Suppose that $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ where $a , b , c , d$ are real numbers with $a \neq 0$. The equation $f ( x ) = 0$ has exactly two distinct real solutions. If $f ^ { \prime } ( x )$ is the derivative of $f ( x )$, then which of the following is a possible graph of $f ^ { \prime } ( x )$?
(A), (B), (C), (D) [graphs as provided in the figure]

The precise interval on which the function $f(x) = \log_{1/2}\left(x^2 - 2x - 3\right)$ is monotonically decreasing, is
(A) $(-\infty, -1)$
(B) $(-\infty, 1)$
(C) $(1, \infty)$
(D) $(3, \infty)$

The graph in the figure, representing the continuous function $y = f(x)$, is the union of the parabolic arc $\Gamma_{1}$, the circular arc $\Gamma_{2}$ and the hyperbolic arc $\Gamma_{3}$.
a) Write an analytical expression of the function $f$ defined piecewise on the interval $[-2; 2]$, using the equations:
$$y = a(x + 2)^{2} \quad x^{2} + y^{2} + b = 0 \quad x^{2} - y^{2} + c = 0$$
and identify the appropriate values for the real parameters $a, b, c$.
Study the differentiability of the function $f$ and write the equations of any tangent lines at the points with abscissa
$$x = -2 \quad x = 0 \quad x = 1 \quad x = 2$$
b) Starting from the graph of the function $f$, deduce that of its derivative $f^{\prime}$ and identify the intervals of concavity and convexity of $F(x) = \int_{-2}^{x} f(t) dt$.
c) Consider the function $y = \frac{1}{4}(x + 2)^{2}$, defined on the interval $[-2; 0]$, of which $\Gamma_{1}$ is the representative graph. Explain why it is invertible and write the analytical expression of its inverse function $h$. Study the differentiability of $h$ and sketch its graph.
d) Let $S$ be the bounded region in the second quadrant, between the graph $\Gamma_{1}$ and the coordinate axes. Determine the value of the real parameter $k$ so that the line with equation $x = k$ divides $S$ into two equivalent regions.

Given a real parameter $a$, with $a \neq 0$, consider the function $f_{a}$ defined as follows:
$$f_{a}(x) = \frac{x^{2} - ax}{x^{2} - a}$$
whose graph will be denoted by $\Omega_{a}$.
a) As the parameter $a$ varies, determine the domain of $f_{a}$, study any discontinuities and write the equations of all its asymptotes.
b) Show that, for $a \neq 1$, all graphs $\Omega_{a}$ intersect their horizontal asymptote at the same point and share the same tangent line at the origin.
c) As $a < 1$ varies, identify the intervals of monotonicity of the function $f_{a}$. Study the function $f_{-1}(x)$ and sketch its graph $\Omega_{-1}$.
d) Determine the area of the bounded region between the graph $\Omega_{-1}$, the line tangent to it at the origin and the line $x = \sqrt{3}$.

5. Determine the values of the real parameters $a$ and $b$ of the function $f ( x ) = \frac { a x ^ { 2 } + b x + 3 } { 2 x ^ { 2 } + 5 x - 1 }$ so that it has the line $y = 2$ as a horizontal asymptote and a stationary point at $x = 1$. For the values found, determine whether $f ( x )$ has further asymptotes.

3. Boccioni's futurist work ``Unique Forms of Continuity in Space'' from 1913, featured on the 20-cent coin, depicts a man advancing rapidly through space. A part of the profile highlighted in the figure, in an appropriate coordinate system, can be approximated by the function
$$f ( x ) = \left\{ \begin{array} { l r } - 4 x ^ { 2 } - 8 x , & - 1 \leq x \leq 0 \\ 1 + \tan \left( x + \frac { 3 } { 4 } \pi \right) , & 0 < x \leq 2 \end{array} \right.$$
Sketch the graph of $f$, after analyzing its continuity and differentiability on the interval $[ - 1 ; 2 ]$. [Figure]

12. The number of values of $x$ where the function $f ( x ) = \cos x + \cos ( \sqrt { } 2 x )$ attains its maximum is :
(A) 0
(B) 1
(C) 2
(D) infinite

27. Let $h ( x ) = \min \{ x ; x 2 \}$, for every real number of $x$. Then :
(A) $h$ is continuous for all $x$
(B) $h$ is differentiable for all $x$
(C) $\mathrm { h } ^ { \prime } ( \mathrm { x } ) = 1$, for all $\mathrm { x } > 1$
(D) $h$ is not differentiable at two values of $x$

20. $\lim x \rightarrow 0 ( x \tan 2 x - 2 x \tan x ) / ( 1 - \cos 2 x ) 2$ is:
(A) $y = 2$
(B) $y = 2 x$
(C) $y = 2 x - 4$
(D) $y = 2 \times 2 - 4$

14. If $f ( x ) = \min \left\{ 1 , x ^ { 2 } , x ^ { 3 } \right\}$, then
(A) $\mathrm { f } ( \mathrm { x } )$ is continuous $\forall \mathrm { x } \in \mathrm { R }$
(B) $\mathrm { f } ^ { \prime } ( \mathrm { x } ) > 0 , \forall \mathrm { x } > 1$
(C) $\mathrm { f } ( \mathrm { x } )$ is not differentiable but continuous $\forall \mathrm { x } \in \mathrm { R }$
(D) $f ( x )$ is not differentiable for two values of $x$
Sol. (A), (C) $\mathrm { f } ( \mathrm { x } ) = \min . \left\{ 1 , \mathrm { x } ^ { 2 } , \mathrm { x } ^ { 3 } \right\}$ $\Rightarrow \mathrm { f } ( \mathrm { x } ) = \begin{cases} \mathrm { x } ^ { 3 } , & \mathrm { x } \leq 1 \\ 1 , & \mathrm { x } > 1 \end{cases}$ $\Rightarrow \mathrm { f } ( \mathrm { x } )$ is continuous $\forall \mathrm { x } \in \mathrm { R }$ and non-differentiable at $\mathrm { x } = 1$. [Figure]

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

Let $a$ and $b$ be non-zero real numbers. Then, the equation $$\left( a x ^ { 2 } + b y ^ { 2 } + c \right) \left( x ^ { 2 } - 5 x y + 6 y ^ { 2 } \right) = 0$$ represents
(A) four straight lines, when $c = 0$ and $a , b$ are of the same sign
(B) two straight lines and a circle, when $a = b$, and $c$ is of sign opposite to that of $a$
(C) two straight lines and a hyperbola, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$
(D) a circle and an ellipse, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$

The locus of the orthocentre of the triangle formed by the lines $$\begin{aligned} &(1+p)x-py+p(1+p)=0\\ &(1+q)x-qy+q(1+q)=0 \end{aligned}$$ and $y=0$, where $p\neq q$, is
(A) a hyperbola
(B) a parabola
(C) an ellipse
(D) a straight line

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The real number $s$ lies in the interval
A) $\left( - \frac { 1 } { 4 } , 0 \right)$
B) $\left( - 11 , - \frac { 3 } { 4 } \right)$
C) $\left( - \frac { 3 } { 4 } , - \frac { 1 } { 2 } \right)$
D) $\left( 0 , \frac { 1 } { 4 } \right)$

The number of points in $( - \infty , \infty )$, for which $x ^ { 2 } - x \sin x - \cos x = 0$, is
(A) 6
(B) 4
(C) 2
(D) 0

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be respectively given by $f(x) = |x| + 1$ and $g(x) = x^2 + 1$. Define $h : \mathbb{R} \rightarrow \mathbb{R}$ by $$h(x) = \begin{cases} \max\{f(x), g(x)\} & \text{if } x \leq 0 \\ \min\{f(x), g(x)\} & \text{if } x > 0 \end{cases}$$ The number of points at which $h(x)$ is not differentiable is

Let $f : [0, 4\pi] \rightarrow [0, \pi]$ be defined by $f(x) = \cos^{-1}(\cos x)$. The number of points $x \in [0, 4\pi]$ satisfying the equation $$f(x) = \frac{10 - x}{10}$$ is

Let $f : \left[ - \frac { 1 } { 2 } , 2 \right] \rightarrow \mathbb { R }$ and $g : \left[ - \frac { 1 } { 2 } , 2 \right] \rightarrow \mathbb { R }$ be functions defined by $f ( x ) = \left[ x ^ { 2 } - 3 \right]$ and $g ( x ) = | x | f ( x ) + | 4 x - 7 | f ( x )$, where $[ y ]$ denotes the greatest integer less than or equal to $y$ for $y \in \mathbb { R }$. Then
(A) $f$ is discontinuous exactly at three points in $\left[ - \frac { 1 } { 2 } , 2 \right]$
(B) $f$ is discontinuous exactly at four points in $\left[ - \frac { 1 } { 2 } , 2 \right]$
(C) $g$ is NOT differentiable exactly at four points in $\left( - \frac { 1 } { 2 } , 2 \right)$
(D) $g$ is NOT differentiable exactly at five points in $\left( - \frac { 1 } { 2 } , 2 \right)$

Let $f ( x ) = \frac { 1 - x ( 1 + | 1 - x | ) } { | 1 - x | } \cos \left( \frac { 1 } { 1 - x } \right)$ for $x \neq 1$. Then
[A] $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 0$
[B] $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ does not exist
[C] $\lim _ { x \rightarrow 1 ^ { + } } f ( x ) = 0$
[D] $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ does not exist

Let $f _ { 1 } : \mathbb { R } \rightarrow \mathbb { R } , f _ { 2 } : \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) \rightarrow \mathbb { R } , f _ { 3 } : \left( - 1 , e ^ { \frac { \pi } { 2 } } - 2 \right) \rightarrow \mathbb { R }$ and $f _ { 4 } : \mathbb { R } \rightarrow \mathbb { R }$ be functions defined by
(i) $\quad f _ { 1 } ( x ) = \sin \left( \sqrt { 1 - e ^ { - x ^ { 2 } } } \right)$,
(ii) $\quad f _ { 2 } ( x ) = \left\{ \begin{array} { c c } \frac { | \sin x | } { \tan ^ { - 1 } x } & \text { if } x \neq 0 \\ 1 & \text { if } x = 0 \end{array} \right.$, where the inverse trigonometric function $\tan ^ { - 1 } x$ assumes values in $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$,
(iii) $\quad f _ { 3 } ( x ) = \left[ \sin \left( \log _ { e } ( x + 2 ) \right) \right]$, where, for $t \in \mathbb { R } , [ t ]$ denotes the greatest integer less than or equal to $t$,
(iv) $\quad f _ { 4 } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } \sin \left( \frac { 1 } { x } \right) & \text { if } x \neq 0 \\ 0 & \text { if } x = 0 \end{array} \right.$.
LIST-I P. The function $f _ { 1 }$ is Q. The function $f _ { 2 }$ is R. The function $f _ { 3 }$ is S. The function $f _ { 4 }$ is
LIST-II

1. NOT continuous at $x = 0$
2. continuous at $x = 0$ and NOT differentiable at $x = 0$
3. differentiable at $x = 0$ and its derivative is NOT continuous at $x = 0$
4. differentiable at $x = 0$ and its derivative is continuous at $x = 0$

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 2 ; } \mathbf { Q } \rightarrow \mathbf { 3 ; } \mathbf { R } \rightarrow \mathbf { 1 ; } \mathbf { S } \rightarrow \mathbf { 4 }$
(B) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 1 } ; \mathbf { R } \rightarrow \mathbf { 2 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(D) $\mathbf { P } \rightarrow \mathbf { 2 } ; \mathbf { Q } \rightarrow \mathbf { 1 } ; \mathbf { R } \rightarrow \mathbf { 4 } ; \mathbf { S } \rightarrow \mathbf { 3 }$

For any real number $t$, let $\lfloor t \rfloor$ be the largest integer less than or equal to $t$. Then the number of points of discontinuity of the function $x \mapsto \lfloor x^2 - 3 \rfloor$ for $x \in (-\infty, 0)$ is ____.
Let $f: [-1, 3] \to \mathbb{R}$ be defined as $$f(x) = \begin{cases} |x| + [x], & -1 \leq x < 1 \\ x + |x|, & 1 \leq x < 2 \\ x + [x], & 2 \leq x \leq 3 \end{cases}$$ where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points of discontinuity of $f$ in the interval $(-1, 3)$ is ____.

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be the function defined as $f ( x ) = [ 4 x ] \left( x - \frac { 1 } { 4 } \right) ^ { 2 } \left( x - \frac { 1 } { 2 } \right)$, where $[ x ]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?
(A) The function $f$ is discontinuous exactly at one point in $( 0,1 )$
(B) There is exactly one point in $( 0,1 )$ at which the function $f$ is continuous but NOT differentiable
(C) The function $f$ is NOT differentiable at more than three points in $( 0,1 )$
(D) The minimum value of the function $f$ is $- \frac { 1 } { 512 }$

For each $x \in R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $\lim_{x \rightarrow 0^-} \frac{x([x] + |x|)\sin[x]}{|x|}$ is equal to
(1) 1
(2) 0
(3) $-\sin 1$
(4) $\sin 1$

Let $f ( x ) = \left\{ \begin{array} { c c } \max \left( | x | , x ^ { 2 } \right) , & | x | \leq 2 \\ 8 - 2 | x | , & 2 < | x | \leq 4 \end{array} \right.$. Let $S$ be the set of points in the interval $( - 4,4 )$ at which $f$ is not differentiable. Then $S$
(1) equals $\{ - 2 , - 1,0,1,2 \}$
(2) equals $\{ - 2,2 \}$
(3) is an empty set
(4) equal $\{ - 2 , - 1,1,2 \}$

Let $f : [ - 1,3 ] \rightarrow \mathrm { R }$ be defined as $$f ( x ) = \begin{cases} |x| + [x], & -1 \leq x < 1 \\ x + |x|, & 1 \leq x < 2 \\ x + [x], & 2 \leq x \leq 3 \end{cases}$$ where $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is discontinuous at:
(1) Only one point
(2) Only two points
(3) Four or more points
(4) Only three points