4. Among the following functions, which one is both an even function and has a zero point?
(A) $y = \ln x$
(B) $y = x ^ { 2 } + 1$
(C) $y = \sin x$
(D) $y = \cos x$

Let $\mathrm { f } ^ { \prime } ( \mathrm { x } )$ be the derivative of the odd function $f ( x ) ( x \in \mathbf { R } )$. Given $\mathrm { f } ( - 1 ) = 0$, and when $\mathrm { x } > 0$, $x f ^ { \prime } ( x ) - f ( x ) < 0$. Then the range of $x$ for which $f ( x ) > 0$ holds is
(A) $( - \infty , - 1 ) \cup ( 0,1 )$
(B) $( - 1,0 ) \cup ( 1 , + \infty )$
(C) $( - \infty , - 1 ) \cup ( - 1,0 )$
(D) $( 0,1 ) \cup ( 1 , + \infty )$

Let the function $f ( x ) = \ln | 2 x + 1 | - \ln | 2 x - 1 |$ . Then $f ( x )$
A． is an even function and monotonically increasing on $\left( \frac { 1 } { 2 } , + \infty \right)$
B． is an odd function and monotonically decreasing on $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$
C． is an even function and monotonically increasing on $\left( - \infty , - \frac { 1 } { 2 } \right)$
D． is an odd function and monotonically decreasing on $\left( - \infty , - \frac { 1 } { 2 } \right)$

If there exists $a \in \mathbb{R}$ with $a \neq 0$ such that for all $x \in \mathbb{R}$, the inequality $f ( x + a ) < f ( x ) + f ( a )$ always holds, then function $f ( x )$ is said to have property $P$. Given: $q _ { 1 }$: $f ( x )$ is monotonically decreasing and $f ( x ) > 0$ always holds; $q _ { 2 }$: $f ( x )$ is monotonically increasing and there exists $x _ { 0 } < 0$ such that $f \left( x _ { 0 } \right) = 0$. Which is a sufficient condition for $f ( x )$ to have property $P$? ( )
A. Only $q _ { 1 }$
B. Only $q _ { 2 }$
C. Both $q _ { 1 }$ and $q _ { 2 }$
D. Neither $q _ { 1 }$ nor $q _ { 2 }$

Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = (x^2 - 3)e^x + 2$, then
A. $f(0) = 0$
B. When $x < 0$, $f(x) = -(x^2 - 3)e^{-x} - 2$
C. $f(x) < 0$ if and only if $x > \sqrt{3}$
D. $x = -1$ is a local maximum point of $f(x)$

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The two endomorphisms $T$ and $M$ of $E$ are defined by $$\forall P \in \mathbb{R}_{2m}[X], \quad T(P) = P' \text{ and } M(P) = P^*$$ where $P^*(X) = P(-X)$. We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$.
What are the spaces $F^+$ and $F^-$ in this case?

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The two endomorphisms $T$ and $M$ of $E$ are defined by $T(P) = P'$ and $M(P) = P^*$ where $P^*(X) = P(-X)$. We set $$\mathbb{R}_k^0[X] = \{P \in \mathbb{R}_k[X] \mid P(-1) = 0 \text{ and } P(1) = 0\}$$ The subspace $G$ consists of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$, where $S(P,Q) = P(1)Q(1) - P(-1)Q(-1)$.
Determine the subspace $G$. Is hypothesis (H5) satisfied?

STATEMENT-1: The curve $y = \frac{-x^2}{2} + x + 1$ is symmetric with respect to the line $x = 1$. because STATEMENT-2: A parabola is symmetric about its axis.
(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

The function $f ( x ) = | \sin 4 x | + | \cos 2 x |$, is a periodic function with a fundamental period
(1) $\pi$
(2) $2 \pi$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$