1. The point of intersection of the lines
\[l_{1}:r(t) = (i - 6j + 2k) + t(i + 2j + k)\]
\[l_{2}:R(u) = (4j + k) + u(2i + j + 2k)\]
a) at the tip of r(7)
b) at the tip of R(4)
c) (8, 8, 9)
d) All of the Above
Explanation: Equation r (t) = R(u) we obtain
The two lines intersect at the tip of this vector
2. If a, b, c are three unit vectors such that \[a\times\left(b\times c\right)=\frac{1}{2}b\] and c being non parallel then
a) angle between a and b is \[\pi/2\]
b) angle between a and c is \[\pi/4\]
c) angle between a and c is \[\pi/3\]
d) Both a and c
Explanation: a × (b × c) = (a \[\cdot\] c)b - (a \[\cdot\] b)c = \frac{1}{2} b
3. If a, b and c be non-coplanar unit vectors
equally inclined to one another at an acute angle \[\theta\] . If
a × b + b × r = pa + qb + rc
then
a) p = r
b) \[p=\frac{1}{\sqrt{1+2\cos\theta}},q=-\frac{2\cos\theta}{\sqrt{1+2\cos\theta}}\]
c) \[r=\frac{1}{\sqrt{1+2\cos\theta}}\]
d) All of the Above
Explanation:
4. The volume of the parallelopiped whose sides are
given by OA = 2i - 3j, OB = i + j - k, OC = 3i - k
is
a) 4/13
b) 4
c) 2/7
d) 1/17
Explanation:
5. The points with position vectors 60i + 3j, 40i - 8j,
ai - 52j are collinear if
a) a = - 40
b) a = 40
c) a = 20
d) a = -20
Explanation:
6. Let p, q, r be three mutually perpendicular vectors
of the same magnitude. If a vector x satisfies the
equation \[p \times ((x - q) \times p) + q \times ((x - r) \times q) + r\times ((x - p) \times
r) = 0\]
then x is given by
a) \[\frac{1}{2}\left(p + q - 2r\right)\]
b) \[\frac{1}{2}\left(p + q + r\right)\]
c) \[\frac{1}{3}\left(p + q + r\right)\]
d) \[\frac{1}{3}\left(2p + q - r\right)\]
Explanation:
7. If |a| = 2, |b| = 3 |c| = 4 and a + b + c = 0 then the
value of b . c + c . a + a . b is equal to
a) 19/2
b) -19/2
c) 29/2
d) -29/2
Explanation: Taking dot product on both sides by a, b and c, we have
8. If A, B, C, D are four points in space and \[\mid AB\times CD + BC \times AD + CA \times BD\mid = \lambda\]
(area of the triangle
ABC). Then the value of \[\lambda\] is
a) 1
b) 2
c) 3
d) 4
Explanation:
9.Given a = i + j - k, b = - i + 2j + k and c = - i +
2j - k. A unit vector perpendicular to both a + b and
b + c is
a) \[\frac{2i+j+k}{\sqrt{6}}\]
b) j
c) k
d) \[\frac{i+j+k}{\sqrt{3}}\]
Explanation:
10. Let P, Q, R be points with position vectors \[r_{1}=3i -2j - k,r_{2}=i +3j + 4k\]
and \[r_{3}=2i +j -2 k\] relative
to an origin 0. The distance of P from the plane OQR
is (magnitude)
a) 2
b) 3
c) 1
d) \[11/\sqrt{3}\]
Explanation:
