Engineering Mathematics GATE-1994

Q 1: The inverse of a matrix $\begin{bmatrix}a&0\\0&b\end{bmatrix}$

A) $\begin{bmatrix}ab&0\\0&1\end{bmatrix}$
B) $\begin{bmatrix}b&0\\a&b\end{bmatrix}$
C) $\begin{bmatrix}1/a&0\\0&1/b\end{bmatrix}$
D) $\begin{bmatrix}1/b&0\\0&1/a\end{bmatrix}$

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Q 2: The limit of $f(x)=\frac x{\sin x}$ as x→0 is

A) 0
B) 1
C) 2
D) ∞

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Q 3: Integrating factor for the differential equation $\frac{dy}{dx}+P(x)y=Q(x)$ is

A) $exp\left[\int Pdx\right]$
B) $exp\left[-\int Pdx\right]$
C) $\int Pdx$
D) $dp/dx$

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Q 4: If $\underline i,\underline j,\underline k$ are the unit vectors in rectangular coordinates, then the curl of the vector $i\underline y+jy+\underline kz$

A) $\underline k$
B) $-\underline k$
C) $\underline j+\underline k$
D) $\underline i+\underline k$

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Q 5: The solution for the differential equation $\frac{d^2y}{dx^2}+5\frac{dy}{dx}+6y=0$ is

A) $C_1e^{-2t}+C_2e^{3t}$
B) $C_1\sin2t+C_2\cos2t$
C) $C_1e^{2t}+C_2e^{-3t}$
D) $C_1e^{-2t}+C_2e^{-3t}$

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Q 6: The Taylor’s series expansion of f(x) around x = a is ______________.

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Q 7: For a differential function f(x) to have a maximum, $\frac{df}{dx}$ should be ________ and $\frac{d^2f}{dx^2}$ should be ____________.

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Q 8: $Mdx+Ndy$ is an exact differential when __________.

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Q 9: The integral of $x\sin x$ is ____________.

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Q 10: The Green’s theorem relates _________ integrals to surface integrals.

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Q 11: If ‘a’ is a scalar and $\underline b$ is a vector, then $\nabla\times a\underline b=$ _________.

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Q 12: The differential equation $\frac{d^2y}{dx^2}+y=0$, with the conditions y(0) = 0 and y(1) = 1 is called a _______ value problem.

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Q 13: State with reasons whether the statement is true or false

The series $1+x+x^2+x^3+$ for x < 1 is divergent.

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Q 14: Match the items in the left column with the appropriate items in the right column.

(I) $\cosh at$	(A) $a/\left(s^2+a^2\right)$
(II) $\sinh at$	(B) $a/\left(s^2-a^2\right)$
	(C) $s/\left(s^2-a^2\right)$
	(D) $s/\left(s^2+a^2\right)$

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Q 15: Match the items in the left column with the appropriate items in the right column.

(I) $\frac{dy}{dx}=x^2+y^2$	(A) linear 1st order ODE with constant coefficient
(II) $\frac{dy}{dx}=x^2+y$	(B) linear ODE with variable coefficient
	(C) 1st order nonlinear ODE
	(D) linear 2nd order ODE

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Q 16: Find the eigenvalues of the matrix $\begin{bmatrix}0&2\\-1&-1\end{bmatrix}$

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