Deduce that if $v\in\mathcal{H}$, then the restriction of $B$ to $v^\perp$ is an inner product.

We identify $M_3(\mathbb{R})$ with the linear endomorphisms of $V$. Let $G$ be the set of endomorphisms $g$ such that $$B(gu,gv) = B(u,v)$$ for all $u,v\in V$.
Show that $G$ is a group under composition of linear maps.

We denote by arcch $:[1,+\infty)\rightarrow\mathbb{R}_+$ the inverse of the hyperbolic cosine. Let $v\in\mathcal{H}$. Show that the set $T_v\mathcal{H}$ of vectors tangent to $\mathcal{H}$ at point $v$ is a vector subspace of $V$ and determine this subspace. Deduce that the restriction of $B$ to $T_v\mathcal{H}$ is an inner product.

Let $d$ be the hyperbolic distance on $\mathcal{H}$ and $G_0$ the subgroup of endomorphisms preserving $B$ and $\mathcal{H}$. Show that $d(gu,gv) = d(u,v)$ for all $g\in G$.

For all $(t,\theta)\in\mathbb{R}_+\times[0,2\pi]$, define $$F(t,\theta) = \begin{pmatrix} \frac{1}{\sqrt{3}}\operatorname{sh}(t)\cos(\theta) \\ \frac{1}{\sqrt{3}}\operatorname{sh}(t)\sin(\theta) \\ \operatorname{ch}(t) \end{pmatrix}.$$ Show that $F$ takes values in $\mathcal{H}$ and that $F:\mathbb{R}_+\times[0,2\pi]\rightarrow\mathcal{H}$ is surjective.

134- From point $A(5,-2,1)$, a line perpendicular to the plane with equation $x = t+1$, $y = -2t+1$, $z = 2t-3$ is drawn. What are the coordinates of the intersection point of this line and the plane?
(1) $(2,-1,-1)$ (2) $(1,1,-3)$ (3) $(4,5,3)$ (4) $(3,-3,1)$

135- The plane passing through the two intersecting lines $(D): \begin{cases} 2x+y=3 \\ 2y-z=0 \end{cases}$ and $(D'): \dfrac{x+1}{2}=\dfrac{y}{1}=\dfrac{z+1}{3}$. Which value does the $z$-axis intercept cut?
(1) $-0.8$ (2) $-0.6$ (3) $0.8$ (4) $1.2$

135- What is the length of the common perpendicular of the two lines $$\frac{x-1}{1} = \frac{y+2}{-1} = \frac{z}{3} \quad \text{and} \quad \begin{cases} x = 2y - 1 \\ z = 3y - 2 \end{cases}$$?
(1) $\sqrt{3}$ (2) $\sqrt{6}$ (3) $2\sqrt{3}$ (4) $2\sqrt{6}$

135- The plane passing through the line with equation $\dfrac{x+1}{2} = \dfrac{y}{3} = \dfrac{z-2}{-1}$ and the point $(0,3,0)$ intersects the $Z$-axis at what elevation?
(1) $-2$ (2) $-3$ (3) $2$ (4) $3$

135- The shortest distance between the two lines $\dfrac{x-1}{3} = -y + 4 = \dfrac{z}{5}$ and $\begin{cases} x = 2 \\ y = 5 \end{cases}$ is which of the following?
(1) $\dfrac{3}{\sqrt{10}}$ (2) $\dfrac{4}{\sqrt{10}}$ (3) $\sqrt{10}$ (4) $2\sqrt{5}$

5. Given the points $A(-2,3,1), B(3,0,-1), C(2,2,-3)$, determine the equation of the line $r$ passing through $A$ and $B$ and the equation of the plane $\pi$ perpendicular to $r$ and passing through $C$.

7. Determine the coordinates of the centres of the spheres with radius $\sqrt{6}$ tangent to the plane $\pi$ with equation:
$$x + 2 y - z + 1 = 0$$
at its point $P$ with coordinates $(1,0,2)$.

9. A plane is parallel to two lines whose direction ratios are $( 1,0 , - 1 )$ and $( - 1,1,0 )$ and it contains the point $( 1,1,1 )$. If it cuts coordinate axis at $\mathrm { A } , \mathrm { B } , \mathrm { C }$, then find the volume of the tetrahedron OABC .
Sol. Let $( l , m , n )$ be the direction ratios of the normal to the required plane so that $l - n = 0$ and $- l + m = 0$ $\Rightarrow \mathrm { l } = \mathrm { m } = \mathrm { n }$ and hence the equation of the plane containing $( 1,1,1 )$ is $\frac { \mathrm { x } } { 3 } + \frac { \mathrm { y } } { 3 } + \frac { \mathrm { z } } { 3 } = 1$. Its intercepts with the coordinate axes are $\mathrm { A } ( 3,0,0 ) ; \mathrm { B } ( 0,3,0 ) ; \mathrm { C } ( 0,0,3 )$. Hence the volume of OABC $= \frac { 1 } { 6 } \left| \begin{array} { l l l } 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array} \right| = \frac { 27 } { 6 } = \frac { 9 } { 2 }$ cubic units.

A line with positive direction cosines passes through the point $P(2,-1,2)$ and makes equal angles with the coordinate axes. The line meets the plane $$2x+y+z=9$$ at point $Q$. The length of the line segment $PQ$ equals
(A) 1
(B) $\sqrt{2}$
(C) $\sqrt{3}$
(D) 2

Equation of the plane containing the straight line $\frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 4 }$ and perpendicular to the plane containing the straight lines $\frac { x } { 3 } = \frac { y } { 4 } = \frac { z } { 2 }$ and $\frac { x } { 4 } = \frac { y } { 2 } = \frac { z } { 3 }$ is
A) $x + 2 y - 2 z = 0$
B) $3 x + 2 y - 2 z = 0$
C) $x - 2 y + z = 0$
D) $5 x + 2 y - 4 z = 0$

If the distance between the plane $\mathrm { Ax } - 2 \mathrm { y } + \mathrm { z } = \mathrm { d }$ and the plane containing the lines $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { 4 }$ and $\frac { x - 2 } { 3 } = \frac { y - 3 } { 4 } = \frac { z - 4 } { 5 }$ is $\sqrt { 6 }$, then $| d |$ is

41. The equation of a plane passing through the line of intersection of the planes $x + 2 y + 3 z = 2$ and $x - y + z = 3$ and at a distance $\frac { 2 } { \sqrt { 3 } }$ from the point $( 3,1 , - 1 )$ is
(A) $5 x - 11 y + z = 17$
(B) $\sqrt { 2 } x + y = 3 \sqrt { 2 } - 1$
(C) $x + y + z = \sqrt { 3 }$
(D) $x - \sqrt { 2 } y = 1 - \sqrt { 2 }$

ANSWER : A

1. Let $P Q R$ be a triangle of area $\triangle$ with $a = 2 , b = \frac { 7 } { 2 }$ and $c = \frac { 5 } { 2 }$, where $a , b$ and $c$ are the lengths of the sides of the triangle opposite to the angles at $P , Q$ and $R$ respectively. Then $\frac { 2 \sin P - \sin 2 P } { 2 \sin P + \sin 2 P }$ equals
   (A) $\frac { 3 } { 4 \Delta }$
   (B) $\frac { 45 } { 4 \Delta }$
   (C) $\left( \frac { 3 } { 4 \Delta } \right) ^ { 2 }$
   (D) $\left( \frac { 45 } { 4 \Delta } \right) ^ { 2 }$

ANSWER : C

1. If $\vec { a }$ and $\vec { b }$ are vectors such that $| \vec { a } + \vec { b } | = \sqrt { 29 }$ and $\vec { a } \times ( 2 \hat { i } + 3 \hat { j } + 4 \hat { k } ) = ( 2 \hat { i } + 3 \hat { j } + 4 \hat { k } ) \times \vec { b }$, then a possible value of $( \vec { a } + \vec { b } ) \cdot ( - 7 \hat { i } + 2 \hat { j } + 3 \hat { k } )$ is
   (A) 0
   (B) 3
   (C) 4
   (D) 8
2. If $P$ is a $3 \times 3$ matrix such that $P ^ { T } = 2 P + I$, where $P ^ { T }$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = \left[ \begin{array} { c } x \\ y \\ z \end{array} \right] \neq \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$ such that
   (A) $P X = \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$
   (B) $P X = X$
   (C) $P X = 2 X$
   (D) $P X = - X$

ANSWER : D

1. Let $\alpha$ (a) and $\beta$ (a) be the roots of the equation $( \sqrt [ 3 ] { 1 + a } - 1 ) x ^ { 2 } + ( \sqrt { 1 + a } - 1 ) x + ( \sqrt [ 6 ] { 1 + a } - 1 ) = 0$ where $a > - 1$. Then $\lim _ { a \rightarrow 0 ^ { + } } \alpha ( a )$ and $\lim _ { a \rightarrow 0 ^ { + } } \beta ( a )$ are
   (A) $- \frac { 5 } { 2 }$ and 1
   (B) $- \frac { 1 } { 2 }$ and - 1
   (C) $- \frac { 7 } { 2 }$ and 2
   (D) $- \frac { 9 } { 2 }$ and 3

ANSWER : B

1. Four fair dice $D _ { 1 } , D _ { 2 } , D _ { 3 }$ and $D _ { 4 }$, each having six faces numbered $1,2,3,4,5$ and 6 , are rolled simultaneously. The probability that $D _ { 4 }$ shows a number appearing on one of $D _ { 1 } , D _ { 2 }$ and $D _ { 3 }$ is
   (A) $\frac { 91 } { 216 }$
   (B) $\frac { 108 } { 216 }$
   (C) $\frac { 125 } { 216 }$
   (D) $\frac { 127 } { 216 }$

ANSWER : A

1. The value of the integral $\int _ { - \pi / 2 } ^ { \pi / 2 } \left( x ^ { 2 } + \ln \frac { \pi + x } { \pi - x } \right) \cos x \mathrm {~d} x$ is
   (A) 0
   (B) $\frac { \pi ^ { 2 } } { 2 } - 4$
   (C) $\frac { \pi ^ { 2 } } { 2 } + 4$
   (D) $\frac { \pi ^ { 2 } } { 2 }$

ANSWER : B

1. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be in harmonic progression with $a _ { 1 } = 5$ and $a _ { 20 } = 25$. The least positive integer $n$ for which $a _ { n } < 0$ is
   (A) 22
   (B) 23
   (C) 24
   (D) 25

SECTION II : Paragraph Type

This section contains $\mathbf { 6 }$ multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Questions 49 and 50

Let $a _ { n }$ denote the number of all $n$-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let $b _ { n } =$ the number of such $n$-digit integers ending with digit 1 and $c _ { n } =$ the number of such $n$-digit integers ending with digit 0 .

Two lines $L _ { 1 } : x = 5 , \frac { y } { 3 - \alpha } = \frac { z } { - 2 }$ and $L _ { 2 } : x = \alpha , \frac { y } { - 1 } = \frac { z } { 2 - \alpha }$ are coplanar. Then $\alpha$ can take value(s)
(A) 1
(B) 2
(C) 3
(D) 4

Perpendiculars are drawn from points on the line $\frac { x + 2 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z } { 3 }$ to the plane $x + y + z = 3$. The feet of perpendiculars lie on the line
(A) $\frac { x } { 5 } = \frac { y - 1 } { 8 } = \frac { z - 2 } { - 13 }$
(B) $\frac { x } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 2 } { - 5 }$
(C) $\frac { x } { 4 } = \frac { y - 1 } { 3 } = \frac { z - 2 } { - 7 }$
(D) $\frac { x } { 2 } = \frac { y - 1 } { - 7 } = \frac { z - 2 } { 5 }$

A line $l$ passing through the origin is perpendicular to the lines $l _ { 1 } : ( 3 + t ) \hat { i } + ( - 1 + 2 t ) \hat { j } + ( 4 + 2 t ) \hat { k } , - \infty < t < \infty$ $l _ { 2 } : ( 3 + 2 s ) \hat { i } + ( 3 + 2 s ) \hat { j } + ( 2 + s ) \hat { k } , - \infty < s < \infty$ Then, the coordinate(s) of the point(s) on $l _ { 2 }$ at a distance of $\sqrt { 17 }$ from the point of intersection of $l$ and $l _ { 1 }$ is (are)
(A) $\left( \frac { 7 } { 3 } , \frac { 7 } { 3 } , \frac { 5 } { 3 } \right)$
(B) $( - 1 , - 1,0 )$
(C) $( 1,1,1 )$
(D) $\left( \frac { 7 } { 9 } , \frac { 7 } { 9 } , \frac { 8 } { 9 } \right)$

Consider the lines $L _ { 1 } : \frac { x - 1 } { 2 } = \frac { y } { - 1 } = \frac { z + 3 } { 1 } , L _ { 2 } : \frac { x - 4 } { 1 } = \frac { y + 3 } { 1 } = \frac { z + 3 } { 2 }$ and the planes $P _ { 1 } : 7 x + y + 2 z = 3 , P _ { 2 } : 3 x + 5 y - 6 z = 4$. Let $a x + b y + c z = d$ be the equation of the plane passing through the point of intersection of lines $L _ { 1 }$ and $L _ { 2 }$, and perpendicular to planes $P _ { 1 }$ and $P _ { 2 }$.
Match List I with List II and select the correct answer using the code given below the lists:
List I

• [P.] $a =$
• [Q.] $b =$
• [R.] $c =$
• [S.] $d =$

List II

1. $13$
2. $-3$
3. $1$
4. $-2$

Codes:

	P	Q	R	S
(A)	3	2	4	1
(B)	1	3	4	2
(C)	3	2	1	4
(D)	2	4	1	3

Let $P$ be the image of the point $( 3,1,7 )$ with respect to the plane $x - y + z = 3$. Then the equation of the plane passing through $P$ and containing the straight line $\frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { 1 }$ is
(A) $x + y - 3 z = 0$
(B) $3 x + z = 0$
(C) $x - 4 y + 7 z = 0$
(D) $2 x - y = 0$

The equation of the plane passing through the point $( 1,1,1 )$ and perpendicular to the planes $2 x + y - 2 z = 5$ and $3 x - 6 y - 2 z = 7$, is
[A] $14 x + 2 y - 15 z = 1$
[B] $14 x - 2 y + 15 z = 27$
[C] $14 x + 2 y + 15 z = 31$
[D] $- 14 x + 2 y + 15 z = 3$

Let $P _ { 1 }$ and $P _ { 2 }$ be two planes given by
$$\begin{aligned} & P _ { 1 } : 10 x + 15 y + 12 z - 60 = 0 \\ & P _ { 2 } : \quad - 2 x + 5 y + 4 z - 20 = 0 \end{aligned}$$
Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P _ { 1 }$ and $P _ { 2 }$ ?
(A) $\frac { x - 1 } { 0 } = \frac { y - 1 } { 0 } = \frac { z - 1 } { 5 }$
(B) $\frac { x - 6 } { - 5 } = \frac { y } { 2 } = \frac { z } { 3 }$
(C) $\frac { x } { - 2 } = \frac { y - 4 } { 5 } = \frac { z } { 4 }$
(D) $\frac { x } { 1 } = \frac { y - 4 } { - 2 } = \frac { z } { 3 }$

Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$$\vec { r } = - ( t + p ) \hat { \imath } + t \hat { \jmath } + ( 1 + p ) \hat { k }$$
where $t , p$ are real parameters and $\hat { \imath } , \hat { \jmath } , \hat { k }$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10 \hat { \imath } + 15 \hat { \jmath } + 20 \hat { k }$ and $\alpha \hat { \imath } + \beta \hat { \jmath } + \gamma \hat { k }$ respectively, then which of the following is/are TRUE ?
(A) $3 ( \alpha + \beta ) = - 101$
(B) $3 ( \beta + \gamma ) = - 71$
(C) $3 ( \gamma + \alpha ) = - 86$
(D) $3 ( \alpha + \beta + \gamma ) = - 121$