P and Q play chess frequently against each other. Of these matches, P has won $80 \%$ of the matches, drawn $15 \%$ of the matches and lost $5 \%$ of the matches. If they play 3 more matches, what is the probability of $P$ winning exactly 2 of these 3 matches?

For the differential equation given below, which one of the following options is correct?

$$ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0,0 \leq x \leq 1,0 \leq y \leq 1 $$

$$ \text { The values of a function } f \text { obtained for different values of } x \text { are shown in the table below. } $$

$$ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 0.25 & 0.5 & 0.75 & 1.0 \\ \hline f(x) & 0.9 & 2.0 & 1.5 & 1.8 & 0.4 \\ \hline \end{array} $$

$$ \text { Using Simpson's one-third rule, } $$

$$ \int_0^1 f(x) d x \approx $$__________[Rounded off to 2 decimal places]

In the closed interval $[0,3]$, the minimum value of the function $f$ given below is $f(x)=2 x^3-9 x^2+12 x$

If $C$ is the unit circle in the complex plane with its center at the origin, then the value of $n$ in the equation given below is _______ (rounded off to 1 decimal place).

$$ \oint_c \frac{z^3}{\left(z^2+4\right)\left(z^2-4\right)} d z=2 \pi i n $$

The directional derivative of the function $f$ given below at the point $(1,0)$ in the direction of $\frac{1}{2}(\hat{i}+\sqrt{3} \hat{j})$ is _______ (Rounded off to 1 decimal place).

$$ f(x, y)=x^2+x y^2 $$

Let $y$ be the solution of the differential equation with the initial conditions given below. If $y(x=2)=A \ln 2$, then the value of $A$ is _________ (rounded off to 2 decimal places).

$$ x^2 \frac{d^2 y}{d x^2}+3 x \frac{d y}{d x}+y=0 \quad y(x=1)=0 \quad 3 x \frac{d y}{d x}(x=1)=1 $$

A truss structure is loaded as shown in the figure below. Among the options given, which member in the truss is a zero-force member?

Consider two identical tanks with a bottom hole of diameter $d$. One tank is filled with water and the other tank is filled with engine oil. The height of the fluid column $h$ is same in both the cases. The fluid exit velocity in the two tanks are $V_1$ and $V_2$. Neglecting all losses, which one of the following options is correct?

A pitot tube connected to a U-tube mercury manometer measures the speed of air flowing in the wind tunnel as shown in the figure below. The density of air is $1.23 \mathrm{~kg} \mathrm{~m}^{-3}$ while the density of water is $1000 \mathrm{~kg} \mathrm{~m}^{-3}$. For the manometer reading of $h=30 \mathrm{~mm}$ of mercury, the speed of air in the wind tunnel is __________ $\mathrm{m} \mathrm{s}^{-1}$ (rounded off to 1 decimal place).

Assume: Specific gravity of mercury $=13.6$

Acceleration due to gravity $=10 \mathrm{~m} \mathrm{~s}^{-2}$

Consider a velocity field $\vec{V}=3 z \hat{i}+0 \hat{j}+C x \hat{k}$, where $C$ is a constant. if the flow is irrotational, the value of C is ________ (rounded off to 1 decimal place).

Let a spherical block of ice at $-7^{\circ} \mathrm{C}$ be exposed to atmospheric air at $30^{\circ} \mathrm{C}$ with the gravitational direction as shown in the figure below. What will be the overall direction of air flow in this situation?

During a welding operation, thermal power of 2500 W is incident normally on a metallic surface. As shown in the figure below (figure is NOT to scale), the heated area is circular. Out of the incident power, $85 \%$ of the power is absorbed within a circle of radius 5 mm while $65 \%$ is absorbed within an inner concentric circle of radius 3 mm . The power density in the shaded area is __________ $\mathrm{Wmm}^{-2}$ (rounded off to 2 decimal places).

Consider a cylindrical furnace of 5 m diameter and 5 m length with bottom, top and curved surfaces maintained at uniform temperatures of $800 \mathrm{~K}, 1500 \mathrm{~K}$ and 500 K , respectively. The view factor between the bottom and top surfaces, $F_{12}$ is 0.2 . The magnitude of net radiation heat transfer rate between the bottom surface and the curved surface is _________ kW (rounded off to 1 decimal place).

All surfaces of the furnace can be assumed as black.

The Stefan-Boltzmann constant, $\sigma=5.67 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}^{-4}$.

Water enters a tube of diameter, $D=60 \mathrm{~mm}$ with mass flow rate of $0.01 \mathrm{~kg} \mathrm{~s}^{-1}$ as shown in the figure below. The inlet mean temperature is $T_{m, i}=293 \mathrm{~K}$ and the uniform heat flux at the surface of the tube is $2000 \mathrm{Wm}^{-2}$. For the exit mean temperature of $T_{m, o}=$ 353 K , the length of the tube, $L$ is ___________ m (rounded off to 1 decimal place). Use the specific heat of water as $4181 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$.

Considering the actual demand and the forecast for a product given in the table below, the mean forecast error and the mean absolute deviation, respectively, are

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { Period } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Actual demand } & 425 & 415 & 420 & 430 & 427 & 418 & 422 & 416 & 426 & 421 \\ \hline \text { Forecast } & 427 & 422 & 416 & 422 & 423 & 420 & 419 & 418 & 430 & 415 \\ \hline \end{array} $$

A company uses 3000 units of a part annually. The units are priced as given in the table below. It costs Rs. 150 to place an order. Carrying costs are 40 percent of the purchase price per unit on an annual basis. The minimum total annual cost is Rs. ____________ (rounded off to 1 decimal place).

A project involves eight activities with the precedence relationship and duration as shown in the table below. The slack for the activity D is __________hours (answer in integer).

The endurance limit of a specific grade of steel is same as its yield strength. The ultimate strength of this grade of steel is twice of its yield strength. A component made of this steel is loaded in tension and unloaded periodically. It is required that the component does NOT fail for at least $10^6$ loading cycles, as per the Soderberg law. Considering a factor of safety of 2 , the maximum applied tensile principal stress is

A pair of spur gears is required to maintain a velocity ratio of $1: 2$. The module of the gears is 10 mm and the addendum is 10 mm . If the operating pressure angle is $15^{\circ}$, the minimum number of teeth required on the pinion to ensure NO interference/undercutting is _________ (answer in integer).

Two plates of thickness 10 mm each are to be joined by a transverse fillet weld on one side and the resulting structure is loaded as shown in the figure below.

If the ultimate tensile strength of the weld material is 150 MPa and the factor of safety to be used is 3 , the minimum length of the weld required to ensure that the weld does NOT fail is __________ mm (rounded off to 2 decimal places).

$$ \text { Match the mold elements in the casting process with the most suitable function } $$

Wire drawing operation is performed on a perfectly plastic metal without any strain hardening. Assuming no friction and no redundant work, the maximum possible percentage reduction in area in a single pass is closest to

$$ \text { In relation to additive manufacturing, match the following } $$

Two metal parts (a cylinder and a cube) of same volume are cast under identical conditions. The diameter of the cylinder is equal to its height. The ratio of the solidification time of the cube to that of the cylinder is ___________ (rounded off to 2 decimal places).

Assume that solidification time follows Chvorinov's rule with an exponent of 2 .

Cylindrical workpieces of diameter 60 mm and length 400 mm are machined on a lathe at a cutting speed of $25 \mathrm{~m} \mathrm{~min}^{-1}$ and a feed of $0.2 \mathrm{~mm} \mathrm{rev}^{-1}$. The Taylor's tool life parameters $C$ and $n$ for this setup are 75 and 0.25 , respectively. The tool changing time is 3 minutes. With a labor and overhead cost of Rs. 5 per minute, the tool changing cost per piece is Rs. ___________ (rounded off to 2 decimal places).

The yield stress of a metal in uniaxial tension is 200 MPa . According to von Mises yield criterion, the yield stress (in MPa) of the metal in pure shear is closest to

A metallic square plate is subjected to a uniform hydrostatic pressure $(P)$. Choose the correct Mohr's circle representing the state of stress at any point in the plate from the options given below.

For the Mohr's circle, normal stress is positive towards right and shear stress is positive in the upward direction.

The state of stress at a point is shown in the figure given below. Under plane stress assumption, the normal strain along the thickness direction $\left(\varepsilon_{z z}\right)$ is _________ (rounded off to 2 decimal places).

Use the Young's Modulus $E=200 \mathrm{GPa}$ and Poisson's ratio $v=0.27$.

A simply supported beam of length 1 m is subjected to a uniformly distributed bending moment of 1 N m per m throughout the length as shown in the figure given below. The bending moment at the mid-point of the beam is __________ N m (rounded off to the nearest integer).

An isotropic brittle material is tested in the universal testing machine. The stress-strain diagram for the material shows a bi-linear elastic behavior as shown in the figure given below. The strain energy density is _________ MJ m³- (rounded off to 2 decimal places).

A rigid circular disc of radius $r$ (in m ) is rolling without slipping on a flat surface as shown in the figure below. The angular velocity of the disc is $\omega$ (in rad s-1). The velocities (in $\mathrm{m} \mathrm{s}^{-1}$ ) at points O and A , respectively, are

The system shown in the figure below consists of a cantilever beam (with flexural rigidity El and negligible mass), a spring (with spring constant $K$ and negligible mass) and a block of mass $m$. Assuming a lumped parameter model for the system, the fundamental natural frequency $\left(\omega_n\right)$ of the system is

An offset slider-crank mechanism is shown in the figure below. The length of the stroke of the slider is ________ mm (rounded off to nearest integer).

Air inside a rigid, thermally-insulated tank undergoes stirring as shown in the figure below. Which one of the following options is correct?

A thermal power plant is running with no reheat or regeneration. The specific enthalpy and specific entropy of steam at the turbine inlet are $3344 \mathrm{~kJ} \mathrm{~kg}^{-1}$ and $6.5 \mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$, respectively. The turbine isentropic efficiency is 0.9 , and the mass flow rate of steam at the turbine inlet is $102 \mathrm{~kg} \mathrm{~s}^{-1}$. The turbine power output is ________ MW (rounded off to 1 decimal place). Properties of saturated liquid and saturated vapor at turbine exit pressure

A thermodynamically closed system contains 1 kg of hydrogen. The system undergoes a reversible polytropic process with polytropic index 1.3. The work output during the process is 400 kJ . During the process, hydrogen behaves as an ideal gas with constant specific heats. The absolute value of heat transfer during the process is _________ kJ (rounded off to 1 decimal place).

Specific heat of hydrogen at constant pressure $=14.56 \mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$

Specific heat of hydrogen at constant volume $=10.4 \mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$

A heat pump, operating in reversed Carnot cycle, maintains a steady air temperature of 300 K inside an auditorium. The heat pump receives heat from the ambient air. The ambient air temperature is 280 K . Heat loss from the auditorium is 15 kW . The power consumption of the heat pump is __________ kW (rounded off to 2 decimal places).

Consider a Pelton wheel of 1 m diameter. The magnitude of relative velocity of water at the bucket inlet is same as the magnitude of relative velocity of water at the bucket exit. The absolute speed of water at the bucket inlet is $125.66 \mathrm{~m} \mathrm{~s}-1$. For maximum power output from the Pelton wheel, the rpm of the Pelton wheel is_______ (rounded off to 1 decimal place).

Two cars, $P$ and $Q$, start from a point $X$ in India at 10 AM . Car $P$ travels North with a speed of $25 \mathrm{~km} / \mathrm{h}$ and car Q travels East with a speed of $30 \mathrm{~km} / \mathrm{h}$. Car P travels continuously but car Q stops for some time after travelling for one hour. If both the cars are at the same distance from $X$ at 11:30 AM, for how long (in minutes) did car Q stop?

The ceiling function of a real number $x$, denoted by $\operatorname{ce}(x)$, is defined as the smallest integer that is greater than or equal to $x$. Similarly, the floor function, denoted by $f l(x)$, is defined as the largest integer that is smaller than or equal to $x$. Which one of the following statements is NOT correct for all possible values of $x$ ?

Identify the option that has the most appropriate sequence such that a coherent paragraph is formed:

P. At once, without thinking much, people rushed towards the city in hordes with the sole aim of grabbing as much gold as they could.

Q. However, little did they realize about the impending hardships they would have to face on their way to the city: miles of mud, unfriendly forests, hungry beasts and inimical local lords - all of which would reduce their chances of getting gold to almost zero.

R. All of them thought that easily they could lay their hands on gold and become wealthy overnight.

S. About a hundred years ago, the news that gold had been discovered in Kolar spread like wildfire and the whole State was in raptures.

The given figure is reflected about the horizontal dashed line and then rotated clockwise by $90^{\circ}$ about an axis perpendicular to the plane of the figure.

Which one of the following options correctly shows the resultant figure?

Note: The figures shown are representative.

Which one of the following options has the correct sequence of objects arranged in the increasing number of mirror lines (lines of symmetry)?