Engineering Mathematics GATE-2007

Q 1: Given $i=\sqrt{-1}$, the ratio $\left(\frac{i+3}{i+1}\right)$ is given by

A) i
B) -2
C) -i+2
D) i+1

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Q 2: The value of ‘a’ for which the following set of equations

$y+2z=0$
$2x+y+z=0$
$ax+2y=0$

have a non-trivial solution, is

A) 0
B) 8
C) -2
D) 3

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Q 3: The initial condition for which the following equation

$(x^2+2x)\frac{dy}{dx}=2(x+1)y;\;\;y(x_0)=y_0$

has infinitely many solutions, is

A) y(x=0) = 5
B) y(x=0) = 1
C) y(x=2) = 1
D) y(x=-2) = 0

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Q4: Given that the Laplace transform of the function below over a single period 0 < t < 2 is $\frac1{s^2}\left(1-e^{-s}\right)^2$, the Laplace transform of the periodic function over 0 < t < ∞ is

A) $\frac1s(1-e^{s})^2$
B) $\frac1s(1-e^{-s})^2$
C) $\frac{1\left(1-e^{-s}\right)}{s^2\left(1+e^{-s}\right)}$
D) $\frac1s\tanh\frac s2$

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Q 5: If $z=x+iy$ is a complex number, where $i=\sqrt{-1}$ then the derivative of $z\overline z$ at 2 + i is

A) 0
B) 2
C) 4
D) does not exist

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Q 6: $\overset=A$ and $\overset=B$ are two 3×3 matrix such that $\overset=A=\begin{bmatrix}-2&4&6\\1&2&1\\0&4&4\end{bmatrix}$, $\overset=B=\overset=0$ and $\overset=A\overset=B=\overset=0$. Then the rank of matrix $\overset=B$ is

A) r = 2
B) r < 3
C) r ≤ 3
D) r = 3

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Q 7: The solution of the following differential equation $x\frac{dy}{dx}+y(y^2-1)=2x^3$ is

A) $0$
B) $2+ce^{-x^2/2}$
C) $c_1x+c_2x^2$
D) $2x+cxe^{-x^2/2}$

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Q 8: The directional derivative of $f=\frac12\sqrt{x^2+y^2}$ at (1, 1) in the direction of $\overrightarrow b=\overrightarrow i-\overrightarrow j$ is

A) $0$
B) $1/\sqrt2$
C) $\sqrt2$
D) $2$

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Q 9: Evaluate the following integral (n ≠ 0) $\int_c\left(-xy^ndx+x^nydy\right)$ within the area of a triangle with vertices (0, 0), (1, 0) and (1, 1) (counter-clockwise).

A) 0
B) 1/(n+1)
C) 1/2
D) n/2

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Q 10: The family of curves that is orthogonal to xy = c is

A) $y=c_1x$
B) $y=c_1/x$
C) $y^2+x^2=c_1$
D) $y^2-x^2=c_1$

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Q 11: The Laplace transform of $f(t)=1/\sqrt t$ is

A) $\sqrt{\frac{\mathrm\pi}s}$
B) $\frac1{\sqrt s}$
C) $\frac1{s^{3/2}}$
D) does not exist

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Q 12: The thickness of a conductive coating in micrometers has a probability density function of $600x^{-2}$ for 100 μm < x < 120 μm. The mean and the variance of the coating thickness is

A) 1 μm, 108.39 μm²
B) 33.83 μm, 1 μm²
C) 105 μm, 11 μm²
D) 109.39 μm, 33.83 μm²

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