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

Let $$\overrightarrow { b } =4\hat { i } +3\hat { j } $$ and $$\overrightarrow{c}$$ be two vector perpendicular to each other in the $$xy$$-plane. Then a vector in the same plane having projections $$1$$ and $$2$$ along $$\overrightarrow{b}$$ and $$\overrightarrow{c}$$, respectively, is
  • $$\hat { i } +2\hat { j } $$
  • $$2\hat { i } -\hat { j } $$
  • $$2\hat { i } +\hat { j } $$
  • 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 $$\triangle ABC$$. If $$\overrightarrow{AB}=\overrightarrow{a}$$ 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
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