# Recent questions tagged gateme-2020-set2

The sum of two normally distributed random variables $X$ and $Y$ is always normally distributed normally distributed, only if $X$ and $Y$ are independent normally distributed, only if $X$ and $Y$ have the same standard deviation normally distributed, only if $X$ and $Y$ have the same mean
A matrix $P$ is decomposed into its symmetric part $S$ and skew symmetric part $V$. If $S= \begin{pmatrix} -4 & 4 & 2 \\ 4 & 3 & 7/2 \\ 2 & 7/2 & 2 \end{pmatrix}, \: \: V= \begin{pmatrix} 0 & -2 & 3 \\ 2 & 0 & 7/2 \\ -3 & -7/2 & 0 \end{pmatrix}$ then matrix $P$ ... $\begin{pmatrix} -2 & 9/2 & -1 \\ -1 & 81/4 & 11 \\ -2 & 45/2 & 73/4 \end{pmatrix}$
Let $I=\displaystyle \int_{x=0}^1 \int_{y=0}^{x^2} xy^2 dy \: dx$. Then, $I$ may also be expressed as $\displaystyle \int_{y=0}^1 \int_{x=0}^{\sqrt{y}} xy^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=\sqrt{y}}^1 yx^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=\sqrt{y}}^1 xy^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=0}^{\sqrt{y}} yx^2 dx \: dy$
The solution of $\dfrac{d^2y}{dt^2}-y=1,$ which additionally satisfies $y \bigg \vert_{t=0} = \dfrac{dy}{dt} \bigg \vert_{t=0}=0$ in the Laplace $s$-domain is $\dfrac{1}{s(s+1)(s-1)} \\$ $\dfrac{1}{s(s+1)} \\$ $\dfrac{1}{s(s-1)} \\$ $\dfrac{1}{s-1} \\$
An attempt is made to pull a roller of weight $W$ over a curb (step) by applying a horizontal force $F$ as shown in the figure. The coefficient of static friction between the roller and the ground (including the edge of the step) is $\mu$ Identify the correct free body diagram (FBD) of the roller when the roller is just about to climb over the step.
A circular disk of radius $r$ is confined to roll without slipping at $P$ and $Q$ as shown in the figure. If the plates have velocities as shown, the magnitude of angular velocity of the disk is $\dfrac{V}{r} \\$ $\dfrac{V}{2r} \\$ $\dfrac{2V}{3r} \\$ $\dfrac{3V}{2r}$
The equation of motion of a spring-mass-damper system is given by $\dfrac{d^2x}{dt^2}+3 \dfrac{dx}{dt} +9x = 10 \sin(5t)$ The damping factor for the system is $0.25$ $0.5$ $2$ $3$
The number of qualitatively distinct kinematic inversions possible for a Grashof chain with four revolute pairs is $1$ $2$ $3$ $4$
The process, that uses a tapered horn to amplify and focus the mechanical energy for machining of glass, is electrochemical machining electrical discharge machining ultrasonic machining abrasive jet machining
Two plates, each of $6$ mm thickness, are to be butt-welded. Consider the following processes and select the correct sequence in increasing order of size of the heat affected zone. Arc welding MIG welding Laser beam welding Submerged arc welding $1-4-2-3$ $3-4-2-1$ $4-3-2-1$ $3-2-4-1$
Which one of the following statements about a phase diagram is INCORRECT? It indicates the temperature at which different phases start to melt Relative amount of different phases can be found under given equillibrium conditions It gives information on transformation rates Solid solubility limits are depicted by it
The figure below shows a symbolic representation of the surface texture in a perpendicular lay orientation with indicative values (I through VI) marking the various specifications whose definitions are listed below. P: Maximum waviness Height (mm); Q: Maximum Roughness Height (mm); R: Minimum Roughness Height (mm); S: Maximum ... -U, II-S, III-Q, IV-T, V-R, VI-P I-Q, II-U, III-R, IV-T, V-S, VI-P
In Materials Requirement Planning, if the inventory holding cost is very high and the set up cost is zero, which one of the following lot sizing approaches should be used? Economic Order Quantity Lot-for-Lot Base Stock Level Fixed Period Quantity, for $2$ periods
Which of the following conditions is used to determine the stable equilibrium of all partially submerged floating bodies? Centre of buoyancy must be above the centre of gravity Centre of buoyancy must be below the centre of gravity Metacentre must be at a higher level than the centre of gravity Metacentre must be at a lower level than the centre of gravity
In the space above the mercury column in a barometer tube, the gauge pressure of the vapour is positive, but more than one atmosphere negative zero positive, but less than one atmosphere
A closed vessel contains pure water, in thermal equilibrium with its vapour at $25^\circ C$(Stage$\#1$), as shown. The vessel in this stage is then kept aside an isothermal oven which is having an atmosphere of hot air maintained at $80^\circ C$ ... through Valve $A$ Nothing will happen - the vessel will continue to remain in equilibrium All the vapor inside the vessel will immediately condense
For an air-standard Diesel cycle, heat addition is at constant volume and heat rejection is at constant pressure heat addition is at constant pressure and heat rejection is at constant pressure heat addition is at constant pressure and heat rejection is at constant volume heat addition is at constant volume and heat rejection is at constant volume
The values of enthalpies at the stator inlet and rotor outlet of a hydraulic turbomachine stage are $h_1$ and $h_3$ respectively. The enthalpy at the stator outlet (or, rotor inlet) is $h_2$. The condition $(h_2-h_1)=(h_3-h_2)$ indicates that the degree of reaction of this stage is $\text{zero}$ $50 \%$ $75 \%$ $100 \%$
Let $\textbf{I}$ be a $100$ dimensional identity matrix and $\textbf{E}$ be the set of its distinct (no value appears more than once in $\textbf{E})$ real eigen values. The number of elements in $\textbf{E}$ is _________
A beam of negligible mass is hinged at support $P$ and has a roller support $Q$ as shown in the figure. A point load of $1200 \: N$ is applied at point $R$. The magnitude of the reaction force at support $Q$ is _________ $N$.
A machine member is subjected to fluctuating stress $\sigma = \sigma_0 \cos (8 \pi t)$. The endurance limit of the material is $350$ MPa. If the factor of safety used in the design is $3.5$ then the maximum allowable value of $\sigma_0$ is _______ MPa. (round off to $2$ decimal places)
A bolt head has to be made at the end of a rod of diameter $d=12$ mm by localized forging (upsetting) operation. The length of the unsupported portion of the rod is $40$ mm. To avoid buckling of the rod, a closed forging operation has to be performed with a maximum die diameter of _______ mm.
Consider the following network of activities, with each activity named $\textbf{A – L}$, illustrated in the nodes of the network. The number of hours required for each activity is shown alongside the nodes. The slack on the activity $\textbf{L}$, is _______ hours.
In a furnace, the inner and outer sides of the brick wall $(k_1=2.5 \: \text{W/m.k})$ are maintained at $1100 ^\circ C$ and $700 ^\circ C$, respectively as shown in figure. The brick wall is covered by an insulating material of thermal conductivity $k_2$. The thickness of the ... the composite walls is $2500 \: W/m^2$. The value of $k_2$ is _______ $\text{W/m.K}$ (round off to one decimal place)
If a reversed Carnot cycle operates between the temperature limits of $27^\circ C$ and $-3^\circ C$, then the ratio of the COP of a refrigerator to that of a heat pump (COP of refrigerator / COP of heat pump) based on the cycle is __________ (round off to $2$ decimal places).
The directional derivative of $f(x,y,z) = xyz$ at point $(-1,1,3)$ in the direction of vector $\hat{i} – 2 \hat{j} +2 \hat{k}$ is $3\hat{i} – 3 \hat{j} - \hat{k} \\$ $- \dfrac{7}{3} \\$ $\dfrac{7}{3} \\$ $7$
The function $f(z)$ of complex variable $z=x+iy$, where $i=\sqrt{-1}$, is given as $f(z)=(x^3-3xy^2)+i \: v(x,y)$. For this function to be analytic, $v(x,y)$ should be $(3xy^2-y^3) +$ constant $(3x^2y^2-y^3) +$ constant $(x^3-3x^2 y) +$ constant $(3x^2y-y^3) +$ constant
A cantilever of length $l$, and flexural rigidity $EI$, stiffened by a spring of stiffness $k$, is loaded by a transverse force $P$, as shown. The transverse deflection under the load is $\dfrac{Pl^3}{3EI} \begin{bmatrix} \dfrac{3EI}{3EI+2kl^3} \end{bmatrix}$ ... $\dfrac{Pl^3}{3EI} \begin{bmatrix} \dfrac{3EI}{3EI+kl^3} \end{bmatrix}$
The sun (S) and the Planet (P) of an epicyclic gear train shown in the figure have identical number of teeth. If the sun (S) and the outer ring (R) gears are rotated in the same direction with angular speed $\omega_S$ and $\omega_R$, respectively, then the angular speed of the arm $AB$ ... $\dfrac{3}{4} \omega _R - \dfrac{1}{4} \omega _S$
A thin-walled cylinder of radius $r$ and thickness $t$ is open at both ends, and fits snugly between two rigid walls under ambient conditions, as shown in the figure. The material of the cylinder has Young's modulus $E$, Poisson's ratio $v$, and coefficient of thermal expansion $\alpha$. What is the ... $\Delta T = \big( v + \dfrac{1}{2} \big) \dfrac{pr}{ \alpha t E}$
A helical spring has spring constant $k$. If the wire diameter, spring diameter and the number of coils are all doubled then the spring constant of the new spring becomes $k/2$ $k$ $8k$ $16k$
Two rollers of diameters $D_1$ (in mm) and $D_2$ (in mm) are used to measure the internal taper angle in the V-groove of a machined component. The heights $H_1$ (in mm) and $H_2$ (in mm) are measured by using a height gauge after inserting the rollers into the same V-groove as shown in the ... $\sin \alpha = \dfrac{(H_1-H_2) }{ (D_1 - D_2)}$
The forecast for the monthly demand of a product is given in the table below: ... is made by using the exponential smoothing method. The exponential smoothing coefficient used in forecasting the demand is $0.10$ $0.40$ $0.50$ $1.00$
One kg of air in a closed system undergoes an irreversible process from an initial state of $p_1=1$ bar (absolute) and $T_1=27^\circ C$, to a final state of $p_2=3$ bar (absolute) and $T_2=127 ^ \circ C$. If the gas constant of air is $287 \: \text{J/kg.K}$ ... specific entropy (in $\text{J/kg.K}$) of the air in the process is $-26.3$ $28.4$ $172.0$ indeterminate, as the process is irreversible
For the integral $\displaystyle \int_0 ^{\pi/2} (8+4 \cos x) dx$, the absolute percentage error in numerical evaluation with the Trapezoidal rule, using only the end points, is ________ (round off to one decimal place).
A fair coin is tossed $20$ times. The probability that ‘head’ will appear exactly $4$ times in the first ten tosses, and ‘tail’ will appear exactly $4$ times in the next ten tosses is _________ (round off to $3$ decimal places)
A hollow spherical ball of radius $20$ cm floats in still water, with half of its volume submerged. Taking the density of water as $1000 \: kg/m^3$, and the acceleration due to gravity as $10 \: m/s^2$, the natural frequency of small oscillations of the ball, normal to the water surface is __________ radians/s (round off to $2$ decimal places)
Uniaxial compression test data for a solid metal bar of length $1$ m is shown in the figure. The bar material has a linear elastic response from $O$ to $P$ followed by a nonlinear response. The point $P$ represents the yield point of the material. The rod is ... the bar so that it does not buckle under axial loading before reaching the yield point is _______ mm ( round off to one decimal place)
The turning moment diagram of a flywheel fitted to a fictitious engine is shown in the figure. The mean turning moment is $2000$ Nm. The average engine speed is $1000$ rpm. For fluctuation in the speed to be within $\pm 2\%$ of the average speed, the mass moment of inertia of the flywheel is ________ $kg \cdot m^2$
A rigid block of mass $m_1=10$ kg having velocity $v_0=2 \: m/s$ strikes a stationary block of mass $m_2=30$ kg after traveling $1$ m along a frictionless horizontal surface shown in the figure. The two masses stick together and jointly move by a distance of $0.25$ m ... having a spring constant $k=10^5 \: N/m$. The maximum deflection of the spring is _________ cm (round off to $2$ decimal places)