11. Let $x$ be a real-valued variable. What is the range of the function $f ( x ) = \sin ^ { 2 } x - \sin x + 2$ ?
(a) $[ 0,2 ]$
(b) $[ 1,2 ]$
(c) $[ 1,4 ]$
(d) $\left[ \frac { 7 } { 4 } , 4 \right]$

For a positive number $a$, on the closed interval $[ - a , a ]$, the function
$$f ( x ) = \frac { x - 5 } { ( x - 5 ) ^ { 2 } + 36 }$$
has maximum value $M$ and minimum value $m$. Find the minimum value of $a$ such that $M + m = 0$. [4 points]

12. Given the function $f ( x ) = \left\{ \begin{array} { l } x ^ { 2 } , x \leq 1 \\ x + \frac { 6 } { x } - 6 , x > 1 \end{array} \right.$ , then $f [ f ( - 2 ) ] =$ $\_\_\_\_$ , and the minimum value of $f ( x )$ is $\_\_\_\_$.

10. Given the function $f ( x ) = \left\{ \begin{array} { l } x + \frac { 2 } { x } - 1 , x \geq 1 \\ \lg \left( x ^ { 2 } + 1 \right) , x < 1 \end{array} \right.$ , then $f ( f ( - 3 ) ) =$ $\_\_\_\_$ , and the minimum value of $f ( x )$ is $\_\_\_\_$ .

The range of the function $f(x) = \sqrt{x^2 - 2x - 3}$ is
A. $[-2, 2]$
B. $[-1, 1]$
C. $[0, 4]$
D. $[1, 3]$

Let $a \in \mathbb{R}$. If there exists a function $f ( x )$ with domain $\mathbb{R}$ that satisfies both ``for any $x _ { 0 } \in \mathbb{R}$, the value of $f \left( x _ { 0 } \right)$ is either $x _ { 0 } ^ { 2 }$ or $x _ { 0 }$'' and ``the equation $f ( x ) = a$ has no real solutions'', find the range of $a$ as $\_\_\_\_$

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0$$ and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Justify the existence of the real number $\Delta _ { n }$ for all $n \in \mathbb { N } ^ { * }$.

Let $f \in \mathcal{C}(\mathbb{R})$ such that $$\lim_{x \rightarrow -\infty} f(x) = +\infty \quad \text{and} \quad \lim_{x \rightarrow +\infty} f(x) = +\infty$$ a) Show that the set $\{x \in \mathbb{R} \mid f(x) \leq f(0)\}$ is closed and bounded. b) Deduce that there exists $x_* \in \mathbb{R}$ such that $f(x_*) = \min\{f(x) \mid x \in \mathbb{R}\}$.

101- For which set of values of $a$, the graph of $f(x) = (a-3)x^2 + ax - 1$ does not pass through the first quadrant?

• [(1)] $a \leq 2$
• [(2)] $0 < a \leq 2$
• [(3)] $2 < a < 3$
• [(4)] $0 < a < 3$

13. If $f ( x ) = ( x 2 - 1 ) / ( x 2 + 1 )$, for every real number $x$, then the minimum value of $f$ :
(A) does not exist because $f$ is unbounded.
(B) is not attained even though $f$ is bounded
(C) is equal to 1
(D) is equal to - 1

9. Let $\theta \in \left( 0 , \frac { \pi } { 4 } \right)$ and $\mathrm { t } _ { 1 } = ( \tan \theta ) ^ { \tan \theta } , \mathrm { t } _ { 2 } = ( \tan \theta ) ^ { \cot \theta } , \mathrm { t } _ { 3 } = ( \cot \theta ) ^ { \tan \theta }$ and $\mathrm { t } _ { 4 } = ( \cot \theta ) ^ { \cot \theta }$, then
(A) $\mathrm { t } _ { 1 } > \mathrm { t } _ { 2 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 4 }$
(B) $\mathrm { t } _ { 4 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 2 }$
(C) $\mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 2 } > \mathrm { t } _ { 4 }$
(D) $\mathrm { t } _ { 2 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 4 }$

Sol. (B)

Given $\theta \in \left( 0 , \frac { \pi } { 4 } \right)$, then $\tan \theta < 1$ and $\cot \theta > 1$. Let $\tan \theta = 1 - \lambda _ { 1 }$ and $\cot \theta = 1 + \lambda _ { 2 }$ where $\lambda _ { 1 }$ and $\lambda _ { 2 }$ are very small and positive. then $\mathrm { t } _ { 1 } = \left( 1 - \lambda _ { 1 } \right) ^ { 1 - \lambda _ { 1 } } , \mathrm { t } _ { 2 } = \left( 1 - \lambda _ { 1 } \right) ^ { 1 + \lambda _ { 2 } }$
$$t _ { 3 } = \left( 1 + \lambda _ { 2 } \right) ^ { 1 - \lambda _ { 1 } } \text { and } t _ { 4 } = \left( 1 + \lambda _ { 2 } \right) ^ { 1 + \lambda _ { 2 } }$$
Hence $\mathrm { t } _ { 4 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 2 }$.

Let $| M |$ denote the determinant of a square matrix $M$. Let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow \mathbb { R }$ be the function defined by
$$g ( \theta ) = \sqrt { f ( \theta ) - 1 } + \sqrt { f \left( \frac { \pi } { 2 } - \theta \right) - 1 }$$
where $$f ( \theta ) = \frac { 1 } { 2 } \left| \begin{array} { c c c } 1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1 \end{array} \right| + \left| \begin{array} { c c c } \sin \pi & \cos \left( \theta + \frac { \pi } { 4 } \right) & \tan \left( \theta - \frac { \pi } { 4 } \right) \\ \sin \left( \theta - \frac { \pi } { 4 } \right) & - \cos \frac { \pi } { 2 } & \log _ { e } \left( \frac { 4 } { \pi } \right) \\ \cot \left( \theta + \frac { \pi } { 4 } \right) & \log _ { e } \left( \frac { \pi } { 4 } \right) & \tan \pi \end{array} \right|$$
Let $p ( x )$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g ( \theta )$, and $p ( 2 ) = 2 - \sqrt { 2 }$. Then, which of the following is/are TRUE ?
(A) $p \left( \frac { 3 + \sqrt { 2 } } { 4 } \right) < 0$
(B) $p \left( \frac { 1 + 3 \sqrt { 2 } } { 4 } \right) > 0$
(C) $p \left( \frac { 5 \sqrt { 2 } - 1 } { 4 } \right) > 0$
(D) $\quad p \left( \frac { 5 - \sqrt { 2 } } { 4 } \right) < 0$

The range of the function $f ( x ) = \frac { x } { 1 + | x | } , x \in R$, is
(1) $R$
(2) $( - 1,1 )$
(3) $R - \{ 0 \}$
(4) $[ - 1,1 ]$

If $f(x) = \frac{\tan^{-1}x + \log_e 123}{x \log_e 1234 - \tan^{-1}x}$, $x > 0$, then the least value of $f(f(x)) + f\!\left(f\!\left(\frac{4}{x}\right)\right)$ is
(1) 0
(2) 8
(3) 2
(4) 4

Let the range of the function $f ( x ) = \frac { 1 } { 2 + \sin 3 x + \cos 3 x } , x \in \mathbb { R }$ be $[ a , b ]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac { \alpha } { \beta }$ is equal to
(1) $\pi$
(2) $\sqrt { \pi }$
(3) 2
(4) $\sqrt { 2 }$

If the domain of the function $\log _ { 5 } \left( 18 x - x ^ { 2 } - 77 \right)$ is $( \alpha , \beta )$ and the domain of the function $\log _ { ( x - 1 ) } \left( \frac { 2 x ^ { 2 } + 3 x - 2 } { x ^ { 2 } - 3 x - 4 } \right)$ is $( \gamma , \delta )$, then $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$ is equal to :
(1) 195
(2) 179
(3) 186
(4) 174

Q72. Let the range of the function $f ( x ) = \frac { 1 } { 2 + \sin 3 x + \cos 3 x } , x \in \mathbb { R }$ be $[ a , b ]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac { \alpha } { \beta }$ is equal to
(1) $\pi$
(2) $\sqrt { \pi }$
(3) 2
(4) $\sqrt { 2 }$

The minimum value of $3 \sin ^ { 2 } \theta + \cos ^ { 2 } \theta - 6 \sin \theta \cos \theta + 2$, where $\theta \in \left( 0 , \frac { \pi } { 2 } \right)$ (A) $\mathbf { 4 } + \sqrt { \mathbf { 1 0 } }$ (B) - 1 (C) 1 (D) $4 - \sqrt { 10 }$

2. For ALL APPLICANTS.

Let
$$f _ { n } ( x ) = \left( 2 + ( - 2 ) ^ { n } \right) x ^ { 2 } + ( n + 3 ) x + n ^ { 2 }$$
where $n$ is a positive integer and $x$ is any real number.
(i) Write down $f _ { 3 } ( x )$.
Find the maximum value of $f _ { 3 } ( x )$. For what values of $n$ does $f _ { n } ( x )$ have a maximum value (as $x$ varies)? [0pt] [Note you are not being asked to calculate the value of this maximum.]
(ii) Write down $f _ { 1 } ( x )$.
Calculate $f _ { 1 } \left( f _ { 1 } ( x ) \right)$ and $f _ { 1 } \left( f _ { 1 } \left( f _ { 1 } ( x ) \right) \right)$. Find an expression, simplified as much as possible, for
$$f _ { 1 } \left( f _ { 1 } \left( f _ { 1 } \left( \cdots f _ { 1 } ( x ) \right) \right) \right)$$
where $f _ { 1 }$ is applied $k$ times. [Here $k$ is a positive integer.]
(iii) Write down $f _ { 2 } ( x )$.
The function
$$f _ { 2 } \left( f _ { 2 } \left( f _ { 2 } \left( \cdots f _ { 2 } ( x ) \right) \right) \right) ,$$
where $f _ { 2 }$ is applied $k$ times, is a polynomial in $x$. What is the degree of this polynomial?