JEE Main > Vectors and ThreeDimensional Geometry

Explore popular questions from Vectors and ThreeDimensional Geometry for JEE Main. This collection covers Vectors and ThreeDimensional Geometry previous year JEE Main questions hand picked by popular teachers.


Q 1.    

Correct4

Incorrect-1

The angle between straight lines whose direction cosines are {tex} \left( \frac { 1 } { 2 } , - \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right) {/tex} and {tex} \left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } \right) {/tex} is

A

{tex} \cos ^ { - 1 } \left( \frac { 2 } { \sqrt { 3 } } \right) {/tex}

B

{tex} \cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 6 } } \right) {/tex}

{tex} \cos ^ { - 1 } \left( - \frac { 1 } { \sqrt { 6 } } \right) {/tex}

D

None of these

Explanation


Q 2.    

Correct4

Incorrect-1

Which one of the following is best condition for the plane {tex} a x + b y + c z + d = 0 {/tex} to intersect the {tex} x {/tex} -and {tex} y {/tex} -axis at equal angle?

{tex} | a | = | b | {/tex}

B

{tex} a = - b {/tex}

C

{tex} a = b {/tex}

D

{tex} a ^ { 2 } + b ^ { 2 } = 1 {/tex}

Explanation


Q 3.    

Correct4

Incorrect-1

The equation of a straight line parallel to the {tex} x {/tex} -axis is given by

A

{tex} \frac { x - a } { 1 } = \frac { y - b } { 1 } = \frac { z - c } { 1 } {/tex}

B

{tex} \frac { x - a } { 0 } = \frac { y - b } { 1 } = \frac { z - c } { 1 } {/tex}

C

{tex} \frac { x - a } { 0 } = \frac { y - b } { 0 } = \frac { z - c } { 1 } {/tex}

{tex} \frac { x - a } { 1 } = \frac { y - b } { 0 } = \frac { z - c } { 0 } {/tex}

Explanation


Q 4.    

Correct4

Incorrect-1

If {tex} P ( 2,3 , - 6 ) {/tex} and {tex} Q ( 3 , - 4,5 ) {/tex} are two points, the direction cosines of the line {tex} P Q {/tex} are

A

{tex} - \frac { 1 } { \sqrt { 171 } } , - \frac { 7 } { \sqrt { 177 } } , - \frac { 11 } { \sqrt { 171 } } {/tex}

{tex} \frac { 1 } { \sqrt { 171 } } , - \frac { 7 } { \sqrt { 171 } } , \frac { 11 } { \sqrt { 171 } } {/tex}

C

{tex} \frac { 1 } { \sqrt { 171 } } , \frac { 7 } { \sqrt { 171 } } , - \frac { 11 } { \sqrt { 171 } } {/tex}

D

{tex} - \frac { 7 } { \sqrt { 171 } } , - \frac { 1 } { \sqrt { 171 } } , \frac { 11 } { \sqrt { 171 } } {/tex}

Explanation


Q 5.    

Correct4

Incorrect-1

The ratio in which yz-plane divides the line joining the points {tex} A ( 3,1 , - 5 ) {/tex} and {tex} B ( 1,4 , - 6 ) {/tex} is

{tex} - 3 : 1 {/tex}

B

{tex} 3 : 1 {/tex}

C

{tex} - 1 : 3 {/tex}

D

{tex} 1 : 3 {/tex}

Explanation


Q 6.    

Correct4

Incorrect-1

A straight line is inclined to the axes of {tex} x {/tex} and {tex} z {/tex} at angles {tex} 45 ^ { \circ } {/tex} and {tex} 60 ^ { \circ } , {/tex} respectively, then the inclination of the line to the {tex} y {/tex} -axis is

A

{tex} 30 ^ { \circ } {/tex}

B

{tex} 45 ^ { \circ } {/tex}

{tex} 60 ^ { \circ } {/tex}

D

{tex} 90 ^ { \circ } {/tex}

Explanation


Q 7.    

Correct4

Incorrect-1

The angle between two diagonals of a cube is

A

{tex} \cos \theta = \sqrt { 3 } / 2 {/tex}

B

{tex} \cos \theta = 1 / \sqrt { 2 } {/tex}

{tex} \cos \theta = 1 / 3 {/tex}

D

None of these

Explanation


Q 8.    

Correct4

Incorrect-1

Given that {tex} A ( 3,2 , - 4 ) , B ( 5,4 , - 6 ) {/tex} and {tex} C ( 9,8 , - 10 ) {/tex} are collinear. The ratio in which {tex} B {/tex} divides {tex} A C {/tex} is

{tex} 1 : 2 {/tex}

B

{tex} 2 : 1 {/tex}

C

{tex} - 1 : 2 {/tex}

D

{tex} - 2:1 {/tex}

Explanation


Q 9.    

Correct4

Incorrect-1

If {tex} P _ { 1 } P _ { 2 } {/tex} is perpendicular to {tex} P _ { 2 } P _ { 3 } , {/tex} then the value of {tex} k , {/tex} where {tex} P _ { 1 } ( k , 1 , - 1 ) , P _ { 2 } ( 2 k , 0,2 ) {/tex} and {tex} P _ { 3 } ( 2 + 2 k , k , 1 ) , {/tex} is

3

B

-3

C

2

D

-2

Explanation


Q 10.    

Correct4

Incorrect-1

{tex} A {/tex} is the point {tex} ( 3,7,5 ) {/tex} and {tex} B {/tex} is the point {tex} ( - 3,2,6 ) {/tex} . The projection of {tex} A B {/tex} on the line that joins the points {tex} ( 7,9,4 ) {/tex} and {tex} ( 4,5 , - 8 ) {/tex} is

A

26

2

C

13

D

4

Explanation


Q 11.    

Correct4

Incorrect-1

The shortest distance of the point from {tex} P \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right) {/tex} on the {tex} x {/tex} -axis is equal to

A

{tex} \sqrt { x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } } {/tex}

B

{tex} \sqrt { x _ { 1 } ^ { 2 } + z _ { 1 } ^ { 2 } } {/tex}

{tex} \sqrt { y _ { 1 } ^ { 2 } + z _ { 1 } ^ { 2 } } {/tex}

D

None of these

Explanation


Q 12.    

Correct4

Incorrect-1

The point of intersection of the {tex} x y {/tex} -plane and the line passing through the points {tex} A = ( 3,4,1 ) {/tex} and {tex} B = ( 5,1,6 ) {/tex} are

A

{tex} ( - \frac { 13 } { 5 } , \frac { 23 } { 5 } , 0 ){/tex}

{tex} ( \frac { 13 } { 5 } , \frac { 23 } { 5 } , 0) {/tex}

C

{tex} ( \frac { 13 } { 5 } , - \frac { 23 } { 5 } , 0) {/tex}

D

{tex} \left( - \frac { 13 } { 5 } , - \frac { 23 } { 5 } , 0 \right) {/tex}

Explanation



Q 13.    

Correct4

Incorrect-1

The equation of the plane containing the line {tex} \frac { x - \alpha } { l } = \frac { y - \beta } { m } = \frac { z - \gamma } { n } {/tex} is {tex} a ( x - \alpha ) + b ( y - \beta ) + c ( z - \gamma ) = 0 , {/tex} where {tex} a l + b m + c n {/tex} is equal to

A

1

B

-1

C

2

0

Explanation


Q 14.    

Correct4

Incorrect-1

The shortest distance between the two straight lines {tex} \frac { x - 4 / 3 } { 2 } = \frac { y + 6 / 5 } { 3 } = \frac { z - 3 / 2 } { 4 } {/tex} and {tex} \frac { 5 y + 6 } { 8 } = \frac { 2 z - 3 } { 9 } = \frac { 3 x - 4 } { 5 } {/tex} is

A

{tex} \sqrt { 29 } {/tex}

B

{tex}3{/tex}

{tex}0{/tex}

D

6{tex} \sqrt { 10 } {/tex}

Explanation


Q 15.    

Correct4

Incorrect-1

A straight line passes through the point {tex} ( 2 , - 1 , - 1 ) {/tex} . It is parallel to the plane {tex} 4 x + y + z + 2 = 0 {/tex} and is perpendicular to the line {tex} \frac { x } { 1 } = \frac { y } { - 2 } = \frac { z - 5 } { 1 } {/tex} . The equation of the straight line is

A

{tex} \frac { x - 2 } { 4 } = \frac { y + 1 } { 1 } = \frac { z + 1 } { 1 } {/tex}

B

{tex} \frac { x + 2 } { 4 } = \frac { y - 1 } { 1 } = \frac { z - 1 } { 1 } {/tex}

{tex} \frac { x - 2 } { - 1 } = \frac { y + 1 } { 1 } = \frac { z + 1 } { 3 } {/tex}

D

{tex} \frac { x + 2 } { - 1 } = \frac { y - 1 } { 1 } = \frac { z - 1 } { 3 } {/tex}

Explanation


Q 16.    

Correct4

Incorrect-1

If centre of a sphere is {tex} ( 1,4 , - 3 ) {/tex} and the radius is 3 units, then the equation of the sphere is

{tex} x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 2 x - 8 y + 6 z + 17 = 0 {/tex}

B

{tex} 2 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) - 2 x - 8 y + 6 z + 17 = 0 {/tex}

C

{tex} x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 16 y + 12 z + 17 = 0 {/tex}

D

{tex} x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 x + 8 y - 6 z - 17 = 0 {/tex}

Explanation


Q 17.    

Correct4

Incorrect-1

If equation of a sphere is {tex} 2 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) - 4 x - 8 y + 12 z - 7 = 0 {/tex} and one extremity of its diameter is {tex} ( 2 , - 1,1 ) , {/tex} then the other extremity of diameter of the sphere will be

A

{tex} ( 2,9 , - 13 ) {/tex}

B

{tex} ( 0,9,7 ) {/tex}

{tex} ( 0,5,7 ) {/tex}

D

{tex} ( 2,5 , - 13 ) {/tex}

Explanation


Q 18.    

Correct4

Incorrect-1

The direction cosines of the line that is perpendicular to the lines with direction cosines proportional to {tex} ( 1 , - 2 , - 2 ) , ( 0,2,1 ) {/tex} are

{tex} \left( \frac { 2 } { 3 } , - \frac { 1 } { 3 } , \frac { 2 } { 3 } \right) {/tex}

B

{tex} \left( \frac { 2 } { 3 } , \frac { - 1 } { 3 } , \frac { - 2 } { 3 } \right) {/tex}

C

{tex} \left( \frac { 2 } { 3 } , \frac { 1 } { 3 } , \frac { - 2 } { 3 } \right) {/tex}

D

{tex} \left( \frac { - 2 } { 3 } , \frac { - 1 } { 3 } , \frac { - 2 } { 3 } \right) {/tex}

Explanation


Q 19.    

Correct4

Incorrect-1

The points {tex} ( 4,7,8 ) , ( 2,3,4 ) , ( - 1 , - 2,1 ) {/tex} and {tex} ( 1,2,5 ) {/tex} are

The vertices of a parallelogram

B

Collinear

C

The vertices of a trapezium

D

Concyclic

Explanation


Q 20.    

Correct4

Incorrect-1

The equation of the plane parallel to the plane {tex} 4 x - 3 y + 2 z + 1 {/tex} {tex} = 0 {/tex} and passing through the point {tex} ( 5,1 , - 6 ) {/tex} is

{tex} 4 x - 3 y + 2 z - 5 = 0 {/tex}

B

{tex} 3 x - 4 y + 2 z - 5 = 0 {/tex}

C

{tex} 4 x - 3 y + 2 z + 5 = 0 {/tex}

D

{tex} 3 x - 4 y + 2 z + 5 = 0 {/tex}

Explanation


Q 21.    

Correct4

Incorrect-1

A plane is passed through the middle point of the segment {tex} A ( - 2,5,1 ) {/tex} and {tex} B ( 6,1,5 ) {/tex} and is perpendicular to this line. Its equation is

{tex} 2 x - y + z = 4 {/tex}

B

{tex} 2 x + y + z = 4 {/tex}

C

{tex} x - 3 y + z = 5 {/tex}

D

{tex} x - 4 y + 2 z = 5 {/tex}

Explanation


Q 22.    

Correct4

Incorrect-1

A plane meets the coordinates axes in {tex} A , B {/tex} and {tex} C {/tex} such that the centroid of the triangle {tex} A B C {/tex} is {tex} ( a , b , c ) . {/tex} The equation of the plane is

{tex} \frac { x } { a } + \frac { y } { b } + \frac { z } { c } = 3 {/tex}

B

{tex} \frac { x } { a } + \frac { y } { b } + \frac { z } { c } = 1 {/tex}

C

{tex} \frac { x } { a } + \frac { y } { b } + \frac { z } { c } = 2 {/tex}

D

None of these

Explanation


Q 23.    

Correct4

Incorrect-1

The radius of the sphere {tex} ( x + 1 ) ( x + 3 ) + ( y - 2 ) ( y - 4 ) + ( z + 1 ) {/tex} {tex} ( z + 3 ) = 0 {/tex} is

A

{tex} \sqrt { 2 } {/tex}

B

{tex} 2{/tex}

{tex} \sqrt { 3 } {/tex}

D

{tex}3{/tex}

Explanation


Q 24.    

Correct4

Incorrect-1

The sum of the direction cosines of a straight line is

A

Zero

B

One

Constant

D

None of these

Explanation


Q 25.    

Correct4

Incorrect-1

The angle between two lines whose direction cosines are given by the equation {tex} l + m + n = 0 , l ^ {2} + m ^ { 2 } + n ^ { 2 } = 0 {/tex} is

{tex} \frac { \pi } { 3 } {/tex}

B

{tex} \frac { 7 \pi } { 3 } {/tex}

C

{tex} \frac { \pi } { 4 } {/tex}

D

None of these

Explanation