ISRO Scientist or Engineer Civil 2014

Instructions

For the following questions answer them individually

Question 1

The minimum and maximum eigen value of the matrix $$\begin{bmatrix}1 & 1& 3\\ 1& 5 &1\\3&1&1\end{bmatrix}$$ ]are —2 and 6 respectively. Whatis the other eigen value?

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Question 2

Evaluate $$\lim_{X \rightarrow 1} \frac{X^{X-}X}{X-1- \log X}$$

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Question 3

If the rank of the matrix A is 2, the rank of 2A is

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Question 4

The degree ofthe differential equation $$\frac{d^{2} X}{dt^{2}}+2x^{3}$$ is

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Question 5

$$\int_{0}^{\frac{\pi}{2}} \sin^{2}xdx$$ equal to

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Question 6

Two coins are simultaneously tossed. The probability of two heads simultaneously appearing is

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Question 7

The $$\lim_{x \rightarrow 0} \sin(\frac{\frac{2}{3x}}{x})$$ is

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Question 8

Evaluate $$\int_{0}^{\frac{\pi}{2}} \frac{\surd (\sin x)}{\surd (\sin x)+\surd (\cos x)}$$ dx

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Question 9

Evaluate $$\lim_{x \rightarrow 0} \frac{e^{x} \sin x -x-x^{2}}{x^{2}+xlog(1-x)}$$

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Question 10

The rank of A=$$\begin{bmatrix}0 & 0& 0&0\\ 4& 2 &3&0\\1&0&0&0\\4&0&3&0\end{bmatrix}$$ is

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