Untitled Document

Question Bank No: 3

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

a)		$45^{\circ}$
b)		$60^{\circ}$
c)		$\cos^{- 1} (\frac{1}{3})$
d)		$\cos^{- 1} (\frac{2}{7})$

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

a)		only x
b)		only y
c)		both x and y
d)		neither x nor y

3. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are unit vectors, then ${| \vec{a} - \vec{b} |}^{2} +$ ${| \vec{b} - \vec{c} |}^{2} + {| \vec{c} - \vec{a} |}^{2}$ does not exceed

a)		4
b)		9
c)		8
d)		6

4. If the vectors $\vec{a}$ , $\vec{b}$ , $\vec{c}$ form the sides BC, CA, AB respectively of triangle ABC, then

a)		$\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} =$ 0
b)		$\vec{a} \times$ $\vec{b} =$ $\vec{b} \times$ $\vec{c} = \vec{c} \times \vec{a}$
c)		$\vec{a} . \vec{b} = \vec{b} . \vec{c} =$ $\vec{c} . \vec{a}$
d)		$\vec{a} \times \vec{b} +$ $\vec{b} \times \vec{c} +$ $\vec{c} \times \vec{a} =$ 0

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

a)		0
b)		$\frac{π}{4}$
c)		$\frac{π}{3}$
d)		$\frac{π}{2}$

6. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are unit coplanar vectors , then the scalar triple product $[2 \vec{a} - \vec{b}, 2 \vec{b} - \vec{c}, 2 \vec{c} - \vec{a}]$ =

a)		0
b)		1
c)		- $\sqrt{3}$
d)		$\sqrt{3}$

7. Let $\vec{a}$ and $\vec{b}$ be two non collinear unit vectors. If $\vec{u}$ = $\vec{a} - (\vec{a} . \vec{b}) \vec{b}$ and $\vec{v}$ = $\vec{a} \times \vec{b}$ , then $| \vec{v} |$ is

a)		$\| \vec{u} \| + \| \vec{a} + \vec{b} \|$
b)		$\| \vec{u} \|$ + $\| \vec{u} . \vec{a} \|$
c)		$\| \vec{u} \|$ + $\| \vec{u} . \vec{b} \|$
d)		$\| \vec{u} \|$ + $\vec{u} . \| \vec{a} + \vec{b} \|$

8. Let $\vec{a}$ = 2 $\hat{i}$ + $\hat{j}$ + $\hat{k}$ , $\vec{b}$ = $\hat{i}$ +2 $\hat{j}$ - $\hat{k}$ and a unit vector $\vec{c}$ be coplanar. If $\vec{c}$ is perpendicular to $\vec{a}$ , then $\vec{c}$ =

a)		$\frac{- \hat{j} + \hat{k}}{\sqrt{2}}$
b)		$\frac{(- \hat{i} - \hat{j} - \hat{k})}{\sqrt{3}}$
c)		$\frac{\hat{i} - 2 \hat{j}}{\sqrt{5}}$
d)		$\frac{(- \hat{i} - \hat{j} - \hat{k}}{\sqrt{3}}$

9. Let $\vec{a}$ = 2 $\hat{i}$ + $\hat{j}$ -2 $\hat{k}$ and $\vec{b}$ = $\hat{i}$ + $\hat{j}$ . If $\vec{c}$ is a vector such that $\vec{a}$ . $\vec{c}$ = $| \vec{c} |$ , $| \vec{c} - \vec{a} |$ = 2 $\sqrt{2}$ and the angle between ( $\vec{a}$ $\times$ $\vec{b}$ ) and $\vec{c}$ is $30^{0}$ , then $| (\vec{a} \times \vec{b}) \times \vec{c} |$ =

a)		$\frac{2}{3}$
b)		$\frac{3}{2}$
c)		2
d)		3

10. If $\vec{a}$ = $\hat{i}$ + $\hat{j}$ + $\hat{k}$ , $\vec{b}$ = 4 $\hat{i}$ +3 $\hat{j}$ +4 $\hat{k}$ , $\vec{c}$ = $\hat{i}$ + $α \hat{j}$ + $β \hat{k}$ are linearly dependent vectors and $| \vec{c} |$ = $\sqrt{3}$ , then

a)		$α$ = 1, $β$ = -1
b)		$α$ = 1, $β$ = $\pm$ 1
c)		$α$ = -1, $β$ = $\pm$ 1
d)		$α$ = $\pm$ 1, $β$ = 1

11. For three vectors $\vec{u}$ , $\vec{v}$ , $\vec{w}$ which of the following expressions is not equal to any of the remaining?

a)		$\vec{u}$ .( $\vec{v}$ $\times$ $\vec{w}$ )
b)		( $\vec{v}$ $\times$ $\vec{w}$ ). $\vec{u}$
c)		$\vec{v}$ .( $\vec{u}$ $\times$ $\vec{w}$ )
d)		( $\vec{u}$ $\times$ $\vec{v}$ ). $\vec{w}$

12. Which of the following expressions are meaningful?

a)		$\vec{u}$ .( $\vec{v}$ $\times \vec{w}$ )
b)		( $\vec{u}$ . $\vec{v}$ ). $\vec{w}$
c)		( $\vec{u}$ $\times \vec{v}$ ) $\vec{w}$
d)		$\vec{u}$ $\times$ ( $\vec{v}$ . $\vec{w}$ )

13. Let $\vec{p}$ , $\vec{q}$ , $\vec{r}$ be three mutually perpendicular vectors of the same magnitude. If a vector $\vec{x}$ satisfies the equation $\vec{p}$ $\times ((\vec{x}$ - $\vec{q}$ ) $\times$ $\vec{p}$ ) + $\vec{q}$ $\times$ (( $\vec{x}$ - $\vec{r}$ ) $\times \vec{q}$ ) + $\vec{r}$ $\times$ (( $\vec{x}$ - $\vec{p}$ ) $\times$ $\vec{r}$ ) = 0 , then $\vec{x}$ is

a)		$\frac{1}{2}$ ( $\vec{p}$ + $\vec{q}$ -2 $\vec{r}$ )
b)		$\frac{1}{2}$ ( $\vec{p}$ + $\vec{q}$ + $\vec{r}$ )
c)		$\frac{1}{3}$ ( $\vec{p}$ + $\vec{q}$ + $\vec{r}$ )
d)		$\frac{1}{3}$ (2 $\vec{p}$ + $\vec{q}$ - $\vec{r}$ )

14. Let $\vec{a}$ , $\vec{b}$ , $\vec{c}$ be three vectors having magnitudes 1, 1, 2 respectively. If $\vec{a}$ $\times$ ( $\vec{a}$ $\times \vec{c}$ ) + $\vec{b}$ =0, then the acute angle between $\vec{a}$ and $\vec{c}$ is

a)		$\frac{π}{6}$
b)		$\frac{π}{4}$
c)		$\frac{π}{3}$
d)		none of these

15. Let $\vec{OA}$ = $\vec{a}$ , $\vec{OB}$ = 10 $\vec{a}$ + 2 $\vec{b}$ , $\vec{OC}$ = $\vec{b}$ where O< A C are noncollinear points. Let p denote the area of the quadrilateral OABC and let q denote the area of the parallelogram with OA and OC as adjacent sides. If p = kq, then k =

a)		2
b)		3
c)		6
d)		4

16. A non zero vector $\vec{a}$ is parallel to the line of intersection of the plane determined by the vectors $\hat{i}$ , $\hat{i}$ + $\hat{j}$ and the plane determined by the vectors $\hat{i}$ - $\hat{j}$ , $\hat{i}$ + $\hat{k}$ . Then the angle between $\vec{a}$ and the vector $\hat{i}$ - 2 $\hat{j}$ +2 $\hat{k}$ is

a)		$\frac{π}{6}$
b)		$\frac{π}{4}$
c)		$\frac{π}{3}$
d)		$\frac{π}{2}$

17. Let $\vec{a}$ , $\vec{b}$ , $\vec{c}$ be three non coplanar vectors such that $\vec{a}$ $\times$ ( $\vec{b}$ $\times \vec{c}$ ) = $\frac{\vec{b}}{2}$ , then the angle between $\vec{a}$ and $\vec{b}$ is

a)		$\frac{3 π}{4}$
b)		$\frac{π}{4}$
c)		$\frac{π}{2}$
d)		$π$

18. Let $\vec{u}$ , $\vec{v}$ , $\vec{w}$ be such that $\vec{u}$ + $\vec{v}$ + $\vec{w}$ = 0, $| \vec{u} |$ = 3, $| \vec{v} |$ = 4, $| \vec{w} |$ =5. The value of $\vec{u}$ . $\vec{v}$ + $\vec{v}$ . $\vec{w}$ + $\vec{w}$ . $\vec{u}$ =

a)		47
b)		-25
c)		0
d)		25

19. Let a, b, c be distinct non negative numbers. If teh vectors ai + aj + ck, i + k, ci + cj +bk lie in a plane, then c is

a)		A.M of a, b
b)		G.M of a, b
c)		H.M of a, b
d)		zero

20. Let $\vec{a}$ =2i - j + k , and $\vec{b}$ =+ 2j -k and $\vec{c}$ =i+ j -2k be three vectors. A vector in the plane of $\vec{b}$ and $\vec{c}$ whose projection on $\vec{a}$ is of magnitude $\sqrt{\frac{2}{3}}$ is

a)		2i +3j -k
b)		2i +3j +3k
c)		-2i - j +5k
d)		2i + j +5k

21. The scalar $\vec{A}$ .( $\vec{B}$ + $\vec{C)}$ $\times$ ( $\vec{A}$ + $\vec{B}$ + $\vec{C}$ ) =

a)		0
b)		$[\vec{A} \vec{B} \vec{C}]$ + $[\vec{B} \vec{C} \vec{A}]$
c)		$[\vec{A} \vec{B} \vec{C}]$
d)		none of these

22. For nonzero vectors $\vec{a}$ , $\vec{b}$ , $\vec{c}$ , $| \vec{a} \times \vec{b} . \vec{c} |$ = $| \vec{a} |$ $| \vec{b} |$ $| \vec{c} |$ holds if and only if

a)		$\vec{a}$ . $\vec{b}$ = $\vec{b}$ . $\vec{c}$ = 0
b)		$\vec{b}$ . $\vec{c}$ = $\vec{c}$ . $\vec{a}$ = 0
c)		$\vec{c}$ . $\vec{a}$ = $\vec{a}$ . $\vec{b}$ =0
d)		$\vec{a}$ . $\vec{b}$ = $\vec{b}$ . $\vec{c}$ = $\vec{c}$ . $\vec{a}$ = 0

23. The points with position vectors 60 $\hat{i}$ +3 $\hat{j}$ , 40 $\hat{i}$ -8 $\hat{j}$ , a $\hat{i}$ -52 $\hat{j}$ are collinear if a is

a)		40
b)		-40
c)		20
d)		none of these

24. The volume of the parallelopiped whose sides are given
by $\vec{OA}$ = 2 $\hat{i}$ -3 $\hat{j}$ ,
$\vec{OB}$ = $\hat{i}$ + $\hat{j}$ - $\hat{k}$ , $\vec{OC}$ =3 $\hat{i}$ - $\hat{k}$ is

a)		$\frac{4}{13}$
b)		4
c)		$\frac{2}{7}$
d)		none of these

25. Let $\vec{a}$ , $\vec{b}$ , $\vec{c}$ be three non coplanar vectors and $\vec{p}$ , $\vec{q}$ , $\vec{r}$ are vectors defined
by $\vec{p}$ = $\frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}$ , $\vec{q}$ = $\frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]}$ , $\vec{r}$ = $\frac{\vec{a} \times \vec{b}}{[\vec{a}, \vec{b}, \vec{c}]}$ . Then the expression $(\vec{a} + \vec{b}) .$ $\vec{p}$ + $(\vec{b} + \vec{c})$ . $\vec{q} +$ $(\vec{c} + \vec{a})$ . $\vec{r}$ =

a)		0
b)		1
c)		2
d)		3

26. The number of unit vectors perpendicular to the vectors $\vec{a}$ = $\hat{i}$ + $\hat{j}$ and $\vec{b}$ = $\hat{j}$ + $\hat{k}$ is

a)		1
b)		2
c)		3
d)		indifinite

27. A vector $\vec{a}$ has components 2p and 1 with respect to a rectangular coordinate system. This system is rotated through a certain angle about the origin in the counterclockwise sense. If with respect to the new system, $\vec{a}$ has components p+1 and 1, then

a)		p = 0
b)		p = 1 or p = - $\frac{1}{3}$
c)		p = -1 or p = $\frac{1}{3}$
d)		p = 1 or p = -1

28. Which of the following is not an example of a vector?

a)		Displacement
b)		Momentum
c)		Mass
d)		Angular velocity

29. The magnitude of a vector can be

a)		negative
b)		zero
c)		positive
d)		none of these

30. Which of the following is not an example of a scalar?

a)		work
b)		force
c)		power
d)		time

31. If cross product of two non-zero vectors is zero, then the vectors are

a)		collinear
b)		co-directional
c)		co-initial
d)		co-terminus