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CBSE Questions for Class 12 Commerce Maths Vector Algebra Quiz 5 - MCQExams.com

A man starts from O moves 500m turns by 60 and moves 500m again turns by 60 and moves 500m and so on. Find the displacement after (i) 5th turn , (ii) 3rd turn.
  • -500m,1000m
  • 500m,5003m
  • 1000m,5003m
  • none of these
The projection of the vector ˆi2ˆj+ˆk on the vector 4ˆi4ˆj+7ˆk is
  • 5610
  • 199
  • 919
  • 619
If the vector product of a constant vector OA with a variable vector OB in a fixed plane OAB be a constant vector, then locus of B is :
  • A straight line perpendicular to OA
  • A circle with centre O radius equal to |OA|
  • A straight line parallel to OA
  • None of these
There are N co-planar vectors each of magnitude VEach vector is inclined to the preceding vector at angle 2π/N. What is the magnitude of their resultant?

  • zero
  • V/N
  • V
  • NV
P,Q have position vectors ab relative to the origin 'O' and X,Y  divide PQ intermally and extemally respectively in the ratio 2:1, then vector XY=
  • 32(ba)
  • 43(ca)
  • 56(ba)
  • 43(ba)
If . and \times represent dot product and cross product respectively then which of the following is meaningless?
  • \left ( \vec{a}\times \vec{b} \right ).\left ( \vec{c}\times \vec{d} \right )
  • \left ( \vec{a}\times \vec{b} \right )\times \left ( \vec{c}\times \vec{d} \right )
  • \left ( \vec{a}.\vec{b} \right ) .\left ( \vec{c}\times \vec{d} \right )
  • \left ( \vec{a}.\vec{b} \right )\times \left ( \vec{c}\times \vec{d} \right )
If \vec{a}.\vec{b}=0 and \vec{a}\times \vec{b}=0 then
  • \vec{a}\parallel \vec{b}
  • \vec{a}\perp \vec{b}
  • \vec{a}=\vec{0} or \vec{b}=\vec{0}
  • None of these
If G and G' be the centroids of the triangles ABC and A'B'C' respectively, then \overrightarrow { AA' } +\overrightarrow { BB' } +\overrightarrow { CC' } =
  • \displaystyle \dfrac { 2 }{ 3 } \overrightarrow { GG' }
  • \overrightarrow { GG' }
  • 2\overrightarrow { GG' }
  • 3\overrightarrow { GG' }
Five coplanar forces of equal magnitudes 10 N each, act at a point such that the angle between any two consecutive forces is same. The magnitude of their resultant is :
  • 0
  • 10 N
  • 20 N
  • 10 \sqrt {2} N
P is a point on the line through the point A whose position vector is \overrightarrow{a} and the line is parallel to the vector \overrightarrow{b}. If PA=6, the position vector of P is
  • \overrightarrow{a}+6\overrightarrow{b}
  • \displaystyle \overrightarrow{a}+\dfrac{6}{\left |\overrightarrow{b} \right |}\overrightarrow{b}
  • \overrightarrow{a}-6\overrightarrow{b}
  • \displaystyle \overrightarrow{b}+\dfrac{6}{\left |\overrightarrow{a} \right |}\overrightarrow{a}
A straight line r=a+\lambda b meets the plane r.n=0 in P. The position vector of P is
  • \displaystyle a+\frac { a.n }{ b.n } b
  • \displaystyle a+\frac { b.n }{ a.n } b
  • \displaystyle a-\frac { a.n }{ b.n } b
  • None of these
Let \overrightarrow{AB}=3\hat{i}+\hat{j}-\hat{k} and \overrightarrow{AC}=\hat{i}-\hat{j}+3\hat{k}. If the point P on the line segment BC is equidistant from AB and AC then \overrightarrow{AP} is
  • 2\hat{i}-\hat{k}
  • \hat{i}-2\hat{k}
  • 2\hat{i}+\hat{k}
  • None of these
If the vectors \vec{a}, \vec{b}, \vec{c} and \vec{d} are coplanar, then \left ( \vec{a}\times \vec{b} \right )\times \left ( \vec{c}\times \vec{d} \right ) is equal to
  • \vec{a}+\vec{b}+\vec{c}+\vec{d}
  • \vec{0}
  • \vec{a}+\vec{b}=\vec{c}+\vec{d}
  • None of these
If \vec{a}, \vec{b}, \vec{c}, \vec{d} are any four vectors then \left ( \vec{a}\times \vec{b} \right )\times \left ( \vec{c}\times \vec{d} \right ) is a vector
  • Perpendicular to \vec{a}, \vec{b}, \vec{c}, \vec{d}
  • Along the line of intersection of two planes, one containing \vec{a}, \vec{b} and the other containing \vec{c}, \vec{d}
  • Equally inclined to both \vec{a}\times \vec{b} and \vec{c}\times \vec{d}
  • None of these
The position vectors of two vertices and the centroid of a triangle are \vec{i}+\vec{j}, 2\vec{i}-\vec{j}+\vec{k} and \vec{k} respectively, then the position vector of the third vertex of the triangle is 
  • -3\vec{i}+2\vec{k}
  • 3\vec{i}-2\vec{k}
  • \vec{i}+\dfrac{2}{3}\vec{k}
  • None of these
\displaystyle R(\bar{r}) is any point on a semi-circle, \displaystyle P(\bar{p}) and \displaystyle Q(\bar{q}) are the position vector of the end point of the diameter of that semi-circle, then \displaystyle \overline{PR}\cdot \overline{QR} is equal to
  • 1
  • 0
  • 3
  • Not defined
In a triangle OAB, E is the mid-point of OB and D is a point on AB such that AD: DB = 2 :If OD and AE intersect at P, determine the ratio OP : PD using vector methods.
  • OP:PD=2:3
  • OP:PD=3:2
  • OP:PD=1:3
  • OP:PD=3:1
In a triangle ABC, D divides BC in the ratio 3 : 2 and E divides CA in the ratio 1 :The lines AD and BE meet at H and CH meets AB in F. Find the ratio in which F divides AB.
  • \displaystyle AF:FB=2:1
  • \displaystyle AF:FB=1:2
  • \displaystyle AF:FB=2:3
  • \displaystyle AF:FB=3:2
\displaystyle \bar{a}= x\hat{i}+y\hat{j}+z\hat{k},\bar{b}= \hat{j} then the vector \displaystyle \bar{c} for which \displaystyle \bar{a},\bar{b},\bar{c} form a right hand triad
  • \displaystyle x(\hat{i}-\hat{k})
  • \displaystyle \bar{0}
  • \displaystyle -z\hat{i}+x\hat{k}
  • \displaystyle y\hat{j}
Let \displaystyle \bar{p} and \displaystyle \bar{q} be two distinct points. Let R and S be the points dividing PQ internally and externally in the ratio 2:3. If \displaystyle \overline{OR} \perp  \overline{OS}, then
  • \displaystyle 9p^{2}= 4q^{2}
  • \displaystyle 4p^{2}= 9q^{2}
  • \displaystyle 9p= 4q
  • \displaystyle 4p= 9q
The difference of the squares on the diagonals is four times the rectangle contained by either of these sides and the projection of the other upon it.
  • \displaystyle = 4 ( rectangle contained by AB and projection of AC on AB).
  • \displaystyle = 1 ( rectangle contained by AB and projection of AC on AB).
  • \displaystyle = 8 ( rectangle contained by AB and projection of AC on AB).
  • \displaystyle = 2 ( rectangle contained by AB and projection of AC on AB).
State true or false:
The four diagonals of a parallelopiped and the joins of the mid-points of opposite edges are concurrent at a common point of bisection.
  • True
  • False
ABC is a \displaystyle \Delta and G is its centroid. If \displaystyle \overline{AB}= \bar{b} and \displaystyle \overline{AC}= \bar{c}, then \displaystyle \overline{AG} is equal to
  • \displaystyle \dfrac{2}{3}\left ( \bar{b}+\bar{c} \right )
  • \displaystyle \bar{b}+\bar{c}
  • \displaystyle \dfrac{1}{3}\left ( \bar{b}+\bar{c} \right )
  • \displaystyle \dfrac{3}{2}\left ( \bar{b}+\bar{c} \right )
In a parallelogram ABCD,\left| AB \right| =a,\left| AD \right| =b and \left| AC \right| =c. Then, DB.AB has the value
  • \displaystyle\frac { 3{ a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 } }{ 2 }
  • \displaystyle\frac { { a }^{ 2 }+3{ b }^{ 2 }-{ c }^{ 2 } }{ 2 }
  • \displaystyle\frac { { a }^{ 2 }-{ b }^{ 2 }+3{ c }^{ 2 } }{ 2 }
  • \displaystyle\frac { { a }^{ 2 }+3{ b }^{ 2 }+{ c }^{ 2 } }{ 2 }
Let \bar{a}, \bar{b} and \bar{c} be vectors with magnitudes 3, 4 and 5 respectively and \bar{a}+\bar{b}+\bar{c}=0, then the value of \bar{a}.\bar{b}+\bar{b}.\bar{c}+\bar{c}.\bar{a} is
  • 48
  • -26
  • 25
  • -25
The sides of a \triangle are in A.P, then the line joining the centroid to the incenter is parallel to
  • The largest side
  • The middle side
  • The smallest side
  • None of these
If \bar{\alpha }, \bar{\beta } and \bar{\gamma } be vertices of a \triangle whose circumcenter is at the origin, then orthocenter is given by
  • \displaystyle \dfrac{\bar{\alpha }+\bar{\beta }+\bar{\gamma }}{4}
  • \displaystyle \dfrac{\bar{\alpha }+\bar{\beta }+\bar{\gamma }}{2}
  • \bar{\alpha }+\bar{\beta }+\bar{\gamma }
  • \displaystyle \dfrac{\bar{\alpha }+\bar{\beta }+\bar{\gamma }}{3}
If S is the circumcenter, O is the orthocenter of \triangle ABC, then \displaystyle \vec{SA}+\vec{SB}+\vec{SC}=
  • 2\vec{OS}
  • 2\vec{SO}
  • \vec{OS}
  • \vec{SO}
For non-zero vectors \bar{a}, \bar{b} and \bar{c}, \left | \left ( \bar{a}\times \bar{b} \right ).\bar{c} \right |=\left | \bar{a} \right |\left | \bar{b} \right |\left | \bar{c} \right | iff
  • \bar{a}.\bar{c}=0, \bar{a}.\bar{b}=0
  • \bar{a}.\bar{b}=0, \bar{b}.\bar{c}=0
  • \bar{c}.\bar{a}=0, \bar{b}.\bar{c}=0
  • \bar{a}.\bar{b}=\bar{b}.\bar{c}=\bar{c}.\bar{a}=0
Two planes are perpendicular to each other,one of them contains vector \bar{a} and \bar{b}, other contains \bar{c} and \bar{d} then \left ( \bar{a}\times \bar{b} \right )\cdot \left (\bar{c}\times \bar{d}  \right )=
  • 1
  • 0
  • \left [ \bar{a}\bar{b}\bar{c} \right ]
  • \left [ \bar{b}\bar{c}\bar{d} \right ]
Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin, then \bar{OA}+\bar{OB}+\bar{OC}+\bar{OD}=
  • 2\bar{OP}
  • 3\bar{OP}
  • \bar{OP}
  • 4\bar{OP}
If a. \displaystyle b\neq 0, find the vector r which satisfies the equations \displaystyle \left ( r-c \right )\times b= 0, r.a= 0
  • \displaystyle r= \frac{\left ( a.b \right )c+\left ( a.c \right )b}{a.b}
  • \displaystyle r= \frac{\left ( a.b \right )c-\left ( a.c \right )b}{a.b}
  • \displaystyle r= \frac{-\left ( a.b \right )c-\left ( a.c \right )b}{a.b}
  • \displaystyle r= \frac{-\left ( a.b \right )c+\left ( a.c \right )b}{a.b}
If \alpha \bar{a}+\beta \bar{b}+\gamma \bar{c}=0, then \left ( \bar{a}\times \bar{b} \right )\times \left [ \left ( \bar{b}\times \bar{c} \right )\times \left ( \bar{c}\times \bar{a} \right ) \right ] is equal to
  • \bar{0}
  • A vector \perp plane of \bar{a}, \bar{b} and \bar{c}
  • A scalar quantity
  • None of these
A unit vector perpendicular to each of the vectors \displaystyle 2\hat{i}+4\hat{j}-\hat{k} and \displaystyle \hat{i}-2\hat{j}+3\hat{k} forming a right handed system is
  • \displaystyle 7\hat{i}-10\hat{j}+8\hat{k}
  • \displaystyle \frac{10\hat{i}-7\hat{j}-8\hat{k}}{\sqrt{213}}
  • \displaystyle -7\hat{i}+10\hat{j}+8\hat{k}
  • \displaystyle \frac{-10\hat{i}-7\hat{j}-8\hat{k}}{\sqrt{213}}
The position vector of the points A and B are respectively \bar{a} and \bar{b}  divides AB in the ratio 3:1 and Q iis the midpoint of AP. The position vector of Q is
  • \displaystyle \dfrac{3\bar{a}+5\bar{b}}{8}
  • \displaystyle\dfrac{3\bar{a}+\bar{b}}{4}
  • \displaystyle\dfrac{\bar{a}+3\bar{b}}{4}
  • \displaystyle\dfrac{5\bar{a}+3\bar{b}}{8}
For three unit vectors \bar{a}\bar{b} and \bar{c}if \bar{a}+\bar{b}+\bar{c}= \bar{0}, then the value of \bar{a}\cdot \left ( \bar{b}+\bar{c} \right )+\bar{b}\cdot \left ( \bar{c}+\bar{a} \right )+\bar{c}\cdot \left ( \bar{a}+\bar{b} \right ) is equal to
  • -\dfrac{3}{2}
  • 0
  • -3
  • None of these
If \displaystyle \bar{a}\times \bar{b}=\bar{b}\times \bar{c}=\bar{c}\times \bar{a} then \displaystyle \bar{a}+\bar{b}+\bar{c}=?
  • abc
  • -1
  • 0
  • 2
The vector \left ( \bar{a }\times\bar{b } \right )\times \left ( \bar{c }\times\bar{b } \right ) is
  • At right angle to \bar{b}
  • Is parallel to \bar{c}
  • Is parallel to \bar{b}
  • None of these
In a trapezium the vector \overline{BC} = \alpha \overline{AD}. We will then find that \bar{p}= \overline{AC}+\overline{BD} is collinear with \overline{AD}. if \bar{p}= \mu \overline{AD} then
  • \mu =\alpha +2
  • \mu +\alpha = 2
  • \alpha = \mu +1
  • \mu = \alpha+1
If C is the mid point of AB and P is any point outside AB, then
  • \displaystyle \overline{PA}+\overline{PB}+\overline{PC}=0
  • \displaystyle \overline{PA}+\overline{PB}+2\overline{PC}=\bar{0}
  • \displaystyle \overline{PA}+\overline{PB}=\overline{PC}
  • \displaystyle \overline{PA}+\overline{PB}=2\overline{PC}
Given A=ai+bj+ck, \ B=di+3j+4k and C=3i+j-2k. If the vectors A,B and C form a triangle such that A=B+C and area(\Delta ABC)=5\sqrt6, then
  • a=-8, \ b=-4, \ c=2, \ d=-11
  • a=-8, \ b=4, \ c=-2, \ d=-11
  • a=-8,\ b=4, \ c=2, \ d=-11
  • None of the above
Two identical particles are located at \overrightarrow{x} and  \overrightarrow{y} with reference to the origin of three dimensional co-ordinate system. The position vector of centre of mass of the system is given by
  • \overrightarrow { x } - \overrightarrow { y }
  • \displaystyle \frac {\overrightarrow { x } + \overrightarrow { y }} {{2}}
  • (\overrightarrow { x } - \overrightarrow { y })
  • \displaystyle \frac{\overrightarrow { x } - \overrightarrow { y }} {{2}}
A vector a can be written as
  • (a\cdot i)i+(a\cdot j)j+(a\cdot k)k
  • (a\cdot j)i+(a\cdot k)j+(a\cdot i)k
  • (a\cdot k)j+(a\cdot i)j+(a\cdot j)k
  • (a\cdot a)i+(i+j+k) k
If b \neq 0, then every vector a can be written in a unique manner as the sum of a vector a_{||} parallel to b and a vector a_{\perp} perpendicular to b. If a is parallel to b, then a_{||} = a and a_{\perp } =0. If a is perpendicular to b, then a_{||} = 0 and a_{\perp} = a. The vector a_{||} is called the projection of a on b and is denoted by proj_{b}a. Since proj_{b} a is parallel to b, it is a scalar multiple of the unit vector in the direction of b, i.e., proj _{b} a =\lambda u_{b}
The scalar \lambda is called the component of a in the direction of b and is denoted by comp _{b} a . In fact, proj _{b} a = (a \cdot u_{b})u_{b} and comp _{b} a = a \cdot u_{b}

If a = -2i + j + k and b =4i -3j + k , then proj_{b} a is equal to
  • 4i -3j + 2k
  • \displaystyle -\frac{ 5}{13} (4i- 3j + k)
  • \displaystyle \frac{ 5}{13} (4i -3j+ k)
  • \displaystyle -\frac{ 4}{11} (2i -j + 2k)
ABCD is a quadrilateral and E the point of intersection of the lines joining the middle points of opposite sides. If O is any point, then the resultant of OA,OB,OC and OD is equal to
  • 2OE
  • OE
  • 4OE
  • none of these
The components of vector i + j + k along vector i+2j+3k is
  • (3/7) (i + 2j + 3k)
  • (i + 2j + 3k)
  • (1/7) (i + 2j + 3k)
  • (4/7) (i + 2j + 3k)
Four forces of magnitude P, 2P, 3P and 4P act along the four sides of a square ABCD in cyclic order. Use the vector method to find the resultant force.
  • \displaystyle 2\sqrt{2}P
  • \displaystyle 3\sqrt{2}P
  • \displaystyle \sqrt{2}P
  • \displaystyle -2\sqrt{2}P
OAB is a given triangle such that \displaystyle \overrightarrow{OA}=\bar a, \displaystyle \overrightarrow{OB}=\bar b.
Also C is a point on AB such that \overrightarrow{AB}=2\overrightarrow{BC}. State which of the following statements are correct?
  • \displaystyle \overrightarrow{AC}=\frac{2}{3}\left ( \bar b-\bar a \right ),
  • \displaystyle \overrightarrow{AC}=\frac{2}{3}\left ( \bar a-\bar b \right ),
  • \displaystyle \overrightarrow{AC}=\frac{2}{3}\left (\bar b+\bar a \right ),
  • \displaystyle \overrightarrow{AC}=\frac{3}{2}\left ( \bar b-\bar a \right ),
ABC is any triangle and D, E, F are the middle points of its sides BC, CA, AB respectively. Express the vectors \displaystyle \overrightarrow{CF} and \overrightarrow{BE} as linear combination of the vectors \displaystyle \overrightarrow{AB} and \overrightarrow{AC}
  • \displaystyle \overrightarrow{BE}= \frac{1}{2}\overrightarrow{AC}-\overrightarrow{AB};\,\,\,\overrightarrow{CF}= \frac{1}{2}\overrightarrow{AB}-\overrightarrow{AC}
  • \displaystyle \overrightarrow{BE}= \frac{1}{2}\overrightarrow{AC}+\overrightarrow{AB};\,\,\,\overrightarrow{CF}= \frac{1}{2}\overrightarrow{AB}+\overrightarrow{AC}
  • \displaystyle \overrightarrow{CF}= \frac{1}{2}\overrightarrow{AC}-\overrightarrow{AB};\,\,\,\overrightarrow{BE}= \frac{1}{2}\overrightarrow{AB}-\overrightarrow{AC}
  • \displaystyle \overrightarrow{CF}= \frac{1}{2}\overrightarrow{AC}+\overrightarrow{AB};\,\,\,\overrightarrow{BE}= \frac{1}{2}\overrightarrow{AB}+\overrightarrow{AC}
If A\left( \overrightarrow { a }  \right) ,B( \overrightarrow { b } ) ,C\left( \overrightarrow { c }  \right) be the vertices of a triangle whose circumcentre is the origin, then orthocenter is given by
  • \displaystyle \dfrac { \overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c }  }{ 3 }
  • \displaystyle \dfrac { \overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c }  }{ 2 }
  • \overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c }
  • None of these
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Practice Class 12 Commerce Maths Quiz Questions and Answers