Mathematics - Three Dimensional Geometry
Exam Duration: 45 Mins Total Questions : 30
The ratio in which the xy-plane divides the join of (-2,4,7) and (3,-5,8) is
- (a)
7:8
- (b)
-7:8
- (c)
-8:7
- (d)
NONE OF THESE
The equation of the plane through the point (1,2,-1) and perpendicular to the line of intersection of the planes, (x+4y-2z)=2 and 3x-y+z=1, is
- (a)
2x-7y+13z-1=0
- (b)
2x-7y-13z=1
- (c)
2x+7y-z-17=0
- (d)
NONE OF THESE
The symmetrical form of the equation of the line of intersection of the planes
x+2y+3z=3
and 2x-y+4z=1
is
- (a)
\(\frac { x-1 }{ 11 } =\frac { y-1 }{ -2 } =\frac { z }{ 5 }\)
- (b)
\(\frac { x-1 }{ 11 } =\frac { y-1 }{ 2 } =\frac { z }{- 5 }\)
- (c)
\(\frac { x-1 }{ 11 } =\frac { y-1 }{ -2 } =\frac { z }{- 5 }\)
- (d)
NONE OF THESE
The equation of the sphere passing througgh the points (0,0,0), (a,0,0), (0,b,0), (0,0,c) is
- (a)
x2+y2+z2+ax+by+cz=0
- (b)
x2+y2+z2-ax+by-cz=0
- (c)
x2+y2+z2-ax+by+cz=0
- (d)
x2+y2+z2-ax-by-cz=0
The pair of lines whose direction cosines are given by the equations 3l+m+5n=0 and 6mn-2nl+5lm-0 are
- (a)
parallel
- (b)
inclined at cos-1(1/3)
- (c)
inclined at cos-1(1/6)
- (d)
None of the above
If the planes x-cy-bz=0, cx-y+az=0 and bx+ay-z=0 pass through a straight line, then the value of a2+b2+c2+2abc is
- (a)
2
- (b)
3
- (c)
0
- (d)
1
The distance of the point (1,-5,9) from the plane x-y+z=5 measured along a straight line x=y=z is
- (a)
\(3\sqrt5\)
- (b)
\(10\sqrt3\)
- (c)
\(5\sqrt3\)
- (d)
\(3\sqrt{10}\)
Let the line \({x-2\over3}={y-1\over{-5}}={z+2\over2}\) lies in the plane x+3y-\(\alpha z\)+\(\beta\)=0. Then \((\alpha,\beta)\) equals
- (a)
(6,-17)
- (b)
(-6,7)
- (c)
(5,-15)
- (d)
(-5,15)
The coordinates of a point on the line \(\frac { x-1 }{ 2 } =\frac { y+1 }{ -3 } =z\) at a distance \(4\sqrt{4}\) from the point ( 1, -1, 0 ) nearer the origin are
- (a)
( 9, -13, 4 )
- (b)
\((8\sqrt{14},-12,-1)\)
- (c)
\((-8\sqrt{14},12,1)\)
- (d)
(-7, 11, -4)
A variable plane which remains at a constant distance p from the origin cuts the coordinate axes in A, B, C. The locus of the centroid of the tetrahedron OABC is
y2z2 + z2x2 + x2y2 = kx2 y2 z2, where k is equal to
- (a)
9p2
- (b)
\(\frac{9}{{p}^{2}}\)
- (c)
\(\frac {7}{{p}^{2}}\)
- (d)
\(\frac {16}{{p}^{2}}\)
The extremities of a diameter of a sphere lie on positive y and positive z-axes at distances 2 and 4 from the origin, respectively, then
- (a)
sphere passes through the origin
- (b)
centre of the sphere is (0, 1, 2)
- (c)
radius of the sphere is \(\sqrt{5}\)
- (d)
equation of a diameter is \(\frac{x}{0}=\frac{y-2}{1}=\frac{z-4}{-2}\)
Let two planes P1 : 2x - y + z = 2 and P2 : x + 2y - z = 3 are given
The equation of the plane through the intersection of P1 and P2 and the point (3,2,1) is
- (a)
3x - y + 2z - 9 = 0
- (b)
x - 3y - 2z + 1 = 0
- (c)
2x - 3y + z - 1 = 0
- (d)
4x - 3y + 2z - 8 = 0
The perpendicular distance of the origin from the plane which makes intercepts 12, 3 and 4 on x,y,z axes respectively, is
- (a)
13
- (b)
11
- (c)
17
- (d)
none of these
Which of the following is false?
- (a)
30o. 45o, 60o can be the direction angles of a line in space.
- (b)
90o. 135o, 45o can be the direction angles of a line in space.
- (c)
120o. 60o, 45o can be the direction angles of a line in space.
- (d)
60o. 45o, 60o can be the direction angles of a line in space.
The lines \(\frac { x-1 }{ 3 } =\frac { y-1 }{ -1 } =\frac { z+1 }{ 0 } \) and \(\frac { x-4 }{ 2 } =\frac { y-0 }{ 0 } =\frac { z+1 }{ 3 } \)
- (a)
(0,1,-1)
- (b)
(1,1,-1)
- (c)
(4,0,-1)
- (d)
none of these
The angle between the lines \(\frac { x-5 }{ 7 } =\frac { y+2 }{ -5 } =\frac { z }{ 1 } and\frac { x }{ 1 } =\frac { y }{ 2 } =\frac { z }{ 3 } \)is
- (a)
0
- (b)
\(\frac{\pi}{2}\)
- (c)
\(\frac{\pi}{3}\)
- (d)
\(\frac{\pi}{4}\)
The equation of the line passing through the point (1,2,-4) and perpendicular to the two lines.\(\frac { x }{ 1 } =\frac { y }{ 2 } =\frac { z }{ -1 } and\frac { x }{ -1 } =\frac { y }{ 1 } =\frac { z }{ -2 } \) will be
- (a)
\(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z+4 }{ 6 } \)
- (b)
\(\frac { x-1 }{ -2 } =\frac { y-2 }{ 3 } =\frac { z+4 }{ 8 } \)
- (c)
\(\frac { x-1 }{ 3 } =\frac { y-2 }{ 2 } =\frac { z+4 }{ 8 } \)
- (d)
None of these
Equation of line passing through (1,2,-3) and parallel to the line .\(\frac { x-2 }{ 1 } =\frac { y+1 }{ 3 } =\frac { z-1 }{ 4 } \)is
- (a)
\(\frac { x-1 }{ 1 } =\frac { y-2 }{ 3 } =\frac { z+3 }{ 4 } \)
- (b)
\(\frac { x-2 }{ 1 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ -3 } \)
- (c)
\(\frac { x-1 }{ 1 } =\frac { y-3 }{ 2 } =\frac { z-1 }{ -3 } \)
- (d)
None of these
If lines \(\frac { x-1 }{-3 } =\frac { y-2 }{ 2k } =\frac { z-3 }{ 2 } \) and\(\frac { x-1 }{ 3k } =\frac { y-5 }{ 1 } =\frac { z-6 }{ -5 } \) are mutually perpendicular, then k is equal to
- (a)
\(-\frac{10}{7}\)
- (b)
\(-\frac{7}{10}\)
- (c)
-10
- (d)
-7
The distence between the lines \(\vec { r } =\hat { i } +\hat { j } +\lambda (\hat { i } -2\hat { j } +3\hat { k } )\)and \(\vec { r } =(2\hat { i } -3\hat { k } )+\mu (\hat { i } -2\hat { j } +3\hat { k } )\quad \)
- (a)
\(\sqrt { \frac { 59 }{ 14 } } \)
- (b)
\(\sqrt { \frac { 59 }{ 7 } } \)
- (c)
\(\sqrt { \frac { 118 }{ 7 } } \)
- (d)
\(\sqrt { \frac { 59 }{ 7 } } \)
Find the length of perpendicular from the origin to the plane \({ \vec { r } }.(3\hat { i } -4\hat { j } +12\hat { k } )=5\)
- (a)
\(\frac{5}{13}\)
- (b)
\(\frac{5}{\sqrt{3}}\)
- (c)
\(\frac{5}{23}\)
- (d)
\(\frac{\sqrt5}{13}\)
Equation of a plane passing through three non-collinear points is
- (a)
\((\vec { r } -\vec { a } ).[(\vec { b } -\vec { a } )\times (\vec { c } -\vec { a } )]=0\)where \(\vec { a } ,\vec { b } ,\vec { c } \) are the position vectors of three non-collinear points
- (b)
\(\left| \begin{matrix} { x-x }_{ 1 } & y{ -y }_{ 1 } & { z-z }_{ 1 } \\ { x }_{ 2 }-{ x }_{ 1 } & { y }_{ 2 }-{ y }_{ 1 } & { z }_{ 2 }-{ z }_{ 1 } \\ { x }_{ 3 }-{ x }_{ 1 } & { y }_{ 3 }-{ y }_{ 1 } & { z }_{ 3 }-{ z }_{ 1 } \end{matrix} \right| =0\)
- (c)
both (a)and(b)
- (d)
None of these
The equation of a plane which cut off intercepts a,b,c on X, Y, Z-axes respectively is
- (a)
ax+by+cz=0
- (b)
ax+by+cz=1
- (c)
\(\frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =0\)
- (d)
\(\frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =1\)
If the lines \(\frac { x-2 }{ 1 } =\frac { y-9 }{ 2 } =\frac { z-13 }{ 3 } and\quad \frac { x-1 }{ 1 } =\frac { y-1 }{ -2 } =\frac { z+2 }{ 3 } \) are coplanar, then a=
- (a)
2
- (b)
-2
- (c)
3
- (d)
-3
Statement -I: The pair of lines given by \(\therefore \vec { r } =\hat { i } -\hat { j } +\lambda (2\hat { i } +\hat { k } )\)and \(\therefore \vec { r } =2\hat { i } -\hat { k } +\mu (2\hat { i } +\hat { j } -\hat { k } )\) intersect.
Statement -II: Two lines intersect each other, if they are not parallel and shortest distance=0.
- (a)
If both Statement -I and Statement -II are true but Statement -II is the correct explanation of Statement -I.
- (b)
If both Statement -I and Statement -II are true and Statement -II is not the correct explanation of Statement -I.
- (c)
If Statement -I is true but Statement -II is false.
- (d)
If Statement -I is false and Statement -II is true.