Given two distinct nonzero vectors $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ in 3 dimensions, define a sequence of vectors by
$$\mathbf { v } _ { n + 2 } = \mathbf { v } _ { n } \times \mathbf { v } _ { n + 1 } \left( \text { so } \mathbf { v } _ { 3 } = \mathbf { v } _ { 1 } \times \mathbf { v } _ { 2 } , \mathbf { v } _ { 4 } = \mathbf { v } _ { 2 } \times \mathbf { v } _ { 3 } \text { and so on } \right) .$$
Let $S = \left\{ \mathbf { v } _ { n } \mid n = 1,2 , \ldots \right\}$ and $U = \left\{ \left. \frac { \mathbf { v } _ { n } } { \left| \mathbf { v } _ { n } \right| } \right\rvert\, n = 1,2 , \ldots \right\}$. (Note: Here $\times$ denotes the cross product of vectors and $| \mathbf { v } |$ denotes the magnitude of the vector $\mathbf { v }$. The vector $\mathbf { 0 }$ with 0 magnitude, if it occurs in $S$, is counted. But in that case of course the $\mathbf { 0 }$ vector is not considered while listing elements of $U$.)
(a) There exist vectors $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$ for which the cardinality of $S$ is 2.
(b) There exist vectors $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$ for which the cardinality of $S$ is 3.
(c) There exist vectors $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$ for which the cardinality of $S$ is 4.
(d) Suppose that for some $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$, the set $S$ is infinite. Then the set $U$ is also infinite.

(A) (5 marks) Consider the euclidean space $\mathbb { R } ^ { n }$ with the usual norm and dot product. Let $\mathrm { x } , \mathrm { y } \in \mathbb { R } ^ { n }$ be such that
$$\| \mathrm { x } + t \mathrm { y } \| \geq \| \mathrm { x } \| , \text { for all } t \in \mathbb { R }$$
Show that $\mathbf { x } \cdot \mathbf { y } = 0$.
(B) (5 marks) Consider the vector field $\vec { v } = \left( v _ { x } , v _ { y } \right)$ (with components $\left( v _ { x } , v _ { y } \right)$ ) on $\mathbb { R } ^ { 2 }$:
$$v _ { x } ( x , y ) = x - y , v _ { y } ( x , y ) = y + x$$
Compute the line integral of $\vec { v }$ along the unit circle (counterclockwise). Is there a function $f$ such that $\vec { v } = \operatorname { grad } f$?

There is a regular tetrahedron OABC with edge length 6. Let $S _ { 1 } , S _ { 2 } , S _ { 3 }$ be the orthogonal projections onto plane ABC of the three circles inscribed in triangles $\triangle \mathrm { OAB } , \triangle \mathrm { OBC } , \triangle \mathrm { OCA }$ respectively. As shown in the figure, let $S$ be the area of the dark region enclosed by the three figures $S _ { 1 } , S _ { 2 } , S _ { 3 }$. Find the value of $( S + \pi ) ^ { 2 }$. [4 points]

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
We consider $u, v \in E^p$. Show that $$\Omega_p(u)(v) = \langle\Omega_p(u), \Omega_p(v)\rangle.$$

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\omega$ and $\omega^{\prime}$ elements of $\mathcal{A}_p(E, \mathbb{R})$: $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha})$$
We consider $u, v \in E^p$. Show that $$\Omega_p(u)(v) = \langle\Omega_p(u), \Omega_p(v)\rangle$$

Let $p \in \llbracket 1, d \rrbracket$. We define a relation on the set of bases of a subspace $V$ of dimension $p$ of $E$ by: $e$ and $e^{\prime}$ are in relation if $\operatorname{det}_e(e^{\prime}) > 0$ where $\operatorname{det}_e(e^{\prime})$ is the determinant of $e^{\prime}$ in the basis $e$. We admit that this relation is an equivalence relation on the set of bases of $V$ for which there exist exactly two equivalence classes called orientations of $V$. An oriented subspace is a pair $(V, C)$ where $C$ is an orientation of $V$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$.
(a) Show that if $e$ and $e^{\prime}$ are two free families of cardinal $p$ of $E$ then $\Omega_p(e)$ and $\Omega_p(e^{\prime})$ are collinear if and only if $\operatorname{Vect}(e) = \operatorname{Vect}(e^{\prime})$.
(b) Show that for all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, there exists a unique $\Psi(V, C) \in \mathscr{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.

Let $p \in \llbracket 1, d \rrbracket$. We define a relation on the set of bases of a subspace $V$ of dimension $p$ of $E$ by: $e$ and $e^{\prime}$ are in relation if $\det_e(e^{\prime}) > 0$ where $\det_e(e^{\prime})$ is the determinant of $e^{\prime}$ in the basis $e$. We admit that this relation is an equivalence relation on the set of bases of $V$ for which there exist exactly two equivalence classes called orientations of $V$. An oriented subspace is a pair $(V, C)$ where $C$ is an orientation of $V$.
We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$.
(a) Show that if $e$ and $e^{\prime}$ are two free families of cardinal $p$ of $E$ then $\Omega_p(e)$ and $\Omega_p(e^{\prime})$ are collinear if and only if $\operatorname{Vect}(e) = \operatorname{Vect}(e^{\prime})$.
(b) Show that for all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, there exists a unique $\Psi(V, C) \in \mathcal{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.

Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathscr{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$. We equip $\mathscr{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
(a) Verify that $\Psi : (V, C) \mapsto \Psi(V, C)$ is an injection from $\widetilde{\operatorname{Gr}}(p, E)$ into the unit sphere of $\mathscr{A}_p(E, \mathbb{R})$.
(b) Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a compact subset of $\mathscr{A}_p(E, \mathbb{R})$.

Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathcal{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.
We equip $\mathcal{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
(a) Verify that $\Psi : (V, C) \mapsto \Psi(V, C)$ is an injection from $\widetilde{\operatorname{Gr}}(p, E)$ into the unit sphere of $\mathcal{A}_p(E, \mathbb{R})$.
(b) Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a compact subset of $\mathcal{A}_p(E, \mathbb{R})$.

Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$. Which one of the following is correct?
(A) $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} = \vec{0}$
(B) $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} \neq \vec{0}$
(C) $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{a} \times \vec{c} \neq \vec{0}$
(D) $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}$ are mutually perpendicular

The number of distinct real values of $\lambda$ for which the vectors $-\lambda^2\hat{i}+\hat{j}+\hat{k}$, $\hat{i}-\lambda^2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\lambda^2\hat{k}$ are coplanar is
(A) 0
(B) 1
(C) 2
(D) 3

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat { a } , \hat { b } , \hat { c }$ such that $$\hat { a } \cdot \hat { b } = \hat { b } \cdot \hat { c } = \hat { c } \cdot \hat { a } = \frac { 1 } { 2 }$$ Then, the volume of the parallelopiped is
(A) $\frac { 1 } { \sqrt { 2 } }$
(B) $\frac { 1 } { 2 \sqrt { 2 } }$
(C) $\frac { \sqrt { 3 } } { 2 }$
(D) $\frac { 1 } { \sqrt { 3 } }$

If $\vec { a } , \vec { b } , \vec { c }$ and $\vec { d }$ are unit vectors such that
$$( \vec { a } \times \vec { b } ) \cdot ( \vec { c } \times \vec { d } ) = 1$$
and $\quad \vec { a } \cdot \vec { c } = \frac { 1 } { 2 }$,
then

If $\overrightarrow { \mathrm { a } }$ and $\overrightarrow { \mathrm { b } }$ are vectors in space given by $\overrightarrow { \mathrm { a } } = \frac { \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } } { \sqrt { 5 } }$ and $\overrightarrow { \mathrm { b } } = \frac { 2 \hat { \mathrm { i } } + \hat { \mathrm { j } } + 3 \hat { \mathrm { k } } } { \sqrt { 14 } }$, then the value of $( 2 \vec { a } + \vec { b } ) \cdot [ ( \vec { a } \times \vec { b } ) \times ( \vec { a } - 2 \vec { b } ) ]$ is

Consider the vectors
$$\vec { x } = \hat { \imath } + 2 \hat { \jmath } + 3 \hat { k } , \quad \vec { y } = 2 \hat { \imath } + 3 \hat { \jmath } + \hat { k } , \quad \text { and } \quad \vec { z } = 3 \hat { \imath } + \hat { \jmath } + 2 \hat { k }$$
For two distinct positive real numbers $\alpha$ and $\beta$, define
$$\vec { X } = \alpha \vec { x } + \beta \vec { y } - \vec { z } , \quad \vec { Y } = \alpha \vec { y } + \beta \vec { z } - \vec { x } , \quad \text { and } \quad \vec { Z } = \alpha \vec { z } + \beta \vec { x } - \vec { y }$$
If the vectors $\vec { X } , \vec { Y }$, and $\vec { Z }$ lie in a plane, then the value of $\alpha + \beta - 3$ is $\_\_\_\_$.

Let $\vec { w } = \hat { \imath } + \hat { \jmath } - 2 \hat { k }$, and $\vec { u }$ and $\vec { v }$ be two vectors, such that $\vec { u } \times \vec { v } = \vec { w }$ and $\vec { v } \times \vec { w } = \vec { u }$. Let $\alpha , \beta , \gamma$, and $t$ be real numbers such that $\vec { u } = \alpha \hat { \imath } + \beta \hat { \jmath } + \gamma \hat { k } , \quad - t \alpha + \beta + \gamma = 0 , \quad \alpha - t \beta + \gamma = 0 , \quad$ and $\alpha + \beta - t \gamma = 0$.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I

(P) $| \vec { v } | ^ { 2 }$ is equal to (Q) If $\alpha = \sqrt { 3 }$, then $\gamma ^ { 2 }$ is equal to (R) If $\alpha = \sqrt { 3 }$, then $( \beta + \gamma ) ^ { 2 }$ is equal to (S) If $\alpha = \sqrt { 2 }$, then $t + 3$ is equal to

List-II

(1) 0
(2) 1
(3) 2
(4) 3
(5) 5

(A)	$( \mathrm { P } ) \rightarrow ( 2 )$	$( \mathrm { Q } ) \rightarrow ( 1 )$	$( \mathrm { R } ) \rightarrow ( 4 )$	$( \mathrm { S } ) \rightarrow ( 5 )$
(B)	$( \mathrm { P } ) \rightarrow ( 2 )$	$( \mathrm { Q } ) \rightarrow ( 4 )$	$( \mathrm { R } ) \rightarrow ( 3 )$	$( \mathrm { S } ) \rightarrow ( 5 )$
(C)	$( \mathrm { P } ) \rightarrow ( 2 )$	$( \mathrm { Q } ) \rightarrow ( 1 )$	$( \mathrm { R } ) \rightarrow ( 4 )$	$( \mathrm { S } ) \rightarrow ( 3 )$
(D)	$( \mathrm { P } ) \rightarrow ( 5 )$	$( \mathrm { Q } ) \rightarrow ( 4 )$	$( \mathrm { R } ) \rightarrow ( 1 )$	$( \mathrm { S } ) \rightarrow ( 3 )$

If the volume of a parallelopiped, whose coterminous edges are given by the vectors $\overrightarrow { \mathrm { a } } = \hat { i } + \hat { j } + n \widehat { k }$, $\overrightarrow { \mathrm { b } } = 2 \hat { \mathrm { i } } + 4 \hat { \mathrm { j } } - n \hat { k }$ and, $\overrightarrow { \mathrm { c } } = \hat { \mathrm { i } } + n \hat { j } + 3 \hat { k } ( \mathrm { n } \geq 0 )$ is 158 cubic units, then :
(1) $\overrightarrow { \mathrm { a } } \cdot \overrightarrow { \mathrm { c } } = 17$
(2) $\overrightarrow { \mathrm { b } } \cdot \overrightarrow { \mathrm { c } } = 10$
(3) $n = 7$
(4) $n = 9$

Three particles $P , Q$ and $R$ are moving along the vectors $\vec { A } = \hat { \mathrm { i } } + \hat { \mathrm { j } } , \overrightarrow { \mathrm { B } } = \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ and $\vec { C } = - \hat { \mathrm { i } } + \hat { \mathrm { j } }$, respectively. They strike on a point and start to move in different directions. Now particle $P$ is moving normal to the plane which contains vector $\vec { A }$ and $\vec { B }$. Similarly particle $Q$ is moving normal to the plane which contains vector $\vec { A }$ and $\vec { C }$. The angle between the direction of motion of $P$ and $Q$ is $\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { x } } \right)$. Then the value of $x$ is $\_\_\_\_$ .

Let $| \vec { a } | = 2 , | \vec { b } | = 3$ and the angle between the vectors $\vec { a }$ and $\vec { b }$ be $\frac { \pi } { 4 }$. Then $| ( \vec { a } + 2 \vec { b } ) \times ( 2 \vec { a } - 3 \vec { b } ) | ^ { 2 }$ is equal to
(1) 441
(2) 482
(3) 841
(4) 882

For three vectors $\vec { A } = ( - x \hat { i } - 6 \hat { j } - 2 \hat { k } ) , \vec { B } = ( - \hat { i } + 4 \hat { j } + 3 \hat { k } )$ and $\vec { C } = ( - 8 \hat { i } - \hat { j } + 3 \hat { k } )$, if $\vec { A } \cdot ( \vec { B } \times \vec { C } ) = 0$, then value of $x$ is $\_\_\_\_$

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 $\overrightarrow{\mathrm{a}} = 2\hat{i} - \hat{j} + 3\hat{k}$, $\overrightarrow{\mathrm{b}} = 3\hat{i} - 5\hat{j} + \hat{k}$ and $\overrightarrow{\mathrm{c}}$ be a vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}} = \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}$ and $(\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168$. Then the maximum value of $|\vec{c}|^2$ is:
(1) 462
(2) 77
(3) 154
(4) 308

In coordinate space, a parallelepiped has three vertices of one base at $( - 1,2,1 ) , ( - 4,1,3 ) , ( 2,0 , - 3 )$ , and one vertex of another face lies on the $xy$-plane at distance 1 from the origin. Among parallelepipeds satisfying the above conditions, the maximum volume is (17-1)(17-2).

It is known that $P$, $Q$, $R$ are three non-collinear points on the plane $2x - 3y + 5z = \sqrt{7}$ in coordinate space. Let $\overrightarrow{PQ} = (a_{1}, b_{1}, c_{1})$, $\overrightarrow{PR} = (a_{2}, b_{2}, c_{2})$. Select the option in which the absolute value of the determinant is the largest.
(1) $\left|\begin{array}{ccc} -1 & 1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(2) $\left|\begin{array}{ccc} 1 & -1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(3) $\left|\begin{array}{ccc} 1 & 1 & -1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(4) $\left|\begin{array}{ccc} -1 & -1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(5) $\left|\begin{array}{ccc} -1 & -1 & -1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$

As shown in the figure, $OABC-DEFG$ is a cube. Which of the following vectors is parallel to the cross product $\overrightarrow{AD} \times \overrightarrow{AG}$?
(1) $\overrightarrow{AE}$
(2) $\overrightarrow{BE}$
(3) $\overrightarrow{CE}$
(4) $\overrightarrow{DE}$
(5) $\overrightarrow{OE}$