GATE CE 2009
GATE CE
1
The area under the curve shown between $$x=1$$ and $$x=5$$ is to be evaluated using the trapezoidal rule. The following points on the curve are given

The evaluated area (In m2) will be
2
Laplace transform of $$f\left( x \right) = \cos \,h\left( {ax} \right)$$ is
3
The analytical function has singularities at, where $$f(z) = {{z - 1} \over {{z^2} + 1}}$$
4
The value of the integral $$\int\limits_C {{{\cos \left( {2\pi z} \right)} \over {\left( {2z - 1} \right)\left( {z - 3} \right)}}} dz$$ where C is a closed curve given by |z| = 1 is
5
Solution of the differential equation $$3y{{dy} \over {dx}} + 2x = 0$$ represents a family of
6
The standard normal probability function can be approximated as
$$$F\left( {{X_N}} \right) = {1 \over {1 + \exp \left( { - 1.7255{X_N}{{\left| {{X_N}} \right|}^{0.12}}} \right)}}\,\,\,$$$
$$\,{X_N} = $$ standard normal deviate. If mean and standard deviation of annual precipitation are $$102$$ cm and $$27$$ cm respectively, the probability that the annual precipitation will be $$b/w$$ $$90$$ cm and $$102$$ cm is
$$\,{X_N} = $$ standard normal deviate. If mean and standard deviation of annual precipitation are $$102$$ cm and $$27$$ cm respectively, the probability that the annual precipitation will be $$b/w$$ $$90$$ cm and $$102$$ cm is
7
For a scalar function $$\,f\left( {x,y,z} \right) = {x^2} + 3{y^2} + 2{z^2},\,\,$$ the directional derivative at the point $$P(1,2,-1)$$ in the direction of a vector $$\widehat i - \widehat j + 2\widehat k\,\,$$ is
8
For a scalar function $$f(x,y,z)=$$ $${x^2} + 3{y^2} + 2{z^2},\,\,$$ the gradient at the point $$P(1,2,-1)$$ is
9
A square matrix $$B$$ is symmetric if ____
10
In the solution of the following set of linear equations by Gauss-elimination using partial pivoting
$$$5x+y+2z=34,$$$
$$$4y-3z=12$$$
and
$$$10x-2y+z=-4.$$$
The pivots for elimination of $$x$$ and $$y$$ are
The pivots for elimination of $$x$$ and $$y$$ are
11
In quadrantal bearing system, bearing of a line varies from
12
An agricultural land of 437ha is to be irrigated for a particular crop. The base
period of the crop is 90 days and the total depth of water required by the crop is
105cm. If a rainfall of 15cm occurs during the base period, the duty of irrigation
water is
13
A rectangular concrete beam of width $$120$$ $$mm$$ and depth $$200$$ $$mm$$ is prestressed by pretensioning to a force of $$150$$ $$kN$$ at an eccentricity of $$20$$ $$mm.$$ The cross sectional area of the prestressing steel is $$187.5\,\,m{m^2}.$$ Take modulus of elasticity of steel and concrete as $$2.1 \times {10^5}\,\,MPa$$ and $$3.0 \times {10^4}\,\,MPa$$ respectively. The percentage loss of stress in the prestressing steel due to elastic deformation of concrete is
14
For limit state of collapse, the partial safety factors recommended by
IS 456:2000 for estimating the design strength of concrete and reinforcing steel
are respectively
15
The square root of the ratio of moment of inertia of the cross-section to its cross-sectional area is called
16
Consider the following statements for a compression member
$${\rm I}.\,\,\,\,$$ The elastic critical stress in compression increases with decrease in slenderness ratio
$${\rm II}.\,\,\,\,$$ The effective length depends on the boundary conditions at its ends
$${\rm III}.\,\,\,\,$$ The elastic critical stress in compression is independent of the slenderness ratio
$${\rm IV}.\,\,\,\,$$ The ratio of the effective length to its radius of gyration is called as slenderness ratio
The TRUE statements are
17
A 12 mm thick plate is connected to two 8 mm plates, on either side through a
16 mm diameter power driven field rivet as shown in the figure below. Assuming
permissible shear stress as 90MPa and permissible bearing stress as 270 MPa in
the rivet, the rivet value of the joint is


18
A vertical rod $$PQ$$ of length $$L$$ is fixed at its top end $$P$$ and has a flange fixed to the bottom end $$Q.A$$ weight $$W$$ is dropped vertically from a height $$h\left( { < L} \right)$$ on to the flange. The axial stress in the rod can be reduced by
19
A hollow circular shaft has an outer diameter of $$100$$ $$mm$$ and a wall thickness of $$25$$ $$mm.$$ The allowable shear stress in the shaft is $$125$$ $$MPa.$$ The maximum torque the shaft can transmit is
20
The point within the cross sectional plane of a beam through which the resultant of the external loading on the beam has to pass through to ensure pure bending without twisting of the cross-section of the beam is called
21
Match List- $${\rm I}$$ (Shear Force Diagrams) beams with List- $${\rm II}$$ (Diagram of beams with supports and loading) and select the correct answer by using the codes given below the lists:


22
Consider the following statements:
$$1.$$ On a principal plane, only normal stress acts
$$2.$$ On a principal plane, both normal and shear stresses act
$$3.$$ On a principal plane, only shear stress acts
$$4.$$ Isotropic state of stress is independent of frame of reference.
$$1.$$ On a principal plane, only normal stress acts
$$2.$$ On a principal plane, both normal and shear stresses act
$$3.$$ On a principal plane, only shear stress acts
$$4.$$ Isotropic state of stress is independent of frame of reference.
Which of the above statements is/are correct ?
23
In the theory of plastic bending of beams, the ratio of plastic moment to yield moment is called
24
The degree of static indeterminacy of a rigidly jointed frame in a horizontal plane
and subjected to vertical loads only, as shown in figure below is


25
The value of lateral friction or side friction used in the design of horizontal curve
as per India Roads Congress guidelines is