Consider the following complex function $$f\left( z \right) = {9 \over {\left( {z - 1} \right)\left( {z + 2} \right)}}.$$ Which of the following is ONE of the residues of the above function ?

The integral $$\,\int_{{x_1}}^{{x_2}} {{x^2}dx\,\,} $$ with $${x_2} > {x_1} > 0$$ is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If $${\rm I}$$ is the exact value of the integral obtained analytically and $$J$$ is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?

The quadratic equation $${x^2} - 4x + 4 = 0$$ is to be solved numerically, starting with the initial guess $${x_0} = 3.$$ The Newton- Raphson method is applied once to get a new estimate and then the Secant method is applied once using in the initial guess and this new estimate. The estimated value of the root after the application of the Secant method is ________.

Consider the following differential equation
$$x\left( {y\,dx + x\,dy} \right)\cos \left( {{y \over x}} \right)$$
$$\,\,\,\,\,\,\,\,\,\, = y\left( {x\,dy - y\,dx} \right)\sin \left( {{y \over x}} \right)$$

Which of the following is the solution of the above equation ($$C$$ is an arbitrary constant)

Consider the following probability mass function (p.m.f) of a random variable $$X.$$ $$$p\left( {x,q} \right) = \left\{ {\matrix{ q & {if\,X = 0} \cr {1 - q} & {if\,X = 1} \cr 0 & {otherwise} \cr } } \right.$$$
$$q=0.4,$$ the variance of $$X$$ is _______.

The directional derivative of the field $$u(x, y, z)=$$ $${x^2} - 3yz$$ in the direction of the vector $$\left( {\widehat i + \widehat j - 2\widehat k} \right)\,\,$$ at point $$(2, -1, 4)$$ is _______.

For what value of $$'p'$$ the following set of equations will have no solutions? $$$2x+3y=5$$$ $$$3x+py=10$$$

The smallest and largest Eigen values of the
following matrix are : $$\left[ {\matrix{ 3 & { - 2} & 2 \cr 4 & { - 4} & 6 \cr 2 & { - 3} & 5 \cr } } \right]$$

In a catchment, there are four rain-gauge stations, P, Q, R, and S. Normal annual precipitation values at these stations are 780 mm, 850 mm, 920 mm, and 980 mm, respectively. In the year 2013, stations Q, R, and S, were operative but P was not. Using the normal ratio method, the precipitation at station P for the year 2013 has been estimated as 860 mm. If the observed precipitation at stations Q and R for the year 2013 were 930 mm and 1010 mm, respectively; what was the observed precipitation (in mm) at station S for that year?

The two columns below show some parameters and their possible values.

Parameters	Value
P-Gross Command Area	I-100 hectares/cumec
Q-Permanent Wilting Point	II-6℃
R-Duty of canal water	III-1000 hectares
S-Delta of wheat	IV-1000 cm
	V-40 cm
	VI-0.12

Which of the following options matches the parameters and the values correctly?

The development length of a deformed reinforcement bar can be expressed as $$\left( {1/k} \right)\left( {\phi {\sigma _s}/{\tau _{bd}}} \right).$$

From the $$IS:$$ $$456$$-$$2000,$$ the value of $$k$$ can be calculated as _________________.

Consider the singly reinforced beam shown in the figure below:

At cross-section $$XX,$$ which of the following statements is TRUE at the limit state?

Consider the singly reinforced beam section given below (left figure). The stress block parameters for the cross-section from $$IS:$$ $$456$$ - $$2000$$ are also given below (right figure). The moment of resistance for the given section by the limit state method is ______ $$kN$$-$$m.$$

Workability of concrete can be measured using slump, compaction factor and Vee bee time. Consider the following statements for workability of concrete:

(i) As the slump increases, the Vebe time increases

(ii) As the slum increases, the compaction factor increases

Which of the following is TRUE?

The composition of an air-entrained concrete is given below:

Water : 184 kg /m³

Ordinary Portland Cement(OPC) : 368 kg /m³

Sand : 606 kg /m³

Coarse aggregate : 1155 kg /m³

Assume the specific gravity of OPC, sand and coarse aggregate to be 3.14, 2.67 and 2.74, respectively. The air content is_________ liters/m³

Consider the following statements for air-entrained concrete:

(i) Air-entrainment reduces the water demand for a given level of workability

(ii) Use of air-entrained concrete is required in environments where cyclic freezing and thawing is expected.

Which of the following is TRUE?

A bracket plate connected to a column flange transmits a load of $$100$$ $$kN$$ as shown in the following figure. The maximum force for which the bolts should be designed is _______ $$kN$$

Two beams are connected by a linear spring as shown in the following figure. For a load $$P$$ as shown in the figure, the percentage of the applied load $$P$$ carried by the spring is ______________.

A tapered circular rod of diameter varying from 20 mm to 10 mm is connected to another uniform circular rod of diameter 10 mm as shown in the following figure. Both bars are made of same material with the modulus of elasticity, E = 2 $$ \times $$ 10⁵ MPa. When subjected to a load P = 30$$\pi $$ kN, the deflection at point A is _______________ mm.

Two triangular wedges are glued together as shown in the following figure. The stress acting normal to the interface, $${\sigma _n}$$ is _____$$MPa.$$

For the beam shown below, the stiffness coefficient $${K_{22}}$$ can be written as

For formation of collapse mechanism in the following figure, the minimum value of $${{P_u}}$$ is $$c{M_P}/L.$$ $${M_P}$$ and $$3{M_P}$$ denote the plastic moment capacities of beam sections as shown in this figure. The value of $$c$$ is ________.

For the beam shown below, the value of the support moment
$$M$$ is ____________ $$KN$$-$$m$$.

For the $$2D$$ truss with the applied loads shown below, the strain energy in the member $$XY$$ is _______________ $$kN$$-$$m$$. For member $$XY$$, assume $$AE = 30$$ $$kN,$$ where $$A$$ is cross-section area and $$E$$ is the modulus of elasticity.