8. Show that there exists a unique family $q _ { 0 } , \ldots , q _ { n - j - 1 }$ of monic polynomials of $\mathbb { R } [ \boldsymbol { X } ]$ such that $\operatorname { deg } \left( q _ { i } \right) = i$ for $0 \leqslant i \leqslant n - j - 1$ and such that for all $0 \leqslant i \neq i ^ { \prime } \leqslant n - j - 1$,
$$\left\langle q _ { i } , q _ { i ^ { \prime } } \right\rangle _ { j } = 0 .$$
8t. We set $q _ { n - j } = \prod _ { i = j + 1 } ^ { n } \left( X - r _ { i } \right)$. Show that $q _ { n - j }$ is the unique monic polynomial of degree $n - j$ satisfying, for all $0 \leqslant i \leqslant n - j - 1$,
$$\left\langle q _ { i } , q _ { n - j } \right\rangle _ { j } = 0$$
Let $2 \leqslant i \leqslant n - j$. Show that there exist real numbers $a _ { i }$ and $b _ { i }$ such that
$$q _ { i } - X q _ { i - 1 } = a _ { i } q _ { i - 1 } + b _ { i } q _ { i - 2 }$$
9b. Show that
9c. Show that $b _ { i } < 0$.
$$b _ { i } \left\langle q _ { i - 2 } , q _ { i - 2 } \right\rangle _ { j } = - \langle \underbrace { X q _ { i - 1 } , q _ { i - 2 } } _ { \geqslant 0 } \rangle _ { j } j .$$

1. For $i \in \{ 0,1 \}$, show that the polynomial $q _ { i }$ has exactly $i$ roots in $\mathbb { R }$ (note that we do not require the roots to belong to the interval $I$ ).

10b. Show that, for all $1 \leqslant i \leqslant n - j$, the polynomial $q _ { i }$ has exactly $i$ distinct real roots, that these roots are simple and that if $x _ { 1 } < x _ { 2 }$ are two consecutive roots of $q _ { i }$, there exists a unique root of $q _ { i - 1 }$ in the interval $] x _ { 1 } , x _ { 2 } [$. 10c. Deduce that, for all $0 \leqslant i \leqslant n - j - 1$, we have $q _ { i } \left( r _ { j + 1 } \right) > 0$. For $0 \leqslant i \leqslant n - j - 1$, there therefore exists a unique real number $\alpha _ { i }$ such that
$$q _ { i + 1 } \left( r _ { j + 1 } \right) + \alpha _ { i } q _ { i } \left( r _ { j + 1 } \right) = 0$$
We fix $0 \leqslant i \leqslant n - j - 1$ and we set
$$p _ { i } = \frac { q _ { i + 1 } + \alpha _ { i } q _ { i } } { X - r _ { j + 1 } }$$
We denote $c _ { 0 } , \ldots , c _ { i } \in \mathbb { R }$ the coordinates of $p _ { i }$ in the basis $\left( q _ { 0 } , \ldots , q _ { i } \right)$ of $\mathbb { R } _ { i } [ X ]$. 11a. Show that, for $0 \leqslant \ell \leqslant i$,
$$\left\langle q _ { i + 1 } + \alpha _ { i } q _ { i } , \frac { q _ { \ell } - q _ { \ell } \left( r _ { j + 1 } \right) } { X - r _ { j + 1 } } \right\rangle _ { j } = 0 .$$
11b. Show that, for every integer $0 \leqslant \ell \leqslant i$, there exists a real $\gamma _ { \ell } > 0$ such that $c _ { \ell } = \gamma _ { \ell } c _ { 0 }$ and deduce that $c _ { \ell } > 0$.

Write, in Python language, a function \texttt{mat\_adj(graph)} which takes as argument a graph of $m \in \mathbb{N}^*$ vertices, directed or undirected, represented by a dictionary having as keys the integers from 0 to $m-1$, and for value associated with such a key the adjacency list of the corresponding vertex, and which returns, respecting the enumeration of the vertices, the adjacency matrix of this graph.
Thus \texttt{mat\_adj(\{0: [1, 2], 1: [0], 2: [0, 1, 2]\})} must return $$[[0,1,1],[1,0,0],[1,1,1]].$$

12. Show that, if $0 \leqslant j \leqslant n - 2$, for all $0 \leqslant i \leqslant n - j - 1$, the polynomial $p _ { i }$ is orthogonal to $\mathbb { R } _ { i - 1 } [ X ]$ for the inner product $\langle \cdot , \cdot \rangle _ { j + 1 }$.

12. Show that, if $0 \leqslant j \leqslant n - 2$, for all $0 \leqslant i \leqslant n - j - 1$, the polynomial $p _ { i }$ is orthogonal to $\mathbb { R } _ { i - 1 } [ X ]$ for the inner product $\langle \cdot , \cdot \rangle _ { j + 1 }$.

13. Let $\mathscr { B } = \left( a _ { 0 } , \ldots , a _ { n } \right)$ be the unique orthogonal basis of $\left( \mathbb { R } _ { n } [ X ] , \langle \cdot , \cdot \rangle \right)$ such that $a _ { i }$ is a monic polynomial of degree $i$ for all $0 \leqslant i \leqslant n$. Show that, for all $0 \leqslant j \leqslant n - 1$, the coefficients of the polynomial $\prod _ { \ell = j + 1 } ^ { n } \left( X - r _ { \ell } \right)$ in the basis $\mathscr { B }$ are strictly positive real numbers. Hint: one may denote $\left( q _ { j , 0 } , \ldots , q _ { j , n - j } \right)$ the basis of $\left( \mathbb { R } _ { n - j } [ X ] , \langle \cdot , \cdot \rangle _ { j } \right)$ obtained in questions $8 a$ and $8 b$ and reason by descending induction on $j$.

Third Part

Let $\lambda$ be a strictly positive real number. For all real $x$ and $r$ such that $| x | < 1$ and $| r | < 1$, we set
$$F _ { \lambda } ( x , r ) = \left( 1 - 2 r x + r ^ { 2 } \right) ^ { - \lambda }$$

1. Show that the function $F _ { \lambda }$ is of class $\mathscr { C } ^ { \infty }$ on $] - 1,1 \left[ ^ { 2 } \right.$.
2. Show that for $x \in ] - 1,1 \left[ \right.$, the function $r \mapsto F _ { \lambda } ( x , r )$ is expandable as a power series in a neighborhood of 0 . For $x \in ] - 1,1 \left[ \right.$, we denote $a _ { n } ^ { ( \lambda ) } ( x )$ the $n$-th coefficient of the expansion of the function $r \mapsto F _ { \lambda } ( x , r )$ so that, for $r$ in a neighborhood of 0 ,

$$F _ { \lambda } ( x , r ) = \sum _ { n \geqslant 0 } a _ { n } ^ { ( \lambda ) } ( x ) r ^ { n }$$
16a. For $x \in ] - 1,1 \left[ \right.$, show that $a _ { 1 } ^ { ( \lambda ) } ( x ) = 2 \lambda x a _ { 0 } ^ { ( \lambda ) } ( x )$ and that, for every integer $n \geqslant 1$,
$$( n + 1 ) a _ { n + 1 } ^ { ( \lambda ) } ( x ) = 2 ( n + \lambda ) x a _ { n } ^ { ( \lambda ) } ( x ) - ( n + 2 \lambda - 1 ) a _ { n - 1 } ^ { ( \lambda ) } ( x )$$
Hint: one may begin by computing $\left( 1 - 2 x r + r ^ { 2 } \right) \frac { \partial F _ { \lambda } } { \partial r } ( x , r )$. 16b. Deduce that, for all $n \geqslant 0$, the function $a _ { n } ^ { ( \lambda ) }$ is a polynomial of degree $n$ whose leading coefficient and parity will be determined. We now assume that $\lambda > \frac { 1 } { 2 }$. For $P , Q \in \mathbb { R } [ X ]$, we set
$$\langle P , Q \rangle = \int _ { - 1 } ^ { 1 } P ( x ) Q ( x ) \left( 1 - x ^ { 2 } \right) ^ { \lambda - \frac { 1 } { 2 } } d x$$

13. Let $\mathscr { B } = \left( a _ { 0 } , \ldots , a _ { n } \right)$ be the unique orthogonal basis of $\left( \mathbb { R } _ { n } [ X ] , \langle \cdot , \cdot \rangle \right)$ such that $a _ { i }$ is a monic polynomial of degree $i$ for all $0 \leqslant i \leqslant n$. Show that, for all $0 \leqslant j \leqslant n - 1$, the coefficients of the polynomial $\prod _ { \ell = j + 1 } ^ { n } \left( X - r _ { \ell } \right)$ in the basis $\mathscr { B }$ are strictly positive real numbers. Hint: one may denote $\left( q _ { j , 0 } , \ldots , q _ { j , n - j } \right)$ the basis of $\left( \mathbb { R } _ { n - j } [ X ] , \langle \cdot , \cdot \rangle _ { j } \right)$ obtained in questions 8a and 8b and reason by descending induction on $j$.

Third Part

Let $\lambda$ be a strictly positive real number. For all real $x$ and $r$ such that $| x | < 1$ and $| r | < 1$, we set
$$F _ { \lambda } ( x , r ) = \left( 1 - 2 r x + r ^ { 2 } \right) ^ { - \lambda }$$
Show that the function $F _ { \lambda }$ is of class $\mathscr { C } ^ { \infty }$ on $] - 1,1 \left[ ^ { 2 } \right.$.

15. Show that for $x \in ] - 1,1 \left[ \right.$, the function $r \mapsto F _ { \lambda } ( x , r )$ is expandable as a power series in a neighborhood of $\mathbf { 0 }$. For $x \in ] - 1,1$, we denote $a _ { n } ^ { ( \lambda ) } ( x )$ the $n$-th coefficient of the expansion of the function $r \mapsto F _ { \lambda } ( x , r )$ so that, for $r$ in a neighborhood of 0 ,
$$F _ { \lambda } ( x , r ) = \sum _ { n \geqslant 0 } a _ { n } ^ { ( \lambda ) } ( x ) r ^ { n } .$$
16a. For $x \in ] - 1,1 \left[ \right.$, show that $a _ { 1 } ^ { ( \lambda ) } ( x ) = 2 \lambda x a _ { 0 } ^ { ( \lambda ) } ( x )$ and that, for every integer $n \geqslant 1$,
$$( n + 1 ) a _ { n + 1 } ^ { ( \lambda ) } ( x ) = 2 ( n + \lambda ) x a _ { n } ^ { ( \lambda ) } ( x ) - ( n + 2 \lambda - 1 ) a _ { n - 1 } ^ { ( \lambda ) } ( x ) .$$
Hint: one may begin by computing $\left( 1 - 2 x r + r ^ { 2 } \right) \frac { \partial F _ { \lambda } } { \partial r } ( x , r )$. 16b. Deduce that, for all $n \geqslant 0$, the function $a _ { n } ^ { ( \lambda ) }$ is a polynomial of degree $n$ whose leading coefficient and parity will be determined. We now assume that $\lambda > \frac { 1 } { 2 }$. For $P , Q \in \mathbb { R } [ X ]$, we set

19. Show that the polynomials $T _ { n }$ are functions of positive type in dimension 2.
Hint: one may use the exponential form of the cosine. We shall admit, in the rest of the problem, that for every integer $n \geqslant 0$ and every integer $N \geqslant 4$, the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ is of positive type in dimension $N$. For an integer $N \geqslant 2$, we say that a polynomial $P \in \mathbb { R } [ X ]$ is $N$-conductive if, for every absolutely monotone function $f$ from $[ - 1,1 ]$ to $\mathbb { R }$, the polynomial $H ( f , P )$ is a function of positive type in dimension $N$.

19. Show that the polynomials $T _ { n }$ are functions of positive type in dimension 2 .
Hint: you may use the exponential form of the cosine. We shall admit, in the rest of the problem, that for every integer $n \geqslant 0$ and every integer $N \geqslant 4$, the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ is of positive type in dimension $N$. For an integer $N \geqslant 2$, we say that a polynomial $P \in \mathbb { R } [ X ]$ is $N$-conductive if, for every absolutely monotone function $f$ from $[ - 1,1 ]$ to $\mathbb { R }$, the polynomial $H ( f , P )$ is a function of positive type in dimension $N$.

20. Let $P _ { 1 }$ and $P _ { 2 }$ be two $N$-conductive polynomials. Show that if $P _ { 1 }$ is of positive type in dimension $N$, then $P _ { 1 } P _ { 2 }$ is $N$-conductive. We fix an integer $N \geqslant 4$ and an integer $n \geqslant 2$. We admit that the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ has $n$ simple real roots $r _ { 1 } > r _ { 2 } > \cdots > r _ { n }$ in $] - 1,1 [$. Let $f : [ - 1,1 ] \rightarrow \mathbb { R }$ be an absolutely monotone function.

20. Let $P _ { 1 }$ and $P _ { 2 }$ be two $N$-conductive polynomials. Show that if $P _ { 1 }$ is of positive type in dimension $N$, then $P _ { 1 } P _ { 2 }$ is $N$-conductive. We fix an integer $N \geqslant 4$ and an integer $n \geqslant 2$. We admit that the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ has $n$ simple real roots $r _ { 1 } > r _ { 2 } > \cdots > r _ { n }$ in $] - 1,1 [$. Let $f : [ - 1,1 ] \rightarrow \mathbb { R }$ be an absolutely monotone function.

21. Show that the polynomial $H \left( f , \prod _ { i = 1 } ^ { n } \left( X - r _ { i } \right) \right)$ is a function of positive type in dimension $N$.

21. Show that the polynomial $H \left( f , \prod _ { i = 1 } ^ { n } \left( X - r _ { i } \right) \right)$ is a function of positive type in dimension $N$.

22. Show that if $Q \in \mathbb { R } [ X ]$ is a polynomial such that $\operatorname { deg } ( Q ) < \operatorname { deg } ( P )$ and, for every integer $1 \leqslant i \leqslant m$ and every integer $0 \leqslant k < k _ { i } , Q ^ { ( k ) } \left( t _ { i } \right) = 0$, then $Q = 0$. 2b. Show that there exists a unique polynomial $H ( f , P ) \in \mathbb { R } [ X ]$ such that $\operatorname { deg } ( H ( f , P ) ) < \operatorname { deg } ( P )$ and such that, for every integer $1 \leqslant i \leqslant m$ and every integer $0 \leqslant k < k _ { i }$,
$$H ( f , P ) ^ { ( k ) } \left( t _ { i } \right) = f ^ { ( k ) } \left( t _ { i } \right) .$$
For $t \in [ a , b ] \backslash \left\{ t _ { 1 } , \ldots , t _ { m } \right\}$. We set
$$Q ( f , P ) ( t ) = \frac { f ( t ) - H ( f , P ) ( t ) } { \left( t - t _ { 1 } \right) ^ { k _ { 1 } } \cdots \left( t - t _ { m } \right) ^ { k _ { m } } } .$$
3a. We set $g = f - H ( f , P )$. Show that, for every integer $1 \leqslant i \leqslant m$ and every real $x \in [ a , b ]$, we have
$$f ( x ) - H ( f , P ) ( x ) = \left( x - t _ { i } \right) ^ { k _ { i } } \int _ { 0 } ^ { 1 } \frac { v ^ { k _ { i } - 1 } } { \left( k _ { i } - 1 \right) ! } g ^ { \left( k _ { i } \right) } \left( t _ { i } v + x ( 1 - v ) \right) d v$$
3b. Show that the function $Q ( f , P )$ extends uniquely to a function of class $\mathscr { C } ^ { \infty }$ from $[ a , b ]$ to $\mathbb { R }$.
4a. Let $s _ { 0 } \in [ a , b ]$ and let an integer $n \geqslant 1$. Show that
$$Q \left( f , \left( X - s _ { 0 } \right) ^ { n } \right) \left( s _ { 0 } \right) = \frac { f ^ { ( n ) } \left( s _ { 0 } \right) } { n ! }$$

• 4b. Let $P _ { 1 } , P _ { 2 } \in \mathbb { R } [ X ]$ be two monic polynomials split in $] a , b [$. Show that

$$H \left( f , P _ { 1 } P _ { 2 } \right) = H \left( f , P _ { 1 } \right) + P _ { 1 } H \left( Q \left( f , P _ { 1 } \right) , P _ { 2 } \right) \quad \text { and } \quad Q \left( f , P _ { 1 } P _ { 2 } \right) = Q \left( Q \left( f , P _ { 1 } \right) , P _ { 2 } \right)$$
We fix $t \in [ a , b ] \backslash \left\{ t _ { 1 } , \ldots , t _ { m } \right\}$. For all $s \in [ a , b ]$, we set
$$Q _ { t } ( s ) = f ( s ) - H ( f , P ) ( s ) - Q ( f , P ) ( t ) \prod _ { i = 1 } ^ { m } \left( s - t _ { i } \right) ^ { k _ { i } }$$
5a. Show that the function $Q _ { t }$ vanishes to order $\operatorname { deg } ( P ) + 1$ in the interval [ $\min \left( t , t _ { 1 } \right) , \max \left( t , t _ { m } \right)$ ]. 5b. Deduce that if $P$ is monic, there exists $\xi \in \left[ \min \left( t , t _ { 1 } \right) , \max \left( t , t _ { m } \right) \right]$ such that
$$f ( t ) - H ( f , P ) ( t ) = \frac { f ^ { ( \operatorname { deg } ( P ) ) } ( \xi ) } { \operatorname { deg } ( P ) ! } P ( t )$$
We say that a function $h$ from $[ a , b ]$ to $\mathbb { R }$ is absolutely monotone on an interval $[ a , b ]$ if it is of class $\mathscr { C } ^ { \infty }$ on $[ a , b ]$ and if, for every integer $n \geqslant 0$, the function $h ^ { ( n ) }$ takes positive values on $[ a , b ]$. In particular $h$ takes positive values.

2. $\tan ( \alpha + B ) = 50 / \alpha = \frac { 10 / \alpha + \tan B } { 1 - 10 / d \tan B }$ solving Yan $B$ and maximizing writ $d$ you'll find $d = 10 \sqrt { 5 }$ feet and corresponding to a 3!
$$\begin{aligned} & 4 \text { SEP; } T \in Q \text { | are tangency points } \\ & C K \text { are mid points of } A B , M N \\ & \triangle P A C \cong \triangle P Q T \text { and } \triangle Q M K \cong \triangle Q P S \\ & y _ { 2 } A B = A C = P A \cdot S T / P Q = \frac { P S Q M } { P Q } = M K = 1 / 2 M N \text { QED } \\ & s = ( x + y + z ) \\ & \triangle = \sqrt { ( x + y + z ) ( x + y + z - x - y ) ( x + y + z - y - z ) ( x + y + z - x - z ) } \\ & = \sqrt { ( x + y + z ) n y z } \\ & \triangle = 1 / 2 r ( ( x + y ) + ( y + z ) + ( x + z ) ) = r ( x - y + z ) \\ & = \sqrt { ( x + y + z ) x y z } \end{aligned}$$
$$\begin{aligned} x & = \lim _ { n \rightarrow \alpha } \frac { 1 } { 2 n } \log \frac { 2 n ! } { ( n ! ) ^ { 2 } } \\ & = \lim _ { n \rightarrow \alpha } \frac { 1 } { 2 n } \lim _ { n \rightarrow \alpha } \log \frac { \frac { ( n + 1 ) ( n + 2 ) \cdots ( n + n ) } { n ^ { n } } } { \frac { 12 \cdot 3 \cdots } { n ^ { n } } } \\ & = \lim _ { n \rightarrow \alpha } \frac { 1 } { 2 n } ( \log ( 1 + 1 / n ) ( 1 + 2 / n ) ( 1 + 3 / n ) \cdots ( 1 + n / n ) - \log ( 1 / n + 2 / n + \cdots + n / n ) ) \\ & = 1 / 2 \int _ { 0 } ^ { 1 } \ln ( 1 + x ) - \ln x d x = 1 / 2 \left[ \left. \ln \log ( x + 1 ) \right| _ { 0 } ^ { 1 } - \int _ { 0 } ^ { 1 } \frac { x d x } { 1 + x } \right] - [ x \ln x \cdot x ] _ { 0 } ^ { 1 } \end{aligned}$$
$$= 1 / 2 ( ( \ln 2 + \ln 2 - 1 ) - ( - 1 ) ) = 1 / 2 \ln 4$$
7
$$\text { a) } \begin{aligned} f ( x ) & = x ^ { 5 } + x - 10 \\ f ^ { \prime } ( x ) & = 4 x ^ { 4 } + 1 > 0 \end{aligned}$$
$$\text { b) } \begin{gathered} f ( 1 ) = - 8 ; f ( 2 ) = 24 \\ f ( 1 ) f ( 2 ) < 0 \end{gathered}$$
[Figure]
$$\text { c) let } \begin{aligned} x = p / q ; & ( p / q ) ^ { 5 } + ( p / q ) = 10 \\ & \Rightarrow p 5 / q = q 10 q - p \end{aligned}$$
fraction =integer contradictian!

8. Number of pairs ew/o anykind of war $= \binom { 8 } { 2 } = 28$
The poirs that harms $= 5 f [ ( 2,4 ) , ( 2,6 ) , ( 4,6 ) , ( 6,8 ) , ( ( 2,4 ) , ( 6,8 ) ) ]$

9. $( a * b ) \quad \mathrm { lcm } = k m ; g c d = m ; a = k m ; b = k n ; c = k p$
$$\begin{array} { l l } ( a * b ) = \frac { k m n } { k } = m n ; b * c = \frac { k n p } { k } = n p & \\ ( a * b ) * c = k m n p / 1 = k m n p & a = k m \quad i = k p \\ ( a * ( b * c ) ) = k m n p / 1 = k m n p & a * i = k m p / k = m p \neq a \end{array}$$

1. 'any two similar figures have an isomerism and homethette velation which takes are figure to the other' a) have segment $A B ( A \neq B )$ and $C D ( C \not \equiv D )$. Have point $P \in A B$ with $P A / A B = \lambda \in [ 0,1 ]$; assocrate to it the point SECD with SC/CD $= \lambda$ b) have cincle $\gamma$ (of the centre $m$ and radius $r > 0$ ) and $\Gamma$ (of centre $m$ and radius $R > 0$ ). Have point $P + \gamma$ making angle $\theta \in [ 0,2 \pi )$ with the horizontal; asscerate to it point $Q \in T ^ { \prime }$ making an angle $\theta$ with the horizontal

Consider an equilateral triangle $ABC$ with side 2.1 cm. You want to place a number of smaller equilateral triangles, each with side 1 cm, over the triangle $ABC$, so that the triangle $ABC$ is fully covered. What is the minimum number of smaller triangles that you need?
(a) 4
(b) 5
(c) 6
(d) 7.

Consider the L-shaped brick in the diagram below. If an ant starts from $A$, find the minimum distance it has to travel along the surface to reach $B$.
(a) $\sqrt{5}$
(b) $2\sqrt{5}$
(c) $( 3 / 2 ) \sqrt{5}$
(d) $3\sqrt{5}$

3. A particle moves in the $\mathrm { X } - \mathrm { Y }$ plane under the influence of a force such that its linear momentum is $\vec { p } ( t ) = A [ \hat { i } \cos ( k t ) - \hat { j } \sin ( k t ) ]$, where $A$ and $k$ are constants. The angle between the force and the momentum is
(A) $0 ^ { \circ }$
(B) $30 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) $90 ^ { \circ }$ Answer
[Figure]
(A)
[Figure]
(A)
[Figure]
C)
[Figure]
(D)
(B)

4. A small object of uniform density rolls up a curved surface with an initial velocity $v$. It reaches up to a maximum height of $\frac { 3 v ^ { 2 } } { 4 g }$ with respect to the initial position. The object is [Figure]
(A) ring
(B) solid sphere
(C) hollow sphere
(D) disc
Answer [Figure]
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D)

5. Water is filled up to a height $h$ in a beaker of radius $R$ as shown in the figure. The density of water is $\rho$, the surface tension of water is $T$ and the atmospheric pressure is $P _ { 0 }$. Consider a vertical section ABCD of the water column through a diameter of the beaker. The force on water on one side of this section by water on the other side of this section has magnitude [Figure]
(A) $\left| 2 P _ { 0 } R h + \pi R ^ { 2 } \rho g h - 2 R T \right|$
(B) $\left| 2 P _ { 0 } R h + R \rho g h ^ { 2 } - 2 R T \right|$
(C) $\left| P _ { 0 } \pi R ^ { 2 } + R \rho g h ^ { 2 } - 2 R T \right|$
(D) $\left| P _ { 0 } \pi R ^ { 2 } + R \rho g h ^ { 2 } + 2 R T \right|$
Answer
[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
[Figure]
(D)

1. A spherical portion has been removed from a solid sphere having a charge distributed uniformly in its volume as shown in the figure. The electric field inside the emptied space is [Figure]
   (A) zero everywhere
   (B) non-zero and uniform
   (C) non-uniform
   (D) zero only at its center Answer [Figure]
   (A)
   (B)
   (C)
   (D)
2. Positive and negative point charges of equal magnitude are kept at $\left( 0,0 , \frac { a } { 2 } \right)$ and $\left( 0,0 , \frac { - a } { 2 } \right)$, respectively. The work done by the electric field when another positive point charge is moved from $( - a , 0,0 )$ to $( 0 , a , 0 )$ is
   (A) positive
   (B) negative
   (C) zero
   (D) depends on the path connecting the initial and final positions Answer

[Figure]
(A)
[Figure]
(A)
[Figure]
(A)
[Figure]
(B)
(C)
(D)

8. A magnetic field $B = B _ { 0 } \bar { j }$ exists in the region $a < x < 2 a$ and $\vec { B } = - B _ { 0 } \hat { j }$, in the region $2 a < x < 3 a$, where $B _ { 0 }$ is a positive constant. A positive point charge moving with a velocity $\vec { v } = v _ { 0 } \hat { i }$, where $v _ { 0 }$ is a positive constant, [Figure] enters the magnetic field at $x = a$. The trajectory of the charge in this region can be like,
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure] Answer
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D)

9. Electrons with de-Broglie wavelength $\lambda$ fall on the target in an X-ray tube. The cut-off wavelength of the emitted X-rays is
(A) $\lambda _ { 0 } = \frac { 2 m c \lambda ^ { 2 } } { h }$
(B) $\quad \lambda _ { 0 } = \frac { 2 h } { m c }$
(C) $\lambda _ { 0 } = \frac { 2 m ^ { 2 } c ^ { 2 } \lambda ^ { 3 } } { h ^ { 2 } }$
(D) $\lambda _ { 0 } = \lambda$ Answer
D [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D)

STATEMENT-1
A block of mass $m$ starts moving on a rough horizontal surface with a velocity $v$. It stops due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of $30^\circ$ with the horizontal and the same block is made to go up on the surface with the same initial velocity $v$. The decrease in the mechanical energy in the second situation is smaller than that in the first situation. because STATEMENT-2 The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True