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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