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question_answer1)
For the line\[\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}\], which of the following is incorrect?
A)
It lines in the plane \[x-2y+z=0\] done
clear
B)
It is same as line\[\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\]. done
clear
C)
It passes through (2, 3, 5). done
clear
D)
It is parallel of the plane \[x-2y+z-6=0\] done
clear
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question_answer2)
The coordinates of the foot of the perpendicular drawn from the origin to the line joining the points (\[-\]9, 4, 5) and (10, 0, \[-\]1) will be
A)
(-3, 2, 1) done
clear
B)
(1, 2, 2) done
clear
C)
(4, 5, 3) done
clear
D)
none of these done
clear
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question_answer3)
The intercept made by the plane \[\vec{r}\cdot \vec{n}=q\]on the x-axis is
A)
\[\frac{q}{\hat{i}\cdot \vec{n}}\] done
clear
B)
\[\frac{\hat{i}\cdot \vec{n}}{q}\] done
clear
C)
\[\frac{\hat{i}\cdot \vec{n}}{q}\] done
clear
D)
\[\frac{q}{\left| {\vec{n}} \right|}\] done
clear
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question_answer4)
If the foot of the perpendicular from the origin to a plane is P (a, b, c), the equation of the plane is
A)
\[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=3\] done
clear
B)
\[ax+by+cz=3\] done
clear
C)
\[ax+by+cz={{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done
clear
D)
\[ax+by+cz=a+b+c\] done
clear
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question_answer5)
The length of projection of the line segment joining the points \[\left( 1,\text{ }0,\,\,-1 \right)\] and \[\left( -1,\,\,2,\,\,2 \right)\] on the plane \[x+3y-5z=6\] is equal to
A)
2 done
clear
B)
\[\sqrt{\frac{271}{53}}\] done
clear
C)
\[\sqrt{\frac{472}{31}}\] done
clear
D)
\[\sqrt{\frac{474}{35}}\] done
clear
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question_answer6)
Let A (1, 1, 1), B (2, 3, 5) and C (\[-\]1,0, 2) be three points, then equation of a plane parallel to the plane ABC which is at distance 2 is
A)
\[2x-3y+z+2\sqrt{14}=0\] done
clear
B)
\[2x-3y+z-\sqrt{14}=0\] done
clear
C)
\[2x-3y+z+2=0\] done
clear
D)
\[2x-3y+z-2=0\] done
clear
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question_answer7)
The point of intersection of the line passing through (0, 0, 1) and intersecting the lines \[x+2y+z=1\], \[-x+y-2z=2\] and \[x+y=2,\,\,x+z=2\] with xy plane is
A)
\[\left( \frac{5}{3},-\frac{1}{3},0 \right)\] done
clear
B)
(1, 1, 0) done
clear
C)
\[\left( \frac{2}{3},-\frac{1}{3},0 \right)\] done
clear
D)
\[\left( -\frac{5}{3},\frac{1}{3},0 \right)\] done
clear
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question_answer8)
The reflection of the point \[\vec{a}\] in the plane \[\vec{r}\].\[\vec{n}\] =q is
A)
\[\vec{a}+\frac{(\vec{q}-\vec{a}\,\cdot \vec{n})}{\left| {\vec{n}} \right|}\] done
clear
B)
\[\vec{a}+2\left( \frac{(\vec{q}-\vec{a}\,\cdot \vec{n})}{{{\left| {\vec{n}} \right|}^{2}}} \right)\vec{n}\] done
clear
C)
\[\vec{a}+\frac{2(\vec{q}-\vec{a}\,\cdot \vec{n})}{\left| {\vec{n}} \right|}\vec{n}\] done
clear
D)
none of these done
clear
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question_answer9)
If a line makes an angle of \[\frac{\pi }{4}\]with the positive direction of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is
A)
\[\frac{\pi }{3}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
\[\frac{\pi }{6}\] done
clear
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question_answer10)
The length of the perpendicular drawn from (1, 2, 3) to the line\[\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\]is
A)
4 done
clear
B)
5 done
clear
C)
6 done
clear
D)
7 done
clear
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question_answer11)
The centre of the circle given by\[\vec{r}\cdot (\hat{i}+2\hat{j}+2\hat{k})=15\] and \[\left| \vec{r}-(\hat{j}+2\hat{k}) \right|=4\]is
A)
(0, 1, 2) done
clear
B)
(1, 3, 4) done
clear
C)
(-1, 3, 4) done
clear
D)
none of these done
clear
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question_answer12)
The radius of the circle in which the sphere \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2z-2y-4z-19=0\] is cut by the plane \[x+2y+2z+7=0\]is
A)
2 done
clear
B)
3 done
clear
C)
4 done
clear
D)
1 done
clear
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question_answer13)
The plane which passes through the point (3, 2, 0) and the line\[\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}\]is
A)
\[x-y+z=1\] done
clear
B)
\[x+y+z=5\] done
clear
C)
\[x+2y-z=1\] done
clear
D)
\[2x-y+z=5\] done
clear
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question_answer14)
The plane \[\vec{r}\,\cdot \vec{n}=q\] will contain the line \[\vec{r}=\vec{a}+\lambda \vec{b}\], If
A)
\[\vec{b}\cdot \vec{n}\ne 0\], \[\vec{a}\,\cdot \vec{n}\ne q\] done
clear
B)
\[\vec{b}\,\cdot \vec{n}=0\],\[\vec{a}\cdot \vec{n}\ne q\] done
clear
C)
\[\vec{b}\cdot \vec{n}=0\],\[\vec{a}\,\cdot \vec{n}=q\] done
clear
D)
\[\vec{b}\,\cdot \vec{n}\ne 0\],\[\vec{a}\,\cdot \vec{n}=q\] done
clear
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question_answer15)
The projection of the line \[\frac{x+1}{1}=\frac{y}{2}=\frac{z-1}{3}\]on the plane \[x-2y+z=6\] is the line of intersection of the plane with the plane
A)
\[2x+y+2=0\] done
clear
B)
\[3x+y-z=2\] done
clear
C)
\[2x-3y+8z=3\] done
clear
D)
none of these done
clear
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question_answer16)
The coordinates of the point p on the line\[\vec{r}=(\hat{i}+\hat{j}+\hat{k})+\lambda (-\hat{i}+\hat{j}-\hat{k})\] which is nearest to the origin is
A)
\[\left( \frac{2}{3},\frac{4}{3},\frac{2}{3} \right)\] done
clear
B)
\[\left( -\frac{2}{3},-\frac{4}{3},\frac{2}{3} \right)\] done
clear
C)
\[\left( \frac{2}{3},\frac{4}{3},-\frac{2}{3} \right)\] done
clear
D)
None of these done
clear
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question_answer17)
The pair of lines whose direction cosines are given by the equations \[3l+m+5n=0\] and \[6mn-2nl+5lm=0\] are
A)
parallel done
clear
B)
perpendicular done
clear
C)
inclined at \[{{\cot }^{-1}}\left( \frac{1}{6} \right)\] done
clear
D)
none of these done
clear
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question_answer18)
Line \[\vec{r}=\vec{a}+\lambda \vec{b}\] will not meet the plane \[\vec{r}\cdot \vec{n}=q\], if
A)
\[\vec{b}\cdot \vec{n}=0,\,\,\vec{a}\cdot \vec{n}=q\] done
clear
B)
\[\vec{b}\cdot \vec{n}\ne 0,\,\,\vec{a}\cdot \vec{n}\ne q\] done
clear
C)
\[\vec{b}\cdot \vec{n}=0,\,\,\vec{a}\cdot \vec{n}\ne q\] done
clear
D)
\[\vec{b}\cdot \vec{n}\ne 0,\,\,\vec{a}\cdot \vec{n}=q\] done
clear
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question_answer19)
What is the equation of the plane which passes through the z-axis and is perpendicular to the line\[\frac{x-a}{\cos \theta }=\frac{y+2}{\sin \theta }=\frac{z-3}{0}\]?
A)
\[x+y\,\text{tan}\,\theta =0\] done
clear
B)
\[y+x\,\text{tan}\,\theta =0\] done
clear
C)
\[x\,\cos \theta -y\,\sin \theta =0\] done
clear
D)
\[x\,\sin \theta -y\,\cos \theta =0\] done
clear
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question_answer20)
The direction ratios (d,r,'s) of the normal to the plane throuth (1, 0, 0) and (0, 1, 0) which makes an angle \[\pi /4\]with the 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_answer21)
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\[\vec{r}\cdot (\hat{i}+5\hat{j}+\hat{k})=5\]is______.
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question_answer22)
\[{{L}_{1}}\] and \[{{L}_{2}}\] are two lines whose vector equations are \[{{L}_{1}}:\vec{r}=\lambda \,((cos\theta +\sqrt{3})\hat{i}+(\sqrt{2}sin\theta )\hat{j}+(cos\theta -\sqrt{3})\hat{k})\]\[{{L}_{2}}:\vec{r}=\mu (a\hat{i}+b\hat{j}+c\hat{k})\], 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 _____.
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question_answer23)
If the plane \[\frac{x}{2}+\frac{y}{3}+\frac{z}{6}=1\] cuts the axes of coordinates at points A, B and C, then the area (sq. units) of the triangle ABC is _____.
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question_answer24)
If the projection of the line \[\frac{x}{2}=\frac{y-1}{2}=\frac{z-1}{1}\] on a pane P is \[\frac{x}{1}=\frac{y-1}{1}=\frac{z-1}{-1}\]. Then the distance of plane P from origin is ______.
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question_answer25)
The radius of the circle in which the sphere \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2x-2y-4z-19=0\]is cut by the plane \[x+2y+2z+7=0\]is _____.
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