Mathematics - Three Dimensional Geometry
Exam Duration: 45 Mins Total Questions : 30
A tetrahedron has vertices O(0,0,0), A(1,2,1), B(2,1,3) and C(-1,1,2). Then the angle between the faces OAB and ABC will be
- (a)
\({ cos }^{ -1 }\left( \frac { 19 }{ 35 } \right) \)
- (b)
\({ cos }^{ -1 }\left( \frac { 17 }{ 31 } \right) \)
- (c)
300
- (d)
900
The plane 2x-3y+4=0, is parallel to the
- (a)
x-axis
- (b)
y-axis
- (c)
z-axis
- (d)
NONE OF THESE
The value of \(\lambda \) for which the lines
\(\frac { x-1 }{ 2 } =\frac { y }{ 3 } =\frac { z+1 }{ 4 } \)
and \(\frac { x+3 }{ 3 } =\frac { y-2 }{ \lambda } =\frac { z-5 }{ 3 } \)
are perpendicular, is
- (a)
6
- (b)
\(\frac { 1 }{ 6 } \)
- (c)
-6
- (d)
\(-\frac { 1 }{ 6 } \)
The angle between the line
\(\frac { x-1 }{ 2 } =\frac { y-1 }{ 3 } =\frac { z-3 }{ 4 }\)
and the plane x+2y-2z=3, is
- (a)
00
- (b)
300
- (c)
600
- (d)
900
The coordinates of the foot of the perpendicular from (1,0,2) to the line
\(\frac { x+1 }{ 3 } =\frac { y-2 }{ -3 } =\frac { z+1 }{ -1 } \) are
- (a)
\(\left( \frac { 8 }{ 19 } ,\frac { -11 }{ 19 } ,\frac { 28 }{ 19 } \right) \)
- (b)
\(\left( \frac { 8 }{ 19 } ,\frac {11 }{ 19 } ,\frac {- 28 }{ 19 } \right) \)
- (c)
\(\left( \frac { -8 }{ 19 } ,\frac { 11 }{ 19 } ,\frac { 28 }{ 19 } \right) \)
- (d)
\(\left( \frac { -8 }{ 19 } ,\frac { -11 }{ 19 } ,\frac { 28 }{ 19 } \right) \)
Equation of the plane containing the lines
\(\frac { x-2 }{ 1 } =\frac { y-3 }{ 1 } =\frac { z-4 }{ -1 } \)
\(\frac { x-2 }{ 1 } =\frac { y-4 }{ 1 } =\frac { z-5 }{ -1 } \)
is
- (a)
3x+2y+z-4=0
- (b)
3x+2y-z-4=0
- (c)
3x-2y+z-4=0
- (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 equation of the sphere which passes through the points (1,0,0), (0,1,0), (0,0,1) and which has radius as small as possible, is
- (a)
3(x2+y2+z2)+4x-4y-4z-4=0
- (b)
3(x2+y2+z2)-2x-2y-2z-1=0
- (c)
3(x2+y2+z2)+4x+4y-4z-4=0
- (d)
NONE OF THESE
The spheres x2+y2+z2=25
and x2+y2+z2-24x-40y+18z+225=0
- (a)
touch internally
- (b)
touch externally
- (c)
intersect
- (d)
NONE OF THESE
The equation of the planes through the intersection of the planes x+3y+6=0 and 3x-4y-4z=0 whose perpendicular distance from the origin is unity, are
- (a)
x+y-2z+3=0; x-2y-2z-3=0
- (b)
2x+y-2z+3=0; x-2y-2z-3=0
- (c)
x-y+2z+3=0; x+2y+2z+3=0
- (d)
ax-y+2z-3=0; x+2y+2z+3=0
If the angle \(\theta\) between the line \({x+1\over1}={y-1\over2}={z-2\over2}\) and the plane 2x-y+\(\lambda\)z+4=0 is such that \(sin\theta={1\over3}\) . The value of \(\lambda \) is
- (a)
\(-{4\over3}\)
- (b)
\(3\over4\)
- (c)
-\(3\over5\)
- (d)
\(5\over3\)
The locus of the point, the sum of squares of whose distances, from the planes x - z = 0, x - 2y + z = 0 and x + y + z = 0 is 36 is
- (a)
x2 + y2 + z2 = 6
- (b)
x2 + y2 + z2 = 36
- (c)
x2 + y2 + z2 = 216
- (d)
x-2 + y-2 + z-2 = \(\frac{1}{36}\)
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)
\(\frac {16}{{p}^{2}}\)
A plane moves such that its distance from the origin is a constant p. If it intersects the coordinate axes at A, B, C then the locus of the centroid of the triangle ABC is
- (a)
\(\frac { 1 }{ { x }^{ 2 } } +\frac { 1 }{ { y }^{ 2 } } +\frac { 1 }{ { z }^{ 2 } } =\frac { 1 }{ { p }^{ 2 } }\)
- (b)
\(\\ \frac { 1 }{ { x }^{ 2 } } +\frac { 1 }{ { y }^{ 2 } } +\frac { 1 }{ { z }^{ 2 } } =\frac { 9 }{ { p }^{ 2 } }\)
- (c)
\(\\ \frac { 1 }{ { x }^{ 2 } } +\frac { 1 }{ { y }^{ 2 } } +\frac { 1 }{ { z }^{ 2 } } =\frac { 2 }{ { p }^{ 2 } }\)
- (d)
\(\frac { 1 }{ { x }^{ 2 } } +\frac { 1 }{ { y }^{ 2 } } +\frac { 1 }{ { z }^{ 2 } } =\frac { 4 }{ { p }^{ 2 } } \)
The distance between two points P and Q is d and the length of their projections of PQ on the coordinate planes are d1, d2, d3. Then \({ d }_{ 1 }^{ 2 }+{ d }_{ 2 }^{ 2 }+{ d }_{ 3 }^{ 2 }={ kd }^{ 2 },\) where k is
- (a)
1
- (b)
5
- (c)
3
- (d)
2
The lines which intersect the skew lines y = mx, z = c ; y = - mx, z = -c and the x-axis lie on the surface
- (a)
cz = mxy
- (b)
cy = mxz
- (c)
xy = cmz
- (d)
none of these
The plane containing the two lines \(\frac { x-3 }{ 1 } =\frac { y-2 }{ 4 } =\frac { z-1 }{ 5 } \) and \(\frac { x-2 }{ 1 } =\frac { y+3 }{ -4 } =\frac { z+1 }{ 5 } \)i is 11x = my + nz = 28, where
- (a)
m = -1, n = 3
- (b)
m = 1, n = - 3
- (c)
m = -1, n = - 3
- (d)
m = 1, n = 3
If OABC is a tetrahedron such that OA2 + BC2 = OB2 + CA2 = OC2 + AB2, then
- (a)
OA \(\bot\) BC
- (b)
OB \(\bot\) CA
- (c)
OC \(\bot\) AB
- (d)
AB \(\bot\) BC
The cosines of the angle between any two diagonals of a cube is
- (a)
\(\frac{1}{3}\)
- (b)
\(\frac{1}{2}\)
- (c)
\(\frac{2}{3}\)
- (d)
\(\frac{1}{\sqrt3}\)
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.
P is a point on the line segment joining the points (3,2,-1) and (6,2,-2). If x-co-ordinate of P is 5, then its y co-ordinate is
- (a)
2
- (b)
1
- (c)
-1
- (d)
-2
The equation of the line joining the points (-3,4,11) and (1,-2,7) is
- (a)
\(\frac { x+3 }{ 2 } =\frac { y-4 }{ 3 } =\frac { z-11 }{ 4 } \)
- (b)
\(\frac { x+3 }{ -2 } =\frac { y-4 }{ 3 } =\frac { z-11 }{ 4 } \)
- (c)
\(\frac { x+3 }{ -2 } =\frac { y+4 }{ 3 } =\frac { z+11 }{ 4 } \)
- (d)
\(\frac { x+3 }{ 2 } =\frac { y+4 }{ -3 } =\frac { z+11 }{ 4 } \)
The angle between a line whose direction ratios are in the ratio 2:2:1 and a line joining (3,1,4) to (7,2,12) is
- (a)
cos-1(2/3)
- (b)
cos-1(-2/3)
- (c)
tan-1(2/3)
- (d)
None of these
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
The vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector \(=3\hat { i } +5\hat { j } -6\hat { k } \) .
- (a)
\(\vec { r } .(3\hat { i } +5\hat { j } -6\hat { k } )=7\)
- (b)
\(\vec { r } .(3\hat { i } +5\hat { j } -6\hat { k } )=\frac { 7 }{ \sqrt { 70 } } \)
- (c)
\(\vec { r } .(\frac { 3 }{ \sqrt { 70 } } \hat { i } +\frac { 5 }{ \sqrt { 70 } } \hat { j } -\frac { 6 }{ \sqrt { 70 } } \hat { k } )=7\sqrt { 70 } \)
- (d)
\(\vec { r } .(\frac { 3 }{ \sqrt { 70 } } \hat { i } +\frac { 5 }{ \sqrt { 70 } } \hat { j } -\frac { 6 }{ \sqrt { 70 } } \hat { k } )=7\)
Four points (0,-1,-1) (-4,4,4)(4,5,1) and (3,9,4) are coplanar. Find the equation of the plane containg them
- (a)
5x+7y+11z-4=0
- (b)
5x-7y-1z+4=0
- (c)
5x-7y-11z-4=0
- (d)
5x+7y-11z+4=0
Two lines \(\vec { r } ={ \vec { a } }_{ 1 }+\lambda { \vec { b } }_{ 1 }and\quad \vec { r } ={ \vec { a } }_{ 2 }+\mu { \vec { b } }_{ 2 }\) are said to be coplanar, if
- (a)
(a2-a1).(b1xb2)=0
- (b)
\(\left| \begin{matrix} { x }_{ 1 } & { y }_{ 1 } & { z }_{ 1 } \\ { a }_{ 1 } & { b }_{ 1 } & c_{ 1 } \\ { a }_{ 2 } & { b }_{ 2 } & { c }_{ 2 } \end{matrix} \right| =0\), where(x1,y1,z1) coordinates of apoint on any of the line and a1,b1bc1 and a2,b2,c2 are the direction ratios of \({ \vec { b } }_{ 1 }\)and\({ \vec { b } }_{2 }\)
- (c)
Both (a) and (b)
- (d)
None of the above
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
The equation of the plane through the point (2,5,-3) perpendicular to the planes x+2y+2z=1 and x-2y+3z=4 is
- (a)
3x-4y+2z-20=0
- (b)
7x-y+5z=30
- (c)
x-2y+z=11
- (d)
10x-y-4z=27
The sine of the angle between the straight line \(\frac { x-2 }{ 3 } =\frac { y-3 }{ 4 } =\frac { z-4 }{ 5 } \)and the plane 2x-2y+z=5 is
- (a)
\(\frac{10}{6\sqrt5}\)
- (b)
\(\frac{4}{5\sqrt2}\)
- (c)
\(\frac { 2\sqrt { 3 } }{ 5 } \)
- (d)
\(\frac { {\sqrt2 } }{ 5 } \)