1. A vector a which is collinear with the vector \[6i -8j -\frac{15}{2}k\] of magnitude 50 making on obtuse angle
with z-axis is
a) 24i - 32j - 30k
b) - 24i + 32j + 30k
c) 24i + 32j - 30k
d) none of these
Explanation:
2. Let there be two points A, B on the curve \[y=x^{2}\] in the plane OXY satisfying OA.i = 1 and OB. i =
- 2 then the length of the vector 2OA - 3OB is
a) \[\sqrt{14}\]
b) \[2\sqrt{51}\]
c) \[3\sqrt{41}\]
d) \[2\sqrt{41}\]
Explanation:
3. If A, B, C, D are four points in space satisfying \[AB . CD = K\left[\mid AD\mid ^{2} + \mid BC\mid^{2} - \mid AC\mid^{2} - \mid BD\mid^{2}\right]\]
then the value of K is
a) 2
b) 1/3
c) 1/2
d) 1
Explanation:
4. For unit vectors b and c and any vector a, the value
of \[\left\{\left\{\left(a+b\right)\times\left(a+c\right)\right\}\times\left(b \times c\right)\right\}.\left(b+c\right)\]
is
a) \[\mid a \mid^{2}\]
b) \[2\mid a \mid^{2}\]
c) \[3\mid a \mid^{2}\]
d) 0
Explanation:
5. The distance of the point B with position vector
i + 2j + k from the line passing through the point A
with position vector 4i + 2j + 2k and parallel to the
vector 2i + 3j + 6k is
a) \[\sqrt{10}\]
b) \[\sqrt{5}\]
c) \[\sqrt{6}\]
d) 2
Explanation:
6. Three non-coplanar vector a, b and c are drawn
from a common initial point. The angle between the
plane passing through the terminal points of these
vectors and the vector \[a\times b +b\times c+c\times a\] is
a) \[\pi/4\]
b) \[\pi/2\]
c) \[\pi/3\]
d) \[\pi/6\]
Explanation:
7. If \[p^{th},q^{th},r^{th}\] term of a G.P. are the positive numbers
a, b, c, then the angle between the vectors
\[\log a^{2} i+\log b^{2}j+\log c^{2}k\] and (q - r)i +
(r - p)j + ( p - q)k is
a) \[\pi/3\]
b) \[\pi/2\]
c) \[\sin^{-1}\frac{1}{\sqrt{a^{2}+b^{2}+c^{2}}}\]
d) \[\frac{\pi}{4}\]
Explanation:
8. Given three unit vectors a, b, c no two which are
collinear satisfying \[a\times \left(b\times c\right)=\frac{1}{2}b\]
The angle
between a and b is
a) \[\pi/3\]
b) \[\pi/4\]
c) \[\pi/2\]
d) \[2\pi/3\]
Explanation:
9. The volume of the tetrahedron with vertices P(- 1,
2, 0), Q(2, 1, - 3), R (1, 0, 1) and S(3, - 2, 3) is
a) 1/3
b) 2/3
c) 9/14
d) 2/4
Explanation:
10. Consider the parallelopiped with sides a =
3i + 2j + k, b = i + j + 2k and c = i + 3j +3k then the
angle between a and the plane containing the face
determined by b and c is
a) \[\sin^{-1}\left(1/3\right)\]
b) \[\cos^{-1}\left(9/14\right)\]
c) \[\sin^{-1}\left(9/14\right)\]
d) \[\sin^{-1}\left(2/3\right)\]
Explanation:
