The product of two complex numbers $$1 + i\,\,\,\,\& \,\,\,\,2 - 5\,i$$ is

What is $$\mathop {Lim}\limits_{\theta \to 0} {{\sin \theta } \over \theta }\,\,$$ equal to ?

Eigen values of a real symmetric matrix are always

Consider the following system of equations
$$2{x_1} + {x_2} + {x_3} = 0,\,\,{x_2} - {x_3} = 0$$ and $${x_1} + {x_2} = 0.$$
This system has

A series expansion for the function $$\sin \theta $$ is _______.

If $$f(x)$$ is even function and a is a positive real number , then $$\int\limits_{ - a}^a {f\left( x \right)dx\,\,} $$ equals ________.

An unbiased coin is tossed five times. The outcome of each loss is either a head or a tail. Probability of getting at least one head is _______.

Consider the differential equation $${{dy} \over {dx}} = \left( {1 + {y^2}} \right)x\,\,.$$ The general solution with constant $$'C'$$ is

The integral $$\,\int\limits_1^3 {{1 \over x}\,\,dx\,\,\,} $$ when evaluated by using simpson's $$1/{3^{rd}}$$ rule on two equal sub intervals each of length $$1,$$ equals to

A stone with mass of $$0.1$$kg is catapulted as shown in the figure. The total force $${F_x}$$ (in N) exerted by the rubber band as a function of distance $$x$$ (in m) is given by $${F_x} = 300{x^2}.$$ If the stone is displayed by $$0.1$$ m from the un-stretched position (x = 0) of the rubber band, the energy stored in the rubber bad is

A 1 kg block is resting on a surface with coefficient of friction $$\mu $$ $$=0.1.$$ A force of $$0.8$$ $$N$$ is applied to the block as shown in the figure. The friction force is

The coefficient of restitution of a perfectly plastic impact is

Figure shows the schematic for the measurement of velocity of air (density $$ = 1.2\,\,kg/{m^3}$$ ) through a constant -area duct using a Pilot tube and a water-tube manometer. The differential head of water (density $$ = 1000\,kg/{m^3}$$) in the two columns of the manometer is $$10$$ $$mm.$$ Take acceleration due to gravity as $$9.8$$ $$m/{s^2}.$$ The velocity of air in $$m/s$$

The ratios of the laminar hydrodynamic boundary layer thickness to thermal boundary layer thickness of flows of two fluids $$P$$ and $$Q$$ on a flat plate are $${1 \over 2}$$ and $$2$$ respectively. The Reynolds number based on the plate length for both the flows is $${10^4}.$$ The Prandtl and Nusselt numbers for $$P$$ are $${1 \over 8}$$ and $$35$$ respectively. The Prandtl and Nusselt number for $$Q$$ are respectively

In a condenser of a power plant, the steam condenses at a temperature of $${60^ \circ }C.$$ The cooling water enters at $${30^ \circ }C$$ and leaves at $${45^ \circ }C.$$ The logarithmic mean temperature difference $$(LMTD)$$ of the condenser is

A spherical steel ball of $$12mm$$ diameter is initially at $$100K.$$ It is slowly cooled in a surrounding of $$300K.$$ The heat transfer coefficient between the steel ball and the surrounding is $$5W/{m^2}K.$$ The thermal conductivity of steel is $$20$$ $$W/mK.$$ The temperature difference between the centre and the surface of steel ball is

A pipe of $$25$$ $$mm$$ outer diameter carries steam. The heat transfer coefficient between the cylinder and surrounding is $$5$$ $$W/{m^2}K.$$ It is propsed to reduce the heat loss from the pipe by adding insulation having a thermal conductivity of $$0.05$$ $$W/mK$$. Which one of the following statements is TRUE?

The word Kanban is most appropriately associated with

Cars arrive at a service station according to Poisson's distribution with a mean rate of $$5$$ per hour. The service time per car is exponential with a mean of $$10$$ minutes. At state, the average waiting time in the queue is

One unit of product $${P_1}$$ requires $$3$$ $$kg$$ of resource $${R_1}$$ and $$1$$ $$kg$$ of resource $${R_2}$$. One unit of product $${P_2}$$ requires $$2$$ $$kg$$ of resource $${R_1}$$ and $$2$$ $$kg$$ of resource $${R_2}$$. The profits per unit by selling product $${P_1}$$ and $${P_2}$$ are Rs. $$2000$$ and Rs. $$3000$$ respectively. The manufacturer has $$90$$ $$kg$$ of resource $${R_1}$$ and $$100$$ $$kg$$ of resource $${R_2}$$.

The unit worth of resource $${R_2}$$. i.e. dual price of resource $${R_2}$$ in Rs. per $$kg$$ is

One unit of product $${P_1}$$ requires $$3$$ $$kg$$ of resource $${R_1}$$ and $$1$$ $$kg$$ of resource $${R_2}$$. One unit of product $${P_2}$$ requires $$2$$ $$kg$$ of resource $${R_1}$$ and $$2$$ $$kg$$ of resource $${R_2}$$. The profits per unit by selling product $${P_1}$$ and $${P_2}$$ are Rs. $$2000$$ and Rs. $$3000$$ respectively. The manufacturer has $$90$$ $$kg$$ of resource $${R_1}$$ and $$100$$ $$kg$$ of resource $${R_2}$$.

The manufacturer can make a maximum profit of Rs.

Match the following criteria of material failure, under biaxial stresses $${\sigma _1}$$and $${\sigma _2}$$ and yield stress $${\sigma _y}$$ with their corresponding graphic representations:
$$P.$$ Maximum normal stress criterion
$$Q.$$ Maximum distortion energy criterion
$$R.$$ Maximum shear stress criterion

Two identical ball bearings $$P$$ and $$Q$$ are operating at loads $$30kN$$ and $$45kN$$ respectively. The ratio of the life of bearing $$P$$ to the life of bearing $$Q$$ is

A cubic casting of $$50mm$$ side undergoes volumetric solidification shrinkage and volumetric solid contraction of $$4\% $$ and $$6\% $$ respectively. No riser is used. Assume uniform cooling in all directions. The side of cube after solidification and contraction is

Green sand mould indicates that

The maximum possible draft in cold rolling of sheet increases with the

Which one among the following welding processes uses non-consumable electrode?

A single-point cutting tool with $${12^0}$$ rake angle is used to machine a steel work-piece. The depth of cut, $$i.e$$ uncut thickness is $$0.81$$ $$mm.$$ The chip thickness under orthogonal machining condition is $$1.8$$ $$mm.$$ The shear angle is approximately

The shear strength of a sheet metal is $$300MPa.$$ The blanking force required to produce a blank of $$100$$ $$mm$$ diameter from a $$1.54$$ $$mm$$ thick sheet is close to

A hole is dimension $$\phi {9^{\matrix{ { + 0.015} \cr { + 0\,\,\,\,\,\,\,\,\,\,} \cr } }}\,mm.$$ The corresponding shaft is of dimension $$\phi {9^{\matrix{ { + 0.010} \cr { + 0.001} \cr } }}\,mm.$$ The resulting assembly has

A simply supported beam $$PQ$$ is loaded by a moment of $$1$$ $$kN-m$$ at the mid-span of the beam as shown in the figure. The reaction forces $${R_P}$$ and $${R_Q}$$ at supports $$P$$ and $$Q$$ respectively are

A torque $$T$$ is applied at the free end of a stepped rod of circular cross-sections as shown in the figure. The shear modulus of the material of the rod is $$G.$$ The expressions for diameter to produce an angular twist $$\theta $$ at the free end is

A column has a rectangular cross-section of 10 mm $$ \times $$ 20 mm and a length of 1 m. The slenderness ratio of the column is close to

A double - parallelogram mechanism is shown in the figure. Note that PQ is a single link. The mobility of the mechanism is

For the four - bar linkage shown in the figure, the angular velocity of link AB is $$1$$ rad/s. The length of link CD is $$1.5$$ times the length of link $$AB$$. In the configuration shown, the angular velocity of link CD in rad/s is

A disc of mass m is attached to a spring of stiffness $$k$$ as shown in the figure. The disc rolls without slipping on a horizontal surface. The natural frequency of vibration of the system is

A mass of $$1$$ kg is attached to two identical springs each with stiffness $$k=20kN/m$$ as shown in the figure. Under frictionless condition, the natural frequency of the system in $$Hz$$ is close to

The temperature and pressure of air in a large reservoir are $$400$$ $$K$$ and $$3$$ bar respectively. A converging diverging nozzle of exit area $$0.005{m^2}$$ is fitted to the wall of the reservoir as shown in the figure. The static pressure of air at the exit section for isentropic flow through the nozzle is $$50$$ $$kPa.$$ The characteristic gas constant and the ratio of specific heats of air are $$0.287$$ $$kJ/kgK$$ and $$1.4$$ respectively.

The density of air in $$kg/{m^3}$$ at the nozzle exit is

The temperature and pressure of air in a large reservoir are $$400$$ $$K$$ and $$3$$ bar respectively. A converging diverging nozzle of exit area $$0.005{m^2}$$ is fitted to the wall of the reservoir as shown in the figure. The static pressure of air at the exit section for isentropic flow through the nozzle is $$50$$ $$kPa.$$ The characteristic gas constant and the ratio of specific heats of air are $$0.287$$ $$kJ/kgK$$ and $$1.4$$ respectively.

The mass flow rate of air through the nozzle in $$kg/s$$ is

The contents of a well-insulated tank are heated by a resistor of $$23\Omega $$ in which $$10A$$ current is flowing. Consider the tank along with its contents as a thermodynamic system. The work done by the system and the heat transfer to the system are positive. The rates of heat $$(Q),$$ work $$(W)$$ and change in internal energy $$\left( {\Delta U} \right)$$ during the process in $$kW$$ are

A pump handling a liquid raises its pressure from $$1$$ bar to $$30$$ $$bar.$$ Take the density of the liquid as $$990$$ $$kg/{m^3}$$. The isentropic specific work done by the pump in $$kJ/kg$$ is

Heat and work are

In an experimental set-up, air flows between two stations $$P$$ and $$Q$$ adiabatically. The direction of flow depends on the pressure and temperature conditions maintained at $$P$$ and $$Q$$. The conditions at station $$P$$ are $$150$$ $$kPa$$ and $$350$$ $$K.$$ The temperature at station $$Q$$ is $$300$$ $$K.$$ The following are the properties and relations pertaining to air:

Specific heat at constant pressure, $$${C_p} = 1.005\,\,kJ/kg\,\,K;$$$
Specific heat at constant volume, $$${C_v} = 0.718\,\,kJ/kg\,\,K;$$$
Characteristic gas constant, $$R=0.287$$ $$kJ/kgK.$$
Enthalpy $$h = {C_p}T,$$
Internal energy, $$u = {C_v}T.$$

If the air has to flow from station $$P$$ to station $$Q,$$ the maximum possible value of pressure in $$kPa$$ at station $$Q$$ is close to

In an experimental set-up, air flows between two stations $$P$$ and $$Q$$ adiabatically. The direction of flow depends on the pressure and temperature conditions maintained at $$P$$ and $$Q$$. The conditions at station $$P$$ are $$150$$ $$kPa$$ and $$350$$ $$K.$$ The temperature at station $$Q$$ is $$300$$ $$K.$$ The following are the properties and relations pertaining to air:

Specific heat at constant pressure, $$${C_p} = 1.005\,\,kJ/kg\,\,K;$$$
Specific heat at constant volume, $$${C_v} = 0.718\,\,kJ/kg\,\,K;$$$
Characteristic gas constant, $$R=0.287$$ $$kJ/kgK.$$
Enthalpy $$h = {C_p}T,$$
Internal energy, $$u = {C_v}T.$$

If the pressure at station $$Q$$ is $$50$$ $$kPa,$$ the change in entropy $$\left( {{S_Q} - {S_p}} \right)$$ in $$kJ/kgK$$ is

The values of enthalpy of steam at the inlet and outlet of a steam turbine in a Rankine cycle are $$2800$$ $$kJ/kg$$ and $$1800$$ $$kJ/kg$$ respectively. Neglecting pump work, the specific steam consumption in $$kg/kW$$-hour is