JEE Main & Advanced Mathematics Vector Algebra Question Bank

done Mock Test - Vector Algebra Total Questions - 25

• question_answer1)
  A point O is the centre of a circle circumscribed about a triangle ABC. Then \[\overrightarrow{OA}\] sin 2A+\[\overrightarrow{OB}\] sin 2B + \[\overrightarrow{OC}\] sin 2C is equal to
  A)
  \[(\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC})sin2A\] done clear
  B)
  \[3\,\overrightarrow{OG}\], where G is the centroid of triangle ABC done clear
  C)
  \[\overrightarrow{0}\] done clear
  D)
  none of these done clear
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• question_answer2)
  If vectors \[\overrightarrow{AB}=-3\hat{i}+4\hat{k}\] and \[\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}\] are the sides of a \[\Delta \]ABC, then the length of the median through A is
  A)
  \[\sqrt{14}\]       done clear
  B)
  \[\sqrt{18}\] done clear
  C)
  \[\sqrt{29}\]       done clear
  D)
  5 done clear
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• question_answer3)
  Given three vectors \[\vec{a}=6\hat{i}-3\hat{j},\hat{b}=2\hat{i}-6\hat{j}\] and \[\vec{c}=-2\hat{i}+21\hat{j}\] such that \[\overrightarrow{\alpha }=\vec{a}+\vec{b}+\vec{c}\]. Then the resolution of the vector \[\overrightarrow{\alpha }\] into components with respect to \[\vec{a}\] and \[\vec{b}\] is given by
  A)
  \[3\vec{a}-2\vec{b}\] done clear
  B)
  \[3\vec{b}\]\[-\]\[2\vec{a}\] done clear
  C)
  \[2\vec{a}-3\vec{b}\] done clear
  D)
  \[\vec{a}-2\vec{b}\] done clear
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• question_answer4)
  If \[\vec{a}\] and \[\vec{b}\] are two unit vectors and \[\theta \] is the angle between them, then the unit vector along the angular bisector of \[\vec{a}\] and \[\vec{b}\] will be given by
  A)
  \[\frac{\vec{a}-\vec{b}}{2\cos (\theta /2)}\] done clear
  B)
  \[\frac{\vec{a}+\vec{b}}{2\cos (\theta /2)}\] done clear
  C)
  \[\frac{\vec{a}-\vec{b}}{\cos (\theta /2)}\] done clear
  D)
  none of these done clear
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• question_answer5)
  If the vectors \[\vec{a}\] and \[\vec{b}\] are linearly independent satisfying\[(\sqrt{3}tan\theta +1)\vec{a}+(\sqrt{3}sec\theta -2)\vec{b}\]=0, then the most general values of \[\theta \] are
  A)
  \[n\pi -\frac{\pi }{6},n\in z\] done clear
  B)
  \[2n\pi \pm \frac{11\pi }{6},n\in z\] done clear
  C)
  \[n\pi \pm \frac{\pi }{6},n\in z\]   done clear
  D)
  \[2n\pi +\frac{11\pi }{6},n\in z\] done clear
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• question_answer6)
  In triangle ABC,\[\angle A=30{}^\circ \], H is the orthocentre and D is the midpoint of BC. Segment HD is produced to T such that HD = DF. The length AT is equal to
  A)
  2BC                  done clear
  B)
  3BC done clear
  C)
  \[\frac{4}{3}\]BC           done clear
  D)
  none of these done clear
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• question_answer7)
  Find the value of \[\lambda \] so that the points P, Q, R and S on the sides OA, OB, OC and AB, respectively, of a regular tetrahedron OABC are coplanar. It is given that \[\frac{OP}{OA}=\frac{1}{3},\frac{OQ}{OB}=\frac{1}{2},\frac{QR}{OC}=\frac{1}{3}\] and\[\frac{OS}{AB}=\lambda \].
  A)
  \[\lambda =\frac{1}{2}\]            done clear
  B)
  \[\lambda =-1\] done clear
  C)
  \[\lambda =0\]   done clear
  D)
  for no value of \[\lambda \] done clear
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• question_answer8)
  If \[4\hat{i}+7\hat{j}+8\hat{k},\,\,2\hat{i}+3\hat{j}+4\hat{k}\] and \[2\hat{i}+5\hat{j}+7\hat{k}\] are the position vectors of the vertices A, B and C, respectively, of triangle ABC, then the position vector of the point where the bisector of angle A meets BC is
  A)
  \[\frac{2}{3}(-6\hat{i}-8\hat{j}-6\hat{k})\] done clear
  B)
  \[\frac{2}{3}(6\hat{i}+8\hat{j}+6\hat{k})\] done clear
  C)
  \[\frac{1}{3}(6\hat{i}+8\hat{j}\,+18\hat{k})\] done clear
  D)
  \[\frac{1}{3}(5\hat{j}+12\hat{k})\] done clear
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• question_answer9)
  If \[\hat{a},\,\,\hat{b}\] and \[\hat{c}\] are three unit vectors inclined to each other at an angle \[\theta \], then the maximum value of \[\theta \] is
  A)
  \[\frac{\pi }{3}\]             done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  \[\frac{2\pi }{3}\]           done clear
  D)
  \[\frac{5\pi }{6}\] done clear
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• question_answer10)
  Locus of the point P, for which \[\overrightarrow{OP}\] represents a vector with direction cosine \[\cos \,\,\alpha =\frac{1}{2}\] (where O is the origin) is
  A)
  a circle parallel to the y-z plane with centre on the x-axis done clear
  B)
  a cone concentric with the positive x-axis having vertex at the origin and the slant height equal to the magnitude of the vector done clear
  C)
  a ray emanating from the origin and making an angle of \[60{}^\circ \]with the x-axis done clear
  D)
  a disc parallel to the y-z plane with centre on the x-axis and radius equal to \[\left| \overrightarrow{OP} \right|\] sin 60° done clear
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• question_answer11)
  Let \[\vec{a}\], \[\vec{b}\] and \[\vec{c}\] be the three vectors having magnitudes 1, 5 and 3, respectively, such that the angle between \[\vec{a}\] and \[\vec{b}\] is \[\theta \] and \[\vec{a}\times (\vec{a}\times \vec{b})=\vec{c}\]  Then tan \[\theta \] is equal to
  A)
  0                      done clear
  B)
  2/3 done clear
  C)
  3/5                    done clear
  D)
  ¾ done clear
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• question_answer12)
  A uni-modular tangent vector on the curve \[x={{t}^{2}}+2\], \[y=4t-5\], \[z=2{{t}^{2}}-6t\] at t=2 is
  A)
  \[\frac{1}{3}(2\hat{i}+2\hat{j}+\hat{k})\] done clear
  B)
  \[\frac{1}{3}(\hat{i}-\hat{j}-\hat{k})\] done clear
  C)
  \[\frac{1}{6}(2\hat{i}+\hat{j}+\hat{k})\]            done clear
  D)
  \[\frac{2}{3}(\hat{i}+\hat{j}+\hat{k})\] done clear
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• question_answer13)
  If \[\hat{a},\] \[\hat{b}\] and \[\hat{c}\] are three unit vectors, such that \[\hat{a}+\hat{b}+\hat{c}\] is also a unit vector and \[{{\theta }_{1}}\], \[{{\theta }_{2}}\] and \[{{\theta }_{3}}\] are angles between the vectors \[\hat{a}\], \[\hat{b}\]; \[\hat{b}\], \[\hat{c}\] and \[\hat{c}\], \[\hat{a}\], respectively, then among \[\theta _{1}^{{}}\],\[\theta _{2}^{{}}\]and \[\theta _{3}^{{}}\]
  A)
  all are acute angles done clear
  B)
  all are right angles done clear
  C)
  at least one is obtuse angle done clear
  D)
  none of these done clear
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• question_answer14)
  Let \[\vec{a}=\hat{i}+\hat{j};\hat{b}=2\hat{i}-\hat{k}\]. Then vector \[\vec{r}\] satisfying the equations \[\vec{r}\times \vec{a}=\vec{b}\times \vec{a}\]and \[\vec{r}\times \vec{b}=\vec{a}\times \vec{b}\] is
  A)
  \[\hat{i}-\hat{j}+\hat{k}\]          done clear
  B)
  \[3\hat{i}-\hat{j}+\hat{k}\] done clear
  C)
  \[3\hat{i}+\hat{j}-\hat{k}\]         done clear
  D)
  \[\hat{i}-\hat{j}-\hat{k}\] done clear
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• question_answer15)
  If \[\vec{a}\] satisfies \[\vec{a}\times (\hat{i}+2\hat{j}+\hat{k})=\hat{i}-\hat{k}\], then \[\vec{a}\] is equal to
  A)
  \[\lambda \hat{i}+(2\lambda -1)\hat{j}\,+\lambda \hat{k},\lambda \in R\] done clear
  B)
  \[\lambda \hat{i}+(1-2\lambda )\hat{j}\,+\lambda \hat{k},\lambda \in R\] done clear
  C)
  \[\lambda \hat{i}+(2\lambda +1)\hat{j}\,+\lambda \hat{k},\lambda \in R\] done clear
  D)
  \[\lambda \hat{i}-(1+2\lambda )\hat{j}\,+\lambda \hat{k},\lambda \in R\] done clear
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• question_answer16)
  Vectors \[3\vec{a}-5\vec{b}\] and \[2\vec{a}+\vec{b}\] are mutually perpendicular. If \[\vec{a}+4\vec{b}\] and \[\vec{b}-\vec{a}\] are also mutually perpendicular, then the cosine of the angle between \[\vec{a}\] and \[\vec{b}\] is
  A)
  \[\frac{19}{5\sqrt{43}}\] done clear
  B)
  \[\frac{19}{3\sqrt{43}}\] done clear
  C)
  \[\frac{19}{2\sqrt{45}}\] done clear
  D)
  \[\frac{19}{6\sqrt{43}}\] done clear
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• question_answer17)
  Let \[\vec{a}\cdot \vec{b}=0\] where \[\vec{a}\] and \[\vec{b}\] are unit vectors and the unit vector \[\vec{c}\] is inclined at an angle \[\theta \] to both \[\vec{a}\] and \[\vec{b}\]. If \[\vec{c}=m\vec{a}+n\vec{b}+p(\vec{a}\times \vec{b}),(m,n,p\in R)\], then
  A)
  \[-\frac{\pi }{4}\le \theta \le \frac{\pi }{4}\]           done clear
  B)
  \[\frac{\pi }{4}\le \theta \le \frac{3\pi }{4}\] done clear
  C)
  \[0\le \theta \le \frac{\pi }{4}\]     done clear
  D)
  \[0\le \theta \le \frac{3\pi }{4}\] done clear
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• question_answer18)
  If vectors \[\vec{a}\] and \[\vec{b}\] are two adjacent sides of a Parallelogram, then the vector representing the altitude of the parallelogram which is perpendicular to \[\vec{a}\] is
  A)
  \[\vec{b}+\frac{\vec{b}\times \vec{a}}{{{\left| {\vec{a}} \right|}^{2}}}\]          done clear
  B)
  \[\frac{\vec{a}\cdot \vec{b}}{{{\left| {\vec{b}} \right|}^{2}}}\] done clear
  C)
  \[\vec{b}-\frac{\vec{b}\cdot \vec{a}}{{{\left| {\vec{a}} \right|}^{2}}}\vec{a}\] done clear
  D)
  \[\frac{\vec{a}\times (\vec{b}\times \vec{a})}{{{\left| {\vec{b}} \right|}^{2}}}\] done clear
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• question_answer19)
  Let the position vectors of the points P and Q be \[4\hat{i}+\hat{j}+\lambda \hat{k}\] and \[2\hat{i}-\hat{j}+\lambda \hat{k}\], respectively. Vector \[\hat{i}-\hat{j}+6\hat{k}\]is perpendicular to the plane containing the origin and the point?s P and Q. then\[\lambda \]equals
  A)
  \[-\,1/2\] done clear
  B)
  1/2 done clear
  C)
  1                      done clear
  D)
  none of these done clear
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• question_answer20)
  If \[3\lambda \vec{c}+2\mu (\vec{a}\times \vec{b})=0\], then
  A)
  \[3\lambda +2\mu =0\]  done clear
  B)
  \[3\lambda =2\mu \] done clear
  C)
  \[\lambda =\mu \]          done clear
  D)
  \[\lambda +\mu =0\] done clear
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• question_answer21)
  \[\vec{a}\],\[\vec{b}\], and \[\vec{c}\] are vectors such that \[\left| \vec{a}+\vec{b}+3\vec{c} \right|=4\]angle between \[\vec{a}\]and \[\vec{b}\]is \[{{\theta }_{1}}\], between \[\vec{b}\]and \[\vec{c}\] is \[{{\theta }_{2}}\] and between \[\vec{a}\] and \[\vec{c}\] varies \[\left[ \pi /6,2\pi /3 \right].\]Then the maximum value of \[\cos {{\theta }_{1}}+3\cos {{\theta }_{2}}\]is ______.
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• question_answer22)
  Points \[\vec{a},\,\,\vec{b},\,\,\vec{c}\] and \[\vec{d}\] are coplanar and \[(\sin \alpha )\]\[\vec{a}+(2sin2\beta ).\vec{b}+(3sin3\gamma )\vec{c}\,-\vec{d}=\vec{0}\].then the least value of \[{{\sin }^{2}}\alpha +{{\sin }^{2}}2\beta +{{\sin }^{2}}3\gamma \]is ______.
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• question_answer23)
  If \[\vec{a},\vec{b},\] and \[\vec{c}\]are vectors such that \[[\vec{a}\,\vec{b}\,\vec{c}]=4\], then \[[\vec{a}\times \vec{b}\,\,\vec{b}\times \vec{c}\,\,\vec{c}\times \vec{a}]\] is equal to_____.
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• question_answer24)
  If \[\left| {\vec{a}} \right|=5,\,\,\left| {\vec{b}} \right|=4,\]and \[\left| {\vec{c}} \right|\]=3, then what will be the value of \[|\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}|\] will be _____ . Given that \[\vec{a}+\vec{b}+\vec{c}=\vec{0}\].
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• question_answer25)
  If \[\left| {\vec{a}} \right|=4,\left| {\vec{b}} \right|=2\] and the angle between \[\vec{a}\] and \[\vec{b}\]is \[\pi /6\], then \[{{(\vec{a}\,\times \vec{b})}^{2}}\]is equal to ______.
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