GATE CE 2002

## GATE CE

Eigen values of the following matrix are $$\left[ {\matrix{
{ - 1} & 4 \cr
4 & { - 1} \cr
} } \right

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Limit of the following sequence as $$n \to \infty $$ $$\,\,\,$$ is $$\,\,\,$$ $${x_n} = {n^{{1 \over n}}}$$

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The following function has local minima at which value of $$x,$$ $$f\left( x \right) = x\sqrt {5 - {x^2}} $$

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The value of the following improper integral is $$\,\int\limits_0^1 {x\,\log \,x\,dx} = \_\_\_\_\_.$$

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The value of the following definite integral in $$\int\limits_{{\raise0.5ex\hbox{$\scriptstyle { - \pi }$}
\kern-0.1em/\

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The directional derivative of the following function at $$(1, 2)$$ in the direction of $$(4i+3j)$$ is : $$f\left( {x,y}

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The Laplace transform of the following function is
$$$f\left( t \right) = \left\{ {\matrix{
{\sin t} & {for\,\,0

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Using Laplace transforms, solve $${a \over {{s^2} - {a^2}}}\,\,\left( {{d^2}y/d{t^2}} \right) + 4y = 12t\,\,$$
given th

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In natural water, hardness is mainly caused by

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During a 6-hour storm the rainfall intensity was 0.8 cm/hr on a catchment of area 8.6 km2. The measured runoff volume du

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Modulus of elasticity of M25 grade concrete in MPa is

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Read the following two statements
$${\rm I}.\,\,\,\,\,\,\,\,\,\,\,\,$$ Maximum strain in concrete at the outermost compr

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As per provision of $$IS$$ $$456$$-$$2000$$, in the limit state method for design of beams, the limiting value of the de

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A $$10m$$ long prestressing bed is used to cast $$4$$ (pretensioned) prestressed concrete beams of $$2.3$$ $$m$$ each. A

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$$ISA$$ $$100 \times 100 \times 10\,\,mm$$ (Cross sectional area $$ = 1903\,\,m{m^2})$$ serves as tensile member. This a

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$$ISA$$ $$100 \times 100 \times 10mm$$ (cross sectional area $$ = 1903\,\,m{m^2}$$) is welded along $$A$$ and $$B$$ (Ref

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Given reason for the following is not more than $$20$$ words
(a) A maximum permissible distance between lacing and batte

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When designing steel structures, one must ensure that local buckling in webs does not take place. This check may not be

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An $$ISMB$$ $$500$$ is used as a beam in a multi-storey construction. From the view point of structural design, it can b

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The shear modulus (G), modulus of elasticity (E) and the Poisson's ratio ($$\mu$$) of a material are related as

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For the loading given in the figure below, two statements ( $${\rm I}$$ and $${\rm II}$$ ) are made.
$${\rm I}.$$ $$\,

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In the propped cantilever beam carrying a uniformly distributed load of $$w$$-$$kN/m,$$ shown in the following figure, t

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A propped cantilever beam of span $$'L'$$ is loaded with $$u.d.l$$ of intensity $$w$$/unit length, all through the span.

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A steel beam (with a constant $$EI,$$ and span $$L$$) is fixed at both ends and carries a uniformly distributed load ($$

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Calculate the shape factor for the $$T$$-section shown in the following figure made up of two plates $$100\,\,mm \times

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SSD and Friction coefficients are

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