Vector Algebra Test

Vector Algebra

This is Vector Algebra Test-04 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
Find the unit vector in the direction of vector \(\overrightarrow {PQ} \) , where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively
  • A
    \(\frac{1}{{\sqrt 3 }}\hat i + \frac{1}{{\sqrt 3 }}\hat j - \frac{1}{{\sqrt 3 }}\hat k\)
  • B
    \(\frac{1}{{\sqrt 3 }}\hat i + \frac{1}{{\sqrt 3 }}\hat j + \frac{1}{{\sqrt 3 }}\hat k\)
    Correct
  • C
    \(\frac{1}{{\sqrt 3 }}\hat i - \frac{1}{{\sqrt 3 }}\hat j + \frac{1}{{\sqrt 3 }}\hat k\)
  • D
    \( - \frac{1}{{\sqrt 3 }}\hat i + \frac{1}{{\sqrt 3 }}\hat j + \frac{1}{{\sqrt 3 }}\hat k\)
2
Find a vector in the direction of the vector \(5\hat i - \hat j + 2\hat k\) which has a magnitude of 8 units
  • A
    \(\frac{{40}}{{\sqrt {30} }}\hat i + \frac{8}{{\sqrt {30} }}\hat j + \frac{{16}}{{\sqrt {30} }}\hat k\)
  • B
    \( - \frac{{40}}{{\sqrt {30} }}\hat i - \frac{8}{{\sqrt {30} }}\hat j + \frac{{16}}{{\sqrt {30} }}\hat k\)
  • C
    \(\frac{{40}}{{\sqrt {30} }}\hat i - \frac{8}{{\sqrt {30} }}\hat j + \frac{{16}}{{\sqrt {30} }}\hat k\)
    Correct
  • D
    \(\frac{{40}}{{\sqrt {30} }}\hat i - \frac{8}{{\sqrt {30} }}\hat j - \frac{{16}}{{\sqrt {30} }}\hat k\)
3
Find the direction cosines of the vector \(\hat i + 2\hat j + 3\hat k\)
  • A
    \(\frac{1}{{\sqrt {14} }},\;\frac{2}{{\sqrt {14} }},\; - \frac{3}{{\sqrt {14} }}\)
  • B
    \( - \frac{1}{{\sqrt {14} }},\;\frac{2}{{\sqrt {14} }},\;\frac{3}{{\sqrt {14} }}\)
  • C
    \(\frac{1}{{\sqrt {14} }},\; - \frac{2}{{\sqrt {14} }},\;\frac{3}{{\sqrt {14} }}\)
  • D
    \(\frac{1}{{\sqrt {14} }},\;\frac{2}{{\sqrt {14} }},\;\frac{3}{{\sqrt {14} }}\)
    Correct
4
Find the direction cosines of the vector joining the points A(1, 2, –3) and B(–1, –2, 1), directed from A to B.
  • A
    \(\frac{1}{3},\; - \frac{2}{3},\frac{2}{3}\)
  • B
    \( - \frac{1}{3},\; - \frac{2}{3},\frac{2}{3}\)
    Correct
  • C
    \(\; - \frac{1}{3},\; - \frac{2}{3}, - \frac{2}{3}\)
  • D
    \( - \frac{1}{3},\;\frac{2}{3},\frac{2}{3}\)
5
Find the angle between two vectors \(\vec a\;and\;\vec b\) with magnitudes \(\sqrt 3 \) and 2, respectively, having \(\vec a\;.\vec b = \sqrt 6 \)
  • A
    \(\frac{\pi }{2}\)
  • B
    \(\frac{\pi }{5}\)
  • C
    \(\frac{\pi }{3}\)
  • D
    \(\frac{\pi }{4}\)
    Correct
6
Find the angle between two vectors \(\hat i - 2\hat j + 3\hat k\;and\;3\hat i - 2\hat j + \hat k\;\)
  • A
    \({\cos ^{ - 1}}\left( {\frac{4}{7}} \right)\)
  • B
    \({\cos ^{ - 1}}\left( {\frac{6}{7}} \right)\)
  • C
    \({\cos ^{ - 1}}\left( {\frac{5}{9}} \right)\)
  • D
    \({\cos ^{ - 1}}\left( {\frac{5}{7}} \right)\)
    Correct
7
Find the projection of the vector \(\hat i - \hat j\;on\;the\;vector\;\hat i + \hat j\)
  • A
    0
    Correct
  • B
    1
  • C
    –1
  • D
    2
8
Find the projection of the vector\(\hat i + 3\hat j + 7\hat k\) on the vector \(7\hat i - \hat j + 8\hat k\)
  • A
    \(\frac{{66}}{{\sqrt {114} }}\)
  • B
    \(\frac{{60}}{{\sqrt {134} }}\)
  • C
    \(\frac{{60}}{{\sqrt {124} }}\)
  • D
    \(\frac{{60}}{{\sqrt {114} }}\)
    Correct
9
Find \(\left| {\vec a} \right|\;and\;\left| {\vec b} \right|,\;if\;\left( {\vec a + \vec b} \right).\left( {\vec a - \vec b} \right) = 8\;and\;\left| {\vec a} \right| = 8\;\left| {\vec b} \right|\)
  • A
    \(\frac{{21\sqrt 2 }}{{3\sqrt 7 }}\),\(\frac{{2\sqrt 6 }}{{3\sqrt 7 }}\)
  • B
    \(\frac{{19\sqrt 2 }}{{3\sqrt 7 }}\),\(\frac{{2\sqrt 5 }}{{3\sqrt 7 }}\)
  • C
    \(\frac{{16\sqrt 2 }}{{3\sqrt 7 }}\),\(\frac{{2\sqrt 2 }}{{3\sqrt 7 }}\)
    Correct
  • D
    \(\frac{{17\sqrt 2 }}{{3\sqrt 7 }}\),\(\frac{{2\sqrt 3 }}{{3\sqrt 7 }}\)
10
Evaluate the product \(\left( {3\vec a - 5\vec b} \right).\left( {2\vec a + 7\vec b} \right)\)
  • A
    6\({\left| {\vec a} \right|^2} + 11\vec a.\vec b - 35{\left| {\vec b} \right|^2}\)
    Correct
  • B
    7\({\left| {\vec a} \right|^2} + 13\vec a.\vec b - 45{\left| {\vec b} \right|^2}\)
  • C
    6\({\left| {\vec a} \right|^2} + 13\vec a.\vec b - 35{\left| {\vec b} \right|^2}\)
  • D
    6\({\left| {\vec a} \right|^2} + 11\vec a.\vec b - 30{\left| {\vec b} \right|^2}\)
11
Find the magnitude of two vectors\(\;\vec a\;and\;\vec b\), having the same magnitude and such that the angle between them is 30° and their scalar product is \(2\sqrt 3 \)
  • A
    \(\left| {\vec a} \right| = 2,\left| {\vec b} \right| = 1\)
  • B
    \(\left| {\vec a} \right| = 1,\left| {\vec b} \right| = 2\)
  • C
    \(\left| {\vec a} \right| = 2,\left| {\vec b} \right| = 2\)
    Correct
  • D
    \(\left| {\vec a} \right| = 1,\left| {\vec b} \right| = 1\)
12
Find \(\left| {\vec x} \right|\), if for a unit vector \(\widehat a,\;\left( {2\vec x - 3\vec a} \right).\left( {2\vec x + 3\vec a} \right) = 91\)
  • A
    \(\sqrt {19} \)
  • B
    \(\sqrt {17} \)
  • C
    \(\sqrt {15} \)
  • D
    5
    Correct
13
If\(\vec a = 2\hat i + 2\hat j + 3\hat k,\;\vec b = - \hat i + 2\hat j + \hat k\;and\;\vec c = 3\hat i + \hat j\;\)are such that \(\vec a + \lambda \vec b\) is perpendicular to \(\vec c\), then the value of \(\lambda \) is
  • A
    9
  • B
    7
  • C
    11
  • D
    8
    Correct
14
If \(\vec a,\vec b,\vec c\) are unit vectors such that \(\vec a + \vec b + \vec c = 0\) then, the value of \(\vec a.\vec b + \vec b.\vec c + \vec c.\vec a\) is
  • A
    \(\frac{3}{2}\)
  • B
    \(\frac{{ - 5}}{2}\)
  • C
    \(\frac{5}{2}\)
  • D
    \(\frac{{ - 3}}{2}\)
    Correct
15
If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find \(\angle ABC.{\text{ }}[\angle ABC\) is the angle between the vectors \(\overrightarrow {BA} \;and\;\overrightarrow {BC} \;\)]
  • A
    \({\cos ^{ - 1}}\left( {\frac{{11}}{{\sqrt {102} }}} \right)\)
  • B
    \({\cos ^{ - 1}}\left( {\frac{{10}}{{\sqrt {102} }}} \right)\)
    Correct
  • C
    \({\cos ^{ - 1}}\left( {\frac{{13}}{{\sqrt {102} }}} \right)\)
  • D
    \({\cos ^{ - 1}}\left( {\frac{{15}}{{\sqrt {102} }}} \right)\)