CBSE Questions for Class 12 Commerce Maths Vector Algebra Quiz 11 - MCQExams.com

Let $$ABC$$ be a triangle, the position vector of whose vertices are $$7 \hat j + 10 \hat k, - \hat i + 6 \hat j + 6 \hat k$$ and $$- 4 \hat i + 9 \hat j + 6 \hat k$$. Then $$\Delta ABC$$ is
  • isosceles
  • equilateral
  • right-angled
  • none of these
Vector $$ \vec{x} $$ is
  • $$\dfrac{1}{|\vec{a} \times \vec{b}|^{2}}[\vec{a} \times(\vec{a} \times \vec{b})]$$
  • $$\dfrac{\gamma}{|\vec{a} \times \vec{b}|^{2}}[\vec{a} \times \vec{b}-\vec{a} \times(\vec{a} \times \vec{b})]$$
  • $$\dfrac{\gamma}{|\vec{a} \times \vec{b}|^{2}}[\vec{a} \times \vec{b}+\vec{b} \times(\vec{a} \times \vec{b})]$$
  • none of these
Vector $$ \vec{z} $$ is
  • $$\dfrac{\gamma}{|\vec{a} \times \vec{b}|^{2}}[\vec{a}+\vec{b} \times(\vec{a} \times \vec{b})] $$
  • $$\dfrac{\gamma}{|\vec{a} \times \vec{b}|^{2}}[\vec{a}+\vec{b}-\vec{a} \times(\vec{a} \times \vec{b})]$$
  • $$\dfrac{\gamma}{|\vec{a} \times \vec{b}|^{2}}[\vec{a} \times \vec{b}+\vec{b} \times(\vec{a} \times \vec{b})]$$
  • none of these
Vector $$ \vec{y} $$ is
  • $$\dfrac{\vec{a} \times \vec{b}}{\gamma}$$
  • $$\vec{a}+\dfrac{\vec{a} \times \vec{b}}{\gamma}$$
  • $$\vec{a}+\vec{b}+\dfrac{\vec{a} \times \vec{b}}{\gamma} $$
  • none of these
Vectors $$\vec{A} \space and \space \vec{B}$$ satisfying the vector equation $$\vec{A} + \vec{B} = \vec{a}, \vec{A} \times \vec{B} = \vec{b} \space and \space \vec{A} . \vec{a} = 1$$ where $$\vec{a} \space and \space \vec{b}$$ are given vectors are
  • $$\vec{A} = \frac{(\vec{a} \times \vec{b}) - \vec{a}}{a^2}$$
  • $$\vec{B} = \frac{(\vec{b} \times \vec{a}) + \vec{a}(a^2 - 1)}{a^2}$$
  • $$\vec{A} = \frac{(\vec{a} \times \vec{b}) + \vec{a}}{a^2}$$
  • $$\vec{B} = \frac{(\vec{b} \times \vec{a}) - \vec{a}(a^2 - 1)}{a^2}$$
$$ (\vec{P} \times \vec{B}) \times \vec{B} $$ is equal to
  • $$\vec{P}$$
  • $$-\vec{P}$$
  • $$2 \vec{B}$$
  • $$\vec{A}$$
If side $$\vec{AB}$$ of an equilateral triangle ABC lying in the x - y plane is $$3\hat{i}$$, then side $$\vec{CB}$$ can be
  • $$-\frac{3}{2}(\hat{i} - \sqrt{3}\hat{j}$$
  • $$\frac{3}{2}(\hat{i} - \sqrt{3}\hat{j})$$
  • $$-\frac{3}{2}(\hat{i} + \sqrt{3}\hat{j})$$
  • $$\frac{3}{2}(\hat{i} - \sqrt{3} \hat{j})$$
Given that $$\vec a, \vec b, \vec p, \vec q$$ are four vectors such that $$\vec a + \vec b = \mu \vec p, \vec b \cdot \vec q = 0$$ and $$(\vec b)^2 = 1,$$ where $$\mu$$ is scalar. Then $$\mid (\vec a \cdot \vec q) \vec p - (\vec p \cdot \vec q)\vec a \mid$$ is equal to
  • $$2 \mid \vec p \cdot \vec q \mid$$
  • $$(1/2) \mid \vec p \cdot \vec q \mid$$
  • $$\mid \vec p \times \vec q \mid$$
  • $$\mid \vec p \cdot \vec q \mid$$
If $$\vec r$$ and $$\vec s$$ are non-zero constant vectors and the scalar $$b$$ is chosen such that $$\mid \vec r + b \vec s \mid$$ is minimum, then the value of $$\mid b \vec s \mid^2 + \mid \vec r + b \vec s \mid^2$$ is equal to
  • $$2 \mid \vec r \mid^2$$
  • $$\mid \vec r \mid^2/2$$
  • $$3 \mid \vec r \mid^2$$
  • $$\mid \vec r \mid^2$$
If $$\overline a$$ and $$\overline b$$ are adjacent sides of a rhombus, then $$\overline a.\overline b=0$$.
  • True
  • False
Let $$\hat{a} \space and \space \hat{b}$$ be mutually perpendicular unit vectors. Then for any arbitrary $$\vec{r}$$.
  • $$\vec{r} = (\vec{r} . \hat{a})\hat{a} + (\vec{r} . \hat{b})\hat{b} + (\vec{r} . (\hat{a} \times \hat{b})) (\hat{a} \times \hat{b})$$
  • $$\vec{r} = (\vec{r} . \hat{a}) - (\vec{r} . \hat{b})\hat{b} + (\vec{r} . (\hat{a} \times \hat{b}))) (\hat{a} \times \hat{b})$$
  • $$\vec{r} = (\vec{r} . \hat{a})\hat{a} - (\vec{r} . \hat{b})\hat{b} + (\vec{r} . (\hat{a} \times \hat{b})) (\hat{a} \times \hat{b})$$
  • none of these
The vector with initial point $$P(2,-3,5)$$ and terminal point $$Q(3,-4,7)$$ is
  • $$\hat i-\hat j+2\hat k$$
  • $$5\hat i-7\hat j+12\hat k$$
  • $$\hat i+\hat j-2\hat k$$
  • $$None\ of\ these$$
If $$ \vec{X} \cdot \vec{A}=0, \vec{X} \cdot \vec{B}=0 $$ and $$ \vec{X} \cdot \vec{C}=0 $$ for some non-zero vector $$ \vec{X}, $$ then $$ [\vec{A} \vec{B} \vec{C}]=0 $$
  • True
  • False
The position vector of the point which divides the join of points with position vectors $$\vec a +\vec b$$ and $$2\vec a-\vec b$$ in the ratio $$1:2$$ is
  • $$\dfrac {3\vec a+2\vec b}{3}$$
  • $$\vec a$$
  • $$\dfrac {5\vec a-\vec b}{3}$$
  • $$\dfrac {4\vec a+\vec b}{3}$$
The projection of vector $$\vec a=2\hat i-\hat j+\hat k$$ along $$\vec b=\hat i+2\hat j+2\hat k$$ is
  • $$\dfrac {2}{3}$$
  • $$\dfrac {1}{3}$$
  • $$2$$
  • $$\sqrt 6$$
Position vector of a point $$P$$ is a vector whose initial point is origin.
  • True
  • False
Let $$ \vec{u}, \vec{v} $$ and $$ \vec{w} $$ be vectors such that $$ \vec{u}+\vec{v}+\vec{w}=0 . $$ If $$ |\vec{u}|=3,|\vec{v}|=4 $$ and $$ |\vec{w}|=5, $$ then $$ \vec{u} \cdot \vec{v}+\vec{v} \cdot \vec{w}+\vec{w} \cdot \vec{u} $$ is
  • $$47$$
  • $$-25$$
  • $$0$$
  • $$25$$
  • $$50$$
Line $$\overrightarrow{r} = \overrightarrow{a} + \lambda \overrightarrow{b}$$ will not meet the plane $$\overrightarrow{r} \cdot \overrightarrow{n} = q$$, if 
  • $$\overrightarrow{b} \cdot \overrightarrow{n} = 0, \overrightarrow{a} \cdot \overrightarrow{n} = q$$
  • $$\overrightarrow{b} \cdot \overrightarrow{n} \neq 0, \overrightarrow{a} \cdot \overrightarrow{n} \neq q$$
  • $$\overrightarrow{b} \cdot \overrightarrow{n} = 0, \overrightarrow{a} \cdot \overrightarrow{n} \neq q$$
  • $$\overrightarrow{b} \cdot \overrightarrow{n} \neq 0, \overrightarrow{a} \cdot \overrightarrow{n} = q$$
If $$\vec \alpha | =4$$ and $$ -3 \le \lambda \le 2$$, then the range of $$ | \lambda \vec \alpha |$$ is 
  • $$[0, 8]$$
  • $$[-12, 8]$$
  • $$[0, 12]$$
  • $$[8, 12]$$
If $$\vec a, \vec b, \vec c$$ are unit vector such that $$\vec a +\vec b +\vec c=\vec 0$$, then the value of $$\vec a \vec b+\vec b. \vec c+\vec c. \vec a$$ is 
  • $$1$$
  • $$3$$
  • $$-\dfrac 32$$
  • $$None\ of\ these$$
The vector having initial and terminal points as $$(2, 5, 0)$$ and $$(-3, 7, 4)$$, respectively is 
  • $$-\hat i +12 \hat j +4\hat k$$
  • $$5\hat i+2\hat j-4\hat k$$
  • $$-5\hat i +2\hat j+4\hat k$$
  • $$\hat i+\hat j+ \hat k$$
If $$\vec a, \vec b, \vec c$$ are three vectors such that $$ \vec a +\vec b+ \vec c=\vec 0$$ and $$ | \vec a| =2, | \vec b|=3, | \vec c| =5$$, then value of $$\vec a. \vec b+ \vec b. \vec c+ \vec c. \vec a$$ is 
  • $$0$$
  • $$1$$
  • $$-19$$
  • $$38$$
The position vector of the point which divides the join of points $$2 \vec a -3\vec b$$ and $$\vec a+\vec b$$ in the ratio $$3:1$$ is 
  • $$\dfrac{3\vec a-2\vec b}{2}$$
  • $$\dfrac{7\vec a-8\vec b}{4}$$
  • $$\dfrac{3\vec a}{4}$$
  • $$\dfrac{5\vec a}{4}$$
If $$\left| \bar { a }  \right|  =2,\ \left| \bar { b }  \right|  = 3, \left| \bar { c }  \right|  =4$$ then $$\left[ \begin{matrix} \bar { a } +\bar { b }  & \bar { b } +\bar { c }  & \bar { c } -\bar { a }  \end{matrix} \right] $$ is 
  • $$24$$
  • $$-24$$
  • $$0$$
  • $$448$$
If $$\left| \bar { a }  \right| =3,\ \left| \bar { b }  \right| =4,$$ then the value of $$\lambda$$ for which $$\bar { a }+\lambda \bar { b },$$ is
  • $$\dfrac{9}{16}$$
  • $$\dfrac{3}{4}$$
  • $$\dfrac{3}{2}$$
  • $$\dfrac{4}{3}$$
Let $$\bar { p }$$ and $$\bar { q }$$ be the position vectors of $$P$$ 
and $$Q$$ respectively, with respect to $$O$$ and $$\left| \bar { p } 
 \right| =p,\ \left| \bar { q }  \right| =q.$$ The points $$R$$ and $$S
$$ divide $$PQ$$ internally and externally in the ratio $$2:3$$ 
respectively. If $$OR$$ and $$OS$$ are perpendicular; then
  • $$9p^2=4q^2$$
  • $$4p^2=9q^2$$
  • $$9p=4q$$
  • $$4p=9q$$
The value of $$\hat { i }. (\hat { j } \times \hat { k }) + \hat { j }. (\hat { 
i } \times \hat { k })+\hat { k }. (\hat { i } \times \hat { j })$$ is
  • $$0$$
  • $$-1$$
  • $$1$$
  • $$3$$
If $$ \overrightarrow {a} $$ is non zero vector of magnitude 'a ' and $$ \lambda $$ a nonzero scalar then $$ \lambda \overrightarrow {a} $$ is unit vector
  • $$ \lambda =1  $$
  • $$ \lambda = -1 $$
  • $$ a = | \lambda| $$
  • $$ a = 1 / | \lambda | $$
In triangle ABC , which of the following is not true.
1860549_16558d808f3a4366ab5aee7338ba81ba.png
  • $$ \bar {AB} + \bar {BC} + \bar {CA} = \bar {0} $$
  • $$ \bar {AB} + \bar {BC} - \bar {AC} = \bar {0} $$
  • $$ \bar {AB} + \bar {BC} - \bar {CA} = \bar {0} $$
  • $$ \bar {AB} - \bar {CB} + \bar {CA} = \bar {0} $$
The value of $$ \hat {i} .( \hat {j} \times \hat {k}) + \hat {j} . ( \hat {i} \times  \hat {k})  + \hat {k} .( \hat {i} \times  \hat {j}) $$
  • $$ 0 $$
  • $$ -1 $$
  • $$ 1 $$
  • $$ 3 $$
Three vectors of magnitudes $$a,\ 2a,3a$$ meeting a point and three directions are along the diagonals of three adjacent faces of a cube. The magnitude of their resultant is
  • $$3a$$
  • $$5a$$
  • $$2a$$
  • $$4a$$
If $$\vec {x}$$ is a vector whose initial point divides the line joining $$5\hat{i}$$, and $$ 5\hat{j}$$ in the ratio $$\lambda :1$$ and  the terminal point is the origin. Also given $$\left | \vec {x} \right |\leq \sqrt{37}$$, then $$\lambda $$ belongs to
  • $$\left [ -\dfrac{1}{6} ,\dfrac{1}{6}\right ]$$
  • $$(-\infty ,-6)\cup \left ( -\dfrac{1}{6} ,\infty \right )$$
  • $$(-\infty ,-8)$$
  • $$(1,\infty )$$
A scooterist follows a track on a ground that turns to his left by an angle 60$$^{0}$$ after every 400 m. Starting from the given point displacement of the scooterist at the third turn and eighth turn are :
  • $$800\mathrm{ m}; 0\mathrm{ m}$$
  • $$800\mathrm{m},\ 800\sqrt{3}\mathrm{m}$$
  • $$800\mathrm{m};400\sqrt{3}\mathrm{m}$$
  • $$800; 800\sqrt{3}\mathrm{m}$$
The 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 triangle $$ABC$$, then the  length of the median through $$A$$ is:
  • $$\sqrt{72}$$
  • $$\sqrt{33}$$
  • $$\sqrt{45}$$
  • $$\sqrt{18}$$
$$\mathrm{l}\mathrm{n}$$ a triangle O$$\mathrm{A}\mathrm{B},\ \mathrm{E}$$ is the mid-point of $$\mathrm{O}\mathrm{B}$$ and $$\mathrm{D}$$ is a point on $$\mathrm{A}\mathrm{B}$$ such that $$\mathrm{A}\mathrm{D}$$: $$\mathrm{D}\mathrm{B}=2: 1$$. lf $$\mathrm{O}\mathrm{D}$$ and $$\mathrm{A}\mathrm{E}$$ interesect at $$\mathrm{P}$$, then the ratio $$\displaystyle\frac{OP}{PD}$$ is
  • $$1:2$$
  • $$3:2$$
  • $$8:3$$
  • $$4:3$$
In a quadrilateral $$PQRS,\ \vec{PQ}=\vec{a}, \vec{QR}=\vec{b}, \vec{SP}=\vec{a} - \vec{b}.\ M$$ is the mid-point of $$QR$$ and $$X$$ is a point on $$SM$$ such that $$\vec{SX}=\dfrac{4}{5}\vec{SM}$$, then $$\vec{PX}$$ is
  • $$\dfrac{1}{5}\vec{PR}$$
  • $$\dfrac{3}{5}\vec{PR}$$
  • $$\dfrac{2}{5}\vec{PR}$$
  • None of these
The position vectors of $$A$$ and $$B$$ are $$2\hat{i}+2\hat{j}+\hat{k}$$ and $$2\hat{i}+4\hat{j}+4\hat{k}.$$ The length of the internal bisector of $$\angle BOA$$ of the triangle $$AOB$$ is
  • $$\displaystyle \sqrt { \dfrac { 136 }{ 9 }  } $$
  • $$\displaystyle \sqrt { \dfrac { 139 }{ 9 }  } $$
  • $$\displaystyle \dfrac { 20 }{ 3 } $$
  • $$\displaystyle \sqrt { \dfrac { 217 }{ 9 }  } $$
$$ABCD$$ is a quadrilateral, $$E$$ is the point of intersection of the line joining the midpoints of the opposite sides. If $$O$$ is any point and $$\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = \vec{x OE},$$ then $$x$$ is equal to
  • $$3$$
  • $$9$$
  • $$7$$
  • $$4$$
If $$\overrightarrow{b}$$ is a vector whose initial point divides the join of $$5\widehat{i}$$ and $$5\widehat{j}$$ in the ratio $$k : 1$$  and whose terminal point is the origin and $$|\vec b| \leq \sqrt{37}$$, then $$k$$ lies in the interval
  • $$\left[-6, -\dfrac{1}{6}\right]$$
  • $$\left(- \infty, -6 \right] \cup \left[-\dfrac{1}{6}, \infty \right) $$
  • $$ \left[0, 6 \right] $$
  • None of these
If  $$\vec{a},\vec{b},\vec{c}$$ are three non-zero vectors such that $$\vec{a}\times \vec{b}=\vec{c}$$ and $$\vec{b}\times \vec{c}=\vec{a}$$ , then choose the incorrect option(s)
  • $$\vec{a}. \vec{b}=\vec{b}. \vec{c}=\vec{c} . \vec{a}=0$$
  • $$\left | \vec{b} \right |=\left | \vec{c} \right |$$
  • $$\vec{a}$$ is a unit vector
  • $$\vec{c}$$ is a unit vector
If the vectors $$\hat i - \hat j, \hat j + \hat k$$ and $$\vec a$$ form a triangle, then $$\vec a$$ may be
  • $$- \hat i - \hat k$$
  • $$\hat i - 2 \hat j - \hat k$$
  • $$2 \hat i + \hat j + \hat k$$
  • $$\hat i + \hat k$$
Vectors $$\vec a = \hat i + 2 \hat j + 3 \hat k, \vec b = 2 \hat i - \hat j + \hat k$$ and $$\vec c = 3 \hat i + \hat j + 4 \hat k$$ are so placed that the end point of one vector is the starting point of the next vector, then the vectors are
  • Not coplanar
  • Coplanar but cannot form a triangle
  • Coplanar and form a triangle
  • Coplanar and can form a right-angled triangle
$$ABCD$$ a parallelogram, $$A_1$$ and $$B_1$$ are the midpoints of sides $$BC$$ and $$CD$$, respectively. If $$\vec{AA_1} + \vec{AB_1} = \lambda \vec{AC}$$, then $$\lambda$$ is equal to
  • $$\displaystyle \dfrac{1}{2}$$
  • $$1$$
  • $$\displaystyle \dfrac{3}{2}$$
  • $$2$$
$$L_{1}and L_{2}$$ are two lines whose vector equations are

$$L_{1}:\vec{r}=\lambda \left ( (\cos \theta+\sqrt{3})\hat{i}+(\sqrt{2}\sin\theta)\hat{j}+(\cos \theta-\sqrt{3})\hat{k} \right )$$

$$L_{2}:\vec{r}=\mu \left ( a \hat{i}+b \hat{j}+c\hat{k} \right ),$$

Where $$\lambda\ and\ \mu $$ are scalars and $$\alpha$$ is the acute angle

between $$L_{1}\ and\ L_{2}$$ . If the angle '$$\alpha$$'is independent

of $$\theta $$ then the value of '$$\alpha$$ ' is
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{\pi}{2}$$
$$\overrightarrow{AR}$$ is
  • $$\dfrac{1}{5}(2\vec {b}+\vec {c})$$
  • $$\dfrac{1}{6}(2\vec {b}+\vec {c})$$
  • $$\dfrac{1}{7}(\vec {b}+2\vec {c})$$
  • $$\dfrac{1}{7}(2\vec {b}+\vec {c})$$
In a parallelogram $$OABC,$$ vectors $$\vec{a}, \vec{b}, \vec{c}$$ are, respectively, the position vectors of vertices $$A, B, C$$ with reference to $$O$$ as origin. A point $$E$$ is taken on the side $$BC$$  which divides it in the ratio of $$2 : 1$$. Also, the line segment  $$AE$$  intersects the line bisecting the angle $$\angle$$AOC internally at point $$P$$. If $$CP$$ when extended meets $$AB$$  in point $$F,$$  then the position vector of point $$P$$  is
  • $$\displaystyle \dfrac{|\vec{a}| |\vec{c}|}{3 |\vec{c}| + 2 |\vec{a}|} \left ( \dfrac{\vec{a}}{|\vec{a}|} + \dfrac{\vec{c}}{|\vec{c}|} \right )$$
  • $$\displaystyle \dfrac{3|\vec{a}| |\vec{c}|}{3 |\vec{c}| + 2 |\vec{a}|} \left ( \dfrac{\vec{a}}{|\vec{a}|} + \dfrac{\vec{c}}{|\vec{c}|} \right )$$
  • $$\displaystyle \dfrac{2|\vec{a}| |\vec{c}|}{3 |\vec{c}| + 2 |\vec{a}|} \left ( \dfrac{\vec{a}}{|\vec{a}|} + \dfrac{\vec{c}}{|\vec{c}|} \right )$$
  • None of these
The ratio $$\displaystyle \dfrac{OX}{XC}$$ is
120334.png
  • $$\dfrac{3}{4}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{2}{5}$$
  • $$\dfrac{1}{2}$$
The projection of the line joining the points $$(3, 4, 5)$$ and $$(4, 6, 3)$$ on the line joining the points $$(-1, 2, 4)$$ and $$(1, 0, 5)$$ is
  • $$\dfrac{4}{3}$$
  • $$\dfrac{2}{3}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{2}$$
A parallelogram is constructed on the vectors $$\bar{\alpha }$$ and $$\bar{\beta }$$. A vector which coincides with the altitude of the parallelogram and perpendicular to the side $$\bar{\alpha }$$ expressed in terms of the vectors $$\bar{\alpha }$$ and $$\bar{\beta }$$ is
  • $$\displaystyle \bar{\beta }+\dfrac{\bar{\beta }-\bar{\alpha }}{\left ( \bar{\alpha } \right )^{2}}\bar{\alpha }$$
  • $$\displaystyle \dfrac{\left ( \bar{\alpha }\times \bar{\beta } \right )\times \bar{\alpha }}{\left ( \bar{\alpha } \right )^{2}}$$
  • $$\displaystyle \dfrac{\bar{\beta }\cdot \bar{\alpha }}{\left ( \alpha \right )^{2}}\bar{\alpha }+\bar{\beta }$$
  • $$\displaystyle \left | \bar{\beta } \right |\dfrac{\bar{\alpha }\times \left ( \bar{\alpha }\times \bar{\beta } \right )}{\left ( \alpha \right )^{2}}$$
$$\vec{a} = 2 \widehat{i} - \widehat{j} + \widehat{k}, \vec{b} = \widehat{i} + 2\widehat{j} - \widehat{k} $$ and $$ \vec{c} = \widehat{i} + \widehat{j} - 2 \widehat{k}$$. A vector coplanar with $$\vec{b}$$ and $$\vec{c}$$ whose projection on $$\vec{a}$$ is magnitude $$\displaystyle \sqrt{\dfrac{2}{3}} $$ is
  • $$2 \widehat{i} + 3 \widehat{j} - 3 \widehat{k}$$
  • $$- 2 \widehat{i} - \widehat{j} + 5 \widehat{k}$$
  • $$2 \widehat{i} + 3 \widehat{j} + 3 \widehat{k}$$
  • $$2 \widehat{i} + \widehat{j} + 5 \widehat{k}$$
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