Practice Questions on Engineering Mathematics

Q11.

Find the direction directive of
\[f(x, y, z)=x^2+y^2+3z^2\]
at P(1, 1, 1) in the direction of (1, 0, 2).

A.

\(\dfrac{11}{\sqrt{5}}\)

B.

\(\dfrac{14}{\sqrt{5}}\)

C.

\(\dfrac{17}{\sqrt{5}}\)

D.

\(\dfrac{21}{\sqrt{5}}\)

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Q12.

A solution for the differential equation $ \dot{x(t)}-5x(t)=\delta (t) $ with initial condition $ x(0^{-}=1) $ is:

A.

$ e^{-t}u(t) $

B.

$ te^{5t}u(t) $

C.

$2e^{5t} u(t)  $

D.

$ te^{-2t}u(t)  $

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Q13.

Given

\(\begin{align*} f(z)=\dfrac{1}{z+3}+\dfrac{1}{z+1}\\ \end{align*}\)

What is the value of \[\dfrac{1}{2\pi j}\oint f(z)dz\] if C is a counter clockwise path in the z-plane such that $ |z+1|=1$ ?

A.

3

B.

2

C.

1

D.

0

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Q14.

Find the Fourier Transform of $ \dfrac{1}{2}e^{-4t}u(t-2) $ 

A.

$ \dfrac{e^{-2j\omega}}{4+j\omega} $

B.

$ \dfrac{1}{2}\dfrac{e^{-2(j\omega-4)}+6}{4+j\omega} $

C.

$ \dfrac{e^{2j\omega}}{4+j\omega} $

D.

$ \dfrac{1}{2}\dfrac{e^{-2(j\omega+4)}}{4+j\omega} $

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Q15.

If   \( f(x)=\sin 3x,\;0\leq x\leq \dfrac{\pi}{2} \)  and  f'(c)=0  for \(c\leftrightarrow ]0,\dfrac{\pi}{2}[\) Then, the value of $ c $ is equal to:

A.

$\dfrac{\pi}{4}$

B.

$\dfrac{\pi}{3}$

C.

$\dfrac{\pi}{6}$

D.

0

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