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question_answer1)
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 done
clear
B)
0 done
clear
C)
1 done
clear
D)
2 done
clear
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question_answer2)
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)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-ax-by-cz=0\] done
clear
B)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-2ax-2by-2cz=0\] done
clear
C)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-4ax-4by-4cz=0\] done
clear
D)
None of these done
clear
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question_answer3)
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 done
clear
B)
1 done
clear
C)
-1 done
clear
D)
\[-\,\sqrt{2}\] done
clear
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question_answer4)
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)
\[1,\sqrt{2},1\] done
clear
B)
\[1,1,\sqrt{2}\] done
clear
C)
1, 1, 2 done
clear
D)
\[\sqrt{2},1,\,1\] done
clear
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question_answer5)
What is the distance between the planes\[x-2y+z-1=0\] and\[-3x+6y-3z+2=0\]?
A)
3 unit done
clear
B)
1 unit done
clear
C)
0 done
clear
D)
None of the above done
clear
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question_answer6)
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)
\[180{}^\circ \] done
clear
B)
\[165{}^\circ \] done
clear
C)
\[150{}^\circ \] done
clear
D)
\[135{}^\circ \] done
clear
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question_answer7)
Under what condition does the equation \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2uc+2uy+2wz+d=0\] represent a real sphere?
A)
\[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}={{d}^{2}}\] done
clear
B)
\[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}>d\] done
clear
C)
\[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}<d\] done
clear
D)
\[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}<{{d}^{2}}\] done
clear
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question_answer8)
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)
\[x\,\,cot\,\theta +y=0\] done
clear
B)
\[x\,\,tan\,\,\theta -y=0\] done
clear
C)
\[x+y\,\,cot\,\theta =0\] done
clear
D)
\[x-y\,\,tan\,\theta =0\] done
clear
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question_answer9)
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 done
clear
B)
2 done
clear
C)
4 done
clear
D)
None of these done
clear
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question_answer10)
\[{{L}_{1}}\] 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)
\[\frac{\pi }{6}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{3}\] done
clear
D)
\[\frac{\pi }{2}\] done
clear
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question_answer11)
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)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=3\] done
clear
B)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=6\] done
clear
C)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=9\] done
clear
D)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=12\] done
clear
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question_answer12)
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)
\[\frac{x}{1}=\frac{y}{-3}=\frac{z}{5}\] done
clear
B)
\[\frac{x}{-1}=\frac{y}{3}=\frac{z}{5}\] done
clear
C)
\[\frac{x}{1}=\frac{y}{3}=\frac{z}{-5}\] done
clear
D)
\[\frac{x}{1}=\frac{y}{4}=\frac{z}{-5}\] done
clear
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question_answer13)
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 done
clear
B)
\[\frac{1}{\sqrt{2}}\] done
clear
C)
\[\frac{1}{\sqrt{3}}\] done
clear
D)
\[\frac{1}{2}\] done
clear
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question_answer14)
Two spheres of radii 3 and 4 cut orthogonally The radius of common circle is
A)
12 done
clear
B)
\[\frac{12}{5}\] done
clear
C)
\[\frac{\sqrt{12}}{5}\] done
clear
D)
\[\sqrt{12}\] done
clear
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question_answer15)
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)
\[a=2,\,b=8\] done
clear
B)
\[a=4,b=6\] done
clear
C)
\[a=6,b=4\] done
clear
D)
\[a=8,b=2\] done
clear
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question_answer16)
The equation of the plane which makes with co-ordinate axes, a triangle with its centroid \[(\alpha ,\beta ,\gamma )\]is
A)
\[\alpha x,\beta y,\gamma z=3\] done
clear
B)
\[\alpha x,\beta y,\gamma z=1\] done
clear
C)
\[\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3\] done
clear
D)
\[\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=1\] done
clear
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question_answer17)
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) done
clear
B)
(14, 0, 7) done
clear
C)
(-2, 0, -1) done
clear
D)
(2, 0, -1) done
clear
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question_answer18)
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 done
clear
B)
5 done
clear
C)
9 done
clear
D)
3 done
clear
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question_answer19)
What is the angle between the planes\[2x-y+z=6\] and\[x+y+2z=3\]?
A)
\[\pi /2\] done
clear
B)
\[\pi /3\] done
clear
C)
\[\pi /4\] done
clear
D)
\[\pi /6\] done
clear
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question_answer20)
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)
\[{{x}^{2}}+{{a}^{2}}=2{{c}^{2}}-{{y}^{2}}-{{z}^{2}}\] done
clear
B)
\[{{x}^{2}}+{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\] done
clear
C)
\[{{x}^{2}}-{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\] done
clear
D)
\[{{x}^{2}}+{{a}^{2}}={{c}^{2}}+{{y}^{2}}+{{z}^{2}}\] done
clear
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question_answer21)
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)
\[\alpha \alpha '+\beta \beta '+1=0\] done
clear
B)
\[(\alpha +\alpha ')+(\beta +\beta ')=0\] done
clear
C)
\[\ell \ell '+mm'+nn'=1\] done
clear
D)
\[\ell \ell '+mm'+nn'=0\] done
clear
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question_answer22)
What are the direction cosines of a line which is equally inclined to the positive directions of the axes?
A)
\[\left\langle \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right\rangle \] done
clear
B)
\[\left\langle -\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right\rangle \] done
clear
C)
\[\left\langle -\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right\rangle \] done
clear
D)
\[\left\langle \frac{1}{3},\frac{1}{3},\frac{1}{3} \right\rangle \] done
clear
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question_answer23)
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) done
clear
B)
(-1, -3, 2) done
clear
C)
(2, 1, 3) done
clear
D)
(2, 3, 2) done
clear
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question_answer24)
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)
\[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+t(\hat{j}-\hat{k})\] done
clear
B)
\[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+t(2\hat{i})\] done
clear
C)
\[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+t(\hat{j}+\hat{k})\] done
clear
D)
None of these done
clear
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question_answer25)
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)
\[{{x}^{-1}}+{{y}^{-1}}+{{z}^{-1}}={{p}^{-1}}\] done
clear
B)
\[{{x}^{-2}}+{{y}^{-2}}+{{z}^{-2}}={{p}^{-2}}\] done
clear
C)
\[x+y+z=p\] done
clear
D)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{p}^{2}}\] done
clear
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question_answer26)
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)
\[\pi /3\] done
clear
B)
\[\pi /4\] done
clear
C)
\[\pi /2\] done
clear
D)
0 done
clear
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question_answer27)
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 done
clear
B)
1 done
clear
C)
-2 done
clear
D)
2 done
clear
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question_answer28)
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)
\[\frac{x-1}{3}=\frac{y-1}{10}=\frac{z-1}{17}\] done
clear
B)
\[\frac{x-1}{3}=\frac{y-1}{3}=\frac{z-1}{-5}\] done
clear
C)
\[\frac{x-1}{-2}=\frac{y-1}{1}=\frac{z-1}{-4}\] done
clear
D)
\[\frac{x-1}{8}=\frac{y-1}{-2}=\frac{z-1}{3}\] done
clear
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question_answer29)
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 done
clear
B)
2 done
clear
C)
4 done
clear
D)
\[2\sqrt{3}\] done
clear
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question_answer30)
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 \[\pi \] done
clear
B)
L lies in \[\pi \] done
clear
C)
L is paralel to \[\pi \] done
clear
D)
None of these done
clear
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question_answer31)
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)
\[\pi /4\] done
clear
B)
\[\pi /6\] done
clear
C)
\[\pi /3\] done
clear
D)
\[\pi /2\] done
clear
View Solution play_arrow
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question_answer32)
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)
\[\frac{{{a}^{2}}(\vec{b}\times \vec{c})+{{b}^{2}}(\vec{c}\times \vec{a})+{{c}^{2}}(\vec{a}\times \vec{b})}{2[\vec{a}\,\vec{b}\,\vec{c}]}\] done
clear
B)
\[\frac{{{b}^{2}}(\vec{b}\times \vec{c})+{{a}^{2}}(\vec{c}\times \vec{a})+{{c}^{2}}(\vec{a}\times \vec{b})}{[\vec{a}\,\vec{b}\,\vec{c}]}\] done
clear
C)
\[\frac{{{b}^{2}}(\vec{b}\times \vec{c})+{{a}^{2}}(\vec{c}\times \vec{a})+{{c}^{2}}(\vec{a}\times \vec{b})}{2[\vec{a}\,\vec{b}\,\vec{c}]}\] done
clear
D)
\[\frac{{{a}^{2}}(\vec{a}\times \vec{b})+{{b}^{2}}(\vec{b}\times \hat{c})+{{c}^{2}}(\vec{c}\times \vec{a})}{2[\vec{a}\,\vec{b}\,\vec{c}]}\] done
clear
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question_answer33)
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)
\[2/\sqrt{3}\] done
clear
B)
\[\sqrt{14/3}\] done
clear
C)
\[16/\sqrt{3}\] done
clear
D)
\[5/\sqrt{3}\] done
clear
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question_answer34)
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)
\[\frac{|({{{\vec{a}}}_{2}}-{{{\vec{a}}}_{1}}).{{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}}|}{|{{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}}|}\] done
clear
B)
\[\frac{\left| ({{{\vec{a}}}_{2}}-{{{\vec{a}}}_{1}}).{{{\vec{a}}}_{2}}\times {{{\vec{b}}}_{2}} \right|}{\left| {{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}} \right|}\] done
clear
C)
\[\frac{\left| ({{{\vec{a}}}_{2}}-{{{\vec{b}}}_{2}}).{{{\vec{a}}}_{1}}\times {{{\vec{b}}}_{1}} \right|}{\left| {{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}} \right|}\] done
clear
D)
\[\frac{\left| ({{{\vec{a}}}_{1}}-{{{\vec{b}}}_{2}}).{{{\vec{b}}}_{1}}\times {{{\vec{a}}}_{2}} \right|}{\left| {{{\vec{b}}}_{1}}\times {{{\vec{a}}}_{2}} \right|}\] done
clear
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question_answer35)
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 done
clear
B)
-1 done
clear
C)
2 done
clear
D)
-2 done
clear
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question_answer36)
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 done
clear
B)
3 done
clear
C)
\[\sqrt{11}\] done
clear
D)
11 done
clear
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question_answer37)
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)
\[\ell +m+n=0\] done
clear
B)
\[a+b+c=0\] done
clear
C)
\[\frac{a}{\ell }+\frac{b}{m}+\frac{c}{n}=0\] done
clear
D)
\[a\ell +bm+cn=0\] done
clear
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question_answer38)
What is the angle between two planes \[2x-y+z=4\] and \[x+y+2z=6?\]
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\frac{\pi }{3}\] done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
\[\frac{\pi }{6}\] done
clear
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question_answer39)
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)
\[{{\cos }^{-1}}\left( \frac{1}{3} \right)\] done
clear
B)
\[{{\cos }^{-1}}\left( \frac{4}{21} \right)\] done
clear
C)
\[{{\cos }^{-1}}\left( \frac{4}{9} \right)\] done
clear
D)
\[{{\cos }^{-1}}\left( \frac{7}{\sqrt{84}} \right)\] done
clear
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question_answer40)
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)
\[\frac{1}{2}\] done
clear
B)
\[2\] done
clear
C)
\[1\] done
clear
D)
\[\lambda \in R-\{0\}\] done
clear
View Solution play_arrow
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question_answer41)
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 done
clear
B)
2 only done
clear
C)
3 only done
clear
D)
2 and 3 done
clear
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question_answer42)
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 done
clear
B)
26 done
clear
C)
\[11\frac{4}{13}\] done
clear
D)
13 done
clear
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question_answer43)
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)
\[\pm 1\] done
clear
B)
\[\pm \frac{1}{3}\] done
clear
C)
\[\pm \sqrt{5}\] done
clear
D)
None done
clear
View Solution play_arrow
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question_answer44)
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)
\[[{{\vec{n}}_{2}}{{\vec{n}}_{3}}{{\vec{n}}_{4}}](\vec{r}.{{\vec{n}}_{1}}-{{\vec{q}}_{1}})=[{{\vec{n}}_{1}}{{\vec{n}}_{3}}{{\vec{n}}_{4}}](\vec{r}.{{\vec{n}}_{2}}-{{\vec{q}}_{2}})\] done
clear
B)
\[[{{\vec{n}}_{1}}{{\vec{n}}_{2}}{{\vec{n}}_{4}}](\vec{r}.{{\vec{n}}_{4}}{{q}_{4}})=[{{\vec{n}}_{4}}{{\vec{n}}_{3}}{{\vec{n}}_{1}}](\vec{r}.{{\vec{n}}_{2}}-{{q}_{2}})\] done
clear
C)
\[[{{\vec{n}}_{4}}{{\vec{n}}_{3}}{{\vec{n}}_{1}}](\vec{r}.{{\vec{n}}_{4}}-{{q}_{4}})=[{{\vec{n}}_{1}}{{\vec{n}}_{2}}\vec{n} 3](\vec{r}.{{\vec{n}}_{2}}={{q}_{2}})\] done
clear
D)
None of these done
clear
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question_answer45)
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)
\[-\frac{13}{4}\] done
clear
B)
\[-\frac{17}{4}\] done
clear
C)
\[4\] done
clear
D)
None of these done
clear
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question_answer46)
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)
\[a+b+c=0\] done
clear
B)
\[{{a}^{-1}}+{{b}^{-1}}+{{c}^{-1}}=0\] done
clear
C)
\[a=b=c\] done
clear
D)
None of these done
clear
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question_answer47)
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)
\[\frac{10}{3\sqrt{3}}\] done
clear
B)
\[\frac{10}{9}\] done
clear
C)
\[\frac{10}{3}\] done
clear
D)
\[\frac{3}{10}\] done
clear
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question_answer48)
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)
\[{{\cos }^{-1}}\left( \frac{\sqrt{41}}{34} \right)\] done
clear
B)
\[{{\cos }^{-1}}\left( \frac{21}{34} \right)\] done
clear
C)
\[{{\cos }^{-1}}\left( \frac{43}{63} \right)\] done
clear
D)
\[{{\cos }^{-1}}\left( \frac{34}{63} \right)\] done
clear
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question_answer49)
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)
\[{{\sin }^{-1}}\left( \frac{1}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}} \right)\] done
clear
B)
\[45{}^\circ \] done
clear
C)
\[60{}^\circ \] done
clear
D)
\[90{}^\circ \] done
clear
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question_answer50)
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)
\[2x-3y=0\] done
clear
B)
\[5x-2z=0\] done
clear
C)
\[5y-3z=0\] done
clear
D)
\[3x-2y=0\] done
clear
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question_answer51)
If \[\theta \] is the acute angle between the diagonals of a cube, then which one of the following is correct?
A)
\[\theta <30{}^\circ \] done
clear
B)
\[\theta =60{}^\circ \] done
clear
C)
\[30{}^\circ <\theta <60{}^\circ \] done
clear
D)
\[\theta >60{}^\circ \] done
clear
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question_answer52)
What is the acute angle between the planes \[x+y+2z=3\] and \[-2x+y-z=11?\]
A)
\[\pi /5\] done
clear
B)
\[\pi /4\] done
clear
C)
\[\pi /6\] done
clear
D)
\[\pi /3\] done
clear
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question_answer53)
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 done
clear
B)
2 only done
clear
C)
3 only done
clear
D)
2 and 3 done
clear
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question_answer54)
Under what condition do \[\left\langle \frac{1}{\sqrt{2}},\frac{1}{2},k \right\rangle \] represent direction cosines of a line?
A)
\[k=\frac{1}{2}\] done
clear
B)
\[k=-\frac{1}{2}\] done
clear
C)
\[k=\pm \frac{1}{2}\] done
clear
D)
K can take any value done
clear
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question_answer55)
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) done
clear
B)
(-1, -1, -1) done
clear
C)
(2, 5, 8) done
clear
D)
(-2, -3, -4) done
clear
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question_answer56)
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)
\[\frac{x-2}{-1}=\frac{y-3}{2}=\frac{z-4}{5}\] done
clear
B)
\[\frac{x-2}{1}=\frac{y-3}{-2}=\frac{z-4}{5}\] done
clear
C)
\[\frac{x-2}{-1}=\frac{y-3}{-2}=\frac{z-4}{5}\] done
clear
D)
\[\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{5}\] done
clear
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question_answer57)
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)
\[\alpha =\beta =\gamma =1only\] done
clear
B)
\[\alpha =\beta =\gamma =\frac{3}{7}only\] done
clear
C)
\[\alpha =\beta =\gamma \] done
clear
D)
None of the above done
clear
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question_answer58)
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 done
clear
B)
Only II done
clear
C)
Both I and II done
clear
D)
Neither I nor II done
clear
View Solution play_arrow
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question_answer59)
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)
\[\alpha =2,\beta =2\] done
clear
B)
\[\alpha =1,\beta =2\] done
clear
C)
\[\alpha =2,\beta =1\] done
clear
D)
\[\alpha =2,\beta =1\] done
clear
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question_answer60)
If the direction consines of a line are \[\left( \frac{1}{c},\frac{1}{c},\frac{1}{c} \right)\] then
A)
\[0<c<1\] done
clear
B)
\[c>2\] done
clear
C)
\[c>0\] done
clear
D)
\[c=\pm \sqrt{3}\] done
clear
View Solution play_arrow
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question_answer61)
Distance of the point \[P(\vec{p})\] from the line \[\vec{r}=\vec{a}+\lambda \vec{b}\] is
A)
\[\left| (\vec{a}-\vec{p})+\frac{((\vec{p}-\vec{a}).\vec{b})\vec{b}}{{{\left| {\vec{b}} \right|}^{2}}} \right|\] done
clear
B)
\[\left| (\vec{b}-\vec{p})+\frac{((\vec{p}-\vec{a}).\vec{b})\vec{b}}{{{\left| {\vec{b}} \right|}^{2}}} \right|\] done
clear
C)
\[\left| (\vec{a}-\vec{p})+\frac{((\vec{p}-\vec{b}).\vec{b})\vec{b}}{{{\left| {\vec{b}} \right|}^{2}}} \right|\] done
clear
D)
None of these done
clear
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question_answer62)
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)
\[\vec{r}=\vec{b}+{{\mu }_{1}}(\vec{a}+\vec{c})\] done
clear
B)
\[\vec{r}=\vec{b}+{{\lambda }_{1}}(\vec{a}-\vec{c})\] done
clear
C)
\[\vec{r}=2\vec{b}+{{\lambda }_{2}}(\vec{a}-\vec{c})\] done
clear
D)
None of these done
clear
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question_answer63)
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)
\[V\le \frac{9}{2}\] done
clear
B)
\[V\ge \frac{9}{2}\] done
clear
C)
\[V=\frac{9}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer64)
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)
\[\sqrt{3}\] done
clear
B)
\[\sqrt{6}\] done
clear
C)
3 done
clear
D)
None of these done
clear
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question_answer65)
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)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=3\] done
clear
B)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=6\] done
clear
C)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=9\] done
clear
D)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=12\] done
clear
View Solution play_arrow
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question_answer66)
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)
\[\frac{1}{3},\frac{2}{3},\frac{2}{3}\] done
clear
B)
\[-\frac{1}{3},\frac{2}{3},\frac{2}{3}\] done
clear
C)
\[-\frac{1}{3},\frac{2}{3},-\frac{2}{3}\] done
clear
D)
\[-\frac{1}{3},-\frac{2}{3},\frac{2}{3}\] done
clear
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question_answer67)
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)
\[(1,2,3)\] done
clear
B)
\[(2,4,6)\] done
clear
C)
\[\left( \frac{4}{3},\frac{8}{3},\frac{12}{3} \right)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer68)
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)
\[\frac{{{l}_{1}}-{{l}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\sin \frac{\theta }{2}}\] done
clear
B)
\[\frac{{{l}_{1}}-{{l}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\cos \frac{\theta }{2}}\] done
clear
C)
\[\frac{{{l}_{1}}-{{l}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\sin \frac{\theta }{2}}\] done
clear
D)
\[\frac{{{l}_{1}}-{{l}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\cos \frac{\theta }{2}}\] done
clear
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question_answer69)
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)
\[x+3y+6z=7\] done
clear
B)
\[2x+6y+12z=-13\] done
clear
C)
\[2x+6y+12z=13\] done
clear
D)
\[x+3y+6z=-7\] done
clear
View Solution play_arrow
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question_answer70)
A variable plane at a distance of 1 unit form the origin cuts the coordinate axes at A, B and C. if the centroid \[D(x,y,z)\] of triangle ABC satisfies the relation \[\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}+\frac{1}{{{z}^{2}}}=k\], then the value of k is
A)
3 done
clear
B)
1 done
clear
C)
1/3 done
clear
D)
9 done
clear
View Solution play_arrow
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question_answer71)
A variable plane passes through a fixed point (1, 2, 3). The locus of the foot of the perpendicular from the origin to this plane is given by
A)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-14=0\] done
clear
B)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+x+2y+3z=0\] done
clear
C)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-x-2y-3z=0\] done
clear
D)
None of these done
clear
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question_answer72)
The plane \[2x-3y+6z-11=0\] makes an angle \[{{\sin }^{-1}}(a)\] with the x-axis. Then the value of a is-
A)
\[\frac{\sqrt{3}}{2}\] done
clear
B)
\[\frac{\sqrt{2}}{3}\] done
clear
C)
\[\frac{3}{7}\] done
clear
D)
\[\frac{2}{7}\] done
clear
View Solution play_arrow
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question_answer73)
If O, P are the points (0, 0, 0), (2, 3, -1) respectively, then what is the equation to the plane through P at right angles to OP?
A)
\[2x+3y+z=16\] done
clear
B)
\[2x+3y-z=14\] done
clear
C)
\[2x+3y+z=14\] done
clear
D)
\[2x+3y-z=0\] done
clear
View Solution play_arrow
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question_answer74)
Which one of the following planes is normal the plane \[3x+y+z=5?\]
A)
\[x+2y+z=6\] done
clear
B)
\[x-2y+z=6\] done
clear
C)
\[x+2y-z=6\] done
clear
D)
\[x-2y-z=6\] done
clear
View Solution play_arrow
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question_answer75)
What is the angle between the lines\[\frac{x-2}{1}=\frac{y+1}{-2}\] and \[\frac{x-1}{1}=\frac{2y+3}{3}=\frac{z+5}{2}?\]
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\frac{\pi }{3}\] done
clear
C)
\[\frac{\pi }{6}\] done
clear
D)
None of the above done
clear
View Solution play_arrow
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question_answer76)
The perpendicular distance of P (1, 2, 3) form the lie \[\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\] is
A)
7 done
clear
B)
5 done
clear
C)
0 done
clear
D)
6 done
clear
View Solution play_arrow
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question_answer77)
Equation of the plane through the mid-point of the line segment joining the points P(4, 5, -10) and Q(-1, 2, 1) and perpendicular to PQ is
A)
\[\vec{r}.\left( \frac{3}{2}\hat{i}+\frac{7}{2}\hat{j}-\frac{9}{2}\hat{k} \right)=45\] done
clear
B)
\[\vec{r}.\left( -\hat{i}+2\hat{j}-\hat{k} \right)=\frac{135}{2}\] done
clear
C)
\[\vec{r}.(5\hat{i}+3\hat{j}-11\hat{k})+\frac{135}{2}=0\] done
clear
D)
\[\vec{r}.(5\hat{i}+3\hat{j}-11\hat{k})=\frac{135}{2}\] done
clear
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question_answer78)
The equation of the plane through (1, 1, 1) and passing through the line of intersection of the planes \[x+2y-z+1=0\] and \[3x-y-4z+3=0\]is
A)
\[8x+5y-11z+8=0\] done
clear
B)
\[8x+5y-11z+8=0\] done
clear
C)
\[8x-5y-11z+8=0\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer79)
Chord AB is a diameter of the sphere \[\left| \vec{r}-2\vec{i}-\vec{j}+6\vec{k} \right|=\sqrt{18.}\] if the coordinates of A are (3, 2,-2), then the coordinates of B are
A)
(1, 0, 10) done
clear
B)
(1, 0, -10) done
clear
C)
(-1, 0, 10) done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer80)
The equation of a sphere is \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-10z=0\]. If one end point of a diameter of the sphere is (-3, -4, 5), what is the other end point?
A)
\[(-3,-4,-5)\] done
clear
B)
\[(3,4,5)\] done
clear
C)
\[(3,4,-5)\] done
clear
D)
\[(-3,4,-5)\] done
clear
View Solution play_arrow