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CBSE Questions for Class 12 Commerce Maths Vector Algebra Quiz 4 - MCQExams.com

Let b=4ˆi+3ˆj and c be two vector perpendicular to each other in the xy-plane. Then a vector in the same plane having projections 1 and 2 along b and c, respectively, is
  • ˆi+2ˆj
  • 2ˆiˆj
  • 2ˆi+ˆj
  • None of these
The angle between the Vectors \vec{a}\times\vec{b} and \vec{b}\times\vec{a} is
  • 0^{0}
  • 45^{0}
  • 90^{0}
  • 180^{0}
The vectors \vec{a},\ \vec{b},\ \vec{a}\times \vec{b} form
  • A right handed system
  • A left handed system
  • A set of coplanar vectors
  • A set of mutually perpendicular vectors
In a triangle ABC, D and E are points on BC and AC respectively, such that BD=2DC, AE=3EC, Let P be the point of intersection of AD and BE. Then \dfrac{BE}{PE}=
  • 2:3
  • 3:8
  • 8:3
  • 1:2
If \overline{a}+\overline{b}+\overline{c}=\alpha\overline{d},\overline{b}+\overline{c}+\overline{d}=\beta\overline{a}, then \overline{a}+\overline{b}+\overline{c}+\overline{d} is 
  • (\beta+1)\overline{a}
  • \alpha\overline{a}
  • (\alpha+1)\overline{b}
  • \alpha\overline{a}+\beta\overline{b}
If OABC is a parallelogram with \vec{OB}=\vec{a},\vec{AB}=\vec{b} then \vec{OA}=
  • \vec{a}+\vec{b}
  • \vec{a}-\vec{b}
  • \displaystyle \dfrac{1}{2}(\vec{a}+\vec{b})
  • \displaystyle \dfrac{1}{2}(\vec{a}-\vec{b})
Let us define, the length of a vector a\overline{i}+b\overline{j}+c\overline{k} as |{a}|+|{b}|+|{c}|. This definition coincides with the usual definition of the length of a vector a\overline{i}+b\overline{j}+c\overline{k} if
  • a=b=c=0
  • Any one of a, b, c, is zero
  • Any two of a, b, c are zero
  • a=b=c\neq 0
In \Delta ABC,\ D,\ E,\ F are midpoints of the sides BC, CA and AB respectively. O' is the circumcentre, G' is the centroid, H' is the orthocentre and P is any point.
Match the following
List IList II
1) \vec{PA} +\vec{PB}+\vec{PC}a) 0
2)\vec {GA}+\vec{GB}+\vec{GC}b) \vec{OH}
3)\displaystyle \vec{AD}+\dfrac{2}{3}\vec{BE}+\dfrac{1}{3}\vec{CF}c)\vec{ PD}+\vec{PE}+\vec{PF}
4)\vec{OA}+\vec{OB}+\vec{OC}d){\displaystyle\dfrac{1}{2}}\vec{AC}
  • 1-a,2- b,3- c,4- d
  • 1-c,2- a,3- b,4- c
  • 1-c,2- a,3- d,4- b
  • 1-a,2- b,3- d,4- c
Vector area is a vector quantity associated with each plane figure whose magnitude is
  • Any quantity and direction parallel to the plane
  • Any quantity and direction perpendicular to the plane
  • Equal to the area and direction parallel to the plane
  • Equal to the area and direction perpendicular to the plane
Let OABC be a parallelogram and D the midpoint of OA. The ratio in which OB divides CD in the ratio
  • 1:2
  • 1:3
  • 1:4
  • 2:1
In \Delta OAB, if \vec{OA}=\vec{a},\ \vec{OB}=\vec{b}.  L is mid point of \vec{OA} and M is point on \vec{OB} such that \vec{OM}:\vec{MB}=2:1. If P is mid point of LM then \vec{AP}=
  • {\dfrac{1}{3}\vec{b}-\dfrac{3}{4}\vec{a}}
  • \displaystyle \dfrac{1}{3}\vec{b}+\dfrac{3}{4}\vec{a}
  • {\dfrac{1}{3}\vec{a}-\dfrac{3}{4}\vec{b}}
  • \displaystyle \dfrac{1}{3}\vec{a}+\dfrac{3}{4}\vec{b}
If the vectors \vec{c},\vec{a}=x\hat{i}+y\hat{j}+z\hat{k} and \vec{b}=\hat{j} are such that \vec{a},\vec{c} and \vec{b} form a right handed system, then \vec{c}=
  • z\hat{i}
  • y\hat{i}
  • z\hat{i}-x\hat{k}
  • {-z\hat{i}+x\hat{k}}
The vector \vec{AB}=3\hat{i}+4\hat{k} and \vec{AC}=5\hat{i}-2\hat{j}+4\hat{k} are the sides of a \Delta ABC where A is the origin. The length of median through {A} is
  • \sqrt{72}
  • \sqrt{33}
  • \sqrt{288}
  • \sqrt{18}
If C is the mid point of AB and P is any point out side AB, then
  • \vec{PA}+\vec{PB}+2\vec{PC}=\vec{0}
  • \vec{PA}+\vec{PB}+\vec{PC}=\vec{0}
  • \vec{PA}+\vec{PB}=2\vec{PC}
  • \vec{PA}+\vec{PB}=\vec{PC}
ABC is a triangle and P is any point on BC. If PQ is the resultant of the vectors \vec {AP},\ \vec {PB} and \vec{PC} then ACQB is
  • Rectangle
  • Square
  • Rhombus
  • Parallelogram
If \vec{b} is the vector whose initial point divides the joining 5\hat{i} and 5\hat{j} in the ratio \lambda : 1 and terminal point is at origin. lf |\vec{b}|\leq\sqrt{37}, then \lambda\in.
  • (-\displaystyle \infty, -6]\cup  \left [-\dfrac{1}{6}, \infty \right)
  • (-\displaystyle \infty, -3)\cup \left [-\dfrac{1}{4}, \infty \right)
  • (-\infty, 0)\cup \left(\dfrac{1}{2} , \infty\right)
  • \left [-6, -\displaystyle \dfrac{1}{6}\right]
lf the Vector \overline{\mathrm{c}},\ \vec{\mathrm{a}}=\mathrm{x}\hat{\mathrm{i}}+\mathrm{y}\hat{\mathrm{j}}+\mathrm{z}\hat{\mathrm{k}},\ \vec{\mathrm{b}}=\hat{\mathrm{j}} are such that \vec{\mathrm{a}},\ \vec{\mathrm{c}},\ \vec{\mathrm{b}} form \mathrm{R}.\mathrm{H}.\ \mathrm{S} then \vec{\mathrm{c}}=
  • \mathrm{z}\hat{\mathrm{i}}-\mathrm{x}\hat{\mathrm{k}}
  • \mathrm{x}\hat{\mathrm{i}}-\mathrm{z}\hat{\mathrm{k}}
  • \mathrm{x}\hat{\mathrm{j}}-\mathrm{y}\hat{\mathrm{k}}
  • \mathrm{y}\hat{\mathrm{j}}-\mathrm{x}\hat{\mathrm{k}}
If \vec{r}=3\hat{i}+2\hat{j}-5\hat{k},\vec{a}=2\hat{i}-\hat{j}+\hat{k}\vec{b}=\hat{i}+3\hat{j}-2\hat{k},\ \vec{c}=-2\hat{i}+\hat{j}-3\hat{k} such that \vec{r}=\lambda\vec{a}+\mu\vec{b}+v\vec{c}, then \mu,\ \displaystyle \frac{\lambda}{2} , v are in
  • H.P
  • G.P
  • A.G.P
  • A.P
\overline{a}=x\hat {i}+y\hat {j}+z\hat {k},\ \overline{b}=\hat {j}, then the vector \overline{c} for which \overline{a},\overline{b},\ \overline{c} form a right hand triangle
  • x(\hat {i}-\hat {k})
  • \hat {0}
  • -z\hat {i}+x\hat {k}
  • y\hat {j}
If I is the center of a circle inscribed in a triangle ABC, then |BC|IA+|CA|IB+|AB|IC
  • 0
  • IA+IB+IC
  • \displaystyle \dfrac{IA+IB+IC}{3}
  • \displaystyle \dfrac{IA+IB+IC}{2}
If \vec a=\hat {i}+2\hat {j}-3\hat {k} and \vec {b}=2\hat {i}-\hat {j}-\hat {k} then the ratio between the projection of \vec b on \vec {a} and the projection of \vec {a} on \vec {b} is
  • \sqrt{5}:\sqrt{7}
  • \sqrt{3}:\sqrt{7}
  • \sqrt{7}:\sqrt{3}
  • \sqrt{7}:\sqrt{5}
Given,  |\vec {a}|=|\vec {b}|=1 and |\vec {a}+\vec {b}|=\sqrt{3}. If \vec {c} be a vector such that \vec {c}-\vec {a}-2\vec {b}=3(\vec {a}\times\vec {b}) , then \vec {c}.\vec {b} is equal to
  • -\displaystyle \dfrac{1}{2}
  • \displaystyle \dfrac{1}{2}
  • \displaystyle \dfrac{3}{2}
  • \displaystyle \dfrac{5}{2}
Which of the following is a true statement. 
  • \vec {a}\times\vec {b})\times\vec {c} is coplanar with \vec {c}
  • (\vec {a}\times\vec {b})\times\vec {c} is perpendicular to \vec {a}
  • (\vec {a}\times\vec {b})\times\vec {c} is perpendicular to \vec {b}
  • (\vec {a}\times\vec {b})\times\vec {c} is perpendicular to \vec {c}
If the position vector of a point A is \overrightarrow { a } +\overrightarrow { 2b } and \overrightarrow { a } divides \overrightarrow{AB} in the ratio 2:3, then the position vector of B is
  • \overrightarrow { 2a } -\overrightarrow { b }
  • \overrightarrow { b } -\overrightarrow { 2a }
  • \overrightarrow { a } -\overrightarrow { 3b }
  • \overrightarrow { b }
In a tetrahedron if two pairs of opposite edges are at a right angles then the third pair is inclined at an angle of
  • 30^{0}
  • 60^{0}
  • 90^{0}
  • 120^{0}
If \overrightarrow { a } .\overrightarrow { b } =0 and also \overrightarrow { a } \times \overrightarrow { b } =0, then
  • \overrightarrow{a} is parallel to \overrightarrow{b}
  • \overrightarrow{a} is perpendicular to \overrightarrow{b}
  • Either \overrightarrow{a} or \overrightarrow{b} is a non-zero vector
  • None of these
Let G be the centroid of \triangle ABCIf \overrightarrow{AB}=\overrightarrow{a} and \overrightarrow{AC}=\overrightarrow{b}, then \overrightarrow{AG}, in terms of \overrightarrow{a} and \overrightarrow{b} is
  • \displaystyle \dfrac { 2 }{ 3 } \left( \overrightarrow { a } +\overrightarrow { b }  \right)
  • \displaystyle \dfrac { 1 }{ 6 } \left( \overrightarrow { a } +\overrightarrow { b }  \right)
  • \displaystyle \dfrac { 1 }{ 3 } \left( \overrightarrow { a } +\overrightarrow { b }  \right)
  • \displaystyle \dfrac { 1 }{ 2 } \left( \overrightarrow { a } +\overrightarrow { b }  \right)
If \overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c } are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then the centroid of the triangle satisfies which of the following relation?
  • \overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c }=0
  • \overrightarrow { { a }^{ 2 } } =\overrightarrow { { b }^{ 2 } } +\overrightarrow { { c }^{ 2 } }
  • \overrightarrow { a } +\overrightarrow { b } -\overrightarrow { c } =0
  • None of these
If \alpha(\vec a \times \vec b)+\beta(\vec b \times \vec c)+\gamma(\vec c \times \vec {a})=\vec{0} and at least one of the scalars \alpha,\ \beta,\gamma is non-zero, then the vectors \vec{a},\vec{b},\vec{c} are
  • Collinear
  • Coplanar
  • Non-coplanar
  • Cannot be determined.
lf the four points \overline{a},\overline{b},\overline{c},\overline{d} are coplanar then [\overline{b}\overline{c}\overline{d}]+[\overline{c}\overline{a}\overline{d}]+[\overline{a}\overline{b}\overline{d}]=
  • 0
  • [\overline{a},\overline{b},\overline{c}]
  • 2[\overline{a},\overline{b},\overline{c}]
  • 3[\overline{a},\overline{b},\overline{c}]
A point O is the centre of a circle circumscribed about a triangle ABC, then \vec{OA}\sin 2A + \vec{OB}\sin 2B + \vec{OC} \sin 2C is equal to
  • (\vec{OA} + \vec{OB} + \vec{OC})\sin 2A
  • 3\vec{OG}, where G is the centroid of triangle ABC
  • \vec 0
  • None of these
If \vec {a}=x\hat {i}+12\hat {j}-\hat {k},\vec {b}=2\hat {i}+2x\hat {j}+\hat {k} and \vec {c}=\hat {i}+\hat {k} and given that the vectors \vec {a},\vec {b},\vec {c} form a right handed system, then the range of x is
  • R-[-3,2]
  • (-4,3)
  • R-(-3,2)
  • (-2,3)
If G is the centroid of a  \Delta ABC, then \vec{GA} + \vec{GB} + \vec{GC} is equal to
  • \vec{0}
  • 3\vec{GA}
  • 3\vec{GB}
  • 3\vec{GC}
In triangle ABC, which of the following is not true.
  • \overrightarrow{AB} + \overrightarrow{BC}+\overrightarrow{CA} = \overrightarrow{0}
  • \overrightarrow{AB} + \overrightarrow{BC}- \overrightarrow{AC} = \overrightarrow{0}
  • \overrightarrow{AB} + \overrightarrow{BC}- \overrightarrow{CA} = \overrightarrow{0}
  • \overrightarrow{AB} - \overrightarrow{CB} + \overrightarrow{CA} = \overrightarrow{0}
Six vectors, \vec a to \vec f , all of magnitude 1 and directions indicated in the figure ( Consider all of them to be originating at origin ). Which of the following statement is true?
118073.png
  • \vec b+\vec e=\vec f
  • \vec b+\vec c=\vec f
  • \vec d+\vec c=\vec f
  • \vec d+\vec e=\vec f
If C is the mid point of AB and P is any point outside AB, then 
  • \vec {\mathrm{P}\mathrm{A}}+\vec {\mathrm{P}\mathrm{B}}=2\vec {\mathrm{P}\mathrm{C}}
  • \vec {\mathrm{P}\mathrm{A}}+\vec {\mathrm{P}\mathrm{B}}=\vec {\mathrm{P}\mathrm{C}}
  • \vec {\mathrm{P}\mathrm{A}}+\vec {\mathrm{P}\mathrm{B}}+2\vec {\mathrm{P}\mathrm{C}}=0
  • \vec {\mathrm{P}\mathrm{A}}+\vec {\mathrm{P}\mathrm{B}}=\vec {\mathrm{P}\mathrm{C}}=0
If \vec a is a non-zero vector of modulus a and m is a non-zero scalar, then m \vec a is a unit vector if
  • m = \pm 1
  • a = |m|
  • a = \dfrac{1}{|m|}
  • a = \dfrac{1}{m}
If vector \vec{a} = 2\hat i - 3\hat j + 6\hat k and vector \vec{b} = - 2\hat i + 2\hat j - \hat k, then ratio of Projection of \vec a  on vector  \vec b to Projection of  \vec b  on \vec a is equal to
  • \displaystyle \dfrac{3}{7}
  • \displaystyle \dfrac{7}{3}
  • 3
  • 7
'I' is the incentre of triangle of ABC whose corresponding sides are a, b, c, respectively, a\vec{IB} + b\vec{IB} + c\vec{IC} is always equal to
  • \vec{0}
  • (a+b+c) \vec{BC}
  • (\vec{a} + \vec{b} + \vec{c}) \vec{AC}
  • (a + b + c)\vec{AB}
Let \vec{a}, \vec{b} and \vec{c} be unit vectors such that \vec{a} + \vec{b} - \vec{c} = 0. If the area of triangle formed by vectors \vec{a} and \vec{b} is A, then what is the value of 4A^2?
  • 3
  • 9
  • \dfrac { 3 }{ 4 }
  • \dfrac { 9 }{ 4 }
In a trapezium, vector \vec{BC} = \alpha \vec{AD}. Also, \vec p = \vec{AC} + \vec{BD} is collinear with \vec{AD} and \vec p = \mu \vec{AD}, then which of the following is true?
  • \mu = \alpha + 2
  • \mu + \alpha =1
  • \alpha = \mu + 1
  • \mu = \alpha +1
In triangle ABC, \angle A = 30^o, H is the orthocentre and D is the midpoint of BC. Segment HD  is produced to T  such that HD = DT. The length AT is equal to
  • 2BC
  • 3BC
  • \displaystyle \dfrac{4}{3}BC
  • None of these
P(\vec{p}) and Q(\vec{q}) are the position vectors of two fixed points and R(\vec{r}) is the position vector of a variable point. If R moves such that (\vec{r} - \vec{p}) \times (\vec{r} - \vec{q}) = \vec{0}, then the locus of R is
  • A plane containing the origin O and parallel to two non-collinear vectors \vec{OP} and \vec{OQ}
  • The surface of a sphere described on PQ as its diameter
  • A line passing through points P and Q
  • A set of lines parallel to line PQ
A, B, C and D have position vectors \vec{a}, \vec{b}, \vec{c} and \vec{d}, respectively, such that \vec{a} - \vec{b} = 2 (\vec{d} - \vec{c}), then
  • AB and CD bisect each other
  • AB and CD trisect each other
  • BD and AC bisect each other
  • BD and AC trisect each other
Let ABC be a triangle whose centroid is G, orthocentre is H and circumcentre is the origin 'O'. If D is any point in the plane of the triangle such that no three of O, A, C and D are collinear satisfying the relation \vec{AD} + \vec{BD} + \vec{CH} + 3 \vec{HG} = \lambda \vec{HD}, then what is the value of the scalar '\lambda'?
  • 2
  • 3
  • -1
  • -2
The projection of the line segment joining the points A(-1, 0, 3) and B(2, 5, 1) on the line whose direction ratios are proportional to 6, 2, 3, is
  • \dfrac{10}{7}
  • \dfrac{22}{7}
  • \dfrac{18}{7}
  • none of these
Let \vec {p} is the position vector of the orthocentre & \vec {g} is the position vector of the centroid of the triangle ABC where circumcentre is the origin. If \vec {p}= K\vec{g}, then K=
  • 3
  • 2
  • \displaystyle\dfrac{1}{3}
  • \displaystyle\dfrac{2}{3}
If \vec{a}, \vec{b}   and \vec {c} are three non- coplanar vectors, then the length of projection of vector \vec{a} in the plane of the vectors \vec{b} and \vec{c} may be given by
  • \dfrac{| \vec{a}.(\vec{b}\times \vec{c})|}{|\vec{b} \times \vec{c}|}
  • \dfrac{| \vec{a}\times(\vec{b}\times \vec{c})|}{|\vec{b} \times \vec{c}|}
  • \dfrac{[\vec{a} \vec{b} \vec{c}]}{(\vec{b}. \vec{c})}
  • none of these
Five coplanar forces (each of magnitude 20N) are acting on a body. The angle between two neighboring forces have  the same value. The resultant of these forces is necessarily equal to
  • 20N
  • 20\sqrt{2}N
  • Zero
  • none of these
If the scalar projection of the vector x\hat i - \hat j + \hat k on the vector 2 \hat i - \hat j + 5\hat k is \dfrac{1}{\sqrt{30}}, then value of x is equal to
  • \dfrac{-5}{2} units
  • 6 units
  • - 6 units
  • 3 units
0:0:1


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Practice Class 12 Commerce Maths Quiz Questions and Answers