1. The angle between the lines whose
direction cosines are given by the equation
\[t^{2}+m^{2}-n^{2}=0, l+m+n=0\]
a) \[\pi/6\]
b) \[\pi/4\]
c) \[\pi/3\]
d) \[\pi/2\]
Explanation:
2. The shortest distance between the skew
lines \[l_{1}: r=a_{1}+\lambda b_{1}\] and \[l_{2}: r=a_{2}+\mu b_{2}\]
is
a) \[\frac{(a_{2}-a_{1}).b_{1}\times b_{2}}{\mid b_{1}\times b_{2}\mid}\]
b) \[\frac{(a_{1}-b_{1}).a_{2}\times b_{2}}{\mid b_{1}\times b_{2}\mid}\]
c) \[\frac{(a_{2}-b_{2}).a_{1}\times b_{1}}{\mid b_{1}\times b_{2}\mid}\]
d) \[\frac{(a_{1}-b_{2}).b_{1}\times a_{2}}{\mid b_{1}\times a_{2}\mid}\]
Explanation: Let PQ be the line of shortest distance between
3. The volume of the tetrahedron included between the plane
3x + 4y – 5z – 60 = 0 and the coordinate
planes in cubic units is
a) 60
b) 600
c) 720
d) none of these
Explanation: Equation of the given plane can be written as
4. If the planes x = cy + bz, y = az + cx and
z = bx + ay pass through a line, then \[a^{2}+b^{2}+c^{2}+2abc\] is equal to
a) -1
b) 0
c) 1
d) none of these
Explanation: Let l, m, n be the direction ratios of the line lying in the three planes, then the line is perpendicular to
5. The lines \[r = i - j +\lambda \left(2i + k\right)\] and
\[r =\left(2i - j\right) + \mu \left(i + j - k\right)\] intersect for
a) \[\lambda =1 ,\mu =1\]
b) \[\lambda =2 ,\mu =3\]
c) all values of \[\lambda \] and \[\mu\]
d) no value of \[\lambda \] and \[\mu\].
Explanation: The given lines intersect, if the shortest distance between the lines is zero. We know that the shortest distance between the lines r =
6. A point moves so that the sum of the
squares of its distance from the six faces of a cube given by
\[x=\pm 1,y=\pm 1,z=\pm 1\] is 10 units. The locus of the point is
a) \[x^{2}+y^{2}+z^{2}=1\]
b) \[x^{2}+y^{2}+z^{2}=2\]
c) x + y + z = 1
d) x + y + z = 2
Explanation: Let P(x, y, z) be any point on the locus, then
7. The reflection of the plane 2x+3y+4z-3=0 in the plane x – y + z – 3 = 0 is the plane
a) 4x – 3y + 2z –15 = 0
b) x – 3 y + 2z – 15 = 0
c) 4x + 3y – 2z + 15 = 0
d) none of these
Explanation:
8. The plane 2x – y + 3z + 5 = 0 is rotated
through \[90^{0}\] about its line of intersection with the plane
5x – 4y – 2z + 1 = 0 . the equation of the plane in the new
position is
a) \[6x-9 y-29 z – 31 = 0\]
b) 27x – 24y – 26z – 13 = 0
c) 43x – 32y – 2z + 27 = 0
d) 26x – 43y – 151z – 165 = 0
Explanation: Equation of a plane passing through the line of intersection of the given planes is
9. If \[\theta\] is the angle between the line
r = 2i + 3j - k + (i + j + k) t and the plane
r . (3i - 4j + 5k) = q, then
a) \[\cos\theta=\frac{2\sqrt{6}}{15}\]
b) \[\sin\theta=\frac{2\sqrt{6}}{15}\]
c) \[\cos\theta=-\frac{11\sqrt{7}}{70}\]
d) \[\sin\theta=-\frac{11\sqrt{7}}{70}\]
Explanation: The line is parallel to the vector i + j + k and
10. Equation of the plane through three points
A, B, C with position vectors - 6i + 3j + 2k ,
3i - 2j + 4k, 5i + 7j + 3k is
a) r.(i - j - 7k) + 23 = 0
b) r.(i + j + 7k) = 23
c) r . (i + j - 7k) + 23 = 0
d) r. (i - j - 7k) = 23
Explanation: Equation of the plane passing through three points A,B,C with position vectors a,b,c is
