GATE Questions & Answers of Calculus Civil Engineering

Let x be continuos variable defined over the interval $\style{font-family:'Times New Roman'}{(-\infty,\;\infty)}$, and $\style{font-family:'Times New Roman'}{f(x)=e^{-x-e^{-x}}}$. The integral $\style{font-family:'Times New Roman'}{g\left(x\right)=\;\int f(x)\;dx}$ is equal to

The reaction rate involving reactants A and B is given by $\style{font-family:'Times New Roman'}{-k\left[A\right]^\alpha\left[B\right]^\beta}$. Which one of the following statement is valid for the reaction to be a first-order reaction?

limx0tan xx2-x is equal to ______________

For the function $\style{font-family:'Times New Roman'}{f(x)=a+bx,\;0\leq x\leq1}$, to be a valid probability density function , which one of the following statements is correct?

A partical of mass 2 kg is travelling at a velocity of 1.5 m/s. A force $\style{font-family:'Times New Roman'}{f\left(t\right)=3t^2}$ (in N) is applied to it in the direction of motion for a duration of 2 seconds, where t denotes time in seconds. The velocity (in m/s, up to one decimal place) of the particle immediatly after the removal of the force is __________

let $\style{font-family:'Times New Roman'}{w=f(x,\;y),}$ where $ x $ and $ y $ are function of $ t $. Then according to the chain rule, $\style{font-family:'Times New Roman'}{\frac{dw}{dt}}$ is equal to

The divergence of the vector field $ \boldsymbol V=x^2\boldsymbol i+2y^3\boldsymbol j+z^4\boldsymbol k\; $ at $ x=1,\;y=2,\;z=3 $ is ____________

The tangent to the curve represented by $\style{font-family:'Times New Roman'}{y=x\ln x}$ is required to have 45º inclination with the x-axis. The coordinates of the tangent point would be

Consider the following definite integral:

$I=\int\limits_0^1\frac{\left(\sin^{-1}x\right)^2}{\sqrt{1-x^2}}dx$

The value of the integral is

The value of 011+x2dx+0sin xxdx is

The area of the region bounded by the parabola y=x2+1 and the straight line x+y=3 is

The optimum value of the function f(x) = x2  4x +2 is

What is the value of $\lim\limits_{\begin{array}{c}x\rightarrow0\\y\rightarrow0\end{array}}\frac{xy}{x^2+y^2}$?

The angle of intersection of the curves x2=4y and y2=4x at point (0, 0) is

The area between the parabola x2=8y and the straight line y = 8 is _________.

The quadratic approximation of fx=x3-3x2-5 at the point x=0 is

The directional derivative of the field $ u(x,y,z)=x^2-3yz $ in the direction of the vector i^+j^-2k^ at point (2, –1, 4) is _________.

While minimizing the function f(x) necessary and sufficient conditions for a point, x0 to be a minima are:

limx1+1x2x is equal to

limxx+sinxx equals to

With reference to the conventional Cartesian (x, y) coordinate system, the vertices of a triangle have the following coordinates: (x1, y1) = (1, 0); (x2, y2) = (2, 2); and (x3, y3) = (4, 3). The area of the triangle is equal to

The expression lima0xa-1a is equal to

The solution for $\int\limits_0^{\pi/6}\cos^43\theta\sin^36\theta d\theta$ is:

For the parallelogram OPQR shown in the sketch, OP=ai^+bj^ and OR=ci^+dj^.The area of the parallelogram is

What should be the value of λ such that the function defined below is continuous at x=π2

fx=λcosxπ2-xifxπ21ifx=π2

what is the value of the definite integral $\int\limits_0^\mathrm a\frac{\sqrt{\mathrm x}}{\sqrt{\mathrm x}+\sqrt{\mathrm a-\mathrm x}}dx$

if a and b are two arbitary vectors with magnitudes a and b,respectively, a×b 2

The limx0sinx32x is

Given a function

f (x, y) = 4x2 + 6y28x−4y+8

The optimal value of f (x, y)

For a scalar function f(x,y,z) = x2 +3y2 +2z2, the gradient at the point P(1,2,-1) is

For a scalar function f(x,y,z) = x2 +3y2 +2z2, the directional derivative at the point P(1,2,-1) in the direction of a vector i-j+2k is

The inner (dot) product of two vectors $\overline{\mathrm P}$ and $\overline{\mathrm Q}$ is zero. The angle (degrees) between the two vectors is

A velocity vector is given as V=5xyi+2y2j+3yz2k. The divergence of this velocity vector at (1,1,1) is

Evaluate $\int\limits_0^\infty\frac{\sin t}t\operatorname{d}t$