hat{i} - \hat{j} + 2\hat{k}$ respectively. Then the points $X$ and $Y$ with respect to the plane $P$ are
(1) on the same side
(2) on opposite sides
(3) $X$ lies on $P$
(4) $Y$ lies on $P$

jee-main 2023 Q77 Find Cartesian Equation of a Plane View

The plane, passing through the points $( 0 , - 1 , 2 )$ and $( - 1 , 2 , 1 )$ and parallel to the line passing through $( 5 , 1 , - 7 )$ and $( 1 , - 1 , - 1 )$, also passes through the point
(1) $( - 2 , 5 , 0 )$
(2) $( 1 , - 2 , 1 )$
(3) $( 2 , 0 , 1 )$
(4) $( 0 , 5 , - 2 )$

jee-main 2023 Q78 Coplanarity and Relative Position of Planes View

The line, that is coplanar to the line $\frac { x + 3 } { - 3 } = \frac { y - 1 } { 1 } = \frac { z - 5 } { 5 }$, is
(1) $\frac { x + 1 } { - 1 } = \frac { y - 2 } { 2 } = \frac { z - 5 } { 4 }$
(2) $\frac { x + 1 } { - 1 } = \frac { y - 2 } { 2 } = \frac { z - 5 } { 5 }$
(3) $\frac { x - 1 } { - 1 } = \frac { y - 2 } { 2 } = \frac { z - 5 } { 5 }$
(4) $\frac { x + 1 } { 1 } = \frac { y - 2 } { 2 } = \frac { z - 5 } { 5 }$

jee-main 2023 Q78 Perpendicular/Orthogonal Projection onto a Plane View

Let the foot of perpendicular of the point $P ( 3 , - 2 , - 9 )$ on the plane passing through the points $( - 1 , - 2 , - 3 ) , ( 9,3,4 ) , ( 9 , - 2,1 )$ be $Q ( \alpha , \beta , \gamma )$. Then the distance of $Q$ from the origin is
(1) $\sqrt { 42 }$
(2) $\sqrt { 38 }$
(3) $\sqrt { 35 }$
(4) $\sqrt { 29 }$

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