Question Bank for JEE Main & Advanced Mathematics Three Dimensional Geometry Self Evaluation Test - Three Dimensional Geometry

1)

If a line in the space makes angle \[\alpha ,\beta \] and \[\gamma \] with the coordinate axes, then \[cos\text{ }2\alpha +cos2\beta \] \[+cos\,2\gamma +{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma \]equals. test- The two given expressions on both the sides of the \['='\] sign will have the same value if two numbers from either side or both sides are interchanged. Select the correct numbers to be interchanged from the given options. \['='\] चिन्‍ह के दोनों ओर दिए गए व्यंजकों का मान तब बराबर होगा जब उनमें किसी एक ओर या दोनों ओर की संख्याओं को आपस में बदला जाएगा। दिए गए विकल्पों में से आपस में बदली जाने वाली सही संख्याओं का चयन करें। \[3+5\times 4-24\div 3=7\times 4-3+36\div 6\]

A)

-1

B)

0

C)

1

D)

2

View Solution

2)

A plane passes through a fixed point (a, b, c). The locus of the foot of the perpendicular to it from the origin is the sphere

A)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-ax-by-cz=0\]

B)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-2ax-2by-2cz=0\]

C)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-4ax-4by-4cz=0\]

D)

None of these

View Solution

3)

From a point \[P(\lambda ,\lambda ,\lambda ),\] perpendiculars PQ and PR are drawn, respectively, on the lines \[y=x,\text{ }z=1\] and \[y=-x,\text{ }z=-1\]. If \[\angle QPR\] is a right angle, then the possible value(s) of \[\lambda \] is/are

A)

2

B)

1

C)

-1

D)

\[-\,\sqrt{2}\]

View Solution

4)

The d. r. of normal to the plane through (1, 0, 0), (0, 1, 0) which makes an angle \[\pi /4\] with plane \[x+y=3\] are

A)

\[1,\sqrt{2},1\]

B)

\[1,1,\sqrt{2}\]

C)

1, 1, 2

D)

\[\sqrt{2},1,\,1\]

View Solution

5)

What is the distance between the planes\[x-2y+z-1=0\] and\[-3x+6y-3z+2=0\]?

A)

3 unit

B)

1 unit

C)

0

D)

None of the above

View Solution

6)

A line makes \[45{}^\circ \] with positive x-axis and makes equal angles with positive y, z axes, respectively. What is the sum of the three angles which the line makes with positive x, y and z axes?

A)

\[180{}^\circ \]

B)

\[165{}^\circ \]

C)

\[150{}^\circ \]

D)

\[135{}^\circ \]

View Solution

7)

Under what condition does the equation \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2uc+2uy+2wz+d=0\] represent a real sphere?

A)

\[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}={{d}^{2}}\]

B)

\[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}>d\]

C)

\[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}<d\]

D)

\[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}<{{d}^{2}}\]

View Solution

8)

What is the equation of the plane through z-axis and parallel to the line\[\frac{x-1}{\cos \theta }=\frac{y+2}{\sin \theta }=\frac{z-3}{0}\]?

A)

\[x\,\,cot\,\theta +y=0\]

B)

\[x\,\,tan\,\,\theta -y=0\]

C)

\[x+y\,\,cot\,\theta =0\]

D)

\[x-y\,\,tan\,\theta =0\]

View Solution

9)

Two system of rectangular axes have the same origin. If a plane cuts them at distances a, b, c and a', b', c' respectively from the origin, then\[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}=k\left( \frac{1}{a{{'}^{2}}}+\frac{1}{b{{'}^{2}}}+\frac{1}{c{{'}^{2}}} \right)\], where k is equal to

A)

1

B)

2

C)

4

D)

None of these

View Solution

10)

\[{{L}_{1}}\] and \[{{L}_{2}}\] are two lines whose vector equations are \[{{L}_{1}}:\overset{\to }{\mathop{r}}\,=\lambda ((cos\,\,\theta +\sqrt{3})\hat{i}+(\sqrt{2}sin\,\,\theta )\hat{j}\]\[+(cos\theta -\sqrt{3})\hat{k}){{L}_{2}}:\overset{\to }{\mathop{r}}\,=\mu \left( a\hat{i}+b\hat{j}+c\hat{k} \right)\], where \[\lambda \] and \[\mu \] are scalars and \[\alpha \] is the acute angle between \[{{L}_{1}}\] and\[{{L}_{2}}\]. If the angle \['\alpha '\] is independent of \[\theta \] then the value of \['\alpha '\] is

A)

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

B)

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

C)

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

D)

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

View Solution

11)

The locus of a point, such that the sum of the squares of its distances from the planes\[x+y+z=0\], \[x\text{-}z=0\] and \[x-2y+z=0\text{ }is\text{ }9,\] is

A)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=3\]

B)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=6\]

C)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=9\]

D)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=12\]

View Solution

12)

The line which passes through the origin and intersect the two lines \[\frac{x-1}{2}=\frac{y+3}{4}=\frac{z-5}{3},\frac{x-4}{2}=\frac{y+3}{3}=\frac{z-14}{4},\] is

A)

\[\frac{x}{1}=\frac{y}{-3}=\frac{z}{5}\]

B)

\[\frac{x}{-1}=\frac{y}{3}=\frac{z}{5}\]

C)

\[\frac{x}{1}=\frac{y}{3}=\frac{z}{-5}\]

D)

\[\frac{x}{1}=\frac{y}{4}=\frac{z}{-5}\]

View Solution

13)

Let L be the line of intersection of the planes \[2x+3y+z=1\] and\[x+3y+2z=2\]. If L makes an angle \[\alpha \] with the positive x-axis, then \[cos\text{ }\alpha \]equals

A)

1

B)

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

C)

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

D)

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

View Solution

14)

Two spheres of radii 3 and 4 cut orthogonally The radius of common circle is

A)

12

B)

\[\frac{12}{5}\]

C)

\[\frac{\sqrt{12}}{5}\]

D)

\[\sqrt{12}\]

View Solution

15)

The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz-plane at the point\[\left( 0,\frac{17}{2},\frac{-13}{2} \right)\]. Then

A)

\[a=2,\,b=8\]

B)

\[a=4,b=6\]

C)

\[a=6,b=4\]

D)

\[a=8,b=2\]

View Solution

16)

The equation of the plane which makes with co-ordinate axes, a triangle with its centroid \[(\alpha ,\beta ,\gamma )\]is

A)

\[\alpha x,\beta y,\gamma z=3\]

B)

\[\alpha x,\beta y,\gamma z=1\]

C)

\[\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3\]

D)

\[\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=1\]

View Solution

17)

The direction ratios of the normal to the plane passing through the points (1, -2, 3), (-1, 2, -1) and parallel to \[\frac{x-2}{2}=\frac{y+1}{3}=\frac{z}{4}\] is

A)

(2, 3, 4)

B)

(14, 0, 7)

C)

(-2, 0, -1)

D)

(2, 0, -1)

View Solution

18)

The plane \[x+3y+13=0\] passes through the line of intersection of the planes \[2x-8y+4z=p\]and\[3x-5y+4z+10=0\]. If the plane is perpendicular to the plane\[3x-y-2z-4=0\], then the value of p is equal to

A)

2

B)

5

C)

9

D)

3

View Solution

19)

What is the angle between the planes\[2x-y+z=6\] and\[x+y+2z=3\]?

A)

\[\pi /2\]

B)

\[\pi /3\]

C)

\[\pi /4\]

D)

\[\pi /6\]

View Solution

20)

If the sum of the squares of the distance of the point (x, y, z) from the points (a, 0, 0) and (-a, 0, 0) is \[2{{c}^{2}}\], then which one of the following is correct?

A)

\[{{x}^{2}}+{{a}^{2}}=2{{c}^{2}}-{{y}^{2}}-{{z}^{2}}\]

B)

\[{{x}^{2}}+{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\]

C)

\[{{x}^{2}}-{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\]

D)

\[{{x}^{2}}+{{a}^{2}}={{c}^{2}}+{{y}^{2}}+{{z}^{2}}\]

View Solution

21)

Under what condition are the two lines\[y=\frac{m}{\ell }x+\alpha ,z=\frac{n}{\ell }x+\beta ;\] and \[y=\frac{m'}{\ell '}x+\alpha ',z=\frac{n'}{\ell '}x+\beta '\] Orthogonal?

A)

\[\alpha \alpha '+\beta \beta '+1=0\]

B)

\[(\alpha +\alpha ')+(\beta +\beta ')=0\]

C)

\[\ell \ell '+mm'+nn'=1\]

D)

\[\ell \ell '+mm'+nn'=0\]

View Solution

22)

What are the direction cosines of a line which is equally inclined to the positive directions of the axes?

A)

\[\left\langle \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right\rangle \]

B)

\[\left\langle -\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right\rangle \]

C)

\[\left\langle -\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right\rangle \]

D)

\[\left\langle \frac{1}{3},\frac{1}{3},\frac{1}{3} \right\rangle \]

View Solution

23)

What are the direction ratios of the line determined by the planes \[x-y+2z=1\] and\[x+y-z=3\]?

A)

(-1, 3, 2)

B)

(-1, -3, 2)

C)

(2, 1, 3)

D)

(2, 3, 2)

View Solution

24)

If \[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}+\hat{j}+\hat{k})\] and \[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+\mu (\hat{i}+\hat{j}-\hat{k})\] are two lines, then the equation of acute angle bisector of two lines is

A)

\[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+t(\hat{j}-\hat{k})\]

B)

\[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+t(2\hat{i})\]

C)

\[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+t(\hat{j}+\hat{k})\]

D)

None of these

View Solution

25)

A variable plane which remains at a constant distance 3p from the origin cut the coordinate axes at A, B and C. The locus of the centroid of triangle ABC is

A)

\[{{x}^{-1}}+{{y}^{-1}}+{{z}^{-1}}={{p}^{-1}}\]

B)

\[{{x}^{-2}}+{{y}^{-2}}+{{z}^{-2}}={{p}^{-2}}\]

C)

\[x+y+z=p\]

D)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{p}^{2}}\]

View Solution

26)

The direction cosines l, m, n of two lines are connected by the relations l + m + n = 0, lm = 0, then the angle between them is:

A)

\[\pi /3\]

B)

\[\pi /4\]

C)

\[\pi /2\]

D)

0

View Solution

27)

If the plane \[2ax-3ay+4az+6=0\] passes through the midpoint of the line joining the centres of the spheres \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+6x-8y-2z=13\] and \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-10x+4y-2z=8\] then a equals

A)

-1

B)

1

C)

-2

D)

2

View Solution

28)

The equation of the line which passes through the point (1, 1, 1) and intersect the lines \[\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\] and \[\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}\] is

A)

\[\frac{x-1}{3}=\frac{y-1}{10}=\frac{z-1}{17}\]

B)

\[\frac{x-1}{3}=\frac{y-1}{3}=\frac{z-1}{-5}\]

C)

\[\frac{x-1}{-2}=\frac{y-1}{1}=\frac{z-1}{-4}\]

D)

\[\frac{x-1}{8}=\frac{y-1}{-2}=\frac{z-1}{3}\]

View Solution

29)

The distance of the point (1, -2, 3) from the plane \[x-y+z=5\] measured parallel to the line \[\frac{x}{2}=\frac{y}{3}=\frac{z-1}{-6}\] is

A)

1

B)

2

C)

4

D)

\[2\sqrt{3}\]

View Solution

30)

Given the line \[L:\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-3}{-1}\] and the plane\[\pi :x-2y=z\]. of the following assertions, the only one that is always true is

A)

L is \[\bot \] to \[\pi \]

B)

L lies in \[\pi \]

C)

L is paralel to \[\pi \]

D)

None of these

View Solution

31)

A line makes the same angle \[\alpha \] with each of the x and y axes. If the angle\[\theta \], which it makes with the z-axis, is such that\[si{{n}^{2}}\theta =2\,{{\sin }^{2}}\alpha \], then what is the value of\[\alpha \]?

A)

\[\pi /4\]

B)

\[\pi /6\]

C)

\[\pi /3\]

D)

\[\pi /2\]

View Solution

32)

If OABC is a tetrahedron where O is the origin and A, B, C are three other vertices with position vectors \[\overset{\to }{\mathop{a}}\,,\text{ }\overset{\to }{\mathop{b}}\,\] and \[\overset{\to }{\mathop{c}}\,\] respectively, then the centre of sphere circumscribing the tetrahedron is given by the position vector

A)

\[\frac{{{a}^{2}}(\vec{b}\times \vec{c})+{{b}^{2}}(\vec{c}\times \vec{a})+{{c}^{2}}(\vec{a}\times \vec{b})}{2[\vec{a}\,\vec{b}\,\vec{c}]}\]

B)

\[\frac{{{b}^{2}}(\vec{b}\times \vec{c})+{{a}^{2}}(\vec{c}\times \vec{a})+{{c}^{2}}(\vec{a}\times \vec{b})}{[\vec{a}\,\vec{b}\,\vec{c}]}\]

C)

\[\frac{{{b}^{2}}(\vec{b}\times \vec{c})+{{a}^{2}}(\vec{c}\times \vec{a})+{{c}^{2}}(\vec{a}\times \vec{b})}{2[\vec{a}\,\vec{b}\,\vec{c}]}\]

D)

\[\frac{{{a}^{2}}(\vec{a}\times \vec{b})+{{b}^{2}}(\vec{b}\times \hat{c})+{{c}^{2}}(\vec{c}\times \vec{a})}{2[\vec{a}\,\vec{b}\,\vec{c}]}\]

View Solution

33)

The distance of point A (-2, 3, 1) from the line PQ through P (- 3, 5, 2), which makes equal angles with the axes is

A)

\[2/\sqrt{3}\]

B)

\[\sqrt{14/3}\]

C)

\[16/\sqrt{3}\]

D)

\[5/\sqrt{3}\]

View Solution

34)

The shortest distance between the skew lines\[{{l}_{1}}:\vec{r}={{\vec{a}}_{1}}+\lambda {{\vec{b}}_{1}}{{l}_{2}}:\vec{r}={{\vec{a}}_{2}}+\mu {{\vec{b}}_{2}}\] is

A)

\[\frac{|({{{\vec{a}}}_{2}}-{{{\vec{a}}}_{1}}).{{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}}|}{|{{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}}|}\]

B)

\[\frac{\left| ({{{\vec{a}}}_{2}}-{{{\vec{a}}}_{1}}).{{{\vec{a}}}_{2}}\times {{{\vec{b}}}_{2}} \right|}{\left| {{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}} \right|}\]

C)

\[\frac{\left| ({{{\vec{a}}}_{2}}-{{{\vec{b}}}_{2}}).{{{\vec{a}}}_{1}}\times {{{\vec{b}}}_{1}} \right|}{\left| {{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}} \right|}\]

D)

\[\frac{\left| ({{{\vec{a}}}_{1}}-{{{\vec{b}}}_{2}}).{{{\vec{b}}}_{1}}\times {{{\vec{a}}}_{2}} \right|}{\left| {{{\vec{b}}}_{1}}\times {{{\vec{a}}}_{2}} \right|}\]

View Solution

35)

If the center of the sphere \[a{{x}^{2}}+b{{y}^{2}}+c{{z}^{2}}-2x+4y+2z-3=0\]is \[(1/2,-1,-1/2)\], what is the value of b ?

A)

1

B)

-1

C)

2

D)

-2

View Solution

36)

The foot of the perpendicular drawn from the origin to a plane is the point (1,-3, 1). What is the intercept cut on the x-axis by the plane?

A)

1

B)

3

C)

\[\sqrt{11}\]

D)

11

View Solution

37)

If the straight line \[\frac{x-{{x}_{0}}}{\ell }=\frac{y-{{y}_{0}}}{m}=\frac{z-{{z}_{0}}}{n}\] is parallel to the plane \[ax+by+cz+d=0\]then which one of the following is correct?

A)

\[\ell +m+n=0\]

B)

\[a+b+c=0\]

C)

\[\frac{a}{\ell }+\frac{b}{m}+\frac{c}{n}=0\]

D)

\[a\ell +bm+cn=0\]

View Solution

38)

What is the angle between two planes \[2x-y+z=4\] and \[x+y+2z=6?\]

A)

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

B)

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

C)

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

D)

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

View Solution

39)

The angle between the pair of planes represented by equation\[2{{x}^{2}}-2{{y}^{2}}+4{{z}^{2}}+6xz+2yz+3xy=0\]is

A)

\[{{\cos }^{-1}}\left( \frac{1}{3} \right)\]

B)

\[{{\cos }^{-1}}\left( \frac{4}{21} \right)\]

C)

\[{{\cos }^{-1}}\left( \frac{4}{9} \right)\]

D)

\[{{\cos }^{-1}}\left( \frac{7}{\sqrt{84}} \right)\]

View Solution

40)

Let \[A(\vec{a})\] and \[B(\vec{b})\] be points on two skew line \[\vec{r}=\vec{a}+\vec{\lambda }\] and \[\vec{r}=\vec{b}+u\vec{q}\] and the shortest distance between the skew line is 1, where \[\vec{p}\] and \[\vec{q}\] are unit vectors forming adjacent sides of a parallelogram enclosing an area of \[\frac{1}{2}\]units. If an angle between AB and the line of shortest distance is \[60{}^\circ \], then \[AB=\]

A)

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

B)

\[2\]

C)

\[1\]

D)

\[\lambda \in R-\{0\}\]

View Solution

41)

A line makes angles, \[\theta ,\phi \] and \[\psi \] with x, y, z axes respectively. Consider the following:

1. \[{{\sin }^{2}}\theta +{{\sin }^{2}}\phi ={{\cos }^{2}}\psi \]

2. \[{{\cos }^{2}}\theta +{{\cos }^{2}}\phi ={{\sin }^{2}}\psi \]

3. \[{{\sin }^{2}}\theta +{{\cos }^{2}}\phi ={{\cos }^{2}}\psi \]

Which of the above is/are correct?

A)

1 only

B)

2 only

C)

3 only

D)

2 and 3

View Solution

42)

The shortest distance from the plane\[12x+4y+3z=327\] To the sphere\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+4x-2y-6z=155\] is

A)

39

B)

26

C)

\[11\frac{4}{13}\]

D)

13

View Solution

43)

The line, \[\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}\] intersects the curve \[xy={{c}^{2}},z=0\] if c is equal to

A)

\[\pm 1\]

B)

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

C)

\[\pm \sqrt{5}\]

D)

None

View Solution

44)

The equation of the plane which passes through the line of intersection of planes \[\vec{r}.{{\vec{n}}_{1}}={{q}_{1}},\vec{r}.{{\vec{n}}_{2}}=q\] And is parallel to the line of intersection of planes \[\vec{r}.{{\vec{n}}_{3}}={{q}_{3}}\] and \[\vec{r}.{{\vec{n}}_{4}}={{q}_{4}}\]is

A)

\[[{{\vec{n}}_{2}}{{\vec{n}}_{3}}{{\vec{n}}_{4}}](\vec{r}.{{\vec{n}}_{1}}-{{\vec{q}}_{1}})=[{{\vec{n}}_{1}}{{\vec{n}}_{3}}{{\vec{n}}_{4}}](\vec{r}.{{\vec{n}}_{2}}-{{\vec{q}}_{2}})\]

B)

\[[{{\vec{n}}_{1}}{{\vec{n}}_{2}}{{\vec{n}}_{4}}](\vec{r}.{{\vec{n}}_{4}}{{q}_{4}})=[{{\vec{n}}_{4}}{{\vec{n}}_{3}}{{\vec{n}}_{1}}](\vec{r}.{{\vec{n}}_{2}}-{{q}_{2}})\]

C)

\[[{{\vec{n}}_{4}}{{\vec{n}}_{3}}{{\vec{n}}_{1}}](\vec{r}.{{\vec{n}}_{4}}-{{q}_{4}})=[{{\vec{n}}_{1}}{{\vec{n}}_{2}}\vec{n} 3](\vec{r}.{{\vec{n}}_{2}}={{q}_{2}})\]

D)

None of these

View Solution

45)

Value of\[\lambda \] such that the line\[\frac{x-1}{2}=\frac{y-1}{3}=\frac{z-1}{\lambda }\]Is perpendicular to normal to the plane\[\vec{r}.(2\vec{i}+3\vec{j}+4\vec{k})=0\] is

A)

\[-\frac{13}{4}\]

B)

\[-\frac{17}{4}\]

C)

\[4\]

D)

None of these

View Solution

46)

The direction consines of two lines are related by\[l+m+n=0\]\[a{{l}^{2}}+b{{m}^{2}}+c{{n}^{2}}=0\]. The lines are parallel if

A)

\[a+b+c=0\]

B)

\[{{a}^{-1}}+{{b}^{-1}}+{{c}^{-1}}=0\]

C)

\[a=b=c\]

D)

None of these

View Solution

47)

The distance between the line \[\vec{r}.2\hat{i}-2\hat{j}+3\hat{k}\]\[+\lambda (\hat{i}-\hat{j}+4\hat{k})\] and the plane \[\vec{r}.(\hat{i}-5\hat{j}+\hat{k})=5\] is

A)

\[\frac{10}{3\sqrt{3}}\]

B)

\[\frac{10}{9}\]

C)

\[\frac{10}{3}\]

D)

\[\frac{3}{10}\]

View Solution

48)

The angle between the straight lines \[\vec{r}=(2-3t)\vec{i}+(1+2t)\vec{j}+(2+6t)\vec{k}\] and \[\vec{r}=(1+4s)\vec{i}+(2-s)\vec{j}+(8s-1)\vec{k}\]is

A)

\[{{\cos }^{-1}}\left( \frac{\sqrt{41}}{34} \right)\]

B)

\[{{\cos }^{-1}}\left( \frac{21}{34} \right)\]

C)

\[{{\cos }^{-1}}\left( \frac{43}{63} \right)\]

D)

\[{{\cos }^{-1}}\left( \frac{34}{63} \right)\]

View Solution

49)

The angle between the line \[\frac{x-2}{a}=\frac{y-2}{b}=\frac{z-2}{c}\] and the plane \[ax+by+cz+6=0\] is

A)

\[{{\sin }^{-1}}\left( \frac{1}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}} \right)\]

B)

\[45{}^\circ \]

C)

\[60{}^\circ \]

D)

\[90{}^\circ \]

View Solution

50)

Which one of the following is the plane containing the lien \[\frac{x-2}{2}=\frac{y-3}{3}=\frac{z-4}{5}\] and parallel to z axis?

A)

\[2x-3y=0\]

B)

\[5x-2z=0\]

C)

\[5y-3z=0\]

D)

\[3x-2y=0\]

View Solution

51)

If \[\theta \] is the acute angle between the diagonals of a cube, then which one of the following is correct?

A)

\[\theta <30{}^\circ \]

B)

\[\theta =60{}^\circ \]

C)

\[30{}^\circ <\theta <60{}^\circ \]

D)

\[\theta >60{}^\circ \]

View Solution

52)

What is the acute angle between the planes \[x+y+2z=3\] and \[-2x+y-z=11?\]

A)

\[\pi /5\]

B)

\[\pi /4\]

C)

\[\pi /6\]

D)

\[\pi /3\]

View Solution

53)

A line makes angles \[\theta ,\phi \] and \[\psi \] with x, y, z axes respectively. Consider the following

1. \[{{\sin }^{2}}\theta +{{\sin }^{2}}\phi ={{\cos }^{2}}\psi \]

2. \[{{\cos }^{2}}\theta +{{\cos }^{2}}\phi ={{\sin }^{2}}\psi \]

3. \[{{\sin }^{2}}\theta +{{\cos }^{2}}\phi ={{\cos }^{2}}\psi \]

Which of the above is/are correct?

A)

1 only

B)

2 only

C)

3 only

D)

2 and 3

View Solution

54)

Under what condition do \[\left\langle \frac{1}{\sqrt{2}},\frac{1}{2},k \right\rangle \] represent direction cosines of a line?

A)

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

B)

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

C)

\[k=\pm \frac{1}{2}\]

D)

K can take any value

View Solution

55)

The foot of the perpendicular from the point (1, 6, 3) to the line \[\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\] is

A)

(1, 2, 5)

B)

(-1, -1, -1)

C)

(2, 5, 8)

D)

(-2, -3, -4)

View Solution

56)

If Q is the image of the point P(2, 3, 4) under the reflection in the plane \[x-2y+5z=6\], then the equation of the line \[PQ\]is

A)

\[\frac{x-2}{-1}=\frac{y-3}{2}=\frac{z-4}{5}\]

B)

\[\frac{x-2}{1}=\frac{y-3}{-2}=\frac{z-4}{5}\]

C)

\[\frac{x-2}{-1}=\frac{y-3}{-2}=\frac{z-4}{5}\]

D)

\[\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{5}\]

View Solution

57)

Under which one of the following condition will the two planes \[x+y+z=7\] and\[\alpha x+\beta y+\gamma z=3\], be parallel (but not coincident)?

A)

\[\alpha =\beta =\gamma =1only\]

B)

\[\alpha =\beta =\gamma =\frac{3}{7}only\]

C)

\[\alpha =\beta =\gamma \]

D)

None of the above

View Solution

58)

Consider the following relations among the angles\[\alpha \], \[\beta \] and \[\gamma \] made by a vector with the coordinate axes

I. \[\cos 2\alpha +\cos 2\beta +\cos 2\gamma =-1\]

II. \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma =1\]

Which of the above is/are correct?

A)

Only I

B)

Only II

C)

Both I and II

D)

Neither I nor II

View Solution

59)

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

A)

\[\alpha =2,\beta =2\]

B)

\[\alpha =1,\beta =2\]

C)

\[\alpha =2,\beta =1\]

D)

\[\alpha =2,\beta =1\]

View Solution

60)

If the direction consines of a line are \[\left( \frac{1}{c},\frac{1}{c},\frac{1}{c} \right)\] then

A)

\[0<c<1\]

B)

\[c>2\]

C)

\[c>0\]

D)

\[c=\pm \sqrt{3}\]

View Solution

61)

Distance of the point \[P(\vec{p})\] from the line \[\vec{r}=\vec{a}+\lambda \vec{b}\] is

A)

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

B)

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

C)

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

D)

None of these

View Solution

62)

The vector equation of the line of intersection of the planes \[\vec{r}=\vec{b}+{{\lambda }_{1}}(\vec{b}-\vec{a})+{{\mu }_{1}}(\vec{a}-\vec{c})\] and \[\vec{r}=\vec{b}+{{\lambda }_{2}}(\vec{b}-\vec{c})+{{\mu }_{2}}(\vec{a}+\vec{c})\vec{a},\vec{b},\vec{c}\] being non-coplanar vectors, is

A)

\[\vec{r}=\vec{b}+{{\mu }_{1}}(\vec{a}+\vec{c})\]

B)

\[\vec{r}=\vec{b}+{{\lambda }_{1}}(\vec{a}-\vec{c})\]

C)

\[\vec{r}=2\vec{b}+{{\lambda }_{2}}(\vec{a}-\vec{c})\]

D)

None of these

View Solution

63)

A plane passing through (1, 1, 1) cuts positive direction of coordinate axes at A, B and C, then the volume of tetrahedron OABC satisfies

A)

\[V\le \frac{9}{2}\]

B)

\[V\ge \frac{9}{2}\]

C)

\[V=\frac{9}{2}\]

D)

None of these

View Solution

64)

What is the value of n so that the angle between the lines having direction ratios (1, 1, 1) and (1, -1, n) is \[60{}^\circ \]?

A)

\[\sqrt{3}\]

B)

\[\sqrt{6}\]

C)

3

D)

None of these

View Solution

65)

The locus of a point, such that the sum of the squares of its distances from the planes \[x+y+z=0,\]\[x-z=0\] And \[x-2y+z=0\]is 9, is

A)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=3\]

B)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=6\]

C)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=9\]

D)

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=12\]

View Solution

66)

A mirror and a source of light are situated at the origin 0 and at a point on OX respectively. A ray of light from the source strikes the mirror and is reflected. If the direction ratios of the normal to the plane are 1, -1, 1, then direction consines of the reflected rays are

A)

\[\frac{1}{3},\frac{2}{3},\frac{2}{3}\]

B)

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

C)

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

D)

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

View Solution

67)

If lines \[x=y=z\] and \[x=\frac{y}{2}=\frac{z}{3}\] and third line passing through (1, 1, 1) form a triangle of area \[\sqrt{6}\] units, then the point of intersection of third line with the second line will be

A)

\[(1,2,3)\]

B)

\[(2,4,6)\]

C)

\[\left( \frac{4}{3},\frac{8}{3},\frac{12}{3} \right)\]

D)

None of these

View Solution

68)

If \[{{l}_{1}},{{m}_{1}},{{n}_{1}}\] and \[{{l}_{2}},{{m}_{2}},{{n}_{2}}\] are direction consines of the two lines inclined to each other at an angle \[\theta \], the direction cosines of the bisector of the angle between these lines are

A)

\[\frac{{{l}_{1}}-{{l}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\sin \frac{\theta }{2}}\]

B)

\[\frac{{{l}_{1}}-{{l}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\cos \frac{\theta }{2}}\]

C)

\[\frac{{{l}_{1}}-{{l}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\sin \frac{\theta }{2}}\]

D)

\[\frac{{{l}_{1}}-{{l}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\cos \frac{\theta }{2}}\]

View Solution

69)

The equation of the plane containing the line \[2x-5y+z=3;x+y+4z=5\], and parallel to the plane, \[x+3y+6z=1\], is:

A)

\[x+3y+6z=7\]

B)

\[2x+6y+12z=-13\]

C)

\[2x+6y+12z=13\]

D)

\[x+3y+6z=-7\]

View Solution