Let $\vec{a} = 2\hat{i} + \hat{j} + \hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a} + \vec{b} + \vec{c}| = |\vec{a} + \vec{b} - \vec{c}|$ and $\vec{b} \cdot \vec{c} = 0$. Consider the following two statements: $A$: $|\vec{a} + \lambda\vec{c}| \geq |\vec{a}|$ for all $\lambda \in \mathbb{R}$. $B$: $\vec{a}$ and $\vec{c}$ are always parallel.
(1) only (B) is correct
(2) neither (A) nor (B) is correct
(3) only (A) is correct
(4) both (A) and (B) are correct.

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}| = \sqrt{14}$, $|\vec{b}| = \sqrt{6}$ and $|\vec{a} \times \vec{b}| = \sqrt{48}$. Then $(\vec{a} \cdot \vec{b})^2$ is equal to $\underline{\hspace{1cm}}$.

Let $\vec { a } = 6 \hat { i } + 9 \hat { j } + 12 \hat { k } , \vec { b } = \alpha \hat { i } + 11 \hat { j } - 2 \hat { k }$ and $\vec { c }$ be vectors such that $\vec { a } \times \vec { c } = \vec { a } \times \vec { b }$. If $\vec { a } \cdot \vec { c } = - 12$, and $\vec { c } \cdot ( \hat { i } - 2 \hat { j } + \hat { k } ) = 5$ then $\vec { c } \cdot ( \hat { i } + \hat { j } + \hat { k } )$ is equal to $\_\_\_\_$

Let a unit vector $\widehat { u } = x \hat { i } + y \hat { j } + z \widehat { k }$ make angles $\frac { \pi } { 2 } , \frac { \pi } { 3 }$ and $\frac { 2 \pi } { 3 }$ with the vectors $\frac { 1 } { \sqrt { 2 } } \hat { i } + \frac { 1 } { \sqrt { 2 } } \widehat { k } , \frac { 1 } { \sqrt { 2 } } \hat { j } + \frac { 1 } { \sqrt { 2 } } \widehat { k }$ and $\frac { 1 } { \sqrt { 2 } } \hat { i } + \frac { 1 } { \sqrt { 2 } } \hat { j }$ respectively. If $\vec { v } = \frac { 1 } { \sqrt { 2 } } \hat { i } + \frac { 1 } { \sqrt { 2 } } \hat { j } + \frac { 1 } { \sqrt { 2 } } \hat { k }$, then $| \hat { u } - \vec { v } | ^ { 2 }$ is equal to
(1) $\frac { 11 } { 2 }$
(2) $\frac { 5 } { 2 }$
(3) 9
(4) 7

Let $\mathrm { P } ( 3,2,3 ) , \mathrm { Q } ( 4,6,2 )$ and $\mathrm { R } ( 7,3,2 )$ be the vertices of $\triangle \mathrm { PQR }$. Then, the angle $\angle \mathrm { QPR }$ is
(1) $\frac { \pi } { 6 }$
(2) $\cos ^ { - 1 } \left( \frac { 7 } { 18 } \right)$
(3) $\cos ^ { - 1 } \left( \frac { 1 } { 18 } \right)$
(4) $\frac { \pi } { 3 }$

If $A ( 3,1 , -1 ) , B \left( \frac { 5 } { 3 } , \frac { 7 } { 3 } , \frac { 1 } { 3 } \right) , C ( 2,2,1 )$ and $D \left( \frac { 10 } { 3 } , \frac { 2 } { 3 } , \frac { -1 } { 3 } \right)$ are the vertices of a quadrilateral $ABCD$, then its area is
(1) $\frac { 2 \sqrt { 2 } } { 3 }$
(2) $\frac { 5 \sqrt { 2 } } { 3 }$
(3) $2 \sqrt { 2 }$
(4) $\frac { 4 \sqrt { 2 } } { 3 }$

If $\vec { a } = \hat { i } + 2 \hat { j } + \hat { k } , \vec { b } = 3 ( \hat { i } - \hat { j } + \hat { k } )$ and $\overrightarrow { \mathrm { c } }$ be the vector such that $\vec { a } \times \vec { c } = \vec { b }$ and $\vec { a } \cdot \vec { c } = 3$, then $\vec { a } \cdot ( ( \vec { c } \times \vec { b } ) - \vec { b } - \vec { c } )$ is equal to
(1) 32
(2) 24
(3) 20
(4) 36

Let $\overrightarrow { \mathrm { a } } = \hat { i } + 2 \hat { j } + 3 \hat { k } , \overrightarrow { \mathrm {~b} } = 2 \hat { i } + 3 \hat { j } - 5 \hat { k }$ and $\overrightarrow { \mathrm { c } } = 3 \hat { i } - \hat { j } + \lambda \hat { k }$ be three vectors. Let $\overrightarrow { \mathrm { r } }$ be a unit vector along $\vec { b } + \vec { c }$. If $\vec { r } \cdot \vec { a } = 3$, then $3 \lambda$ is equal to: (1) 21 (2) 30 (3) 25 (4) 27

Let $\vec { a } = 2 \hat { i } + \alpha \hat { j } + \hat { k } , \vec { b } = - \hat { i } + \hat { k } , \vec { c } = \beta \hat { j } - \hat { k }$, where $\alpha$ and $\beta$ are integers and $\alpha \beta = - 6$. Let the values of the ordered pair ( $\alpha , \beta$ ), for which the area of the parallelogram of diagonals $\vec { a } + \vec { b }$ and $\vec { b } + \vec { c }$ is $\frac { \sqrt { 21 } } { 2 }$, be ( $\alpha _ { 1 } , \beta _ { 1 }$ ) and $\left( \alpha _ { 2 } , \beta _ { 2 } \right)$. Then $\alpha _ { 1 } ^ { 2 } + \beta _ { 1 } ^ { 2 } - \alpha _ { 2 } \beta _ { 2 }$ is equal to
(1) 19
(2) 17
(3) 24
(4) 21

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, then $|\vec{c}|$ is equal to
(1) $\sqrt{\frac{11}{6}}$
(2) $\frac{1}{3\sqrt{2}}$
(3) 16
(4) 18

Let $\vec{\mathrm{a}} = 3\hat{i} - \hat{j} + 2\hat{k}$, $\vec{\mathrm{b}} = \vec{\mathrm{a}} \times (\hat{i} - 2\hat{k})$ and $\vec{\mathrm{c}} = \vec{\mathrm{b}} \times \hat{k}$. Then the projection of $\vec{\mathrm{c}} - 2\hat{j}$ on $\vec{a}$ is:
(1) $2\sqrt{14}$
(2) $\sqrt{14}$
(3) $3\sqrt{7}$
(4) $2\sqrt{7}$

Let $\hat { a }$ be a unit vector perpendicular to the vector $\overrightarrow { \mathrm { b } } = \hat { i } - 2 \hat { j } + 3 \hat { k }$ and $\overrightarrow { \mathrm { c } } = 2 \hat { i } + 3 \hat { j } - \hat { k }$, and makes an angle of $\cos ^ { - 1 } \left( - \frac { 1 } { 3 } \right)$ with the vector $\hat { i } + \hat { j } + \hat { k }$. If $\hat { a }$ makes an angle of $\frac { \pi } { 3 }$ with the vector $\hat { i } + \alpha \hat { j } + \hat { k }$, then the value of $\alpha$ is :
(1) $\sqrt { 6 }$
(2) $- \sqrt { 6 }$
(3) $- \sqrt { 3 }$
(4) $\sqrt { 3 }$

Q78. Let a unit vector which makes an angle of $60 ^ { \circ }$ with $2 \hat { i } + 2 \hat { j } - \hat { k }$ and angle $45 ^ { \circ }$ with $\hat { i } - \hat { k }$ be $\overrightarrow { \mathrm { C } }$. Then $\overrightarrow { \mathrm { C } } + \left( - \frac { 1 } { 2 } \hat { i } + \frac { 1 } { 3 \sqrt { 2 } } \hat { j } - \frac { \sqrt { 2 } } { 3 } \hat { k } \right)$ is :
(1) $\frac { \sqrt { 2 } } { 3 } \hat { i } - \frac { 1 } { 2 } \hat { k }$
(2) $\left( \frac { 1 } { \sqrt { 3 } } + \frac { 1 } { 2 } \right) \hat { i } + \left( \frac { 1 } { \sqrt { 3 } } - \frac { 1 } { 3 \sqrt { 2 } } \right) \hat { j } + \left( \frac { 1 } { \sqrt { 3 } } + \frac { \sqrt { 2 } } { 3 } \right) \hat { k }$
(3) $\frac { \sqrt { 2 } } { 3 } \hat { i } + \frac { 1 } { 3 \sqrt { 2 } } \hat { j } - \frac { 1 } { 2 } \hat { k }$
(4) $- \frac { \sqrt { 2 } } { 3 } \hat { i } + \frac { \sqrt { 2 } } { 3 } \hat { j } + \left( \frac { 1 } { 2 } + \frac { 2 \sqrt { 2 } } { 3 } \right) \hat { k }$

Q79. For $\lambda > 0$, let $\theta$ be the angle between the vectors $\vec { a } = \hat { i } + \lambda \hat { j } - 3 \hat { k }$ and $\vec { b } = 3 \hat { i } - \hat { j } + 2 \hat { k }$. If the vectors $\vec { a } + \vec { b }$ and $\vec { a } - \vec { b }$ are mutually perpendicular, then the value of $( 14 \cos \theta ) ^ { 2 }$ is equal to
(1) 50
(2) 40
(3) 25
(4) 20

Q65. If $\mathrm { A } ( 1 , - 1,2 ) , \mathrm { B } ( 5,7 , - 6 ) , \mathrm { C } ( 3,4 , - 10 )$ and $\mathrm { D } ( - 1 , - 4 , - 2 )$ are the vertices of a quadrilateral $A B C D$, then its area is :
(1) $48 \sqrt { 7 }$
(2) $12 \sqrt { 29 }$
(3) $24 \sqrt { 7 }$
(4) $24 \sqrt { 29 }$

Q65. If $A ( 3,1 , - 1 ) , B \left( \frac { 5 } { 3 } , \frac { 7 } { 3 } , \frac { 1 } { 3 } \right) , C ( 2,2,1 )$ and $D \left( \frac { 10 } { 3 } , \frac { 2 } { 3 } , \frac { - 1 } { 3 } \right)$ are the vertices of a quadrilateral $A B C D$, then its area is
(1) $\frac { 2 \sqrt { 2 } } { 3 }$
(2) $\frac { 5 \sqrt { 2 } } { 3 }$
(3) $2 \sqrt { 2 }$
(4) $\frac { 4 \sqrt { 2 } } { 3 }$

Q89. Let $\vec { a } = 2 \hat { i } - 3 \hat { j } + 4 \hat { k } , \vec { b } = 3 \hat { i } + 4 \hat { j } - 5 \hat { k }$ and a vector $\vec { c }$ be such that $\vec { a } \times ( \vec { b } + \vec { c } ) + \vec { b } \times \vec { c } = \hat { i } + 8 \hat { j } + 13 \hat { k }$. If $\vec { a } \cdot \vec { c } = 13$, then $( 24 - \vec { b } \cdot \vec { c } )$ is equal to $\_\_\_\_$

Q77. Let $\overrightarrow { \mathrm { a } } = 2 \hat { i } + \hat { j } - \hat { k } , \overrightarrow { \mathrm {~b} } = ( ( \overrightarrow { \mathrm { a } } \times ( \hat { i } + \hat { j } ) ) \times \hat { i } ) \times \hat { i }$. Then the square of the projection of $\overrightarrow { \mathrm { a } }$ on $\overrightarrow { \mathrm { b } }$ is:
(1) $\frac { 1 } { 3 }$
(2) $\frac { 2 } { 3 }$
(3) 2
(4) $\frac { 1 } { 5 }$

Q78. Let $\overrightarrow { \mathrm { a } } = 6 \hat { i } + \hat { j } - \hat { k }$ and $\overrightarrow { \mathrm { b } } = \hat { i } + \hat { j }$. If $\overrightarrow { \mathrm { c } }$ is a is vector such that $| \overrightarrow { \mathrm { c } } | \geq 6 , \overrightarrow { \mathrm { a } } \cdot \overrightarrow { \mathrm { c } } = 6 | \overrightarrow { \mathrm { c } } | , | \overrightarrow { \mathrm { c } } - \overrightarrow { \mathrm { a } } | = 2 \sqrt { 2 }$ and the angle between $\vec { a } \times \vec { b }$ and $\vec { c }$ is $60 ^ { \circ }$, then $| ( \vec { a } \times \vec { b } ) \times \vec { c } |$ is equal to:
(1) $\frac { 9 } { 2 } ( 6 - \sqrt { 6 } )$
(2) $\frac { 3 } { 2 } \sqrt { 6 }$
(3) $\frac { 9 } { 2 } ( 6 + \sqrt { 6 } )$
(4) $\frac { 3 } { 2 } \sqrt { 3 }$

Q77. The set of all $\alpha$, for which the vectors $\vec { a } = \alpha t \hat { i } + 6 \hat { j } - 3 \hat { k }$ and $\vec { b } = t \hat { i } - 2 \hat { j } - 2 \alpha t \hat { k }$ are inclined at an obtuse angle for all $t \in \mathbb { R }$, is
(1) $\left( - \frac { 4 } { 3 } , 1 \right)$
(2) $[ 0,1 )$
(3) $\left( - \frac { 4 } { 3 } , 0 \right]$
(4) $( - 2,0 ]$

Q79. Let $P ( x , y , z )$ be a point in the first octant, whose projection in the $x y$-plane is the point $Q$. Let $O P = \gamma$; the angle between $O Q$ and the positive $x$-axis be $\theta$; and the angle between $O P$ and the positive $z$-axis be $\phi$, where $O$ is the origin. Then the distance of $P$ from the $x$-axis is
(1) $\gamma \sqrt { 1 - \sin ^ { 2 } \phi \cos ^ { 2 } \theta }$
(2) $\gamma \sqrt { 1 - \sin ^ { 2 } \theta \cos ^ { 2 } \phi }$
(3) $\gamma \sqrt { 1 + \cos ^ { 2 } \phi \sin ^ { 2 } \theta }$
(4) $\gamma \sqrt { 1 + \cos ^ { 2 } \theta \sin ^ { 2 } \phi }$

Q89. Let $\vec { a } = 9 \hat { i } - 13 \hat { j } + 25 \hat { k } , \vec { b } = 3 \hat { i } + 7 \hat { j } - 13 \hat { k }$ and $\vec { c } = 17 \hat { i } - 2 \hat { j } + \hat { k }$ be three given vectors. If $\vec { r }$ is a vector such that $\vec { r } \times \vec { a } = ( \vec { b } + \vec { c } ) \times \vec { a }$ and $\vec { r } \cdot ( \vec { b } - \vec { c } ) = 0$, then $\frac { | 593 \vec { r } + 67 \vec { a } | ^ { 2 } } { ( 593 ) ^ { 2 } }$ is equal to $\_\_\_\_$

Q78. Let $\overrightarrow { \mathrm { a } } = \hat { i } + 2 \hat { j } + 3 \hat { k } , \overrightarrow { \mathrm {~b} } = 2 \hat { i } + 3 \hat { j } - 5 \hat { k }$ and $\overrightarrow { \mathrm { c } } = 3 \hat { i } - \hat { j } + \lambda \hat { k }$ be three vectors. Let $\overrightarrow { \mathrm { r } }$ be anit vector along $\vec { b } + \vec { c }$. If $\vec { r } \cdot \vec { a } = 3$, then $3 \lambda$ is equal to:
(1) 21
(2) 30
(3) 25
(4) 27

Q77. Let three vectors $\overrightarrow { \mathrm { a } } = \alpha \hat { i } + 4 \hat { j } + 2 \hat { k } , \overrightarrow { \mathrm {~b} } = 5 \hat { i } + 3 \hat { j } + 4 \hat { k } , \overrightarrow { \mathrm { c } } = x \hat { i } + y \hat { j } + z \hat { k }$ form a triangle such that $\vec { c } = \vec { a } - \vec { b }$ and the area of the triangle is $5 \sqrt { 6 }$. If $\alpha$ is a positive real number, then $| \vec { c } | ^ { 2 }$ is equal to:
(1) 16
(2) 14
(3) 12
(4) 10

Q78. Let $\overrightarrow { O A } = 2 \vec { a } , \overrightarrow { O B } = 6 \vec { a } + 5 \vec { b }$ and $\overrightarrow { O C } = 3 \vec { b }$, where $O$ is the origin. If the area of the parallelogram with adjacent sides $\overrightarrow { \mathrm { OA } }$ and $\overrightarrow { \mathrm { OC } }$ is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to :
(1) 32
(2) 40
(3) 38
(4) 35