Solve your IGNOU Doubts

Question:

नीचे दिए गए $\text{Fe}_2(\text{Cp})_2(\text{CO})_4$ के $\text{IR}$ स्पेक्ट्रम से क्या सूचना मिलती है?

Image ignouassignments-ignouacademy-com--p-ignou-72203

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Question:

. कुछ धातु कार्बोनिल स्पीशीज़ के $\text{IR}$ स्पेक्ट्रम में $\text{C}-\text{O}$ तनन आवृत्तियाँ इस प्रकार हैं:

$\text{Ni}(\text{CO})_4 \quad 2056\text{ cm}^{-1}$

$[\text{Fe}(\text{CO})_6]^{2+} \quad 2200\text{ cm}^{-1}$

$\text{CO(g)}$ का मान $2143\text{ cm}^{-1}$ दिया गया है। व्याख्या कीजिए।

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Question:

Find the points where the function $f(z) = \frac{\operatorname{Log}(z + 4)}{z^2 + i}$ is not analytic.

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Question:

b) Consider $f(z) = z^2 - z$ and the closed circular region $R = \{z : |z| \leq 1\}$ . Find points in R where |f(z)| has its maximum and minimum values.

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Question:

नाइट्रिक ऑक्साइड की सहायता से कार्बोनिल संकुल से नाइट्रोसिल संकुल का निर्माण कैसे हो सकता है? इसे दर्शाने के लिए तीन उपयुक्त समीकरण दीजिए।

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Question:

a) If $f = u + iv$ is entire such that $u_x + v_y = 0$ in $\mathbb{C}$ then show that f has the form $f(z) = az + b$ where $a, \mathbf{b}$ are constants with $\operatorname{Re} a = 0$ .

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Question:

संकुल $\text{Fe}_2(\text{CO})_9$ की संरचना को सयोंजकता आबंध दृष्टिकोण से व्याख्या कीजिए।

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Question:

$[\text{Pd}(\eta^3-\text{C}_3\text{H}_5)\text{Cl}]$ जैसे ऐलिल संकुल के संदर्भ में हैप्टिसिटी की अवधारणा को स्पष्ट कीजिए। आबंधन और संरचना के संदर्भ में $\eta^3$ -समन्वय, $\eta^1$ -समन्वय से किस प्रकार भिन्न है?

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Question:

डाइमेथिल बेरेलियम की संरचना दीजिए।

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Question:

Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.
i) If $z = a + ib$ , where a and b are integers, then $|1 + z + z^2 + \dots + z^n| \geq |z|^n$ if a > 0.
ii) If f(z) and $\overline{f(\bar{z})}$ are analytic functions in a domain, then f is necessarily a constant.
iii) A real-valued function u(x, y) is harmonic in D iff u(x, -y) is harmonic in D.
iv) $\lim_{n \to \infty} (n!)^{1/n} = \infty$ .
v) The inequality $|e^a - e^b| \leq |a - b|$ holds for $a, b \in D = \{w : \text{Re } w \leq 0\}$ .
vi) If $f(z) = \sum_{n=0}^{\infty} a_n(z - a)^n$ has the property that $\sum_{n=0}^{\infty} f^{(n)}(a)$ converges, then f is necessarily an entire function.
vii) If a power series $\sum_{n=0}^{\infty} a_n z^n$ converges for |z| < 1 and if $b_n \in \mathbb{C}$ is such that |b_n| < n² |a_n| for all $n \geq 0$ , then $\sum_{n=0}^{\infty} b_n z^n$ converges for |z| < 1.
viii) If f is entire and $f(z) = f(-z)$ for all z, then there exists an entire function g such that $f(z) = g(z^2)$ for all $z \in \mathbb{C}$ .
ix) A mobius transformation which maps the upper half plane $\{z : \text{Im } z > 0\}$ onto itself and fixing $0, \infty$ and no other points, must be of the form $Tz = \alpha z$ for some $\alpha > 0$ and $\alpha \neq 1$ .
x) If f is entire and $\text{Re } f(z)$ is bounded as $|z| \to \infty$ , then f is constant.

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Question:

क्रोमिल क्लोराइड परीक्षण क्या है? उपयुक्त अभिक्रिया दीजिए।

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Question:

Find the SVD of the following matrices:

i) $\begin{bmatrix} -1 & 1 & 1 \\ 1 & 1 & 0 \end{bmatrix}$

ii) $\begin{bmatrix} -1 & 1 \\ 1 & 1 \\ 1 & 2 \end{bmatrix}$

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Question:

Find the QR decomposition of the matrix

$$\begin{bmatrix} 2 & -2 & 1 \\ 2 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}$

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Question:

Check which of the following matrices is positive definite and which is positive semi-definite:
$$A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1 \end{bmatrix}, B = \begin{bmatrix} 2 & 0 & 1 \\ 0 & 2 & -1 \\ 1 & -1 & 3 \end{bmatrix}$
Also, find the square root of the positive definite matrix.

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Question:

Use least squares method to find a quadratic polynomial that fits the following data:

(-2, 15.7), (-1, 6.7), (0, 2.7), (1, 3.7), (2, 9.7).

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Question:

Let
$$A = \begin{bmatrix} 2 & 2 & 1 \\ -1 & -1 & 2 \\ 0 & 0 & -2 \end{bmatrix}.$
Find a unitary matrix U such that U^*AU is upper triangular.

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Question:

Solve the following system of differential equations:
$\quad \frac{dy(t)}{dt} = Ay(t)$ with $y(0) = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ , where $A = \begin{bmatrix} 2 & -5 & -11 \\ 0 & -2 & -9 \\ 0 & 1 & 4 \end{bmatrix}$

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Question:

Let M and T be a metro city and a nearby district town, respectively. Our government is trying to develop infrastructure in T so that people shift to T. Each year $15\%$ of T's population moves to M and $10\%$ of M's population moves to T. What is the long term effect on the population of M and T? Are they likely to stabilise?

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Question:

Find the Jordan canonical form J for

$\quad B = \begin{bmatrix} -1 & 0 & -2 & -4 \\ 2 & 1 & 2 & 4 \\ -4 & 2 & -1 & -4 \\ 2 & -1 & 1 & 3 \end{bmatrix}$ .
$\quad$ Also, find a matrix P such that $J = P^{-1}BP$ .

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Question:

If C and D are $n \times n$ matrices such that $CD = -DC$ and D^-1 exists, then show that C is similar to -C. Hence show that the eigenvalues of C must come in plus-minus pairs.