1. The vectors ai + 2a j - 3ak, (2a + 1)i + (2a + 3)j
+ (a + 1)k and (3a + 5)i + (a + 5)j + (a + 2)k are
non-coplanar for a in
a) {0}
b) \[\left(0 ,\infty\right)\]
c) \[\left(1 ,\infty\right)\]
d) Both b and c
Explanation:
2. For non zero, non-collinear vectors p and q, the
value of [i p q]i +[j p q]j + [k p q]k is
a) 0
b) \[2\left(p\times q\right)\]
c) \[\left(q\times p\right)\]
d) \[\left(p\times q\right)\]
Explanation:
3. The position vectors a, b, c and d of four distinct
points A, B, C and D lie on a plane are such that |a - d| = |b - d| = |c - d| then the point D is the
a) centroid of \[\triangle ABC\]
b) orthocentre of \[\triangle ABC\]
c) circumcentre of \[\triangle ABC\]
d) incenter of \[\triangle ABC\]
Explanation:
4. The components of vector i + j + k along vector,
i + 2j + 3k is
a) (3/7) (i + 2j + 3k)
b) (2/7) (i + 2j + 3k)
c) (1/7) (i + 2j + 3k)
d) (4/7) (i + 2j + 3k)
Explanation:
5. The vector \[i \times \left(\left(a \times b\right)\times i\right) + j \times \left[\left(a \times b\right) \times j\right] + k \times
\left[\left(a \times b\right) \times k\right]\]
equals
a) 0
b) \[\left(a . b\right)b\]
c) b
d) \[2\left(a \times b\right)\]
Explanation:
6.The vectors AB = 3i - 2j + 2k and BC = - i - 2k
are the adjacent sides of a parallelogram. The angle
between its diagonals is
a) \[\pi/4\]
b) \[\pi/3\]
c) \[3\pi/4\]
d) Both a and c
Explanation:
7. Let the unit vectors A and B be perpendicular and
the unit vector C be inclined at an angle \[\theta\] to both A
and B. If \[C = \alpha A + \beta B + \gamma \left(A \times B\right)\]
then
a) \[\beta^{2}=\frac{1+\cos 2\theta}{2}\]
b) \[\gamma^{2}=1-2\alpha^{2}\]
c) \[\gamma^{2}=-\cos2\theta \]
d) All of the Above
Explanation:
8. If the unit vectors a and b are inclined at an angle \[ 2\theta\]
and |a - b| < 1, then if \[ 0\leq \theta \leq\pi ,\theta\]
lies in the interval
a) \[\left[0, \pi/6\right)\]
b) \[\left(5\pi/6, \pi\right]\]
c) \[\left[\pi/6, \pi/2\right]\]
d) Both a and b
Explanation:
9. For non-coplanar vectors A, B and C, | \[\left(A\times B\right)\] . C|
= |A| |B| |C| holds if and only if
a) A . B = B . C = C . A = 0
b) A . B = 0 = B . C
c) A . B = 0 = C . A
d) All of the Above
Explanation:
10. If K is the length of any edge of a regular tetrahedron
then the distance of any vertex from the opposite
face is
a) 3/2 K
b) \[\frac{2}{3}K^{2}\]
c) \[\sqrt{\frac{2}{3}}K\]
d) \[\sqrt{3}K\]
Explanation:
