Question 1 : <p>If the vectors $\overrightarrow{\mathbf{a}} = \widehat{\dot{\mathbf{i}}} + a\widehat{\dot{\mathbf{j}}} + a^{2}\widehat{\mathbf{k}},\ \overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} + b\widehat{\dot{\mathbf{j}}} + b^{2}\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}} = \widehat{\dot{\mathbf{i}}} + c\widehat{\dot{\mathbf{j}}} + c^{2}\widehat{\mathbf{k}}$ are three non-coplanar vectors and</p> <p>$\left| \begin{matrix} a & a^{2} & 1 + a^{3} \\ b & b^{2} & 1 + b^{3} \\ c & c^{2} & 1 + c^{3} \\ \end{matrix} \right| = 0,\ \text{then\ the\ value\ of\ }\text{abc}\text{\ is}$</p>
Question 2 :
If $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{c}}$ are vectors such that $\overrightarrow{\mathbf{c}}\mathbf{=}\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} = 0$, then
Question 3 :
The direction cosines l, m, n of two lines are connected by the relation l + m + n = 0, lm = 0, then the angles between them is
Question 4 :
If r⃗ • a⃗ = r⃗ • b⃗ = r⃗ • c⃗ = 0 for some non-zero vector r⃗, then the value of [a⃗b⃗c⃗], is
Question 5 :
A vector c⃗ of magnitude $5\sqrt{6}$ directed along the bisector of the angle between a⃗ = 7î − 4ĵ − 4k̂ and b⃗ = − 2î − ĵ + 2k̂, is
Question 6 :
If {tex} \vec { x } = 3 \hat { i } - 6 \hat { j } - \hat { k } , \vec { y } = \hat { i } + 4 \hat { j } - 3 \hat { k } {/tex} and {tex} \vec { z } = 3 \hat { i } - 4 \hat { j } - 12 \hat { k } , {/tex} then the magnitude of the projection of {tex} \vec { x } \times \vec { y } {/tex} on {tex} \overline { z } {/tex} is
Question 8 :
If $\overrightarrow{\mathbf{a}}\mathbf{=}2\widehat{\mathbf{i}}\mathbf{-}3\widehat{\mathbf{j}}\mathbf{+}5\widehat{\mathbf{k}}$<strong>,</strong> $\overrightarrow{\mathbf{b}}\mathbf{=}3\widehat{\mathbf{i}}\mathbf{-}4\widehat{\mathbf{j}}\mathbf{+}5\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}}\mathbf{=}5\widehat{\mathbf{i}}\mathbf{-}3\widehat{\mathbf{j}}\mathbf{-}2\widehat{\mathbf{k}}$<strong>,</strong> then the volume of the parallelopiped with coterminous edges $\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}}\mathbf{,}\overrightarrow{\mathbf{c}}\mathbf{+}\overrightarrow{\mathbf{a}}$ is
Question 9 : <p>The coordinate the point of intersection of the line</p> <p>$\frac{x - 1}{1} = \frac{y + 2}{2} = \frac{z - 2}{2}\ $with the plane 3x + 4y + 5z − 25 = 0 is </p>
Question 10 :
The equation of the plane containing the line r⃗ = î + ĵ + λ(2î+ĵ+4k̂), is
Question 11 :
If a⃗, b⃗, c⃗ are vectors such that a⃗.b⃗ = 0 and a⃗ + b⃗ = c⃗, then
Question 12 :
Let $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{c}}$ be three non-coplanar vectors and $\overrightarrow{\mathbf{r}}$ be any vector in space such that $\overrightarrow{\mathbf{r}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} = 1,\ \overrightarrow{\mathbf{r}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} = 2\ and\mathbf{\ }\overrightarrow{\mathbf{r}}\mathbf{\bullet}\overrightarrow{\mathbf{c}} = 3$. If $\left\lbrack \ \overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\ \right\rbrack = 1$, then $\overrightarrow{\mathbf{r}}$ is equal to
Question 13 :
In a right angled triangle ABC, the hypotenuse Ab = p, then A⃗B.A⃗C + B⃗C.B⃗A + C⃗A.C⃗B is equal to
Question 14 :
If |a⃗| = |b⃗| = |a⃗+b⃗| = 1, then |a⃗ − b⃗| is equal to
Question 15 :
In a parallelogram ABCD, |A⃗B| = a, |A⃗D| = b and |A⃗C| = c. The value of D⃗B.A⃗B is
Question 16 :
Let the pairs, a⃗, b⃗ and c⃗, d⃗ each determines a plane. Then the planes are parallel, if
Question 17 :
The unit vector perpendicular to vectors î − ĵ and î + ĵ forming a right handed system is
Question 18 :
If the position vector of a point a⃗ + 2b⃗ and a⃗ divides AB in the ratio 2 : 3, then the position vector of B, is
Question 19 :
If the planes x = cy + bz , y = az + cx, z = bx + ay pass through a line, then a<sup>2</sup> + b<sup>2</sup> + c<sup>2</sup> + 2abc is
Question 21 :
If a⃗ is a unit vector such that a⃗ × (î+2ĵ+k̂) = î − k̂, then a⃗=
Question 22 :
If (a⃗×b⃗) + (a⃗.b⃗)<sup>2</sup> = 144 and |a⃗| = 4, then |b⃗|=
Question 23 :
$\overrightarrow{\mathbf{a}} \bullet \lbrack\left( \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right) \times \left( \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right)\rbrack$ equals
Question 24 :
If $\overrightarrow{\mathbf{x}}\ $and $\overrightarrow{\mathbf{y}}$ are unit vectors and $\overrightarrow{\mathbf{x}} \bullet \overrightarrow{\mathbf{y}} = 0,$ then
Question 25 :
The vector equation of the sphere whose centre is the point (1,0,1)and radius is 4, is
Question 26 :
If $\overrightarrow{\mathbf{a}}\mathbf{=}\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\text{\ and\ }\overrightarrow{\mathbf{b}}\mathbf{=}2\widehat{\mathbf{i}}\mathbf{-}\widehat{\mathbf{k}}$ are two vectors, then the point of intersection of two lines $\overrightarrow{\mathbf{r}}\mathbf{\times}\overrightarrow{\mathbf{a}}\mathbf{=}\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{a}}\text{\ and\ }\overrightarrow{\mathbf{r}}\mathbf{\times}\overrightarrow{\mathbf{b}}\mathbf{=}\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}}$ is
Question 27 : <p>The value of λ, for which the four points</p> <p>$2\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}},\ \ \widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}}$, $3\widehat{\dot{\mathbf{i}}} + 4\widehat{\dot{\mathbf{j}}} - 2\widehat{\mathbf{k}}\ ,$ $\widehat{\dot{\mathbf{i}}} - 6\widehat{\dot{\mathbf{j}}} + \lambda\widehat{\mathbf{k}}\ $ are coplanar, is</p>
Question 28 :
$\left( \overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}} \right) \bullet \{\left( \overrightarrow{\mathbf{b}} - \overrightarrow{\mathbf{c}} \right) \times \left( \overrightarrow{\mathbf{c}} - \overrightarrow{\mathbf{a}}\ \right)\}$ is equal to
Question 29 :
If $\overrightarrow{\mathbf{a}} + 2\overrightarrow{\mathbf{b}} + 4\overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{0}}\ \text{and\ }\left( \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right) + \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) + \left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right) = \lambda\left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right),$ then λ is equal to
Question 30 : <p>Let $\overrightarrow{a},\overrightarrow{\ b},\ \overrightarrow{c}$ are three non-coplanar vectors such that $\overrightarrow{r_{1}} = \overrightarrow{a} - \overrightarrow{b} + \overrightarrow{c},\ \overrightarrow{r_{2}} = \overrightarrow{b} + \overrightarrow{c} - \overrightarrow{a},\ \overrightarrow{r_{3}} = \overrightarrow{c} + \overrightarrow{a} + \overrightarrow{b},\ \overrightarrow{r} = 2\overrightarrow{a} - 3\overrightarrow{b} + 4\overrightarrow{c}$</p> <p>If $\overrightarrow{r} = \lambda_{1}\overrightarrow{r_{1}} + \lambda_{2}\overrightarrow{r_{2}} + \lambda_{3}\overrightarrow{r_{3}}$, then</p>
Question 31 :
A non-zero vector $\overrightarrow{\mathbf{a}}$ is parallel to the line of intersection of the plane determined by the vectors $\widehat{\dot{\mathbf{i}}},\ \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}}$ and the plane determined by the vectors $\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}},\ \widehat{\dot{\mathbf{i}}} + \widehat{\mathbf{k}}.$ The angle between $\overrightarrow{\mathbf{a}}$ and $\widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$ is
Question 32 :
If a⃗ = 2î + 2ĵ + 3k̂, b⃗ = − î + 2ĵ + k̂, c⃗ = 3î + ĵ and a⃗ + tb⃗ is normal to the vector c⃗, then the vector of t is
Question 33 :
If θ is the angle between the lines AB and AC where A, B and C are the three points with coordinates (1,2,−1), (2,0,3), (3, − 1, 2) respectively, then $\sqrt{462}\cos\theta$ is equal to
Question 34 :
If r⃗ • a⃗ = r⃗ • b⃗ = r⃗ • c⃗ = 0 where a⃗, b⃗, c⃗ are non-coplanar, then
Question 35 :
Forces acting on a particle have magnitude 5, 3 and 1 unit and act in the direction of the vectors 6$\widehat{\mathbf{i}} + 2\widehat{\mathbf{j}} + 3\widehat{\mathbf{k}}$, $3\widehat{\mathbf{i}} - 2\widehat{\mathbf{j}} + 6\widehat{\mathbf{k}}$ and $2\widehat{\mathbf{i}} - 3\widehat{\mathbf{j}} - 6\widehat{\mathbf{k}}$ respectively. They remain constant while the particle is displaced from the points A(2, − 1, − 3) to B(5, − 1, 1). The work done is
Question 36 :
Vectors $\overrightarrow{\mathbf{a}}\text{\ and\ }\overrightarrow{\mathbf{b}}$ are inclined at an angle θ = 120<sup>∘</sup>. If $\left| \overrightarrow{\mathbf{a}} \right| = 1,\ |\overrightarrow{\mathbf{b}}| = 2,$ then $\lbrack(\overrightarrow{\mathbf{a}} + 3\overrightarrow{\mathbf{b}}) \times (3\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}}\rbrack^{2}$ is equal to
Question 37 :
If $\overrightarrow{a},\overrightarrow{\ b},\ \overrightarrow{c}$ are three vectors such that a⃗ + b⃗ + c⃗ = 0 and |a⃗| = 2, |b⃗| = 3, |c⃗| = 4, then the value of a⃗ • b⃗ + b⃗ • c⃗ + c⃗ • a⃗ is equal to
Question 38 :
For any three vectors a⃗, b⃗, c⃗ the expression (a⃗−b⃗) • {(b⃗−c⃗)×(c⃗−a⃗)} equals
Question 39 :
If ABCDEF is a regular hexagon, then A⃗D + E⃗B + F⃗C equals
Question 40 :
If three points A, B and C have position vectors î + xĵ + 3k̂, 3î + 4ĵ + 7k̂ and yî − 2ĵ − 5k̂ respectively are collinear, then (x,y)=
Question 41 :
If position vector of point A is $\overrightarrow{\mathbf{a}} + 2\overrightarrow{\mathbf{b}}$ and any point $P(\overrightarrow{\mathbf{a})}$ divides $\overrightarrow{\mathbf{\text{AB}}}$ in the ratio of 2 : 3, then position vector of B is
Question 42 :
Vectors a⃗ and b⃗ are inclined at angle θ = 120<sup>∘</sup>. If |a⃗| = 1, |b⃗| = 2, then [(a⃗+3b⃗)×(3a⃗−b⃗)]<sup>2</sup> is equal to
Question 43 :
Let a⃗ = î + ĵ − k̂, b⃗ = î − ĵ + k̂ and c⃗ be a unit vector perpendicular to a⃗ and coplanar with a⃗ and b⃗, then it is given by
Question 44 :
If $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{c}}$ are unit coplanar vectors, then $\lbrack 2\ \overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}}\ 2\overrightarrow{\mathbf{b}}\ - \overrightarrow{\mathbf{c}}\ 2\overrightarrow{\mathbf{c}} - \overrightarrow{\mathbf{a}}\ \rbrack$ is equal to
Question 45 :
If $\overrightarrow{\mathbf{p}},\overrightarrow{\mathbf{q}}$ and $\overrightarrow{\mathbf{r}}$ are perpendicular to $\overrightarrow{\mathbf{q}} + \overrightarrow{\mathbf{r}},\ \overrightarrow{\mathbf{r}} + \overrightarrow{\mathbf{p}}$ and $\overrightarrow{\mathbf{p}} + \overrightarrow{\mathbf{q}}$ respectively and if $\left| \overrightarrow{\mathbf{p}} + \overrightarrow{\mathbf{q}} \right| = 6,\ \left| \overrightarrow{\mathbf{q}} + \overrightarrow{\mathbf{r}} \right| = 4\sqrt{3}$ and $\left| \overrightarrow{\mathbf{r}} + \overrightarrow{\mathbf{p}} \right| = 4,$ then $|\overrightarrow{\mathbf{p}} + \overrightarrow{\mathbf{q}} + \overrightarrow{\mathbf{r}}|$ is
Question 46 : <p>If $\overrightarrow{\mathbf{a}},\ \overrightarrow{\mathbf{b}},\ \overrightarrow{\mathbf{c}}$ are the three vectors mutually perpendicular to each other and</p> <p>$\left| \overrightarrow{\mathbf{a}} \right| = 1,\ |\overrightarrow{\mathbf{b}}| = 3\ $and$\ |\overrightarrow{\mathbf{c}}| = 5,\ $then $\lbrack\overrightarrow{\mathbf{a}} - 2\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{b}} - 3\overrightarrow{\mathbf{c}}\text{\ \ }\overrightarrow{\mathbf{c}} - 4\overrightarrow{\mathbf{a}}\ \rbrack$ is equal to</p>
Question 47 :
Let a⃗ = 2î − ĵ + k̂, b⃗ = î + 2ĵ − k̂ and c⃗ = î + ĵ − 2k̂ be three vectors. A vector in the plane of b⃗ and c⃗ whose projection on a⃗ is of magnitude $\sqrt{2/3}$ is
Question 48 :
The two vectors a⃗ = 2î + ĵ + 3k̂, b⃗ = 4î − λĵ + 6k̂ are parallel if λ=
Question 49 :
The value of λ for which the lines $\frac{x - 1}{1} = \frac{y - 2}{\lambda} = \frac{z + 1}{- 1}$ and $\frac{x + 1}{- \lambda} = \frac{y + 1}{2} = \frac{z - 2}{1}$ are perpendicular to each other is
Question 50 :
If a⃗ is any vector, then (a×î)<sup>2</sup> + (a×ĵ)<sup>2</sup> + (a×k̂)<sup>2</sup>=
