Engineering Mathematics GATE-1997

Q 1: For the matrix A given below

$A=\begin{bmatrix}2&0&0\\1&4&0\\3&5&6\end{bmatrix}$

(a) calculate its eigen values, and (b) determine the eigen vector corresponding to the lowest eigen value. 

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Q 2: The sum of the infinite series $3+1+\frac13+\left(\frac13\right)^2+\cdots+\left(\frac13\right)^n$ is

A) 9
B) 9/2
C) 15/2
D) ∞

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Q 3: $\lim\limits_{x\rightarrow\infty}\frac{x^3+1}{2x^2+80x+1}$ is

A) 0
B) 1/2
C) 1
D) ∞

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Q 4: The value of $\int_0^{5\mathrm\pi}\left(2-\sin x\right)dx$ is

A) > 0
B) < 0
C) 0
D) undefined

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Q 5: The cubic equation $x^3-x+10=0$ has a root in the interval

A) (-1, 0)
B) (0, 1)
C) (-3, -1)
D) (3, 4)

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Q 6: The Fourier series of the function

$f\left(x\right)=\begin{pmatrix}1&0\leq x\leq\mathrm\pi\\-1&-\mathrm\pi<\mathrm x<0\end{pmatrix}$

extended periodically, $f(x+2\pi)=f(x)$, is

A) a sine series
B) a cosine series
C) a mixed series
D) a power series

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Q 7: Given $f(x,y)=x^2+y^2,\;\nabla^2f$ is

A) 4
B) 2
C) 0
D) 4(x + y)²

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