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JEE Advanced > Vectors and ThreeDimensional Geometry

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

Q 1.

Correct4

Incorrect-1

The points with position vectors {tex} 60 i + 3 j , 40 i - 8 j , a i - 52j {/tex} are collinear if

{tex} a = - 40 {/tex}

B

{tex} a = 40 {/tex}

C

{tex} a = 20 {/tex}

D

none of these

Explanation

Q 2.

Correct4

Incorrect-1

Let {tex} \vec { a } = \hat { i } - \hat { j } , \vec { b } = \hat { j } - \hat { k } , \vec { c } = \hat { k } - \hat { i } . {/tex} If {tex} \vec { d } {/tex} isa unit vector such that {tex} \vec { a } \vec { d } = 0 = [ \vec { b } \vec { c } \vec { d } ] , {/tex} then {tex} \vec { d } {/tex} equals

{tex} \pm \frac { \hat { i } + \hat { j } - 2 \hat { k } } { \sqrt { 6 } } {/tex}

B

{tex} \pm \frac { \hat { i } + \hat { j } - \hat { k } } { \sqrt { 3 } } {/tex}

C

{tex} \pm \frac { \hat { i } + \hat { j } + \hat { k } } { \sqrt { 3 } } {/tex}

D

{tex} \pm \hat { k } {/tex}

Explanation


Q 3.

Correct4

Incorrect-1

If {tex} \vec { a } , \vec { b } , \vec { c } {/tex} are non coplanar unit vectors such that {tex} \vec { a } \times ( \vec { b } \times \vec { c } ) = \frac { ( \vec { b } + \vec { c } ) } { \sqrt { 2 } } , {/tex} then the angle between {tex} \vec { a } {/tex} and {tex} \vec { b } {/tex} is

{tex} \frac { 3 \pi } { 4 } {/tex}

B

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

C

{tex} \pi / 2 {/tex}

D

{tex} \pi {/tex}

Explanation

Q 4.

Correct4

Incorrect-1

Let {tex} \vec { u } , \vec { v } {/tex} and {tex} \vec { w } {/tex} be vectors such that {tex} \vec { u } + \vec { v } + \vec { w } = 0 . {/tex} If {tex} | \vec { u } | = 3 , | \vec { v } | = 4 {/tex} and {tex} | \vec { w } | = 5 , {/tex} then {tex} \vec { u }\cdot \vec { v } + \vec { v } \cdot \vec { w } + \vec { w } \cdot \vec { u } {/tex} is

A

{tex}47{/tex}

{tex} - 25 {/tex}

C

{tex}0{/tex}

D

{tex}25{/tex}

Explanation

Q 5.

Correct4

Incorrect-1

Let {tex} a = 2 i + j - 2 k {/tex} and {tex} b = i + j . {/tex} If {tex} c {/tex} is a vector such that {tex} a {/tex}. {tex} c = | \mathbf { c } | , | c - a | = 2 \sqrt { 2 } {/tex} and the angle between {tex} ( a \times b ) {/tex} and {tex} c {/tex} is {tex} 30 ^ { \circ } , {/tex} then {tex} | ( a \times b ) \times c | = {/tex}

A

{tex} 2 / 3 {/tex}

{tex} 3 / 2 {/tex}

C

{tex}2{/tex}

D

{tex}3{/tex}

Explanation

Q 6.

Correct4

Incorrect-1

If the vectors {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } {/tex} form the sides {tex} B C , C A {/tex} and {tex} A B {/tex} respectively of a triangle {tex} A B C , {/tex} then

A

{tex} \vec { a } \cdot \vec { b } + \vec { b } \cdot \vec { c } + \vec { c } \cdot \vec { a } = 0 {/tex}

{tex} \vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a } {/tex}

C

{tex} \vec { a } \cdot \vec { b } = \vec { b } \cdot \vec { c } = \vec { c } \cdot \vec { a } {/tex}

D

{tex} \vec { a } \times \vec { b } + \vec { b } \times \vec { c } + \vec { c } \times \vec { a } = \overrightarrow { 0 } {/tex}

Explanation

Q 7.

Correct4

Incorrect-1

Let the vectors {tex} \vec { a } , \vec { b } , \vec { c } {/tex} and {tex} \vec { \mathrm { d } } {/tex} be such that {tex} ( \vec { \mathrm { a } } \times \vec { \mathrm { b } } ) \times ( \vec { \mathrm { c }} \times \vec {\mathrm { d } } ) = 0 . {/tex} Let {tex}\mathrm P _ { 1 } {/tex} and {tex} \mathrm { P } _ { 2 } {/tex} be planes determined by the pairs of vectors {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } , \vec { d } {/tex} respectively. Then the angle between {tex} P _ { 1 } {/tex} and {tex} P _ { 2 } {/tex} is

{tex}0{/tex}

B

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

C

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

D

{tex} \frac { \pi } { 2 } {/tex}

Explanation


Q 8.

Correct4

Incorrect-1

If {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } {/tex} are unit coplanar vectors, then the scalar triple product {tex} [ 2 \vec { a } - \vec { b } , 2 \vec { b } - \vec { c } , 2 \vec { c } - \vec { a } ] = {/tex}

{tex}0{/tex}

B

{tex}1{/tex}

C

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

D

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

Explanation

Q 9.

Correct4

Incorrect-1

Let {tex} \vec { a } = \vec { i } - \vec { k } , \vec { b } = x \vec { i } + \vec {j}+( 1 - x ) \vec { k } {/tex} and {tex} \vec { c } = y \vec { i } + x \vec { j } + ( 1 + x - y ) \vec { k } . {/tex} Then {tex} [ \vec { a } \vec { b } \vec { c } ] {/tex} depends on

A

only {tex} x {/tex}

B

only {tex} y {/tex}

Neither {tex} x {/tex} Nor {tex} y {/tex}

D

both {tex} x {/tex} and {tex} y {/tex}

Explanation

Q 10.

Correct4

Incorrect-1

If {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } {/tex} are unit vectors, then
{tex} | \vec { a } - \vec { b } | ^ { 2 } + | \vec { b } - \vec { c } | ^ { 2 } + | \vec { c } - \vec { a } | ^ { 2 } {/tex} does NOT exceed

A

{tex}4{/tex}

{tex}9{/tex}

C

{tex}8{/tex}

D

{tex}6{/tex}

Explanation


Q 11.

Correct4

Incorrect-1

If {tex} \vec { a } {/tex} and {tex} \vec { b } {/tex} are two unit vectors such that {tex} \vec { a } + 2 \vec { b } {/tex} and {tex} 5 \vec { a } - 4 \vec { b } {/tex} are perpendicular to each other then the angle between {tex} \vec { a } {/tex} and {tex} \vec { b } {/tex} is

A

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

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

C

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

D

{tex} \cos ^ { - 1 } \left( \frac { 2 } { 7 } \right) {/tex}

Explanation

Q 12.

Correct4

Incorrect-1

Let {tex} \vec { V } = 2 \vec { i } + \vec { j } - \vec { k } {/tex} and {tex} \vec { W } = \vec { i } + 3 \vec { k } . {/tex} If {tex} \vec { U } {/tex} is a unit vector, then the maximum value of the scalar triple product {tex} | \vec { U } \vec { V } \vec { W } | {/tex} is

A

{tex} - 1 {/tex}

B

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

{tex} \sqrt { 59 } {/tex}

D

{tex} \sqrt { 60 } {/tex}

Explanation


Q 13.

Correct4

Incorrect-1

If {tex} \vec { a } = ( \hat { i } + \hat { j } + \hat { k } ) , \vec { a } \cdot \vec { b } = 1 {/tex} and {tex} \vec { a } \times \vec { b } = \hat { j } - \hat { k } , {/tex} then {tex} \vec { b } {/tex} is

A

{tex} \hat { i } - \hat { j } + \hat { k } {/tex}

B

{tex} 2 \hat { j } - \hat { k } {/tex}

{tex} \hat { i } {/tex}

D

{tex} 2 \hat { i } {/tex}

Explanation

Q 14.

Correct4

Incorrect-1

A line with positive direction cosines passes through the point {tex} P ( 2 , - 1,2 ) {/tex} and makes equal angles with the coordinate axes. The line meets the plane {tex} 2 x + y + z = 9 {/tex} at point {tex} Q . {/tex} The length of the line segment {tex} P Q {/tex} equals

A

{tex}1{/tex}

B

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

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

D

{tex}2{/tex}

Explanation

Q 15.

Correct4

Incorrect-1

If the system of equations {tex} x - k y - z = 0 , k x - y - z = 0 , x + y - z {/tex} {tex} = 0 {/tex} has a non-zero solution, then the possible values of {tex} k {/tex} are

A

{tex} - 1,2 {/tex}

B

{tex} 0,1 {/tex}

C

{tex} 1,2 {/tex}

{tex} - 1,1 {/tex}

Explanation

Q 16.

Correct4

Incorrect-1

A plane {tex} 2 x + 3 y + 5 z = 1 {/tex} has a point {tex} P {/tex} which is at minimum distance from line joining {tex} A ( 1,0 , - 3 ) {/tex} and {tex} B ( 1 , - 5,7 ) {/tex} then distance {tex} A P {/tex} is equal to

A

{tex} 3 \sqrt { 5 } {/tex}

{tex} 2 \sqrt { 5 } {/tex}

C

{tex} 4 \sqrt { 5 } {/tex}

D

None of these

Explanation

Q 17.

Correct4

Incorrect-1

If {tex} A = ( p , q , r ) {/tex} and {tex} B = \left( p ^ { \prime } , q ^ { \prime }, r ^ { \prime } \right) {/tex} are two points on the line {tex} \lambda x = \mu y = \gamma z , {/tex} such that {tex} O A = 3 , O B = 4 , {/tex} then {tex} p p ^ { \prime } + q q ^ { \prime } + r r ^ { \prime } {/tex} is equal to

A

7

12

C

5

D

None of these

Explanation

Q 18.

Correct4

Incorrect-1

Let a plane passes through origin and is parallel to line {tex} \frac { x - 1 } { 2 } = \frac { y + 3 } { - 1 } = \frac { z + 1 } { - 2 } {/tex} such that the distance between plane and line is {tex} 5 / 3 , {/tex} then the equation of plane is

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

B

{tex} x - 2 y - 2 z = 0 {/tex}

C

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

D

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

Explanation


Q 19.

Correct4

Incorrect-1

If the shortest distance between lines {tex} \vec { r } = \hat { i } + 2 \hat { j } + 3 \hat { k } + \lambda _ { 1 } {/tex}{tex} ( 2 \hat { i } + 3 \hat { j } + 4 \hat { k } ) {/tex} and {tex} \vec { r } = 2 \hat { i } + 4 \hat { j } + 5 \hat { k } + \lambda _ { 2 } ( 3 \hat { i } + 4 \hat { j } + 5 \hat { k } ) {/tex} is {tex} x {/tex}, then {tex} \cos ^ { - 1 } \cos \sqrt { 6 } x {/tex} is equal to

A

{tex} \frac { 1 } { 2 } {/tex}

B

{tex} 0{/tex}

{tex}1{/tex}

D

{tex} \pi {/tex}

Explanation

Q 20.

Correct4

Incorrect-1

If {tex} \vec { r } = \lambda ( \vec { a } \times \vec { b } ) + \mu ( \vec { b } \times \vec { c } ) + \gamma ( \vec { c } \times \vec { a } ) {/tex} and {tex} [ \vec { a } \vec { b } \vec { c } ] = \frac { 1 } { 8 } , {/tex} then {tex} \lambda + \mu {/tex} {tex} + \gamma {/tex} is

A

{tex} 8 ( \vec { r } \cdot \vec { a } ) {/tex}

B

{tex} 8 ( \vec { r } \cdot \vec { b } ) {/tex}

C

{tex} 8 ( \vec { r } \cdot \vec { c } ) {/tex}

{tex} 8 \vec { r } \cdot ( \vec { a } + \vec { b } + \vec { c } ) {/tex}

Explanation

Q 21.

Correct4

Incorrect-1

In a triangle {tex} O A B , E {/tex} is the mid-point of {tex} O B {/tex} and {tex} D {/tex} is a point on {tex} A B {/tex} such that {tex} A D: D B = 2: 1 . {/tex} If {tex} O D {/tex} and {tex} A E {/tex} intersect at {tex} P {/tex}, then ratio of {tex} \frac { O P } { P D } {/tex} is equal to

{tex}3:2{/tex}

B

{tex}2:3{/tex}

C

{tex}3:4{/tex}

D

{tex}4:3{/tex}

Explanation


Q 22.

Correct4

Incorrect-1

If {tex} \vec { x } {/tex} and {tex} \vec { y } {/tex} be unit vectors and {tex} | \vec{z} | = \frac { 2 } { \sqrt { 7 } } {/tex} such that {tex}\vec z + \vec z \times \vec { x } = \vec { y } {/tex}, then the angle {tex} \theta {/tex} between {tex} \vec { x } {/tex} and {tex} \vec { z } {/tex} is

A

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

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

C

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

D

None of these

Explanation

Q 23.

Correct4

Incorrect-1

If a non-zero vector {tex} \vec { a } {/tex} is parallel to the line of intersection of the plane {tex} P _ { 1 } {/tex} determined by {tex} \hat { i } + \hat { j } {/tex} and {tex} \hat { i } - 2 \hat { j } {/tex} and plane {tex} P _ { 2 } {/tex} determined by vector {tex} 2 \hat { i } + \hat { j } {/tex} and {tex} 3 \hat { i } + 2 \hat { k } {/tex}, then angle between {tex} \vec { a } {/tex} and vector {tex} \hat { i } - 2 \hat { j } + 2 \hat { k } {/tex} is

A

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

{tex} \frac { \pi } { 2 } {/tex}

C

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

D

None of these

Explanation

Q 24.

Correct4

Incorrect-1

{tex} \vec { a } {/tex} and {tex} \vec { b } {/tex} are non-zero, non-collinear vectors such that {tex} | \vec { a } | = {/tex} {tex} 2 ,\ \vec { a } \cdot \vec { b } = 1 {/tex} and angle between {tex} \vec { a } {/tex} and {tex} \vec { b } {/tex} is {tex} \frac { \pi } { 3 } {/tex}. If {tex} \vec { r } {/tex} is any vector satisfying {tex} \vec { r } \cdot \vec { a } = 2 ,\ \vec { r } \cdot \vec { b } = 8 ,\ ( \vec { r } + 2 \vec { a } - 10 \vec { b } ) ( \vec { a } \times \vec { b } ) = 4 \sqrt { 3 } {/tex} and is equal to {tex} \vec { r } + 2 \vec { a } - 10 \vec { b } = \lambda ( \vec { a } \times \vec { b } ) , {/tex} then {tex} \lambda = {/tex}

A

{tex} \frac { 1 } { 2 } {/tex}

2

C

{tex} \frac { 1 } { 4 } {/tex}

D

4

Explanation

Q 25.

Correct4

Incorrect-1

Let {tex} \hat { a } , \hat b , \hat { c } {/tex} be unit vectors such that {tex} \hat { a } \times \hat { b } = \hat { c } {/tex} and {tex} \hat { a } \cdot \hat { b } = 0 {/tex}. Also, {tex} \vec { x } {/tex} is any vectors such that {tex} [\vec x\ \hat { b }\ \hat { c } ] = 3 , [\vec x\ \hat { c }\ \hat { a } ] = 4 {/tex} and {tex} [ \vec { x }\ \hat { a }\ \hat { b } ] = 2 {/tex}. Then {tex} \vec { x } {/tex} is equal to

A

{tex} 2 \hat { a } + 3 \hat { b } + \hat { c } {/tex}

{tex} 3 \hat { a } + 4 \hat { b } + 2 \hat { c } {/tex}

C

{tex} \hat { a } + 2 \hat { b } + 3 \hat { c } {/tex}

D

None of these

Explanation