# GATE Questions & Answers of Calculus Civil Engineering

#### Calculus 34 Question(s)

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

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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?

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is equal to ______________

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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?

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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 __________

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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

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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 ____________

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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

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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 is

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The area of the region bounded by the parabola $y={x}^{2}+1$ and the straight line $x+y=3$ is

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The optimum value of the function is

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

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The angle of intersection of the curves ${x}^{2}=4y$ and ${y}^{2}=4x$ at point (0, 0) is

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The area between the parabola ${x}^{2}=8y$ and the straight line y = 8 is _________.

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The quadratic approximation of $f\left(x\right)={x}^{3}-3{x}^{2}-5$ at the point $x=0$ is

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

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

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$\underset{x\to \infty }{\mathrm{lim}}{\left(1+\frac{1}{x}\right)}^{2x}$ is equal to

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$\underset{x\to \infty }{\mathrm{lim}}\left(\frac{x+\mathrm{sin}x}{x}\right)$ equals to

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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

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The expression $\underset{a\to 0}{\mathrm{lim}}\frac{{x}^{a}-1}{a}$ is equal to

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

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For the parallelogram OPQR shown in the sketch, $\stackrel{\to }{\mathrm{OP}}=a\stackrel{^}{i}+b\stackrel{^}{j}$ and $\stackrel{\to }{\mathrm{OR}}=c\stackrel{^}{i}+d\stackrel{^}{j}$.The area of the parallelogram is ##### Show Answer

What should be the value of λ such that the function defined below is continuous at x=$\frac{\mathrm{\pi }}{2}$

$f\left(x\right)=\left\{\begin{array}{ll}\frac{\lambda \mathrm{cos}x}{\frac{\pi }{2}-x}& ifx\ne \pi }{2}\\ 1& ifx=\pi }{2}\end{array}\right\$

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

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if $\stackrel{\to }{\mathrm{a}}$ and $\stackrel{\to }{\mathrm{b}}$ are two arbitary vectors with magnitudes a and b,respectively,

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The $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}\left[{}_{3}{}^{2}x\right]}{x}$ is

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Given a function

f (x, y) = 4x2 + 6y28x−4y+8

The optimal value of f (x, y)

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

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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 $\stackrel{\to }{i}-\stackrel{\to }{j}+2\stackrel{\to }{k}$ is

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

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A velocity vector is given as $\stackrel{\to }{V}=5xy\stackrel{\to }{i}+2{y}^{2}\stackrel{\to }{j}+3y{z}^{2}\stackrel{\to }{k}$. The divergence of this velocity vector at (1,1,1) is

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