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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 experienced teachers.

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Vectors and ThreeDimensional Geometry

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Q 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


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Q 2. 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


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Q 3. The shortest distance of the plane {tex} 12 + 4 y + 3 z = 327 {/tex} , from the sphere {tex} x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 4 x - 2 y - 6 z = 155 {/tex} , is equal to

A

39 units

B

26 {tex}{ sq }{/tex}. units

13 units

D

None of these

Explanation


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Q 4. 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


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Q 5. The direction ratios of a normal to the plane passing through {tex} ( 1,0,0 ) {/tex} and {tex} ( 0,1,0 ) {/tex} and making an angle {tex} \frac { \pi } { 4 } {/tex} with the plane {tex} x + y = 3 {/tex} are

A

{tex} ( 1 , \sqrt { 2 } , 1 ) {/tex}

{tex} ( 1,1 , \sqrt { 2 } ) {/tex}

C

{tex} ( 1,1,2 ) {/tex}

D

{tex} ( \sqrt { 2 } , 1,1 ) {/tex}

Explanation


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Q 6. 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


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Q 7. 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


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Q 8. {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


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Q 9. 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


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Q 10. 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



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Q 11. The equation of a plane passing through {tex} ( 1,2 , - 3 ) {/tex} and {tex} ( 0,0,0 ) {/tex} and perpendicular to the plane {tex} 3 x - 5 y + 2 z = 11 {/tex} is

A

{tex} 3 x + y + \frac { 5 } { 3 } z = 0 {/tex}

B

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

C

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

{tex} x + y + z = 0 {/tex}

Explanation



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Q 12. 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


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Q 13. The locus represented by {tex} x y + y z = 0 {/tex} is

A

A pair of perpendicular lines

B

A pair of parallel lines

C

A pair of parallel planes

A pair of perpendicular planes

Explanation


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Q 14. 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


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Q 15. 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


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Q 16. 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


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Q 17. A variable plane passes through a fixed point {tex} ( a , b , c ) {/tex} and meets the coordinate axes in {tex} A , B {/tex} and {tex} C {/tex} . The locus of the point common to the plane through {tex} A , B {/tex} and {tex} C {/tex} parallel to the coordinate planes is

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

B

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

C

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

D

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

Explanation


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Q 18. Centroid of the tetrahedon OABC, where A {tex}\equiv (a,2,3), B \equiv (1,b,2),C \equiv (2,1,c){/tex} and O is the origin of (1,2,3). The value of {tex}a^{2} + b^{2} + c^{2}{/tex} is equal to

75

B

80

C

121

D

None of these

Explanation


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Q 19. The equation of the plane passing through the points {tex} ( 2 , - 1,0 ) {/tex} and {tex} ( 3 , - 4,5 ) {/tex} and parallel to the line {tex} 2 x = 3 y = 4 z {/tex} is

A

{tex} 125 x - 90 y - 79 z = 340 {/tex}

B

{tex} 32 x - 21 y - 36 z = 85 {/tex}

C

{tex} 73 x + 61 y - 22 z = 85 {/tex}

{tex} 29 x - 27 y - 22 z = 85 {/tex}

Explanation



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Q 20. The equation of the straight line through the origin parallel to the line {tex} ( b + c ) x + ( c + a ) y + ( a + b ) z = k = ( b - c ) x + ( c - a ) y {/tex} {tex} + ( a - b ) z {/tex} is

A

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

B

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

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

D

None of these

Explanation


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Q 21. The resultant of two forces {tex} P\mathrm { N }{/tex} and {tex}3\mathrm { N } {/tex} is a force of {tex} \mathrm7 { N } {/tex} . If the direction of {tex}3 \mathrm { N } {/tex} force were reversed, the resultant would be {tex} \sqrt { 19 } \mathrm { N } {/tex} . The value of {tex} P {/tex} is

{tex}5 \mathrm { N } {/tex}

B

{tex}6 \mathrm { N } {/tex}

C

{tex}3 N {/tex}

D

{tex}4 \mathrm { N } {/tex}

Explanation


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Q 22. Let {tex} \vec { a } = \hat { i } + \hat { j } + \hat { k } , \vec { b } = \hat { i } - \hat { j } + 2 \hat { k } {/tex} and {tex} \vec { c } = x \hat { i } + ( x - 2 ) \hat { j } - \hat { k } . {/tex} If the vector {tex} \vec { c } {/tex} lies in the plane of {tex} \vec { a } {/tex} and {tex} \overline { b } , {/tex} then {tex} x {/tex} equals

A

0

B

1

C

- 4

- 2

Explanation



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Q 23. If {tex} \hat { u } {/tex} and {tex} \hat { v } {/tex} are unit vectors and {tex} \theta {/tex} is the acute angle between them, then {tex} 2 \hat { u } \times 3 \hat { v } {/tex} is a unit vector for

A

exactly two values of {tex} \theta {/tex}

B

more than two values of {tex} \theta {/tex}

C

no value of {tex} \theta {/tex}

exactly one value of {tex} \theta {/tex}

Explanation


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Q 24. The vector {tex} \vec { a } = \alpha \hat { i } + 2 \hat { j } + \beta \hat { k } {/tex} lies in the plane of the vectors {tex} \overline { b } = \hat { i } + \hat { j } {/tex} and {tex} \vec { c } = \hat { j } + \hat { k } {/tex} and bisects the angle between {tex} \vec { b } {/tex} and {tex} \vec { c } . {/tex} Then which one of the following gives possible values of {tex} \alpha {/tex} and {tex} \beta {/tex} ?

A

{tex} \alpha = 2 , \beta = 2 {/tex}

B

{tex} \alpha = 1 , \beta = 2 {/tex}

C

{tex} \alpha = 2 , \beta = 1 {/tex}

{tex} \alpha = 1 , \beta = 1 {/tex}

Explanation


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Q 25. The non-zero vectors {tex} \vec { a } {/tex}, {tex} \vec { b } {/tex} and {tex} \vec { c } {/tex} are related by {tex} \vec { a } = 8\vec{b} {/tex} and {tex} \vec { c } = - 7 \overline { b } . {/tex} Then the angle between {tex} \vec { a } {/tex} and {tex} \vec { c } {/tex} is

A

0

B

{tex} \pi / 4 {/tex}

C

{tex} \pi / 2 {/tex}

{tex} \pi {/tex}

Explanation