Engineering Mathematics GATE-1996

Q 1: The ratio

$\frac{\int_{x_1}^{x_2}{\displaystyle\frac{dy}x}}{{\displaystyle\frac{x_2-x_1}y}\;at\;x=x_2}(x_2>x_1),$

Where is 1/y a monotonically increasing function of x, is

A) less than unity
B) equal to unity
C) greater than unity
D) less than zero

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Q 2: Solve $\frac{d^2y}{dx^2}-m^2y=0$ subject to y = 1 at x = 0 and $\frac{dy}{dx}=0$ at x = 1.

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Q 3: Given the matrix $\overset=A=\begin{bmatrix}-1&-2\\3&4\end{bmatrix}$ (a) write down the characteristic equation, (b) compute $\left(\overset=A\right)^4$ without direct multiplication.

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Q 4: Solve $\frac{dy}{dx}+0.6y=6e^{-0.5x}$ using the integrating factor method, given y = 1 at x = 0.

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