Engineering Mathematics GATE-2012

Q 1: Consider the following set of linear algebraic equations

x_1+2x_2+3x_3=2 x_2+x_3=-1 2x_2+2x_3=0

The system has

A) a unique solution
B) no solution
C) an infinite number of solutions
D) only the trivial solution

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Q 2: If a and b are arbitrary constants, then the solution to the ordinary differential equation \frac{d^2y}{dx^2}-4y=0 is

A) y=ax+b
B) y=ae^{-x}
C) y=a\sin\left(2x\right)+b\cos\left(2x\right)
D) y=a\cosh\left(2x\right)+b\sinh\left(2x\right)

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Q 3: For the function f(t)=e^{-t/\tau} , the Taylor series approximation for t << t is

A) 1+\frac t\tau
B) 1-\frac t\tau
C) 1-\frac{t^2}{2\tau^2}
D) 1+t

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Q 4: A box containing 10 identical compartments has 6 red balls and 2 blue balls. If each compartment can hold only one ball, then the number of different possible arrangements are

A) 1026
B) 1062
C) 1260
D) 1620

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Q 5: Consider the following (2×2) matrix \begin{pmatrix}4&0\\0&4\end{pmatrix} . Which one of the following vectors is NOT a valid eigenvector of the above matrix?

A) \begin{pmatrix}1\\0\end{pmatrix}
B) \begin{pmatrix}-2\\1\end{pmatrix}
C) \begin{pmatrix}4\\-3\end{pmatrix}
D) \begin{pmatrix}0\\0\end{pmatrix}

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Q 6: If a is a constant, then the value of the integral a\int_0^\infty xe^{-ax}\operatorname dx is

A) 1/a
B) a
C) 1
D) 0

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Q 7: The Newton – Raphson method is used to find the roots of the equation

f(x)=x-\cos\pi x\;\;\;0\leq x\leq1

If the initial guess for the root is 0.5, then the value of x after the first iteration is

A) 1.02
B) 0.62
C) 0.55
D) 0.38

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Q 8: If i=\sqrt{-1} , the value of the integral

\oint_c\frac{7z+i}{z\left(z^2+1\right)}dz\;\;\;\left|z\right|<2,

using the Cauchy residue theorem is

A) 2πi
B) 0
C) -6π
D) 6π

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