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Objective Questions Type I [Only one correct option]

Q1. The points with position vectors 60i + 3j, 40i – 8j, ai – 52j are collinear, if

(1983, 1M) (a) a = – 40

(b) a = 40 (c) a = 20

(d) None of these Q2. A vector has components 2p and 1 with respect to a rectangular Cartesian system.

This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, has components p + 1 and 1, then (1987, 2M)

(a) p = 0

(b) p = 1 or p = –

(c) p = – or p =

(d) p = 1 or p = – 1 Q3. The number of vectors of unit length perpendicular to vectors = (1, 1, 0) and

= (0, 1, 1) is (1987, 2M) (a) one

(b) two (c) three

(d) infinite

Q4. If a, b, c are non-coplanar unit vectors such that

,2

b   ca b  c

then the angle

between and is (1995, 2M)

3π π(a) (b)

4 4

π(c) (d) π

2

Class: XII Subject: Mathematics Topic: Vectors No. of Questions: 30 Duration: 30 Min Maximum Marks: 90

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Q5. are three non-coplanar vectors, then ( ) [( ) ( )]

equals (1995, 2M) (a) 0

( ) , -

( ) , -

( ) , -

Q6. are linearly dependent vectors

and | | √ (1998, 2M) ( ) 1 – 1 ( ) 1 ± 1 ( ) – 1m ß = ± 1 ( ) ± 1 1

Q7. form the sides BC, CA and AB respectively of a triangle ABC, then (2000, 2M)

( )

( )

( )

( )

Q8. are unit coplanar vectors, then the scalar triple product

[ ] (2000, 2M)

(a) 0 (b) 1

( ) √

( ) √

Q9. (1 ) (1 ) [ ]

(2001, 2M) (a) only x (b) only y (c) neither x nor y (d) both x and y

Q10. are two unit vectors such that are perpendicular to each

other, then the angle between and is (2002, 1M) (a) 45° (b) 60°

( ) .

/

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( ) .

/

Q11. is a unit vector, then the maximum value of

the scalar triple product [ ] (2002, 1M)

(a) – 1

( ) √1 √

( ) √

( ) √

Q12. ‘ ’ m m

m m m m (2003, 1M) (a) – 3 (b) 3

( ) 1 √

( ) √

Q13. ( ) 1 (2003, 1M)

( )

( ) ( ) ( )

Q14. The unit vector which is orthogonal to the vector and is coplanar with the

vectors (2004, 1M)

2i - 6 j + k(a)

41

2i - 3j(b)

13

3j - k(c)

10

4i + 3j - 3k(d)

34

Q15. A vector coplanar to and has a

projection along of magnitude

√ , then vector is (2006, 3M)

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(a) 4i - j + 4k

(b) 4i + j - 4k

(c) 2i + j + k

(d) None of these

Q16. m λ λ λ

λ are coplanar, is (2007, 3M) (a) 0

(b) 1 (c) 2 (d) 3

Q17. Two adjacent sides of a parallelogram ABCD are given by 1

11 g g m m ’ ’ m g g g g (2010)

8(a)

9

17(b)

9

1(c)

9

4 5(d)

9

Q18. be three vectors. A vector in the

plane of and , whose projection on

√ g (2011)

(a) 3 3

(b) 3 3

(c)3 3

(d) 3 3

i j k

i j k

i j k

i j k

Objective Questions II [One or more than one correct option]

Q19. The vector(s) which is/are coplanar with vectors are

perpendicular to the vector (2011)

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(a)

(b)

(c)

(d)

j k

i j

i j

j k

Q20. If and are vectors in space given by

√

√ then the value of

( ) [( ) ( – )] (2010)

Fill in the Blanks

Q21. be vectors of length 3, 4, 5 respectively. Let be perpendicular to

Then the length of vector (1981, 2M)

Q22. | 1

1

1

| and the vectors (1 ) (1 ) (1 )

are non-coplanar, then the product abc = . . . . . . . (1985, 2M)

Q23. (1 1 1) ( 1 1) are given vectors, then a vector satisfying the equation

(1985, 91, 2M)

Q24. A unit vector coplanar with and perpendicular to

(1992, 2M)

True/False

Q25. The point with position vectors are collinear for all real values of k. (1984, 1M)

Q26. For any three vectors ,

( ) {( ) ( )} ( ) (1989, 1M)

Analytical/Descriptive

Q27. The position vectors of the points A, B, C and D are

, respectively. If the points A, B, C and D lie on a plane, find the

λ (

)

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Q28. If vectors are coplanar, show that

|

| (1989, 2M)

Q29.

m g

(1990, 3M)

Q30. are four distinct vectors satisfying the conditions

. (2004, 2M)

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Answer Key and Explanations Sol. 1 - (a)

Three points A, B, C are collinear, if 11 and ( ) , then

||

⟹

⟹ a = – 40

Sol. 2 - (b)

Here, ( ) when a system is rotated, the new component of are (p + 1) and 1.

ie, ( 1)

⟹ | | | |

or 4 p² + 1 = (p + 1)² + 1

⟹ 4p² = p² + 2 p + 1

⟹ 3p² – 2p – 1 = 0

⟹ (3 p + 1) (p – 1) = 0

⟹ p = 1, – 1 / 3

Sol. 3 - (b)

A vector perpendicular to and is,

±( )

| |

Therefore, (b) is the answer. Sol. 4 - (a)

Since, ( )

√

⟹ ( ) ( )

√

√

On equating the coefficient of , we get

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1

√

⟹ | || |

√

∴ θ

√ ⟹ θ

Sol. 5 - (d)

( ) [( ) ( )]

( ) [ ]

= { ( ) ( ) ( )}

{ ( ) ( ) ( )}

+ { ( ) ( ) ( )}

[ ] [ ] [ ] , -

Sol. 6 - (d)

Since, are linearly dependent vectors.

⟹ [ ]

⟹ |1 1 1 1

|

Applying C2 → C2 - C1, C3 → 3 – C1

⟹ |1 1 1 1 1

|

⟹ – (ß – 1) = 0 ⟹ ß = 1

Also, | | √ (given)

⟹ 1 ² ² (given c = ) ⟹ 1 ² 1 ⟹ ² 1 ⟹ ± 1

Sol. 7 – (b)

By triangle law,

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Taking cross product by respectively

( )

⟹

⟹ [ ]

Similarly,

∴ Sol. 8 - (a)

If are coplanar vectors, then are also coplanar vectors.

ie, [ ]

Sol. 9 - (c)

[ ] |1 1 1 1 1

|

Applying C3 → 1 + C3

|1 1 1 1

| 1

Therefore, it neither depends on x nor y.

Sol. 10 - (b)

Since, ( ) ( )

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⟹ | | | |

⟹ [ | | | | 1]

⟹ θ

⟹ θ °

Sol. 11 - (c)

Given,

∴ [ ] [( ) ( )]

( ) | || | θ

Which is maximum, if angle between is 0 and maximum value

| | √ .

Sol. 12 - (c)

We know volume of parallelepiped whose edges are [ ]

∴[ ] |1 1 1 1

| 1

Let ( ) 1

⟹ ( ) – 1 ⟹ ( ) F m m m m m m ’ ( ) = 0

⟹ ±

√ ( ) m m m

√ and maximum at

√ .

Sol. 13 - (c)

We know, ( ) ( ) ( )

∴ ( ) ( ) ( ) (√ )

⟹

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⟹

⟹ Sol. 14- (c)

As we know, a vector coplanar to and orthogonal to λ {( ) }.

∴ A vector coplanar to ( ) ( ) and orthogonal to

λ [{( ) ( )} ( )]

λ [( ) ( )]

λ ( 1 )

∴ ( )

√( ) ( )

( )

√

Sol. 15 - (a)

Let vector be coplanar to and .

∴

⟹ ( ) ( )

(1 ) ( ) (1 )

The projection of on =

√ , (given)

⟹

| |

√

⟹ | ( ) ( ) ( )|

√

√

⟹ (2 – t) = ± 1 ⟹ t = 1 or 3

When, t = 1 we have

When, t = 3 we have

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Sol. 16 - (c) Since, given vectors are coplanar

∴ | 1 11 11 1

|

⟹ λ6 – λ² – 2 = 0

⟹ (1 λ²)² (λ² – 2) = 0 ⟹ λ ± √ . Sol. 17 - (b)

1 11

g ‘θ’ is

(θ) |

| | | || |

(1 )( )|

sin θ = √

S ∝ θ ° H ∝ ( °- θ)

This gives sin θ = √

Sol. 18 - (c)

Let

(1 λ) (1 λ) (1 λ)

Projection of

√

⟹

| |

√

⟹ ( ) ( ) ( )

√

√

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⟹ 1 λ – 1 λ – 1 – λ 1 ⟹ λ – 1 = 1 ⟹ λ

∴ Sol. 19 - (a), (d)

Let, and

∴ and , and perpendicular to

⟹ λ ( )

⟹ λ {( ) ( ) }

⟹ λ {(1 1 ) ( ) (1 1) ( )}

⟹ λ { }

⟹ λ * +

⟹ 4λ { }

λ

⟹ (a) is correct.

λ –

⟹ (d) is correct.

Sol. 20 – (5)

From the given information, it is clear that

√

⟹ | | 1 | | 1

Now, ( ) [( ) ( )]

= ( ) , ( ) ( ) -

= [ ] [ ]

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1 1 , - Sol. 21 – (5√ )

Given, | | | | | |

Since, ( ) ( ) ( ) . . . . . . . . (i)

∴ | |

| | | |

| |

( )

= 9 + 16 + 25 + 0

| |

| | √

Sol. 22 – (-1)

Since, | 1

1

1

|

⟹ | 1 1 1

| |

|

⟹ (1 ) |1

1

1

|

⟹ either (1 + abc) = 0 or |1

1

1

|

But (1, a, a²), (1, b, b²), (1, c, c²) are non-coplanar.

⟹ |1

1

1

|

∴ abc = – 1

Sol. 23 (5/3, 2/3, 2/3)

Let

Given,

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Also, given

⟹ ( z – y ) – ( z – x ) + ( y – x ) ⟹ z – y = 0, x – z = 1, y – x = – 1

Also, ⟹ x + y + z = 3 On solving above equations, we get

x =

, y = z =

∴ .

/

Sol. 24

Any vector coplanar with and is given by

( ) ( )

= ( ) ( ) ( )

This vector is perpendicular to (x + y) 1 + (x + 2y)1 + (2x + y) 1 = 0 ⟹ 4x + 4y = 0 ⟹ – x = y ∴ ( )

⟹ | | √ | | Hence, the required unit vector is

±

√ ( )

Sol. 25 (False)

Let position vectors of point and be respectively.

∴ ( ) ( ) ( ) ( )

⟹ – 2 ( 1)

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∴ k + 1 = – 2 ⟹ k = – 3. Hence, it is false statement.

Sol. 26 (True)

( ) {( ) ( )}

( ) ( )

( ) ( ) [ ] [ ]

[ ] ( )

Hence, it is a true statement.

Sol. 27 (-146/17)

Here,

and (λ 1)

We know that, A, B, C, D lie in a plane if are coplanar ie,

[ ] ⟹ | 1 1 1

|

⟹ – 1 ( λ – 21) – 5 (– λ – 4 – 3) – 3 (– 28 – 3) = 0 ⟹ 1 λ 1

⟹ λ

Sol. 28 Given that, are coplanar vectors.

∴

x . . . (i)

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Taking dot with and respectively, we get

x ( ) ( ) ( ) . . . (ii)

and x( ) ( ) ( ) . . . (iii)

Since, Eqs. ( ) ( ) ( ) m g ( ) ( ) ⟹ Non-trivial solutions ∴ ∆

⟹ |

|

Sol. 29 Let

∴

⟹ | 1 1 1

| | 1 1 1

|

⟹ (y – z) – (x – z) + (x – y) = – 10 + 3 + 7 ⟹ y – z = – 10 z – x = 3 x – y = 7

and ⟹ 2x + z = 0, on solving above equations x = – 1, y = – 8 and z = 2

∴

Sol. 30 Given,

and

⟹

⟹ ( ) ( )

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⟹ ( ) ( )

⟹ ( ) ( )

⟹ ( ) ( )

⟹ ( ) || ( )

∴ ( ) ( )

⟹