Let $a, b, r_1$ and $r_2$ be four real numbers such that $0 < r_1 < r_2$. Determine a function $f$ of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$ such that $$\begin{cases} \Delta f = 0 \\ f(x,y) = a & \text{if } \|(x,y)\| = r_1 \\ f(x,y) = b & \text{if } \|(x,y)\| = r_2 \end{cases}$$

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables.
Show that, if $f$ is not identically zero, then $v$ is $2\pi$-periodic.

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables.
Show that, if $f$ is harmonic and not identically zero on $\mathbb{R}^2 \setminus \{(0,0)\}$, then there exists a real number $\lambda$ such that $u$ is a solution of the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ and $v$ is a solution of the differential equation (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$

123- If $G(x) = x^2 \int_{2}^{\sqrt{x}} \dfrac{\ln(t+2)}{t^2}\, dt$, and $G'(4)$ equals how many times $\ln 2$?

• [(1)] $1$
• [(2)] $2$
• [(3)] $1.5$
• [(4)] $3$

121. The volume of a sphere is increasing at a constant rate of $3$ cubic centimeters per second. At the moment when the radius of the sphere is $8$ centimeters, the surface area of the sphere increases how many square centimeters per second?
(1) $1/2$ (2) $1/25$ (3) $1/5$ (4) $1/6$

112- Point $M(x,2)$ lies on the curve $y=2$. It is a variable point. The line segment connecting point $M$ to the origin, makes an angle $\alpha$ with the positive $x$-axis. The rate of change of $\alpha(x)$ with respect to $x$, at the moment $x=4$, is which of the following?
(1) $-0/2$ (2) $-0/1$ (3) $0/05$ (4) $0/15$

124- If $F(x) = x\displaystyle\int_{2}^{x^2} \dfrac{dx}{\sqrt[3]{x^2-1}}$, and $F'(\sqrt{2})$ is given, what is its value?
(1) $3$ (2) $4$ (3) $4/5$ (4) $6$
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A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change in the length of his shadow is
(a) $3.6 \mathrm { ft } . / \mathrm { sec }$.
(b) $2.4 \mathrm { ft } . / \mathrm { sec }$.
(c) $3 \mathrm { ft } . / \mathrm { sec }$.
(d) $12 \mathrm { ft } . / \mathrm { sec }$.

A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change in the length of his shadow is
(a) $3.6 \mathrm { ft } . / \mathrm { sec }$.
(b) $2.4 \mathrm { ft } . / \mathrm { sec }$.
(c) $3 \mathrm { ft } . / \mathrm { sec }$.
(d) $12 \mathrm { ft } . / \mathrm { sec }$.

A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change (in ft./sec.) in the length of his shadow is
(A) 2.4
(B) 3
(C) 3.6
(D) 12

A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change (in ft./sec.) in the length of his shadow is
(A) 2.4
(B) 3
(C) 3.6
(D) 12

A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base. The depth of the pond is 6 m. The square at the bottom has side length 2 m and the top square has side length 8 m. Water is filled in at a rate of $\frac { 19 } { 3 }$ cubic meters per hour. At what rate is the water level rising exactly 1 hour after the water started to fill the pond?

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous odd function, which vanishes exactly at one point and $f ( 1 ) = \frac { 1 } { 2 }$. Suppose that $F ( x ) = \int _ { - 1 } ^ { x } f ( t ) d t$ for all $x \in [ - 1,2 ]$ and $G ( x ) = \int _ { - 1 } ^ { x } t | f ( f ( t ) ) | d t$ for all $x \in [ - 1,2 ]$. If $\lim _ { x \rightarrow 1 } \frac { F ( x ) } { G ( x ) } = \frac { 1 } { 14 }$, then the value of $f \left( \frac { 1 } { 2 } \right)$ is

Let $F ( x ) = \int _ { x } ^ { x ^ { 2 } + \frac { \pi } { 6 } } 2 \cos ^ { 2 } t \, d t$ for all $x \in \mathbb { R }$ and $f : \left[ 0 , \frac { 1 } { 2 } \right] \rightarrow [ 0 , \infty )$ be a continuous function. For $a \in \left[ 0 , \frac { 1 } { 2 } \right]$, if $F ^ { \prime } ( a ) + 2$ is the area of the region bounded by $x = 0 , y = 0 , y = f ( x )$ and $x = a$, then $f ( 0 )$ is

If $g ( x ) = \int _ { \sin x } ^ { \sin ( 2 x ) } \sin ^ { - 1 } ( t ) d t$, then
[A] $g ^ { \prime } \left( \frac { \pi } { 2 } \right) = - 2 \pi$
[B] $g ^ { \prime } \left( - \frac { \pi } { 2 } \right) = 2 \pi$
[C] $g ^ { \prime } \left( \frac { \pi } { 2 } \right) = 2 \pi$
[D] $g ^ { \prime } \left( - \frac { \pi } { 2 } \right) = - 2 \pi$

If the volume of a spherical ball is increasing at the rate of $4 \pi \mathrm { cc } / \mathrm { sec }$ then the rate of increase of its radius (in $\mathrm { cm } / \mathrm { sec }$), when the volume is $288 \pi \mathrm { cc }$ is
(1) $\frac { 1 } { 9 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 24 }$
(4) $\frac { 1 } { 36 }$

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$. Water is poured into it at a constant rate of $5$ cubic $\mathrm { m } / \mathrm { min }$. Then the rate (in $\mathrm { m } / \mathrm { min }$), at which the level of water is rising at the instant when the depth of water in the tank is $10 m$; is:
(1) $\frac { 1 } { 10 \pi }$
(2) $\frac { 1 } { 15 \pi }$
(3) $\frac { 1 } { 5 \pi }$
(4) $\frac { 2 } { \pi }$

Let $f : (0,2) \rightarrow R$ be a twice differentiable function such that $f''(x) > 0$, for all $x \in (0,2)$. If $\phi(x) = f(x) + f(2-x)$, then $\phi$ is
(1) decreasing on $(0,2)$
(2) increasing on $(0,2)$
(3) increasing on $(0,1)$ and decreasing on $(1,2)$
(4) decreasing on $(0,1)$ and increasing on $(1,2)$

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at a rate of $50 \mathrm {~cm} ^ { 3 } / \mathrm { min }$. When the thickness of ice is 5 cm , then the rate (in $\mathrm { cm } / \mathrm { min }$.) at which of the thickness of ice decreases, is:
(1) $\frac { 5 } { 6 \pi }$
(2) $\frac { 1 } { 54 \pi }$
(3) $\frac { 1 } { 36 \pi }$
(4) $\frac { 1 } { 18 \pi }$

Let $f$ be a non-negative function in $[ 0,1 ]$ and twice differentiable in $( 0,1 )$. If $\int _ { 0 } ^ { x } \sqrt { 1 - \left( f ^ { \prime } ( t ) \right) ^ { 2 } } \mathrm { dt } = \int _ { 0 } ^ { x } f ( \mathrm { t } ) \mathrm { dt } , 0 \leq x \leq 1$ and $f ( 0 ) = 0$, then $\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 2 } } \int _ { 0 } ^ { x } f ( \mathrm { t } ) \mathrm { dt } :$
(1) does not exist
(2) equals 0
(3) equals 1
(4) equals $\frac { 1 } { 2 }$

Water is being filled at the rate of $1 \mathrm{~cm}^3 \mathrm{sec}^{-1}$ in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in $\mathrm{cm}^2 \mathrm{sec}^{-1}$) at which the wet conical surface area of the vessel increases is
(1) 5
(2) $\frac{\sqrt{21}}{5}$
(3) $\frac{\sqrt{26}}{5}$
(4) $\frac{\sqrt{26}}{10}$

$\lim _ { x \rightarrow \frac { \pi } { 2 } } \left( \frac { \int _ { x ^ { 3 } } ^ { ( \pi / 2 ) ^ { 3 } } \left( \sin \left( 2 t ^ { 1 / 3 } \right) + \cos \left( t ^ { 1 / 3 } \right) \right) d t } { \left( x - \frac { \pi } { 2 } \right) ^ { 2 } } \right)$ is equal to
(1) $\frac { 5 \pi ^ { 2 } } { 9 }$
(2) $\frac { 9 \pi ^ { 2 } } { 8 }$
(3) $\frac { 11 \pi ^ { 2 } } { 10 }$
(4) $\frac { 3 \pi ^ { 2 } } { 2 }$

If $\lim _ { \mathrm { t } \rightarrow 0 } \left( \int _ { 0 } ^ { 1 } ( 3 x + 5 ) ^ { \mathrm { t } } \mathrm { d } x \right) ^ { \frac { 1 } { t } } = \frac { \alpha } { 5 \mathrm { e } } \left( \frac { 8 } { 5 } \right) ^ { \frac { 2 } { 3 } }$, then $\alpha$ is equal to $\_\_\_\_$