Suppose $F : \mathbb { R } \rightarrow \mathbb { R }$ is a continuous function which has exactly one local maximum. Then which of the following is true?
(A) $F$ cannot have a local minimum.
(B) $F$ must have exactly one local minimum.
(C) $F$ must have at least two local minima.
(D) $F$ must have either a global maximum or a local minimum.

The limit $$\lim _ { n \rightarrow \infty } \left( 2 ^ { - 2 ^ { n + 1 } } + 2 ^ { - 2 ^ { n - 1 } } \right) ^ { 2 ^ { - n } }$$ equals
(A) 1.
(B) $\frac { 1 } { \sqrt { 2 } }$.
(C) 0.
(D) $\frac { 1 } { 4 }$.

In the following figure, $O A B$ is a quarter-circle. The unshaded region is a circle to which $O A$ and $C D$ are tangents. If $C D$ is of length 10 and is parallel to $O A$, then the area of the shaded region in the above figure equals
(A) $25 \pi$.
(B) $50 \pi$.
(C) $75 \pi$.
(D) $100 \pi$.

Suppose $f : \mathbb { Z } \rightarrow \mathbb { Z }$ is a non-decreasing function. Consider the following two cases: $$\begin{aligned} & \text { Case 1. } f ( 0 ) = 2 , f ( 10 ) = 8 \\ & \text { Case 2. } f ( 0 ) = - 2 , f ( 10 ) = 12 \end{aligned}$$ In which of the above cases it is necessarily true that there exists an $n$ with $f ( n ) = n$?
(A) In both cases.
(B) In neither case.
(C) In Case 1. but not necessarily in Case 2.
(D) In Case 2. but not necessarily in Case 1.

Let $T$ be a right-angled triangle in the plane whose side lengths are in a geometric progression. Let $n(T)$ denote the number of sides of $T$ that have integer lengths. Then the maximum value of $n(T)$ over all such $T$ is
(A) 0
(B) 1
(C) 2
(D) 3

For every increasing function $b : [1, \infty) \rightarrow [1, \infty)$ such that $$\int_1^\infty \frac{\mathrm{d}x}{b(x)} < \infty$$ we must have
(A) $\sum_{k=1}^{\infty} \frac{\sqrt{\log k}}{b(k)} < \infty$
(B) $\sum_{k=3}^{\infty} \frac{\log k}{b(\log k)} < \infty$
(C) $\sum_{k=1}^{\infty} \frac{e^k}{b\left(e^k\right)} < \infty$
(D) $\sum_{k=3}^{\infty} \frac{1}{\sqrt{b(\log k)}} < \infty$

Consider the following two statements: (I) There exists a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g\left(x^3 + x^5\right) = e^x - 100$. (II) There exists a continuous function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g\left(e^x\right) = x^3 + x^5$. Then
(A) Only (I) is correct.
(B) Only (II) is correct.
(C) Both (I) and (II) are correct.
(D) Neither (I) nor (II) is correct.

Let $\psi : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with $\int_{-1}^{1} \psi(x)\,\mathrm{d}x = 1$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. Then $$\lim_{\varepsilon \rightarrow 0} \frac{1}{\varepsilon} \int_{1-\varepsilon}^{1+\varepsilon} f(y)\,\psi\!\left(\frac{1-y}{\varepsilon}\right) \mathrm{d}y$$ equals
(A) $f(1)$
(B) $f(1)\psi(0)$
(C) $f'(1)\psi(0)$
(D) $f(1)\psi(1)$

4) Identify this regular polygon, justifying your answer.

PROBLEM 2

Let us consider the function $f : \mathbb { R } \rightarrow \mathbb { R }$, periodic with period $T = 4$ whose graph, in the interval $[0; 4]$, is as follows: [Figure]
As can be seen from Figure 1, the sections $OB, BD, DE$ of the graph are line segments whose endpoints have coordinates: $O(0,0), B(1,1), D(3,-1), E(4,0)$.
1) Establish at which points of its domain the function $f$ is continuous and at which it is differentiable, and verify the existence of the limits: $\lim_{x \rightarrow +\infty} f(x)$ and $\lim_{x \rightarrow +\infty} \frac{f(x)}{x}$; if they exist, determine their value.
Also represent, for $x \in [0; 4]$, the graphs of the functions:
$$\begin{gathered} g ( x ) = f ^ { \prime } ( x ) \\ h ( x ) = \int _ { 0 } ^ { x } f ( t ) d t \end{gathered}$$
2) Consider the function:
$$s ( x ) = \sin ( b x )$$
with $b$ a positive real constant; determine $b$ so that $s(x)$ has the same period as $f(x)$.

Ministry of Education, University and Research

Prove that the square portion of the plane $OABC$ in Figure 1 is subdivided by the graphs of $f(x)$ and $s(x)$ into 3 distinct parts and determine the probabilities that a point chosen at random inside the square $OABC$ falls in each of the 3 identified parts.
3) Now considering the functions:
$$f ( x ) ^ { 2 } \quad \text { and } \quad s ( x ) ^ { 2 }$$
discuss, also with qualitative arguments, the variations (increase or decrease) of the 3 probability values determined in the previous point.
4) Finally, determine the volume of the solid generated by the rotation around the $y$-axis of the portion of the plane between the graph of the function $h$ for $x \in [0; 3]$ and the $x$-axis.

QUESTIONNAIRE

1. Defined the number $E$ as:
$$E = \int _ { 0 } ^ { 1 } x e ^ { x } d x$$
prove that:
$$\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { x } d x = e - 2 E$$
and express
$$\int _ { 0 } ^ { 1 } x ^ { 3 } e ^ { x } d x$$
in terms of $e$ and $E$.
2. A cake in the shape of a cylinder is placed under a plastic dome in the shape of a hemisphere. Prove that the cake occupies less than $3/5$ of the volume of the hemisphere.
3. Knowing that:
$$\lim _ { x \rightarrow 0 } \frac { \sqrt { a x + 2 b } - 6 } { x } = 1$$
determine the values of $a$ and $b$.
4. To draw real numbers in the interval $[0,2]$ a random number generator is created that provides numbers distributed, in that interval, with probability density given by the function:
$$f ( x ) = \frac { 3 } { 2 } x ^ { 2 } - \frac { 3 } { 4 } x ^ { 3 }$$
What will be the mean value of the generated numbers? What is the probability that the first number drawn is $4/3$? What is the probability that the second number drawn is less than 1?

Ministry of Education, University and Research

4. Provide a justified estimate of the number of tiles that, having a stain in the non-coloured part, will be damaged at the end of the production cycle.

Ministry of Education, University and Research

PROBLEM 2

Let us consider the function $f_k : \mathbb{R} \rightarrow \mathbb{R}$ defined as:
$$f_k(x) = -x^3 + kx + 9$$
with $k \in \mathbb{Z}$.
1. Let $\Gamma_k$ be the graph of the function, verify that for any value of the parameter $k$ the line $r_k$, tangent to $\Gamma_k$ at the point with abscissa 0, and the line $s_k$, tangent to $\Gamma_k$ at the point with abscissa 1, meet at a point $M$ with abscissa $\frac{2}{3}$.
2. After verifying that $k = 1$ is the maximum positive integer for which the ordinate of point $M$ is less than 10, study the behaviour of the function $f_1(x)$, determining its stationary and inflection points and sketching its graph.
3. Let $T$ be the triangle bounded by the lines $r_1$, $s_1$ and the $x$-axis, determine the probability that, taking at random a point $P(x_p, y_p)$ inside $T$, it lies above $\Gamma_1$ (that is, that $y_p > f_1(x)$ for such point $P$).
4. In the figure a point $N \in \Gamma_1$ and a portion of the graph $\Gamma_1$ are highlighted. The normal line to $\Gamma_1$ at $N$ (that is, the perpendicular to the tangent line to $\Gamma_1$ at that point) passes through the origin of the axes $O$. The graph $\Gamma_1$ has three points with this property. Prove, more generally, that the graph of any polynomial of degree $n > 0$ cannot have more than $2n - 1$ points at which the normal line to the graph passes through the origin. [Figure]

Ministry of Education, University and Research

QUESTIONNAIRE

1. Prove that the volume of a cylinder inscribed in a cone is less than half the volume of the cone.
2. There are two identical unbalanced dice in the shape of a regular tetrahedron with faces numbered from 1 to 4. When rolling each of the two dice, the probability of getting 1 is twice the probability of getting 2, which in turn is twice the probability of getting 3, which in turn is twice the probability of getting 4. If the two dice are rolled simultaneously, what is the probability that two equal numbers come out?
3. Determine the values of $k$ such that the line with equation $y = -4x + k$ is tangent to the curve with equation $y = x^3 - 4x^2 + 5$.
4. Considering the function $f(x) = \frac{3x - e^{\sin x}}{5 + e^{-x} - \cos x}$, determine, if they exist, the values of $\lim_{x \rightarrow +\infty} f(x)$, $\lim_{x \rightarrow -\infty} f(x)$, justifying the answers provided adequately.

Realize the given Bass on two staves, with the chord parts arranged in close or open position, developing, where possible, a melody coherent with the harmonic content you have developed on the basis of the assigned outline, inserting passing notes and turning notes.
Fedele Fenaroli, Partimenti, book I, 1780

7. Since it seemed to me that by pure chance some facts had occurred just as they had been predicted by the soothsayers, you spoke at length about chance, and you said, for example, that the ``throw of Venus'' could be obtained by randomly throwing four dice [...]
Cicero, De divinatione, II, 21, 48 – translation and edited by S. Timpanaro, Garzanti, Milan 1999. Original text – Nam cum mihi quaedam casu viderentur sic evenire ut praedicta essent a divinantibus, dixisti multa de casu, ut Venerium iaci posse casu quattuor talis iactis [...].
Cicero, in the dialogue with his brother Quintus, speaks of the throw of Venus, which consists of throwing 4 four-sided dice obtaining 4 different results. Assuming that the faces of each die are equally probable, determine: – the probability of obtaining the throw of Venus in the roll of 4 dice; – the probability of obtaining 4 numbers all equal.

6. T is a parallelopiped in which A, B, C and D are vertices of one face. And the face just above it has corresponding vertices $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime } , \mathrm { D } ^ { \prime }$. T is now compressed to S with face ABCD remaining same and $\mathrm { A } ^ { \prime }$, $B ^ { \prime } , C ^ { \prime } , D ^ { \prime }$ shifted to $A ^ { \prime \prime } , B ^ { \prime \prime } , C ^ { \prime \prime } , D ^ { \prime \prime }$ in $S$. The volume of parallelopiped $S$ is reduced to $90 \%$ of T. Prove that locus of $\mathrm { A } ^ { \prime \prime }$ is a plane.
Sol. Let the equation of the plane ABCD be $\mathrm { ax } + \mathrm { by } + \mathrm { cz } + \mathrm { d } = 0$, the point $\mathrm { A } ^ { \prime \prime }$ be $( \alpha , \beta , \gamma )$ and the height of the parallelopiped ABCD be h . $\Rightarrow \frac { | \mathrm { a } \alpha + \mathrm { b } \beta + \mathrm { c } \gamma + \mathrm { d } | } { \sqrt { \mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } + \mathrm { c } ^ { 2 } } } = 0.9 \mathrm {~h} . \Rightarrow \mathrm { a } \alpha + \mathrm { b } \beta + \mathrm { c } \gamma + \mathrm { d } = \pm 0.9 \mathrm {~h} \sqrt { \mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } + \mathrm { c } ^ { 2 } }$ ⇒ the locus of $\mathrm { A } ^ { \prime \prime }$ is a plane parallel to the plane ABCD .

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)

10. STATEMENT-1
If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant. because STATEMENT-2 The linear momentum of an isolated system remains constant.
(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
Answer [Figure]
(A) [Figure]
(B)
◯
(C) [Figure]
(D)

11. STATEMENT-1
A cloth covers a table. Some dishes are kept on it. The cloth can be pulled out without dislodging the dishes from the table. because STATEMENT-2 For every action there is an equal and opposite reaction.
(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
Answer
[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
[Figure]
(D)

1. STATEMENT-1

A vertical iron rod has a coil of wire wound over it at the bottom end. An alternating current flows in the coil. The rod goes through a conducting ring as shown in the figure. The ring can float at a certain height above the coil. because STATEMENT-2 [Figure]
In the above situation, a current is induced in the ring which interacts with the horizontal component of the magnetic field to produce an average force in the upward direction.
(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 Answer
(A) [Figure]
(B) ◯
(C) [Figure]
(D)

13. STATEMENT-1
The total translational kinetic energy of all the molecules of a given mass of an ideal gas is 1.5 times the product of its pressure and its volume. because STATEMENT-2 The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.
(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 Answer
[Figure]
(A)
[Figure]
(A)
[Figure] [Figure]
(B)
(C)
(D)

14. The speed of sound of the whistle is
(A) $340 \mathrm {~m} / \mathrm { s }$ for passengers in A and $310 \mathrm {~m} / \mathrm { s }$ for passengers in B
(B) $360 \mathrm {~m} / \mathrm { s }$ for passengers in A and $310 \mathrm {~m} / \mathrm { s }$ for passengers in B
(C) $310 \mathrm {~m} / \mathrm { s }$ for passengers in A and $360 \mathrm {~m} / \mathrm { s }$ for passengers in B
(D) $340 \mathrm {~m} / \mathrm { s }$ for passengers in both the trains Answer
[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
[Figure]
(D)

1. The distribution of the sound intensity of the whistle as observed by the passengers in $\operatorname { train } \mathrm { A }$ is best represented by

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

16. The spread of frequency as observed by the passengers in train B is
(A) 310 Hz
(B) 330 Hz
(C) 350 Hz
(D) 290 Hz Answer [Figure] ◯ [Figure]
(A)
(B)
(C)
(D)

17. Light travels as a
(A) parallel beam in each medium
(B) convergent beam in each medium
(C) divergent beam in each medium
(D) divergent beam in one medium and convergent beam in the other medium Answer
(A) [Figure]
[Figure]
(B)
[Figure]
(C)
(D)

18. The phases of the light wave at $c , d , e$ and $f$ are $\phi _ { c } , \phi _ { d } , \phi _ { e }$ and $\phi _ { f }$ respectively. It is given that $\phi _ { c } \neq \phi _ { f }$.
(A) $\phi _ { c }$ cannot be equal to $\phi _ { d }$
(B) $\phi _ { d }$ can be equal to $\phi _ { e }$
(C) $\left( \phi _ { d } - \phi _ { f } \right)$ is equal to $\left( \phi _ { c } - \phi _ { e } \right)$
(D) $\left( \phi _ { d } - \phi _ { c } \right)$ is not equal to $\left( \phi _ { f } - \phi _ { e } \right)$ Answer [Figure] [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D)