IISER Mathematics - Three Dimensional Geometry
Exam Duration: 45 Mins Total Questions : 30
A line make angles \(\alpha ,\beta ,\gamma ,\delta \) with the diagonals of a cube, then \({ cos }^{ 2 }\alpha +{ cos }^{ 2 }\beta +{ cos }^{ 2 }\gamma +{ cos }^{ 2 }\delta \), is
- (a)
\(\frac { 1 }{ 3 } \)
- (b)
\(\frac { 2}{ 3 } \)
- (c)
1
- (d)
\(\frac { 4}{ 3 } \)
The number of lines equally inclined to the three axes are
- (a)
2
- (b)
4
- (c)
6
- (d)
8
The equation of the plane passing through the point (2,3,-1) and perpendicular to the lines whose direction ratios are proportional to 3,-4,7, is
- (a)
3x+4y+7z = 13
- (b)
3x-4y-7z = 13
- (c)
3x-4y+7z+13 = 0
- (d)
NONE OF THESE
A plane meets the coordinate axes in A,B,C such that the centroid of the triangle is (a,b,c). Then equation of the plane, is
- (a)
\(\frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =3\)
- (b)
\(\frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =2\)
- (c)
\(\frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =1\)
- (d)
NONE OF THESE
The plane 2x-3y+4=0, is parallel to the
- (a)
x-axis
- (b)
y-axis
- (c)
z-axis
- (d)
NONE OF THESE
Locus of the point, the sum of the square of whose distances from the planes
x+y+z=0,x-z=0,x-2y+z=0 is 9, is
- (a)
x2+y2+z2=9
- (b)
x2-y2+z2=9
- (c)
x2+y2-z2=9
- (d)
x2-y2-z2=9
The equation of the tangent plane to the sphere x2+y2+z2+2x-y+z-14=0 at the point (2,1,2) is
- (a)
6x-y+5z-23=0
- (b)
6x+y+5z-23=0
- (c)
x+6y-5z-23=0
- (d)
NONE OF THESE
If the straight line \({x-\alpha\over l}={y-\beta\over m}={z-r\over n}\) intersect the curve ax2+bx2=1, z=0 then the value of a \((\alpha n-\gamma l)^2+b(\beta n-\gamma m)^2\) is
- (a)
n2
- (b)
m2
- (c)
l2
- (d)
0
A line with positive direction cosines passes through the point P(2,-1,2) and makes equal angles with the coordinate axes. The line meet the plane 2x+y+z=9 at point Q. The length of the line segment PQ equals.
- (a)
\(\sqrt5\)
- (b)
\(3\sqrt2\)
- (c)
\(\sqrt3\)
- (d)
\(\sqrt7\)
A variable plane passes through a fixed point (a,b,c) and meets the axis of referene in A,B and C. The locus of the point of intersection of the planes through A,B and C parallel to the coordinate plane is
- (a)
ax-1+by-1-cz-1=2
- (b)
ax-1+by-1+cz-1=2
- (c)
ax-1-by-1-cz-1=2
- (d)
ax-1-by-1+cz-1=1
The vector equation of the plane passing through the intersection of the plane \(r.(\hat i+\hat j+\hat k)=6\) and \(r.(2\hat i+3\hat j+4\hat k)=-5\) and the point (1,1,1) is
- (a)
\(r.(20\hat i+21\hat j+26\hat k)=69\)
- (b)
\(r.(19\hat i+23\hat j+26\hat k)=68\)
- (c)
\(r.(20\hat i+23\hat j+26\hat k)=69\)
- (d)
\(r.(20\hat i+22\hat j+26\hat k)=69\)
A line segment has length 63 and direction ratios 3,-2,6. If the line makes an obtuse angle with X-axis, then the components of vector are
- (a)
-27,20,18
- (b)
-27,18,-54
- (c)
18,-54,27
- (d)
-27,-18,54
The equation of the planes through the line \({x-1\over2}={y-2\over3}={z-3\over4}\) are parallel to the X-axis and Z-axis, are
- (a)
4y-3z-1=0, 3x+2y-1=0
- (b)
4y-3z+1=0, 3x-2+1=0
- (c)
4y-5z+1=0, 3x-5y+2=0
- (d)
5y-3z+1=0, 3x-3y+4=0
If a line makes an angle of \(\pi\over4\) with the positive directions of each of X-axis and Y-axis, then the angle that the line makes with the positive direction of Z-axis is
- (a)
\(\pi\over6\)
- (b)
\(\pi\over3\)
- (c)
\(\pi\over4\)
- (d)
\(\pi\over2\)
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 = \(\frac{1}{36}\)
If from the point p (a, b, c) perpendiculars PL, PM be drawn to YOZ and ZOX planes, then the equation of the plane OLM is
- (a)
\(\frac { x }{ a } +\frac { y }{ b } +\frac { z }{ c } =0\)
- (b)
\(\frac { x }{ a } -\frac { y }{ b } +\frac { z }{ c } =0\)
- (c)
\(\frac { x }{ a } -\frac { y }{ b } +\frac { z }{ c } =0\)
- (d)
\(\frac { x }{ a } -\frac { y }{ b } -\frac { z }{ c } =0\)
If P, Q, R, S are the points (4,5,3) (6,3,4), (2,4, -1), (0,5,1) the length of projetcion RS on PQ is
- (a)
4/3
- (b)
2/3
- (c)
4
- (d)
6
The plane lx + my = 0 is rotated about its line of intersection with the x O y plane through an angle \(\alpha\) . Then the equation of the plane is lx + my + nz = 0, where n is
- (a)
\(\pm \sqrt { \left( { l }^{ 2 }+{ m }^{ 2 } \right) } \cos { \alpha } \)
- (b)
\(\pm \sqrt { \left( { l }^{ 2 }+{ m }^{ 2 } \right) } \sin { \alpha } \)
- (c)
\(\pm \sqrt { \left( { l }^{ 2 }+{ m }^{ 2 } \right) } \tan { \alpha } \)
- (d)
none of these
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)
do not intersect
- (b)
intersect
- (c)
intersect at (4,0, - 1)
- (d)
intersect at (1,1, - 1)
Let two planes P1 : 2x - y + z = 2 and P2 : x + 2y - z = 3 are given
The equation of the bisector of angle of the planes P1 and P2 which not containing origin is
- (a)
x - 3y + 2z + 1 = 0
- (b)
x + 3y = 5
- (c)
x + 3y + 2z + 2 = 0
- (d)
3x + y = 5
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)
\(\frac{1}{\sqrt3}\)
The vector and the cartesian equations of the line that passes through the origin and (5,-2,3) are A and respectively. Here, A and B refer to
- (a)
\(\vec { r } =5\hat { i } -2\hat { j } +3\hat { k } ;5x=-2y=3z\)
- (b)
\(\vec { r } =\lambda (5\hat { i } -2\hat { j } +3\hat { k } );5x=-2y=3z\)
- (c)
\(\vec { r } =\lambda (5\hat { i } -2\hat { j } +3\hat { k } );\frac { x }{ 5 } =\frac { y }{ -2 } =\frac { z }{ 3 } \)
- (d)
None of these
The equation of the line in vector and cartesian form that passes through the point with position vector \(2\hat { i } +\hat { j } +4\hat { k } \) and is in the direction \(\hat { i } +2\hat { j } -\hat { k } \)are
- (a)
\({ r } =(2\hat { i } -\hat { j } +4\hat { k } )+\lambda (\hat { i } -2\lambda \hat { j } +\hat { k } );\frac { x-1 }{ 2 } =\frac { y-2 }{ -1 } =\frac { z+1 }{ -4 } \)
- (b)
\( { r } =(\hat { i } +2\hat { j } -\hat { k } )+\lambda (2\hat { i } -\hat { j } +4\hat { k } );\frac { x-1 }{ 2 } =\frac { y-2 }{ -1 } =\frac { z+1 }{ 4 } \)
- (c)
\({ r } =(\hat { i } +2\hat { j } -\hat { k } )+\lambda (2\hat { i } -\hat { j } +4\hat { k } );\frac { x-2 }{ 1 } =\frac { y+1 }{ 2 } =\frac { z-4 }{ -1 } \)
- (d)
\( { r } =(2\hat { i } -\hat { j } +4\hat { k } )+\lambda (\hat { i } -2\hat { j } -\hat { k } );\frac { x-2 }{ 1 } =\frac { y+1 }{ 2 } =\frac { z-4 }{ -1 } \)
The vector equation of the line through the points A(3,4,-7) and B(1,-1 6) is
- (a)
\(\vec { r } =(3\hat { i } -4\hat { j } -7\hat { k } )+\lambda (\hat { i } -\hat { j } +6\hat { k } )\)
- (b)
\(\vec { r } =(\hat { i } -\hat { j } +6\hat { k } )+\lambda (3\hat { i } -4\hat { j } -7\hat { k } )\)
- (c)
\(\vec { r } =(3\hat { i } -4\hat { j } -7\hat { k } )+\lambda (-2\hat { i } -5\hat { j } +13\hat { k } )\)
- (d)
\(\vec { r } =(\hat { i } -\hat { j } +6\hat { k } )+\lambda (4\hat { i } +3\hat { j } -\hat { k } )\)
The angle between the lines \(\vec { r } =(4\hat { i } +\hat { j } )+s(2\hat { i } +\hat { j } -3\hat { k } )\)and \(\vec { r } =(\hat { i } +\hat { j } +2\hat { k } )+t(\hat { i } +3\hat { j } +2\hat { k } )\) is
- (a)
\(\frac{3\pi}{2}\)
- (b)
\(\frac{\pi}{3}\)
- (c)
\(\frac{2\pi}{3}\)
- (d)
\(\frac{\pi}{6}\)
Find the angle between the pair of lines given by \(\vec { r } =3\hat { i } +2\hat { j } -4\hat { k } +\lambda (\hat { i } +2\hat { j } +2\hat { k } )\)and \(\vec { r } =5\hat { i } -2\hat { j } +\mu (3\hat { i } +2\hat { j } +6\hat { k } )\)
- (a)
\({ cos }^{ -1 }\left( \frac { 19 }{ 21 } \right) \)
- (b)
\({ cos }^{ -1 }\left( \frac { 23 }{ 19 } \right) \)
- (c)
\({ cos }^{ -1 }\left( \frac { 17 }{ 13 } \right) \)
- (d)
\({ cos }^{ -1 }\left( \frac { 13 }{ 9 } \right) \)
Shortest distance between the two lines \(\vec { r } =(8+3\lambda )\hat { i } -(9+16\lambda )\hat { j } +(10+7\lambda )\hat { k } and\vec { r } =15\hat { i } +29\hat { j } +5\hat { k } +\mu (3\hat { i+8\hat { j } -5\hat { k } ) } \)
- (a)
84
- (b)
14
- (c)
21
- (d)
16
The area of the quadrilateral ABCD, where A(0,4,1), B(2,3,-1), C(4,5,0), and D(2,6,2), is equal to
- (a)
9sq. units
- (b)
18sq.units
- (c)
27sq.units
- (d)
81sq.units
The locus represented by xy+yz=0 is
- (a)
A pair of perpendicular line
- (b)
A pair of parallel lines
- (c)
A pair of parallel planes
- (d)
A pair of perpendicular planes
Consider the lines
\({ L }_{ 1 }:\frac { x+1 }{ 3 } =\frac { y+2 }{ 1 } =\frac { z+1 }{ 2 } ,{ L }_{ 2 }:\frac { x-2 }{ 1 } =\frac { y+2 }{ 2 } =\frac { z-3 }{ 3 } \)
Statement -I: The distance of point (1,1,1) from the plane passing through the point (-1,-2,-1) and whose normal is perpendicular to both the lines L1 and L2 is \(\frac{13}{5\sqrt3}\)
Statement -II: The unit vector perpendicular to both the lines L1 and L2 is \(\frac { -\hat { i } -7\hat { j } +5\hat { k } }{ 5\sqrt { 3 } } \).
- (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.