IGNOU MTE 13 SOLVED ASSIGNMENT 2025

Ques 1.

जाँच कीजिए कि निम्नजिक्षित कथन सत्य है या असमय। अपने अतों की पुष्टि एक लघु उपपरित या प्रतिउदाहरण देकर कीजिए।

i) यदि किसी कथा का प्रतिस्थितकस्य है, तो वह कथन स्वयं भी साय होगा।

ii)  एक रैखिक समधात पुनरावृणित संबंध है।

iii) पुनरावृतिसंबंध के एक विशेष हारका रूप है।

iv) चरी x₁,x₂ और x₃ में एक ऐसा बूलीय व्यंजक है जिसका CNF

 है।

v) यदि एक पासे को तीन बार कैका जाता है तो प्रत्येक बार प्राप्त होने की प्रायिकता  है।

vi) प्रत्येक विषम चक्र की वर्णिक संख्या और कोर वर्णिक संख्यामा होती है।

vii) प्रत्येक ऑयलरीय ग्राफ हैमिल्टोनीय है।

viii)  उन तरीकों की संख्या को दर्शाता है जिनमें किन्हीं 3 वस्तुओं को 4 संदूकों में रखा जाना है।

ix) 5 या अधिक शीर्षों पर एक स्वपूरक समतलीय याक का अभिनव है।

x) 6 के विभाजनों की संख्या 10 है।

Ques 2.

सूरीय व्यंजक  कार्क परिपथ बनाइए।

Ques 3.

निम्नलिखित कराको प्रकरूप में लिखिए।

i) बगीचे में एक नीली आँख वाला आदमी है।

ii) बगीचे में नीली आँखों वाले प्रायेक आद‌मी ने एक लाल टोपी पहनी हुई है।

iii)यदि किसी आद‌मी ने कोई भी टोपी नहीं पहनी है, तो उसकी आँकाली है।

Ques 4.

जनक फलों के प्रयोग से जातकीजिए।

Ques 5.

साद "COMBINSTORICS" के अको व्यवस्थित करने के ग 77 करोड़ तरीके हैं। इन मरीकी की ठीक-ठीक संख्या जात कीजिए।

Ques 6.

पुनरावृ‌त्ति संबंध

को हम कीजिए।

Ques 7.

समुच्चय  से (1,2,3,4) पर सभी आच्छादक फसनी की सूची बनाइए। समुच्चय से (1,2,3,4,5) पर किन आउाद‌क फलन हैं?

Ques 8.

किन्हीं कथनी  p,q और r के लिए, सिद्ध कीजिए।

Ques 9.

नीचे दिए ग्राफ के तीन अतुल्यकारी प्रेरित उपया बाहर, जिनमें प्रत्येक के शीर्षों की संख्या समान हो। अपने चयन की पुष्टि कीजिए।

Ques 10.

क्या पिटर्सन साफ का पूरक समीय है? अपने उत्तर की पुष्टि कीजिए।

Ques 11.

एक ग्राफ के उपविभाजन से आप क्या समोर का एक हैमिल्टनी की उपविभाजन हैमिल्टीनीया होता है? पुष्टि कीजिए।

Ques 12.

MTE-13 की जून, 2021 संयंत्र परीक्षा में निम्नलिखित कथन की एक फायक्ष और एक परीक्षा उपपन्ति देने के लिए कहा गया था।

यदि इस प्रकार है कि सम है और  भी सम है, तो b एक सम संख्या होगी

एक छात्र ने परीक्ष उपपति इस प्रकार दी।

"मान लीजिए b बराबरm+1 है, जोकि एक विषम संख्या है। हम पहले से ही जानते हैं कि a और  सम संख्याएँ हैं। यदि हम  में b =m+1 रख दें तो यह a + m + 1 हो जाती है, जो कि विषम संख्या है। इससे दिए गए कथन का विरोध उत्पन होता है। इसलिए, ब एक सम संख्या है।"

उपरोक्त उपपन्ति में क्या गलत है? साथ ही, एक सही प्रत्यक्ष एवं एक परोक्ष उपपन्ति दीजिए।

Ques 13.

संख्या 7 के सभी विभाजन को लिखिए और उनकी गणना कीजिए। अपनी उत्तर की जाँच के लिए खंड 2 की इकाई 5 के प्रमेया 5 में  लेकर p_n जनक नाका प्रयोग कीजिए। 

Ques 14.

x⁵को पतली कमणिली के रूप में लिखिए और इस प्रकार  के लिए  का मान

Ques 15.

यदि हमें केवल मारुति 800, टाटा सफारी या स्कोर्पियो को ही व्यवस्थित करना है तो इन माडलों की कारों को ॥ स्थानों वाली एक पंक्ति में व्यवस्थित करने के तरीकों की संख्या व, के लिए एक पुनरावृत्ति संबंध ज्ञात कीजिए। एक टाटा सफारी या स्कोर्पियो को दो स्थानों की जरूरत होती है जबकि एक मारुति 800 को केवल एक स्थान की जरूरत होती है। मान कर चलिए कि आपके पास प्रत्येक माडल की अनगिनत कारें हैं, और हम एक ही माडल की दो कारों में फर्क नहीं करते हैं।

Ques 16.

यदि  के लिए Km,n हैमिल्टोनी है पुष्टि कीजिए। और किस प्रकार संबंधित हैं? अपने उत्तर की 

Ques 17.

यदि 50 साइकिली की रंगने के लिए रंग का प्रयोग किया गया है और प्रत्येक साइकिल को एक ही रंग से रंगा गया है, जो दिखाइए कि कम से कम साइकिलों को एक ही रंग से रंगा गया है।

Ques 18.

एक संदूक में 6 साल और 4 ही गई हैं। क्या प्रायिकता है कि संदूक से याद‌छया चुनी हुई चार गौदों में से दो गर्दै जाम और दोहरी हैं?

Ques 19.

किसी ग्राफ G के लिए शीर्ष-संबद्वतांक और काट शीर्ष समुच्चय परिभाषित कीजिए। नीचे दिए गए ग्राफ

के लिए शीर्ष - संबद्वतांक और काट शीर्ष समुच्चय ज्ञात कीजिए।

Ques 20.

0 से 759 तक की संख्या में से किसी संसाली या से विभाजन नहीं है? 

क) पुनरावृ‌नित संबंध

की जनक फना विधि से हम कीजिए। साथ ही अपनी उन्तार से जात कीजिए।

Ques 21.

क्या 7 सीधी पर कोई 4-नियमित साफ है अपने उत्तम की पुष्टि कीजिए। 

Ques 22.

नीचे दी हुई तालिका में परिभाषित करन के लिए खूनीयाव्यांजक ज्ञात कीजिए।

Ques 23.

Check whether the following statements are true or not. Justify your answers with a short proof or a counter example.

i) If the contrapositive of a statement is true, then the statement itself is also true.

ii) a_n+3a_n-1+2a_n-2= 2" is a linear homogeneous recurrence relation.

iii) A particular solution of the recurrence relation a_n-2a_n-1 + a_n-2=1 has the form Cn².

iv) There exists a boolean expression in variables x,x, and x, with CNF as

v) If a dice is rolled thrice, then the probability of getting a 6 each time is 

vi) Every odd cycle has the same chromatic and edge chromatic numbers.

vii) Every Eulerian graph is Hamiltonian.

viii) gives the number of ways in which any 3 objects can be placed in any 4 boxes.

ix) There exists a self-complementary planar graph on 5 or more vertices.

x) The number of partitions of 6 is 10.

Ques 24.

Draw the logic circuit for the Boolean expression

Ques 25.

Express the following statements in symbolic form.

i) There is a man in the park with blue eyes.

ii) Every blue-eyed man in the park is wearing a red hat.

iii) If a man wears no hat, then he has black eyes.

Ques 26.

Using generating functions find 

Ques 27.

There are about 77 crore ways to arrange the letters of the word “COMBINATORICS”. Count the exact number of such ways.

Ques 28.

Solve the recurrence relation:

Ques 29.

List all the onto mappings from the set {a,b,c,d} to {1,2,3,4}. How many onto mappings are there form {a,b,c,d} to {1,2,3,4,5}?

Ques 30.

For any statements p,q and r, prove that

Ques 31.

Draw three nonisomorphic induced subgraphs of the following graph, each having the same number of vertices. Justify your choice.

Ques 32.

Is the complement of the Peterson graph planar? Justify your answer.

Ques 33.

What do you understand by a subdivision of a graph? Is every subdivision of a Hamiltonian graph Hamiltonian? Justify.

Ques 34.

In the June, 2021 Term-End Examination of MTE-13, it was asked to give a direct and an indirect proof of the following statement.

"If a,b ∈ Z such that a is even and a+b is even, then b is even."

One student gave an indirect proof as follows:

"Let b be m+1, which is an odd number. We already know a and a+b are even. If we substitute b=m+1 in a + b, it becomes a+m+1, which is an odd number. This contradicts the given statement. Hence b is an even number."

What is wrong with the above proof? Also give a correct direct and an indirect proof.

Ques 35.

write down and count all the partitions of the number 7. To verify your answer use the generating function for P, taking n =7 in Theorem 5 (of Unit 5, Block2).

Ques 36.

Express x⁵ in terms of falling factorials and hence evaluate for m = 0,1,2,3,4,5.

Ques 37.

Find a recurrence relation for a_n, the number of ways to arrange cars in a row with n spaces if we can use Maruti 800, Tata Safari or Scorpio. A Tata Safari or Scorpio requires two spaces, whereas a Maruti 800 requires just one space. Assume that you have unlimited number of each type of car and we do not distinguish between 2 cars of the same type.

Ques 38.

If K_m,n for m, n ≥ 2 is Hamiltonian, how are m and n related? Justify your answer.

Ques 39.

Show that if 7 colours are used to paint 50 bicycles and each bicycle is coloured with a single colour, at least 8 bicycles will have the same colour.

Ques 40.

A box contains 6 red and 4 green balls. Four balls are selected from the box at random. What is the probability that two of the selected balls will be red and two will be green?

Ques 41.

Define vertex connectivity and cut vertex set of any graph G. Find the vertex connectivity and cut vertex set for the following graph:

Ques 42.

How many numbers from 0 to 759 are not divisible by either 3 or 7?

Ques 43.

Solve the recurrence relation:

using generating function technique. Also find a₅ using your answer.

Ques 44.

Is there a 4-regular graph on 7 vertices? Justify your answer.

Ques 45.

Find the Boolean expression in the DNF form for the function defined in tabular form below:

Ques 46.

Check whether the following statements are true or not. Justify your answers with a short proof or a counter example

i) If the contrapositive of a statement is true, then the statement itself is also true.

ii)   is a linear homogeneous recurrence relation.

iii) A particular solution of the recurrence relationhas the form 

iv) There exists a boolean expression in variables 

v) If a dice is rolled thrice, then the probability of getting a 6 each time is 

vi) Every odd cycle has the same chromatic and edge chromatic numbers.

vii) Every Eulerian graph is Hamiltonian.

viii) 3 4 S gives the number of ways in which any 3 objects can be placed in any 4 boxes.

ix) There exists a self-complementary planar graph on 5 or more vertices.

x) The number of partitions of 6 is 10.

Ques 47.

Draw the logic circuit for the Boolean expression

Ques 48.

Express the following statements in symbolic form.

i) There is a man in the park with blue eyes.

ii) Every blue-eyed man in the park is wearing a red hat.

iii) If a man wears no hat, then he has black eyes

Ques 49.

Using generating functions find 

Ques 50.

There are about 77 crore ways to arrange the letters of the word “COMBINATORICS”. Count the exact number of such ways.

Ques 51.

Solve the recurrence relation:

Ques 52.

List all the onto mappings from the set {a,b,c,d} to {1,2,3,4}. How many onto

mappings are there form {a,b,c,d} to {1,2,3,4,5}?

Ques 53.

For any statements p,q and r, prove that

Ques 54.

Draw three nonisomorphic induced subgraphs of the following graph, each having the same number of vertices. Justify your choice.

Ques 55.

Is the complement of the Peterson graph planar? Justify your answer.b) Is the complement of the Peterson graph planar? Justify your answer.

Ques 56.

Is the complement of the Peterson graph planar? Justify your answer.

Ques 57.

What do you understand by a subdivision of a graph? Is every subdivision of a Hamiltonian graph Hamiltonian? Justify.

Ques 58.

In the June, 2021 Term-End Examination of MTE-13, it was asked to give a direct and an indirect proof of the following statement.

such that a is even and a +b is even, then   b is even."

One student gave an indirect proof as follows:

“Let b be m +1, which is an odd number. We already know a and a +b are even. If we substitute b + m +1 in a + b, it becomes a + m +1, which is an odd number. This contradicts the given statement. Hence b is an even number.”

What is wrong with the above proof ? Also give a correct direct and an indirect proof.

Ques 59.

write down and count all the partitions of the number 7. To verify your answer use the generating function for , Pn taking n= 7 in Theorem 5 (of Unit 5, Block2).

Ques 60.

Express 5 x in terms of falling factorials and hence evaluate m S5 for m = 0,1,2,3,4,5.

Ques 61.

Find a recurrence relation for an , the number of ways to arrange cars in a row with n spaces if we can use Maruti 800, Tata Safari or Scorpio. A Tata Safari or Scorpio requires two spaces, whereas a Maruti 800 requires just one space. Assume that you have unlimited number of each type of car and we do not distinguish between 2 cars of the same type.

Ques 62.

  is Hamiltonian, how are m and n related? Justify your answer

Ques 63.

Show that if 7 colours are used to paint 50 bicycles and each bicycle is coloured with a single colour, at least 8 bicycles will have the same colour.

Ques 64.

A box contains 6 red and 4 green balls. Four balls are selected from the box at random. What is the probability that two of the selected balls will be red and two will be green? 

Ques 65.

Define vertex connectivity and cut vertex set of any graph G. Find the vertex connectivity and cut vertex set for the following graph:

Ques 66.

How many numbers from 0 to 759 are not divisible by either 3 or 7?

Ques 67.

Solve the recurrence relation:

using generating function technique. Also find 5 a using your answer.

Ques 68.

Is there a 4-regular graph on 7 vertices? Justify your answer.

Ques 69.

Find the Boolean expression in the DNF form for the function defined in tabular form below:

x	y	z
1	0	1	1
0	1	0	0
0	0	1	1
1	1	1	1
1	0	0	0
0	1	1	1
1	1	0	1
0	0	0	0

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