Question 1 |

Given f(z)=g(z)+h(z), where f, g, h are complex valued functions of a complex variable z. Which one of the following statements is TRUE?

If f(z) is differentiable at z_0, then g(z) and g(z) are also differentiable at z_0. | |

If g(z) and h(z) are differentiable at z_0, then f(z) is also differentiable at z_0. | |

If f(z) is continuous at z_0, then it is differentiable at z_0. | |

If f(z) is differentiable at z_0, then so are its real and imaginary parts. |

Question 2 |

We have a set of 3 linear equations in 3 unknowns. 'X \equiv Y' means X and Y are equivalent statements and 'X\not\equiv Y' means X and Y are not equivalent statements.

P: There is a unique solution.

Q: The equations are linearly independent.

R: All eigenvalues of the coefficient matrix are nonzero.

S: The determinant of the coefficient matrix is nonzero.

Which one of the following is TRUE?

P: There is a unique solution.

Q: The equations are linearly independent.

R: All eigenvalues of the coefficient matrix are nonzero.

S: The determinant of the coefficient matrix is nonzero.

Which one of the following is TRUE?

P\equiv Q\equiv R\equiv S | |

P\equiv R\not\equiv Q\equiv S | |

P\equiv Q\not\equiv R\equiv S | |

P\not\equiv Q\not\equiv R\not\equiv S |

Question 2 Explanation:

\begin{aligned} a_{11}x_1+a_{12}x_2+a_{13}x_3 &=b_1 \\ a_{21}x_1+a_{22}x_2+a_{23}x_3 &=b_2 \\ a_{31}x_1+a_{32}x_2+a_{33}x_3 &=b_3 \\ \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22}& a_{21}\\ a_{31}& a_{32} &a_{33} \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}&= \begin{bmatrix} b_1\\ b_2\\ b_3 \end{bmatrix} \end{aligned}

if |A|\neq 0 then AX=B can be written as X=A^{-1}B

It leads unique solutions.

If |A|\neq 0 then \lambda _1 \cdot \lambda _2\cdot \lambda _3\neq 0 each \lambda _i is non-zero.

If |A|\neq 0 then all the row (column) vectors of A are linearly independent.

if |A|\neq 0 then AX=B can be written as X=A^{-1}B

It leads unique solutions.

If |A|\neq 0 then \lambda _1 \cdot \lambda _2\cdot \lambda _3\neq 0 each \lambda _i is non-zero.

If |A|\neq 0 then all the row (column) vectors of A are linearly independent.

Question 3 |

Match the following.

P-2 Q-1 R-4 S-3 | |

P-4 Q-1 R-3 S-2 | |

P-4 Q-3 R-1 S-2 | |

P-3 Q-4 R-2 S-1 |

Question 3 Explanation:

Stokes theorem \oint \vec{A}\cdot dl=\int \int (\triangledown \times A)\cdot \hat{n}ds

Gauss theorem \int \int D\cdot ds=Q

Divergence theorem \oint \oint A\cdot \hat{n}ds=\int \int \int \triangledown \cdot \bar{A}dV

Cauchy integral theorem \oint _cf(z)dz=0

Gauss theorem \int \int D\cdot ds=Q

Divergence theorem \oint \oint A\cdot \hat{n}ds=\int \int \int \triangledown \cdot \bar{A}dV

Cauchy integral theorem \oint _cf(z)dz=0

Question 4 |

The Laplace transform of f(t)=2\sqrt{t/\pi } is s^{-3/2}. The Laplace transform of g(t)=\sqrt{1/\pi t} is

3s^{-5/2}/2 | |

s^{-1/2} | |

s^{1/2} | |

s^{3/2} |

Question 4 Explanation:

Given that,

f(t)=2\sqrt{\frac{t}{\pi}}\rightleftharpoons F(s)=s^{-3/2}

By using property of differentiation In time,

\begin{aligned} \frac{df(t)}{dt}&\rightleftharpoons sF(s) \\ \Rightarrow \; \frac{2}{\sqrt{\pi}}\cdot \frac{1}{2}t^{-1/2}&\rightleftharpoons sF(s) \\ \Rightarrow \; \frac{1}{\sqrt{\pi t}}&\rightleftharpoons s\cdot s^{-3/2} \\ \Rightarrow \; \frac{1}{\sqrt{\pi t}}&\rightleftharpoons s^{-1/2} \end{aligned}

f(t)=2\sqrt{\frac{t}{\pi}}\rightleftharpoons F(s)=s^{-3/2}

By using property of differentiation In time,

\begin{aligned} \frac{df(t)}{dt}&\rightleftharpoons sF(s) \\ \Rightarrow \; \frac{2}{\sqrt{\pi}}\cdot \frac{1}{2}t^{-1/2}&\rightleftharpoons sF(s) \\ \Rightarrow \; \frac{1}{\sqrt{\pi t}}&\rightleftharpoons s\cdot s^{-3/2} \\ \Rightarrow \; \frac{1}{\sqrt{\pi t}}&\rightleftharpoons s^{-1/2} \end{aligned}

Question 5 |

Match the following.

P-1 Q-2 R-1 S-3 | |

P-1 Q-2 R-1 S-3 | |

P-1 Q-2 R-3 S-3 | |

P-3 Q-1 R-2 S-1 |

There are 5 questions to complete.