Engineering Mathematics GATE-2008

Q 1: Which one of the following is not an integrating factor for the differential equation $xdy-ydx=0$?

A) $\frac1{x^2}$
B) $\frac1{y^2}$
C) $\frac1{xy}$
D) $\frac1{\left(x+y\right)}$

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Q 2: Which one of the following is not a solution of the differential equation $\frac{d^2y}{dx^2}+y=1$?

A) $y=1$
B) $y=1+\cos x$
C) $y=1+\sin x$
D) $y=2+\sin x+\cos x$

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Q 3: The limit of $\frac{\sin x}x$ as x→∞ is

A) -1
B) 0
C) 1
D) ∞

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Q 4: The unit normal vector to the surface of the sphere $x^2+y^2+z^2=1$ at the point $\left(\frac1{\sqrt2},0,\frac1{\sqrt2}\right)$ is ($\widehat i,\widehat j,\widehat k$ are unit normal vectors in the cartesian coordinate system)

A) $\frac1{\sqrt2}\widehat i+\frac1{\sqrt2}\widehat j$
B) $\frac1{\sqrt2}\widehat i+\frac1{\sqrt2}\widehat k$
C) $\frac1{\sqrt2}\widehat j+\frac1{\sqrt2}\widehat k$
D) $\frac1{\sqrt3}\widehat i+\frac1{\sqrt3}\widehat j+\frac1{\sqrt3}\widehat k$

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Q 5: A non-linear function f (x) is defined in the interval – 1.2 < x < 4 as illustrated in the figure below. The equation f(x) = 0 is solved for x within this interval by using the Newton-Raphson iterative scheme. Among the initial guesses (I₁, I₂, I₃ and I₄), the guess that is likely to lead to the root most rapidly is

A) I₁
B) I₂
C) I₃
D) I₄

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Q 6: Which one of the following transformations $u=f\left(y\right)$ reduces $\frac{dy}{dx}+Ay^3+By=0$ to a linear differential equation?  (A and B are positive constants)

A) $u=y^{-3}$
B) $u=y^{-2}$
C) $u=y^{-1}$
D) $u=y^2$

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Q 7: The Laplace transform of the function $f(t)=t\;\sin t$ is

A) $\frac{2s}{\left(s^2+1\right)^2}$
B) $\frac1{s^2\left(s^2+1\right)}$
C) $\frac1{s^2}+\frac1{\left(s^2+1\right)}$
D) $\frac1{\left(s-1\right)^2+1}$

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Q 8: The value of the surface integral $\oiint_s\left(x\widehat i+y\widehat j\right).\widehat ndA$ evaluated over the surface of a cube having sides of length a is ($\widehat n$ is a unit normal vector)

A) 0
B) a³
C) 2a³
D) 3a³

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Q 9: The first four terms of the Taylor series expansion of cos x about the point x = 0 are

A) $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}$
B) $1-x+\frac{x^2}{2!}-\frac{x^3}{3!}$
C) $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}$
D) $x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}$

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Q 10: If $\begin{bmatrix}1&2\\2&1\end{bmatrix}$, then the eigen values of A³ are

A) 5, 4
B) 3, -1
C) 9, -1
D) 27, -1

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Q 11: An analytic function w(z) is defined as $w=u+iv$, where $i=\sqrt{-1}$ and $z=x+iy$. If the real part is given by $u=\frac y{x^2+y^2}$, w(z) is

A) $\frac1z$
B) $\frac1{z^2}$
C) $\frac iz$
D) $\frac1{iz}$

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Q 12: The normal distribution is given by

$f(x)=\frac1{\sigma\sqrt{2\mathrm\pi}}exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right),\;\;-\infty<x<\infty$

The point of inflexion to the normal curve is

A) $x=-\sigma,+\sigma$
B) $x=\mu+\sigma,\;\mu-\sigma$
C) $x=\mu+2\sigma,\;\mu-2\sigma$
D) $x=\mu+3,\;\mu-3\sigma$

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Q 13: Using Simpson’s 1/3 rule and four equally spaced intervals (n = 4), estimate the value of the integral

$\int_0^{\mathrm\pi/4}\frac{\sin x}{\cos^3x}\operatorname dx$

A) 0.3887
B) 0.4384
C) 0.5016
D) 0.5527

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Q 14: The following differential equation is to be solved numerically by Euler’s explicit method.

$\frac{dy}{dx}=x^2y-1.2y\;\;with\;y(0)=1$

A step size of 0.1 is used. The solution for y at x = 0.1 is

A) 0.880
B) 0.905
C) 1.000
D) 1.100

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Q 15: The Poisson distribution is given by $P(r)=\frac{m^r}{r!}exp\left(-m\right)$. The first moment about the origin for the distribution is

A) 0
B) m
C) 1/m
D) m²

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