IGNOU PHE 14 SOLVED ASSIGNMENT 2025

Ques 1.

Determine the eigenvalues and eigenvectors of the following matrix A:

Ques 2.

Show that the eigenvectors belonging to distinct eigenvalues of a hermitian matrix are orthogonal to each other.

Ques 3.

For the quadratic equation 2x² + 4xy-y² = 24, write down the matrix of coefficients and diagonalize it. Recast it in new variables and identify the conic section it represents. 

Ques 4.

Define covariant and contravariant vectors. Show that velocity and acceleration are contravariant vectors.

Ques 5.

Show that the set of all real numbers excluding zero is a group under multiplication. 

Ques 6.

Using the method of residues, evaluate the contour integral,

      where C is defined by |z| < 4.

Ques 7.

Evaluate the integral

Ques 8.

Using the Cauchy-Riemann equations, show that w = log z is analytic when z is not zero.

Ques 9.

Write Laurent series for the function

about the pole z =-2.

Ques 10.

Determine the Fourier transform of the function f(t) = e ^-|t|. Hence deduce that:

Ques 11.

Using the definition determine the Laplace transform of the function f(t) = e ^3t if it exists.

Ques 12.

Using the method of Laplace transform, solve the following initial value problem:

y"-10y'+9y=5t;    y(0)=-1, y'(0) = 2

Ques 13.

Obtain the first three terms in the expansion of the following function in a series of the form  :

Ques 14.

Using the following expression for a Bessel function of order m:

show that

i) 

ii) 

Ques 15.

Using Rodrigue’s formula, obtain the expression for L₃ (x).

Ques 16.

निम्नलिखित आव्यूह 4 के आइगेन मान और आइगेन सदिष ज्ञात कीजिए।

Ques 17.

सिद्ध करें कि हर्मिटी आव्यूह के भिन्न आइगेन मानों के संगत आइगेन सदिष एक दूसरे के प्रति लांबिक होते हैं।

Ques 18.

द्विघात समीकरण 2x²+4xy-y² = 24, के गुणांकों का आव्यूह लिखिए और उसका विकर्णन कीजिए। उसे नए चरों में लिखिए और बताइए कि यह किस शांकव परिच्छेद को निरूपित करता है।

Ques 19.

सहपरिवर्ती सदिष और प्रतिपरिवर्ती सदिष की परिभाषा दें। सिद्ध करें कि वेग और त्वरण प्रतिपरिवर्ती सदिष हैं।

Ques 20.

अवयव शून्य को छोड़कर सभी वास्तविक संख्याओं का समुच्चय गुणन के अधीन एक समूह होता है।

Ques 21.

अवषिष्ट विधि का उपयोग कर, कंटूर समाकल को परिकलित करें :

                जहां C, |=| < 4 द्वारा परिभाषित है।

Ques 22.

निम्नलिखित समाकल को परिकलित कीजिए :

Ques 23.

कौषी-रीमॉन समीकरणों का उपयोग कर सिद्ध करें कि w = logz विष्लेषिक है जब शून्य नहीं है।

Ques 24.

z=-2 अनंतक के इर्द-गिर्द फलन     की लौरों श्रेणी लिखें।

Ques 25.

फलन f(t) = e ^-|t| का फूरिए रूपांतर प्राप्त करें। इससे सिद्ध करें कि :

Ques 26.

यदि फलन f(t) = e^3t के लाप्लास रूपांतर का अस्तित्व है, तो परिभाषा का प्रयोग करते हुए उसका लाप्लास रूपांतर ज्ञात करें।

Ques 27.

लाप्लास रूपांतर लागू करके आदि मान समस्या :

का हल करें।

Ques 28.

निम्नलिखित फलन :

का प्रसार के रूप की श्रेणी में करते हुए उसके प्रथम तीन पद प्राप्त करें। 

Ques 29.

कोटि के बेसल फलन के लिए निम्नलिखित व्यंजक का प्रयोग करते हुए

सिद्ध करें कि :

i) j₀ (x) = j₁(x)

ii) 

Ques 30.

रोड्रिगेज़ सूत्र की सहायता से L₃(x) का व्यंजक प्राप्त करें।

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