Vector Algebra Test

Vector Algebra

This is Vector Algebra Test-05 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
If \(\vec a\) is a non zero vector of magnitude ‘a’ and \(\lambda \) a non zero scalar, then \(\lambda \vec a\) is a unit vector if
  • A
    a = \(\frac{1}{{\left| \lambda \right|}}\)
    Correct
  • B
    a = \(\left| \lambda \right|\)
  • C
    \(\lambda = 1\)
  • D
    \(\lambda = \; - 1\)
2
Find\(\left| {\vec a \times \vec b} \right|\), if \(\vec a = 3\hat i + \hat j + 2\hat k\;and\;\vec b = 2\hat i - 2\hat j + 4\hat k\)
  • A
    \(19\sqrt 5 \)
  • B
    \(19\sqrt 3 \)
  • C
    \(8\sqrt 3 \)
    Correct
  • D
    \(17\sqrt 2 \)
3
Find a unit vector perpendicular to each of \(\vec a + \vec b\;and\;\vec a - \vec b\) , where \(\vec a = 3\hat i + 2\hat j + 2\hat k\;and\;\vec b = \hat i + 2\hat j - 2\hat k\)
  • A
    \( \pm \frac{2}{3}\hat i \pm \frac{2}{3}\hat j \mp \frac{1}{3}\hat k\)
  • B
    \( \pm \frac{2}{3}\hat i \mp \frac{2}{3}\hat j \mp \frac{1}{3}\hat k\)
    Correct
  • C
    \( \mp \frac{2}{3}\hat i \mp \frac{2}{3}\hat j \mp \frac{1}{3}\hat k\)
  • D
    \( \pm \frac{2}{3}\hat i \pm \frac{2}{3}\hat j \pm \frac{1}{3}\hat k\)
4
If a unit vector \(\vec a\) makes angles \( \frac{\pi }{3}\;with\;{\hat i }\; \frac{{\pi }}{{4}}\;with\; {\hat j}\; and\; an\; acute \;angle\; \theta\; with\; {\hat k}\) , then find \(\theta \)
  • A
    \(\frac{\pi }{5}\)
  • B
    \(\frac{\pi }{{10}}\)
  • C
    \(2\frac{\pi }{3}\)
  • D
    \(\frac{\pi }{3}\;\)
    Correct
5
If a unit vector \(\vec a\) makes angles \(\frac{\pi }{3}\;with\;\hat i,\;\frac{\pi }{4}\;with\;\hat j\;and\;an\;acute\;angle\;\theta \;with\;\hat k\;\) , then the components of \(\vec a\) are
  • A
    \(\frac{1}{3},\;\frac{1}{{\sqrt 2 }},\;\frac{1}{2}\)
  • B
    \(\frac{1}{2},\;\frac{1}{{\sqrt 2 }},\;\frac{1}{3}\)
  • C
    \(\frac{1}{2},\;\frac{1}{{\sqrt 2 }},\;\frac{1}{2}\)
    Correct
  • D
    \(\frac{1}{3},\;\frac{1}{{\sqrt 3 }},\;\frac{1}{2}\)
6
Find \(\lambda \;and\;\mu \;if\;\left( {2\hat i + 6\hat j + 27\hat k} \right) \times \left( {\hat i + \lambda \hat j + \mu \hat k} \right) = \vec 0\)
  • A
    4, \(\frac{{27}}{2}\)
  • B
    3, \(\frac{{27}}{5}\)
  • C
    5, \(\frac{{27}}{2}\)
  • D
    3, \(\frac{{27}}{2}\)
    Correct
7
Let the vectors \(\vec a\;and\;\vec b\) be such that \(\left| {\vec a} \right| = 3\;and\;\left| b \right| = \frac{{\sqrt 2 }}{3},\;then\;\vec a \times \vec b\) is a unit vector if the angle between vectors \(\vec a\;and\;\vec b\;\) is
  • A
    \(\frac{\pi }{6}\)\(\pi /6\)
  • B
    \(\frac{\pi }{2}\)\(\pi /2\)
  • C
    \(\frac{\pi }{3}\)\(\pi /3\)
  • D
    \(\frac{\pi }{4}\)
    Correct
8
If \(\theta \) is the angle between two vectors\(\;\vec a\;and\;\vec b\) , then \(\vec a.\vec b \geqslant 0\) only when
  • A
    \(0 < \theta < \pi \)
  • B
    \(0 \leqslant \theta \leqslant \pi \)
  • C
    \(0 < \theta < \frac{\pi }{2}\)
  • D
    \(0 \leqslant \theta \leqslant \frac{\pi }{2}\)
    Correct
9
Let \(\vec a\;and\;\vec b\) be two unit vectors and θ is the angle between them. Then Let \(\vec a + \vec b\) is a unit vector if
  • A
    \(\theta = \frac{\pi }{4}\)
  • B
    \(\theta = \frac{\pi }{3}\)
  • C
    \(\theta = \frac{\pi }{2}\)
  • D
    \(\theta = \frac{{2\pi }}{3}\)
    Correct
10
If \(\theta \) is the angle between any two vectors \(\vec a\;and\;\vec b\), then \(\left| {\vec a.\vec b} \right| = \left| {\vec a \times \vec b} \right|\) when θ is equal to
  • A
    \(\frac{\pi }{3}\)
  • B
    \(\frac{\pi }{2}\)
  • C
    \(\frac{\pi }{4}\)
    Correct
11
Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).
  • A
    \(\frac{{\sqrt {61} }}{3}\)
  • B
    \(\frac{{\sqrt {65} }}{3}\)
  • C
    \(\frac{{\sqrt {65} }}{2}\)
  • D
    \(\frac{{\sqrt {61} }}{2}\)
    Correct
12
Find the area of the parallelogram whose adjacent sides are determined by the vectors\(\vec a = \hat i - \hat j + 3\hat k\;and\;\vec b = 2\hat i - 7\hat j\; + \hat k\)
  • A
    \(11\sqrt 3 \)
  • B
    \(15\sqrt 2 \)
    Correct
  • C
    \(15\sqrt 3 \)
  • D
    \(11\sqrt 2 \)
13
Area of a rectangle having vertices A, B, C and D with position vectors\( - \hat i + \frac{1}{2}\hat j + 4\hat k,\;\,\,\,\,\hat i + \frac{1}{2}\hat j + 4\hat k,\;\,\,\,\,\hat i - \frac{1}{2}\hat j + 4\hat k\;\,\,\,{\text{and}}\,\, - \hat i - \frac{1}{2}\hat j + 4\hat k\) respectively is
  • A
    2
    Correct
  • B
    4
  • C
    1
  • D
    1/2
14
A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.
  • A
    \(\frac{{ - 5}}{2}\hat i + \frac{{3\sqrt 3 }}{2}\hat j\)
    Correct
  • B
    \(\frac{5}{2}\hat i + \frac{{3\sqrt 3 }}{2}\hat j\)
  • C
    \(\frac{5}{2}\hat i - \frac{{3\sqrt 3 }}{2}\hat j\)
  • D
    \(\frac{{ - 5}}{2}\hat i - \frac{{3\sqrt 3 }}{2}\hat j\)
15
The two adjacent sides of a parallelogram are \(2\hat i - 4\hat j + 5\hat k\;and\;\hat i - 2\hat j - 3\hat k\) .Find the unit vector parallel to its diagonal. Also, find its area.
  • A
    \(\frac{1}{7}\left( {3\hat i - 6\hat j - 7\hat k} \right);15\sqrt 5 \)
  • B
    \(\;\frac{1}{7}\left( { - 3\hat i - 6\hat j + 2\hat k} \right);13\sqrt 5 \)
  • C
    \(\frac{1}{7}\left( {3\hat i - 6\hat j + 2\hat k} \right);11\sqrt 5 \)
    Correct
  • D
    \(\frac{1}{7}\left( {3\hat i - 6\hat j - 2\hat k} \right);12\sqrt 5 \)