We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$ The purpose of Part III is to construct a function of class $C ^ { \infty }$ on $\mathbb { R }$, non-zero, whose all moments of order $p$ ($p \in \mathbb { N }$) are zero. Using the results of questions 36 and 37, conclude.

Express the derivatives $f^{\prime}, f^{\prime\prime}$ and $f^{(3)}$ using usual functions, where $f$ is defined on $I = ]-\pi/2, \pi/2[$ by $$\forall x \in I, \quad f(x) = \frac{\sin x + 1}{\cos x}.$$

Let $f$ be defined on $I = ]-\pi/2, \pi/2[$ by $f(x) = \frac{\sin x + 1}{\cos x}$. Show that there exists a sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ with real coefficients such that $$\forall n \in \mathbb{N}, \forall x \in I, \quad f^{(n)}(x) = \frac{P_n(\sin x)}{(\cos x)^{n+1}}$$ Make explicit the polynomials $P_0, P_1, P_2, P_3$ and, for every natural integer $n$, express $P_{n+1}$ as a function of $P_n$ and $P_n^{\prime}$.

Let $g : \mathbb{R}_+ \rightarrow \mathbb{R}$ be the function defined by $g(x) = \ln\left(1 - p + pe^x\right)$ for all $x \geq 0$.
a. Show that $g$ is well defined and of class $C^2$ on $\mathbb{R}_+$. For $x \geq 0$, express $g''(x)$ in the form $\frac{\alpha\beta}{(\alpha+\beta)^2}$ where $\alpha$ and $\beta$ are positive reals that may depend on $x$.
b. Show that $g''(x) \leq \frac{1}{4}$ for all $x \geq 0$.
c. Show that $$\ln\left(1 - p + pe^x\right) \leq px + \frac{x^2}{8} \text{ for all } x \geq 0$$

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ Prove that, for every positive real $x$, the function $\phi_x$ is of class $\mathcal{C}^2$ on $\mathbb{R}$ and that $$\forall t \in \mathbb{R}, \quad 0 \leqslant \phi_x'(t) \leqslant \frac{x}{\mathrm{e}}.$$

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Justify that $F_{a,b,c}$ is of class $\mathcal{C}^1$ on $]-1,1[$. Calculate its derivative and express it using a Gauss hypergeometric function.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Justify that $F_{a,b,c}$ is of class $\mathcal{C}^\infty$ on $]-1,1[$ and express its $n$-th derivative using a Gauss hypergeometric function.

Let $n \in \mathbb{N}$. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Determine $L_0, L_1, L_2$ and $L_3$.

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Using Leibniz's formula, prove that the function $L_n$ is polynomial of degree $n$. Determine the coefficients $c_{n,k}$ such that $L_n(x) = \sum_{k=0}^{n} c_{n,k} x^k$.

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ For any real number $x$, express $\Phi_n^{(n)}(x)$ and $\Phi_n^{(n+1)}(x)$ in terms of $L_n(x)$ and $L_n'(x)$.

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Use the equality $\Phi_{n+1}^{(n+1)}(x) = \frac{\mathrm{d}^{n+1} x\Phi_n(x)}{\mathrm{d}x^{n+1}}$, which we will justify, to prove the equality $$L_{n+1}(x) = \left(1 - \frac{x}{n+1}\right) L_n(x) + \frac{x}{n+1} L_n'(x)$$ valid for any real number $x$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Show that the function $h$ is of class $C^\infty$ on $\mathbb{R}$ and that, for all $n \in \mathbb{N}^*$, we have $$h^{(2n)}(0) = \frac{(-1)^{n-1}(2n)!}{\pi^{2n} 2^{2n-1}} \zeta(2n)$$

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. If $n \in \mathbf{N}$, we denote by $D_n$ the improper integral $\int_0^{\pi/2} (\ln(\sin(t)))^n \mathrm{~d}t$.
Calculate $f'(0)$ and $f'(1)$.

Let $x > 0$. Using the study of a well-chosen function, show that $$\frac { x } { x ^ { 2 } + 1 } \varphi ( x ) \leqslant \int _ { x } ^ { + \infty } \varphi ( t ) \mathrm { d } t$$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.

We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$
Show that $\Psi$ is of class $\mathcal{C}^1$ on $\mathbf{R}$, then that for all $x \in \mathbf{R}$, $$\Psi'(x) = 4\sum_{k=1}^{+\infty} \rho^k \sin(2kx)$$

Exercise II

II-A- The function $f$ defined on $\mathbb { R } ^ { * }$ by $f ( x ) = e ^ { \frac { 1 } { x } }$ has derivative $f ^ { \prime } ( x ) = e ^ { \frac { 1 } { x } }$. II-B- The function $F$ defined on $[ 0 ; + \infty [$ by $F ( x ) = x \sqrt { x }$ is an antiderivative of the function $f$ defined by $f ( x ) = \frac { 3 } { 2 } \sqrt { x }$. II-C- The function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = ( \ln ( 3 x ) ) ^ { 2 }$ has derivative $f ^ { \prime } ( x ) = \frac { 2 } { 3 x } \ln ( 3 x )$. II-D- $\quad \lim _ { x \rightarrow 0 } ( x \ln ( x ) - x ) = - \infty$. II-E- $\quad \lim _ { x \rightarrow + \infty } \left( x e ^ { x } - \ln ( x ) \right) = 0$.
For each statement, indicate whether it is TRUE or FALSE.

Let $\log x = g(x) = x f(x)$. Find $f^{(n)}(1)$, the $n$-th derivative of $f$ evaluated at $x = 1$.

If $f ( x ) = e ^ { x } \sin x$, then $\left. \frac { d ^ { 10 } } { d x ^ { 10 } } f ( x ) \right| _ { x = 0 }$ equals
(a) 1 .
(B) - 1 .
(C) 10 .
(D) 32 .

7. If $x 2 + y 2 = 1$, then :
(A) yy'" - 2(y ' )2+1=0
(B) $y y ^ { \prime \prime } + \left( y ^ { \prime } \right) 2 + 1 = 0$

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(C) $y y \prime \prime = \left( y ^ { \prime } \right) 2 - 1 = 0$
(D) $y y ^ { \prime \prime } + 2 \left( y ^ { \prime } \right) 2 + 1 = 0$

Let $f$ be a real-valued function defined on the interval $( 0 , \infty )$ by $\mathrm { f } ( \mathrm { x } ) = \ell n \mathrm { x } + \int _ { 0 } ^ { \mathrm { x } } \sqrt { 1 + \sin \mathrm { t } } \mathrm { dt }$. Then which of the following statement(s) is (are) true?
A) $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } )$ exists for all $\mathrm { x } \in ( 0 , \infty )$
B) $f ^ { \prime } ( x )$ exists for all $x \in ( 0 , \infty )$ and $f ^ { \prime }$ is continuous on $( 0 , \infty )$, but not differentiable on $( 0 , \infty )$
C) there exists $\alpha > 1$ such that $\left| \mathrm { f } ^ { \prime } ( \mathrm { x } ) \right| < | \mathrm { f } ( \mathrm { x } ) |$ for all $\mathrm { x } \in ( \alpha , \infty )$
D) there exists $\beta > 0$ such that $| f ( x ) | + \left| f ^ { \prime } ( x ) \right| \leq \beta$ for all $x \in ( 0 , \infty )$

List I P. Let $y(x) = \cos\left(3\cos^{-1}x\right)$, $x \in [-1,1]$, $x \neq \pm\frac{\sqrt{3}}{2}$. Then $\frac{1}{y(x)}\left\{\left(x^2-1\right)\frac{d^2y(x)}{dx^2} + x\frac{dy(x)}{dx}\right\}$ equals Q. Let $A_1, A_2, \ldots, A_n$ $(n > 2)$ be the vertices of a regular polygon of $n$ sides with its centre at the origin. Let $\overrightarrow{a_k}$ be the position vector of the point $A_k$, $k = 1,2,\ldots,n$. If $\left|\sum_{k=1}^{n-1}\left(\overrightarrow{a_k} \times \overrightarrow{a_{k+1}}\right)\right| = \left|\sum_{k=1}^{n-1}\left(\overrightarrow{a_k} \cdot \overrightarrow{a_{k+1}}\right)\right|$, then the minimum value of $n$ is R. If the normal from the point $P(h,1)$ on the ellipse $\frac{x^2}{6} + \frac{y^2}{3} = 1$ is perpendicular to the line $x + y = 8$, then the value of $h$ is S. Number of positive solutions satisfying the equation $\tan^{-1}\left(\frac{1}{2x+1}\right) + \tan^{-1}\left(\frac{1}{4x+1}\right) = \tan^{-1}\left(\frac{2}{x^2}\right)$ is
List II
1. 1
2. 2
3. 3
4. 4
P Q R S
(A) 4321
(B) 2431
(C) 4312
(D) 2413

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only CORRECT combination?
[A] (I) (i) (P)
[B] (II) (ii) (Q)
[C] (III) (iii) (R)
[D] (IV) (iv) (S)

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only CORRECT combination?
[A] (I) (ii) (R)
[B] (II) (iii) (S)
[C] (III) (iv) (P)
[D] (IV) (i) (S)

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only INCORRECT combination?
[A] (I) (iii) (P)
[B] (II) (iv) (Q)
[C] (III) (i) (R)
[D] (II) (iii) (P)

Let $f(x) = \sin(\pi\cos x)$ and $g(x) = \cos(2\pi\sin x)$ be two functions defined for $x > 0$. Define the following sets whose elements are written in the increasing order: $$\begin{array}{ll} X = \{x : f(x) = 0\}, & Y = \{x : f'(x) = 0\} \\ Z = \{x : g(x) = 0\}, & W = \{x : g'(x) = 0\} \end{array}$$
List-I contains the sets $X$, $Y$, $Z$ and $W$. List-II contains some information regarding these sets.
List-I: (I) $X$ (II) $Y$ (III) $Z$ (IV) $W$
List-II: (P) $\supseteq \left\{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\right\}$ (Q) an arithmetic progression (R) NOT an arithmetic progression (S) $\supseteq \left\{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\right\}$ (T) $\supseteq \left\{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\right\}$ (U) $\supseteq \left\{\frac{\pi}{6}, \frac{3\pi}{4}\right\}$
Which of the following is the only CORRECT combination?
(A) (I), (P), (R)
(B) (II), (Q), (T)
(C) (I), (Q), (U)
(D) (II), (R), (S)