Question Bank for JEE Main & Advanced Mathematics Vector Algebra Self Evaluation Test - Vector Algebra

1)

ABCD a parallelogram,\[{{A}_{1}}\] and \[{{B}_{1}}\] are the midpoints of sides BC and CD, respectively. If \[\overrightarrow{A{{A}_{1}}}+\,\overrightarrow{A{{B}_{1}}}=\lambda \overrightarrow{AC}\], then \[\lambda \], is equal to

A)

\[\frac{1}{2}\]

B)

1

C)

\[\frac{3}{2}\]

D)

2

View Solution

2)

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity \[\vec{u}\] and the other from rest with uniform acceleration\[\vec{f}\]. Let \[\alpha \] be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time

A)

\[\frac{u\,\,\cos \,\,\alpha }{f}\]

B)

\[\frac{u\,\,\sin \,\,\alpha }{f}\]

C)

\[\frac{f\,\,\cos \,\,\alpha }{u}\]

D)

\[u\,\,\sin \,\,\alpha \]

View Solution

3)

A force \[\vec{F}=3\hat{i}+2\hat{j}-4\hat{k}\] is applied at the point (1, -1, 2). What is the moment of the force about the point (2, -1, 3)?

A)

\[\hat{i}+4\hat{j}+4\hat{k}\]

B)

\[2\hat{i}+\hat{j}+2\hat{k}\]

C)

\[2\hat{i}-7\hat{j}-2\hat{k}\]

D)

\[2\hat{i}+4\hat{j}-\hat{k}\]

View Solution

4)

What is a vector of unit length orthogonal to both the vectors \[\hat{i}+\hat{j}+\hat{k}\] and\[2\hat{i}+3\hat{j}-\hat{k}\]?

A)

\[\frac{-4\hat{i}+3\hat{j}-\hat{k}}{\sqrt{26}}\]

B)

\[\frac{-4\hat{i}+3\hat{j}+\hat{k}}{\sqrt{26}}\]

C)

\[\frac{-3\hat{i}+2\hat{j}-\hat{k}}{\sqrt{14}}\]

D)

\[\frac{-3\hat{i}+2\hat{j}+\hat{k}}{\sqrt{14}}\]

View Solution

5)

If the vectors \[\alpha \hat{i}+\alpha \hat{j}+\gamma \hat{k},\text{ }\hat{i}+\hat{k}\] and \[\gamma \hat{i}+\gamma \hat{j}+\beta \hat{k}\] lie on a plane, where \[\alpha ,\beta \] and \[\gamma \] are distinct non-negative numbers, then \[\gamma \] is

A)

Arithmetic mean of \[\alpha \] and \[\beta \]

B)

Geometric mean of \[\alpha \] and \[\beta \]

C)

Harmonic mean of \[\alpha \] and \[\beta \]

D)

None of the above

View Solution

6)

The adjacent sides AB and AC of a triangle ABC are represented by the vectors \[-2\hat{i}+3\hat{j}+2\hat{k}\] and \[-4\hat{i}+5\hat{j}+2\hat{k}\] respectively. The area of the triangle ABC is

A)

6 square units

B)

5 square units

C)

4 square units

D)

3 square units

View Solution

7)

Resolved part of vector \[\vec{a}\] along vector \[\vec{b}\] is \[{{\vec{a}}_{1}}\] and that perpendicular to \[\vec{b}\] is \[{{\vec{a}}_{2}}\] then \[{{\vec{a}}_{1}}\times {{\vec{a}}_{2}}\] is equal to

A)

\[\frac{(\vec{a}\times \vec{b})\cdot \vec{b}}{{{\left| {\vec{b}} \right|}^{2}}}\]

B)

\[\frac{(\vec{a}\cdot \vec{b})\vec{a}}{{{\left| {\vec{a}} \right|}^{2}}}\]

C)

\[\frac{(\vec{a}\cdot \vec{b})(\vec{b}\times \vec{a})}{{{\left| {\vec{b}} \right|}^{2}}}\]

D)

\[\frac{(\vec{a}\cdot \vec{b})(\vec{b}\times \vec{a})}{\left| \vec{b}\times \vec{a} \right|}\]

View Solution

8)

The components of a vector \[\vec{a}\] along and perpendicular to a non-zero vector \[\vec{b}\] are

A)

\[\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{b}} \right|}^{2}}} \right)\vec{b}\And \vec{a}-\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{b}} \right|}^{2}}} \right)\vec{b}\]

B)

\[\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{a}} \right|}^{2}}} \right)\vec{b}\And \vec{a}+\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{a}} \right|}^{2}}} \right)\vec{b}\]

C)

\[\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{a}} \right|}^{2}}} \right)\vec{a}-\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{b}} \right|}^{2}}} \right)\vec{a}\]

D)

None of these

View Solution

9)

If ABCDEF is a regular hexagon and\[\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}=k\overrightarrow{AD}\], then find the value of k.

A)

2

B)

3

C)

4

D)

5

View Solution

10)

In a right angle \[\Delta ABC,\text{ }\angle A=90{}^\circ \] and sides a, b, c are respectively, 5 cm, 4 cm and 3 cm. If a force \[\vec{F}\] has moments 0, 9 and 16 in N cm. units respectively about vertices A, B and C, then magnitude of \[\vec{F}\] is

A)

9

B)

4

C)

5

D)

3

View Solution

11)

Let \[\alpha ,\beta ,\gamma \] be distinct real numbers. The points with position vectors \[\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k},\beta \hat{i}+\gamma \hat{j}+\alpha \hat{k}\] and \[\gamma \hat{i}+\alpha \hat{j}+\beta \hat{k}\]

A)

Are collinear

B)

Form an equilateral triangle

C)

Form a scalene triangle

D)

Form a right-angled triangle

View Solution

12)

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

A)

\[\pm \frac{\hat{i}+\hat{j}-2\hat{k}}{\sqrt{6}}\]

B)

\[\pm \frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}\]

C)

\[\pm \frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}\]

D)

\[\pm \,\hat{k}\]

View Solution

13)

If \[\overrightarrow{OA}=\vec{a};\overrightarrow{OB}=\vec{b};\overrightarrow{OC}=2\vec{a}+3\vec{b}\,;\] \[\overrightarrow{OD}=\vec{a}-2\vec{b}\], the length of \[\overrightarrow{OA}\] is three times the length of \[\overrightarrow{OB}\] and \[\overrightarrow{OA}\] is perpendicular to \[\overrightarrow{DB}\] then \[\left( \overrightarrow{BD}\times \overrightarrow{AC} \right).\left( \overrightarrow{OD}\times \overrightarrow{OC} \right)\] is

A)

\[7{{\left| \vec{a}\times \vec{b} \right|}^{2}}\]

B)

\[42|\vec{a}\times \vec{b}{{|}^{2}}\]

C)

0

D)

None of these

View Solution

14)

If \[\vec{r}\cdot \vec{a}=\vec{r}\cdot b=\vec{r}\cdot \vec{c}=\frac{1}{2}\] for some non-zero vector \[\vec{r}\], then the area of the triangle whose vertices are \[A(\vec{a}),B(\vec{b})\] and \[C\left( {\vec{c}} \right)\] is (\[\vec{a},\text{ }\vec{b},\text{ }\vec{c}\] are non-coplanar)

A)

\[\left| [\vec{a}\,\vec{b}\,\vec{c}] \right|\]

B)

\[\left| {\vec{r}} \right|\]

C)

\[\left| [\vec{a}\,\vec{b}\,\vec{c}]\vec{r} \right|\]

D)

None of these

View Solution

15)

If the \[{{p}^{th}},{{q}^{th}}\] and \[{{r}^{th}}\] terms of a G.P. are positive numbers a, b and c respectively, then find the angle between the vectors \[log\,{{a}^{2}}\hat{i}+log\,{{b}^{2}}\hat{j}+log\,{{c}^{2}}\hat{k}\] and \[(q-r)\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}\]

A)

\[\frac{\pi }{6}\]

B)

\[\frac{\pi }{4}\]

C)

\[\frac{\pi }{3}\]

D)

\[\frac{\pi }{2}\]

View Solution

16)

If \[\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=4\hat{i}+3\hat{j}+4\hat{k}\] and\[\vec{c}=\hat{i}+\alpha \hat{j}+\beta \hat{k}\] are coplanar and \[\left| {\vec{c}} \right|=\sqrt{3}\], then

A)

\[\alpha =\sqrt{2},\beta =1\]

B)

\[\alpha =1,\beta =\pm 1\]

C)

\[\alpha =\pm 1,\beta =1\]

D)

\[\alpha =\pm 1,\beta =-1\]

View Solution

17)

Consider the parallelepiped with side \[\vec{a}=3\hat{i}+2\hat{j}+\hat{k},\text{ }\vec{b}=\hat{i}+\hat{j}+2\hat{k}\] and \[\vec{c}=\hat{i}+3\hat{j}+3\hat{k}\] then the angle between \[\vec{a}\]and the plane containing the face determined by \[\vec{b}\] and \[\vec{c}\] is

A)

\[\sin {{\,}^{-1}}\frac{1}{3}\]

B)

\[\cos {{\,}^{-1}}\frac{1}{14}\]

C)

\[sin{{\,}^{-1}}\frac{9}{14}\]

D)

\[sin{{\,}^{-1}}\frac{2}{3}\]

View Solution

18)

If \[(\vec{a}\times \vec{b})\times \vec{c}=\vec{a}\times (\vec{b}\times \vec{c})\] where \[\vec{a},\vec{b}\] and \[\vec{c}\] are any three vectors such that \[\vec{a}.\vec{b}\ne 0,\vec{b}.\vec{c}\ne 0\] then \[\vec{a}\] and \[\vec{c}\] are

A)

Inclined at an angle of \[\frac{\pi }{3}\] between them

B)

Inclined at an angle of \[\frac{\pi }{6}\] between them

C)

Perpendicular

D)

Parallel

View Solution

19)

What is the area of the parallelogram having diagonals \[3\hat{i}+\hat{j}-2\hat{k}\] and \[\hat{i}-3\hat{j}+4\hat{k}\]?

A)

\[5\sqrt{5}\] square units

B)

\[4\sqrt{5}\] square units

C)

\[5\sqrt{3}\] square units

D)

\[15\sqrt{2}\] square units

View Solution

20)

The vectors \[\vec{a},\vec{b},\vec{c}\] and \[\vec{d}\] are such that \[\vec{a}\times \vec{b}=\vec{c}\times d\] and\[\vec{a}\times \vec{c}=\vec{b}\times \vec{d}\]. Which of the following is/ are correct?

1. \[(\vec{a}-\vec{d})\times (\vec{b}-\vec{c})=\vec{0}\]

2. \[(\vec{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \vec{d})=\vec{0}\]

Select the correct answer using the code given below:

A)

1 only

B)

2 only

C)

Both 1 and 2

D)

Neither 1 nor 2

View Solution

21)

A force \[\vec{F}=3\hat{i}+4\hat{j}-3\hat{k}\] is applied at the point P, whose position vector is \[\overrightarrow{r}=\widehat{2i}-2\hat{j}-3\hat{k}\]. What is the magnitude of the moment of the force about the origin?

A)

23 units

B)

19 units

C)

18 units

D)

21 units

View Solution

22)

What is the interior acute angle of the parallelogram whose sides are represented by the vectors \[\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\hat{k}\] and \[\frac{1}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j}+\hat{k}\]?

A)

\[60{}^\circ \]

B)

\[45{}^\circ \]

C)

\[30{}^\circ \]

D)

\[15{}^\circ \]

View Solution

23)

Let \[{{x}^{2}}+3{{y}^{2}}=3\] be the equation of an ellipse in the x-y plane. A and B are two points whose position vectors are \[-\sqrt{3}\hat{i}\] and\[-\sqrt{3}\hat{i}+2\hat{k}\]. Then the position vector of a point P on the ellipse such that \[\angle APB=\pi /4\] is

A)

\[\pm \hat{j}\]

B)

\[\pm (\hat{i}+\hat{j})\]

C)

\[\pm \,\hat{i}\]

D)

None of these

View Solution

24)

A)

\[\frac{1}{2}(\overset{\to }{\mathop{p}}\,+\overset{\to }{\mathop{q}}\,-2\overset{\to }{\mathop{r}}\,)\]

B)

\[\frac{1}{2}(\overset{\to }{\mathop{p}}\,+\overset{\to }{\mathop{q}}\,+\overset{\to }{\mathop{r}}\,)\]

C)

\[\frac{1}{3}(\overset{\to }{\mathop{p}}\,+\overset{\to }{\mathop{q}}\,+\overset{\to }{\mathop{r}}\,)\]

D)

\[\frac{1}{3}(2\overset{\to }{\mathop{p}}\,+\overset{\to }{\mathop{q}}\,-\overset{\to }{\mathop{r}}\,)\]

View Solution

25)

If \[\vec{p}\] and \[\vec{q}\] are non-collinear unit vectors and \[\left| \vec{p}+\vec{q} \right|=\sqrt{3}\], then \[(2\vec{p}-3\vec{q})\cdot (3\vec{p}+\vec{q})\] is equal to

A)

0

B)

\[\frac{1}{3}\]

C)

\[-\frac{1}{3}\]

D)

\[-\frac{1}{2}\]

View Solution

26)

ABCDEF is a regular hexagon where centre O is the origin. If the position vectors of A and B are \[\hat{i}-\hat{j}+2\hat{k}\] and \[2\hat{i}+\hat{j}-\hat{k}\] respectively then \[\overrightarrow{BC}\] is equal to

A)

\[\hat{i}+\hat{j}-2\hat{k}\]

B)

\[-\hat{i}+\hat{j}-2\hat{k}\]

C)

\[3\hat{i}+3\hat{j}-4\hat{k}\]

D)

None of these

View Solution

27)

A)

0

B)

2

C)

1

D)

None of these

View Solution

28)

For any vector \[\overset{\to }{\mathop{p}}\,\], the value of\[\frac{3}{2}\left\{ |\overset{\to }{\mathop{p}}\,\times \hat{i}{{|}^{2}}+|\overset{\to }{\mathop{p}}\,\times \hat{j}{{|}^{2}}+|\overset{\to }{\mathop{p}}\,\times \hat{k}{{|}^{2}} \right\}\] is

A)

\[\overset{\to 2}{\mathop{p}}\,\]

B)

\[2\overset{\to 2}{\mathop{p}}\,\]

C)

\[3\overset{\to 2}{\mathop{p}}\,\]

D)

\[4\overset{\to 2}{\mathop{p}}\,\]

View Solution

29)

\[\hat{i}\times (\vec{A}\times \hat{i})+\hat{j}\times (\vec{A}\times \hat{j})+\hat{k}\times (\vec{A}\times \hat{k})\] is equal to

A)

\[\overset{\to }{\mathop{A}}\,\]

B)

\[2\overset{\to }{\mathop{A}}\,\]

C)

\[3\overset{\to }{\mathop{A}}\,\]

D)

0

View Solution

30)

The vectors \[\overrightarrow{AB}=3\hat{i}+5\hat{j}+4\hat{k}\] and \[\overrightarrow{AC}=5\hat{i}-5\hat{j}+2\hat{k}\] are the sides of a triangle ABC. The length of the median through A is:

A)

\[\sqrt{13}\]units

B)

\[2\sqrt{5}\] units

C)

5 units

D)

10 units

View Solution

31)

A force \[F=2i+j-k\] acts at a point A, whose position vector is \[2i-j\]. The moment of F about the origin is

A)

\[i+2j-4k\]

B)

\[i-2j-4k\]

C)

\[i+2j+4k\]

D)

\[i-2j+4k\]

View Solution

32)

If \[\vec{a},\text{ }\vec{b}\] and \[\vec{c}\] are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?

A)

\[\vec{a}+\vec{b}+\vec{c}=\vec{0}\]

B)

\[\vec{a}+\vec{b}+\vec{c}=\,\,unit\,\,vector\]

C)

\[\vec{a}+\vec{b}=\vec{c}\]

D)

\[\vec{a}=\vec{b}+\vec{c}\]

View Solution

33)

Which one of the following is the unit vector perpendicular to both \[\vec{a}=-\hat{i}+\hat{j}+\hat{k}\] and\[\vec{b}=\hat{i}-\hat{j}+\hat{k}\]?

A)

\[\frac{\hat{i}+\hat{j}}{\sqrt{2}}\]

B)

\[\hat{k}\]

C)

\[\frac{\hat{j}+\hat{k}}{\sqrt{2}}\]

D)

\[\frac{\hat{i}-\hat{j}}{\sqrt{2}}\]

View Solution

34)

If \[\vec{a}=2\hat{i}+2\hat{j}+3\hat{k},\vec{b}=-\hat{i}+2\hat{j}+\hat{k}\] and \[\overrightarrow{c}=3\hat{i}+\hat{j}\]are three vectors such that \[\vec{a}+t\vec{b}\] is perpendicular to \[\vec{c}\], then what is t equal to?

A)

8

B)

6

C)

4

D)

2

View Solution

35)

A)

9

B)

6

C)

5

D)

4

View Solution

36)

A vector \[\overset{\to }{\mathop{a}}\,=(x,y,z)\] of length \[2\sqrt{3}\] which makes equal angles with the vectors \[\overset{\to }{\mathop{b}}\,=(y,\,\,-2z,\,\,3x)\] and \[\overset{\to }{\mathop{c}}\,=(2z,\,\,3x,\,\,-y)\] is perpendicular to \[\overset{\to }{\mathop{d}}\,=(1,-1,2)\] and makes an obtuse angle with y-axis is

A)

(- 2, 2, 2)

B)

\[(1,\,\,1,\,\,\sqrt{10})\]

C)

(2, - 2, - 2)

D)

None of these

View Solution

37)

Force \[\hat{i}+2\hat{j}-3\hat{k},2\hat{i}+3\hat{j}+4\hat{k}\] and \[-\hat{i}-\hat{j}+\hat{k}\] are acting at the point P (0, 1, 2). The moment of these forces about the point A (1, - 2, 0) is

A)

\[2\hat{i}-6\hat{j}+10\hat{k}\]

B)

\[-2\hat{i}+6\hat{j}-10\hat{k}\]

C)

\[2\hat{i}+6\hat{j}-10\hat{k}\]

D)

None of these

View Solution

38)

If vector \[a=2i-3j+6k\] and vector \[b=-2i+2j-k\],\[\text{then}\,\,\,\frac{\text{Projection}\,\,\text{of}\,\,\text{vector}\,\,\text{a}\,\,\text{on}\,\,\text{vector}\,\,\text{b}}{\text{Projection}\,\,\text{of}\,\,\text{vector}\,\,\text{b}\,\,\text{on}\,\,\text{vector}\,\,\text{a}}\text{=}\]

A)

\[\frac{3}{7}\]

B)

\[\frac{7}{3}\]

C)

3

D)

7

View Solution

39)

A vector of magnitude 3, bisecting the angle between the vectors \[\overset{\to }{\mathop{a}}\,=2\hat{i}+\hat{j}-\hat{k}\] and \[\overset{\to }{\mathop{b}}\,=\hat{i}-2\hat{j}+\hat{k}\] and making an obtuse angle with \[\overset{\to }{\mathop{b}}\,\] is

A)

\[\frac{3\hat{i}-\hat{j}}{\sqrt{6}}\]

B)

\[\frac{\hat{i}+3\hat{j}-2\hat{k}}{\sqrt{14}}\]

C)

\[\frac{3(\hat{i}+3\hat{j}-2\hat{k})}{\sqrt{14}}\]

D)

\[\frac{3\hat{i}-\hat{j}}{\sqrt{10}}\]

View Solution

40)

A)

0

B)

1

C)

2

D)

3

View Solution

41)

If \[\vec{u},\vec{v},\vec{w}\] are non-coplanar vectors and p, q are real numbers, then the equality \[[3\vec{u}\,\,p\vec{v}\,\,p\vec{w}]-[p\vec{v}\,\vec{\omega }\,q\vec{u}]-[2\vec{\omega }\,q\vec{v}\,q\vec{u}]=0\] holds for:

A)

Exactly two values of (p, q)

B)

More than two but not all values of (p, q)

C)

All values of (p, q)

D)

Exactly one value of (p, q)

View Solution

42)

The vector \[\overset{\to }{\mathop{c}}\,\] directed along the bisectors of the angle between the vectors \[\overset{\to }{\mathop{a}}\,=7\hat{i}-4\hat{j}-4\hat{k},\] \[\overset{\to }{\mathop{b}}\,=-2\hat{i}-\hat{j}+2\hat{k},\] and \[|\overset{\to }{\mathop{c}}\,|=3\sqrt{6}\] is given by

A)

\[\hat{i}-7\hat{j}+2\hat{k}\]

B)

\[\hat{i}+7\hat{j}-2\hat{k}\]

C)

\[\hat{i}+7\hat{j}+2\hat{k}\]

D)

\[\hat{i}+7\hat{j}+3\hat{k}\]

View Solution

43)

Let \[\vec{r}=(\vec{a}\times \vec{b})\sin \,x+(\vec{b}\times \vec{c})\cos \,y+2(\vec{c}\times \vec{a})\] where \[\vec{a},\vec{b},\vec{c}\]three non-coplanar vectors are. If \[\vec{r}\] is perpendicular to \[\vec{a}+\vec{b}+\vec{c},\] the minimum value of \[{{x}^{2}}+{{y}^{2}}\] is

A)

\[{{\pi }^{2}}\]

B)

\[\frac{{{\pi }^{2}}}{4}\]

C)

\[\frac{5{{\pi }^{2}}}{4}\]

D)

None of these

View Solution

44)

Let \[\vec{a},\vec{b}\] and \[\vec{c}\] be three non-zero vectors such that no two of these are collinear. If the vector \[\vec{a}+2\vec{b}\] is collinear with \[\vec{c}\] and \[\vec{b}+3\vec{c}\] is collinear with \[\vec{a}\] (\[\lambda \] being some non-zero scalar) then \[\vec{a}+2\vec{b}+6\vec{c}\] equals

A)

0

B)

\[\lambda \vec{b}\]

C)

\[\lambda \vec{c}\]

D)

\[\lambda \vec{a}\]

View Solution

45)

The value of 'x' for which the angle between the vectors \[\overset{\to }{\mathop{a}}\,=2{{x}^{2}}\hat{i}+4x\hat{j}+\hat{k}\] and \[\overset{\to }{\mathop{b}}\,=7\hat{i}-2\hat{j}+x\hat{k}\] is obtuse are

A)

\[\operatorname{x} < 0\]

B)

\[x>\frac{1}{2}\]

C)

\[0<x<\frac{1}{2}\]

D)

\[x\in R\]

View Solution

46)

If \[\vec{a},\text{ }\vec{b},\text{ }\vec{c},\text{ }\vec{d}\] are the position vectors of points A, B, C and D respectively such that \[(\vec{a}-\vec{d}).(\vec{b}-\vec{c})=(\vec{b}-\vec{d}).(\vec{c}-\vec{a})=0\] then D is the

A)

Centroid of \[\Delta \text{ }ABC\]

B)

Circumcentre of \[\Delta \,ABC\]

C)

Orthocentre of \[\Delta \,ABC\]

D)

None of these

View Solution

47)

A)

0

B)

-3

C)

3

D)

-9

View Solution

48)

Let \[{{\vec{r}}_{1}},{{\vec{r}}_{2}},{{\vec{r}}_{3}},.....{{\vec{r}}_{n}},\] be the position vectors of points \[{{P}_{1}},{{P}_{2}},{{P}_{3}},....,{{P}_{n}}\] relative to the origin O. If the vector equation \[{{a}_{1}}{{\vec{r}}_{1}}+{{a}_{2}}{{\vec{r}}_{2}}+....+{{a}_{n}}{{\vec{r}}_{n}}=0\] holds, then a similar equation will also hold w.r.t. to any other origin provided

A)

\[{{a}_{1}}+{{a}_{2}}+....+{{a}_{n}}=n\]

B)

\[{{a}_{1}}+{{a}_{2}}+....+{{a}_{n}}=1\]

C)

\[{{a}_{1}}+{{a}_{2}}+....+{{a}_{n}}=0\]

D)

\[{{a}_{1}}={{a}_{2}}={{a}_{3}}=....={{a}_{n}}=0\]

View Solution

49)

If the middle points of sides BC, CA & AB of triangle ABC are respectively D, E, F then position vector of centre of triangle DEF, when position vector of A, B, C are respectively \[\hat{i}+\hat{j},\hat{j}+\hat{k},\hat{k}+\hat{i}\]is

A)

\[\frac{1}{3}(\hat{i}+\hat{j}+\hat{k})\]

B)

\[(\hat{i}+\hat{j}+\hat{k})\]

C)

\[2(\hat{i}+\hat{j}+\hat{k})\]

D)

\[\frac{2}{3}(\hat{i}+\hat{j}+\hat{k})\]

View Solution

50)

If \[\overrightarrow{OA}=\overrightarrow{a};\,\overrightarrow{OB}=\overrightarrow{b};\,\overrightarrow{OC}=2\overrightarrow{a}+3\overrightarrow{b};\] \[\overrightarrow{OD}=\overset{\to }{\mathop{a}}\,-2\text{ }\overset{\to }{\mathop{b}}\,,\]the length of \[\overrightarrow{OA}\] is three times the length of \[\overrightarrow{OB}\] and \[\overrightarrow{OA}\] is perpendicular to \[\overrightarrow{DB}\] then \[(\overrightarrow{BD}\times \overrightarrow{AC}).(\overrightarrow{OD}\times \overrightarrow{OC})\] is

A)

\[7|\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,{{|}^{2}}\]

B)

\[42|\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,{{|}^{2}}\]

C)

0

D)

None of these

View Solution

51)

Let \[\overrightarrow{A}={{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k},\text{ }\overrightarrow{B}={{b}_{1}}\hat{i}+{{b}_{2}}\hat{j}+{{b}_{3}}\hat{k}\] and \[\overrightarrow{C}={{c}_{1}}\hat{i}+{{c}_{2}}\hat{j}+{{c}_{3}}\hat{k}\] be three non-zero vectors such that \[\overrightarrow{C}\] is a unit vector perpendicular to both the vectors \[\overrightarrow{A}\] and \[\overrightarrow{B}\] .If the angle between \[\overrightarrow{A}\] and \[\overrightarrow{B}\] is \[\frac{\pi }{6}\], then.

A)

0

B)

1

C)

\[\frac{1}{4}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})(b_{1}^{2}+b_{3}^{2})\]

D)

\[\frac{3}{4}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2})(c_{1}^{2}+c_{3}^{2})\]

View Solution

52)

If a, b, c are the \[{{p}^{th}},\text{ }{{q}^{th}}.\text{ }{{\text{r}}^{th}}\] terms of an HP and \[\vec{u}=(q-r)\vec{i}+(r-p)\vec{j}+(p-q)\vec{k},\vec{v}=\frac{{\vec{i}}}{a}+\frac{{\vec{j}}}{b}+\frac{{\vec{k}}}{c}\] then

A)

\[\vec{u},\vec{v}\] are parallel vectors

B)

\[\vec{u},\vec{v}\] are orthogonal vectors

C)

\[\vec{u}.\vec{v}=1\]

D)

\[\vec{u}\times \vec{v}=\vec{i}+\vec{j}+\vec{k}\]

View Solution

53)

A)

\[x=2,y=1\]

B)

\[x=1,y=2\]

C)

\[x=-2,y=1\]

D)

\[x=-2,y=-1\]

View Solution

54)

The vectors \[\hat{i}-2x\hat{j}-3y\hat{k}\] and \[\hat{i}+3x\hat{j}+2y\hat{k}\] are orthogonal to each other. Then the locus of the point (x, y) is

A)

Hyperbola

B)

Ellipse

C)

Parabola

D)

Circle

View Solution

55)

If \[\overset{\to }{\mathop{a}}\,=2\hat{i}-2\hat{j}+\hat{k}\] and \[\overset{\to }{\mathop{c}}\,=-\hat{i}+2\hat{k}\] then \[|\overset{\to }{\mathop{c}}\,|.\overset{\to }{\mathop{a}}\,\] is equal to:

A)

\[2\sqrt{5}\hat{i}+2\sqrt{5}\hat{j}+\sqrt{5}\hat{k}\]

B)

\[2\sqrt{5}\hat{i}-2\sqrt{5}\hat{j}+\sqrt{5}\hat{k}\]

C)

\[\sqrt{5}\hat{i}+\sqrt{5}\hat{j}+\sqrt{5}\hat{k}\]

D)

\[\sqrt{5}\hat{i}+2\sqrt{5}\hat{j}+\sqrt{5}\hat{k}\]

View Solution

56)

The upper \[\frac{3}{4}\]th portion of a vertical pole subtends an angle \[{{\tan }^{-1}}\frac{3}{5}\] at a point in the horizontal plane through its foot and at a distance 40 m from the foot. A possible height of the vertical pole is

A)

80 m

B)

20 m

C)

40 m

D)

60 m

View Solution

57)

If \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=1\] where \[a,\text{ }b,\text{ }c\text{ }\in \,\,R\text{ }\], then the maximum value of\[{{(4a-3b)}^{2}}+{{(5b-4c)}^{2}}+{{(3c-5a)}^{2}}\] is

A)

25

B)

50

C)

144

D)

None of these

View Solution

58)

If \[\vec{a}.\,\vec{b}=0\] and \[\vec{a}+\vec{b}\] makes an angle of \[60{}^\circ \] with \[\vec{a}\], then

A)

\[|\vec{a}|=2|\vec{b}|\]

B)

\[2|\vec{a}|=|\vec{b}|\]

C)

\[|\vec{a}|=\sqrt{3}|\vec{b}|\]

D)

\[|\vec{b}|=\sqrt{3}|\vec{a}|\]

View Solution

59)

If \[\overset{\to }{\mathop{a}}\,,\text{ }\overset{\to }{\mathop{b}}\,,\text{ }\overset{\to }{\mathop{c}}\,\] are the position vectors of corners A, B, C of a parallelogram ABCD, then what is the position vector of the corner D?

A)

\[\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,\]

B)

\[\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,-\overset{\to }{\mathop{c}}\,\]

C)

\[\overset{\to }{\mathop{a}}\,-\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,\]

D)

\[-\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,\]

View Solution

60)

For any vector\[\vec{\alpha }\], what is\[\left( \overset{\to }{\mathop{\alpha }}\,.\widehat{i} \right)\widehat{i}+\left( \overset{\to }{\mathop{\alpha }}\,.\widehat{j} \right)\widehat{j}+\left( \overset{\to }{\mathop{a}}\,.\widehat{k} \right)\widehat{k}\] equal to?

A)

\[\overset{\to }{\mathop{\alpha }}\,\]

B)

\[3\overset{\to }{\mathop{\alpha }}\,\]

C)

\[-\overset{\to }{\mathop{\alpha }}\,\]

D)

\[\overset{\to }{\mathop{0}}\,\]

View Solution

61)

If \[\vec{a}=\vec{i}+2\hat{j}-3\hat{k}\] and \[\vec{b}=3\hat{i}-\hat{j}+\lambda \hat{k},\] and \[(\vec{a}+\vec{b})\] is perpendicular to \[\vec{a}-\vec{b}\], then what is the value of \[\lambda \]?

A)

-2 only

B)

\[\pm 2\]

C)

3 only

D)

\[\pm 3\]

View Solution

62)

If \[\overrightarrow{p}=\lambda (\overrightarrow{u}\times \overrightarrow{v})+\mu (\overrightarrow{v}\times \overrightarrow{w})+v(\overrightarrow{w}\times \overrightarrow{u})\] and \[[\overset{\to }{\mathop{u}}\,\,\overset{\to }{\mathop{v}}\,\,\overset{\to }{\mathop{w}}\,]=\frac{1}{5}\], then \[\lambda +\mu +v\] is equal to

A)

5

B)

10

C)

15

D)

None of these

View Solution

63)

The angles of a triangle, two of whose sides are represented by the vectors \[\sqrt{3}(\vec{a}\times \vec{b})\] and \[\vec{b}-(\vec{a}.\vec{b})\vec{a}\] where \[\vec{b}\] is a non-zero vector and \[\vec{a}\]is a unit vector are

A)

\[\tan {{\,}^{-1}}\left( \frac{1}{\sqrt{3}} \right);\,\tan {{\,}^{-1}}\left( \frac{1}{2} \right);\,\tan {{\,}^{-1}}\left( \frac{\sqrt{3}+2}{1-2\sqrt{3}} \right)\]

B)

\[\tan {{\,}^{-1}}\left( \sqrt{3} \right);\,\tan {{\,}^{-1}}\left( \frac{1}{\sqrt{3}} \right);\,\cot {{\,}^{-1}}\left( 0 \right)\]

C)

\[\tan {{\,}^{-1}}\left( \sqrt{3} \right);\,\tan {{\,}^{-1}}\left( 2 \right);\,\tan {{\,}^{-1}}\left( \frac{\sqrt{3}+2}{2\sqrt{3}-1} \right)\]

D)

\[\tan {{\,}^{-1}}\left( \sqrt{3} \right);\,\tan {{\,}^{-1}}\left( \sqrt{2} \right);\,\tan {{\,}^{-1}}\left( \frac{\sqrt{2}+3}{3\sqrt{2}-1} \right)\]

View Solution

64)

The vectors \[(2\hat{i}-m\hat{j}+3m\hat{k})\And \]\[\{(1+m)\hat{i}-3m\hat{j}+\hat{k}\}\] include an acute angle for

A)

All values of m

B)

\[m<-2\]or \[m>-1/2\]

C)

\[m=-1/2\]

D)

\[m\in \left[ -2,-\frac{1}{2} \right]\]

View Solution

65)

If \[\vec{a}\] is a position vector of a point (1, -3) and A is another point (-1, 5), then what are the coordinates of the point B such that\[\overrightarrow{AB}=\vec{a}\]?

A)

(2, 0)

B)

(0, 2)

C)

(-2, 0)

D)

(0, -2)

View Solution

66)

If \[\overset{\to }{\mathop{{{r}_{1}}}}\,,\overset{\to }{\mathop{{{r}_{2}}}}\,,\overset{\to }{\mathop{{{r}_{3}}}}\,\] are the position vectors of three collinear points and scalars m and n exist such that\[\overset{\to }{\mathop{{{r}_{3}}}}\,=m\overset{\to }{\mathop{{{r}_{1}}}}\,+n\overset{\to }{\mathop{{{r}_{2}}}}\,\], then what is the value of (m+n)?

A)

0

B)

1

C)

-1

D)

2

View Solution

67)

If \[\overset{\to }{\mathop{c}}\,\] is the unit vector perpendicular to both the vectors \[\overset{\to }{\mathop{a}}\,\] and \[\overset{\to }{\mathop{b}}\,\], then what is another unit vector perpendicular to both the vectors \[\overset{\to }{\mathop{a}}\,\] and \[\overset{\to }{\mathop{b}}\,?\]

A)

\[\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{a}}\,\]

B)

\[\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{b}}\,\]

C)

\[-\frac{\left( \overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\, \right)}{\left| \overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\, \right|}\]

D)

\[\frac{\left( \overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\, \right)}{\left| \overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\, \right|}\]

View Solution

68)

What is the vector equally inclined to the vectors\[\hat{i}+3\hat{j}\] and\[3\hat{i}+\hat{j}\]?