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BMTC 103: Discrete Mathematics

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Title Name	IGNOU BMTC 103 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSCFMT
Course Name	Bachelor of Science (Mathematics)
Subject Code	BMTC 103
Subject Name	Discrete Mathematics
Year	2026
Session	-
Language	English Medium
Assignment Code	BMTC 103/Assignment-1/2026
Product Description	Assignment of BSCFMT (Bachelor of Science (Mathematics)) 2026. Latest BMTC 103 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Format	Ready-to-Print PDF (.soft copy)

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January 2026 Session: 31st March, 2026
July 2026 Session: 30th September, 2026
January 2026 Session: 31st March, 2026
July 2026 Session: 30th September, 2026

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BMTC 103 (January 2026 - July 2026) - ENGLISH

Assignment
(To be done after reading the course material)

Course Code: BMTC-103
Assignment Code: BMTC-103/TMA/2026
Maximum Marks: 100

1. Which of the following statements are true and which are false? Give reasons for your answer.

(i) ‘x² + y² - 3 is not divisible by 4.’ is a mathematical statement.

(ii) The number of onto functions from $\{1, 2, 3, 4, 5, 6\}$ to $\{a, b, c, d\}$ is 4!S₆⁴.

(iii) The generating function associated with a sequence can never be a polynomial.

(iv) $n \log n \in \mathcal{O}(n^2)$ .

(v) Selection sort is faster than insertion sort in worst case.

(vi) $a_n = a_{\frac{n}{2}} + n, a_1 = 0$ , where n is a power of 2, is a linear recurrence relation.

(vii) The generating function of the sequence $\{1, 2, 3, 4, \dots, n \dots\}$ is (1 - z)^-2.

(viii) If g(x) is the generating function for $\{a_n\}_{n \ge 1}$ , then (1 - x)g(x) is the generating function for the sequence $\{b_n\}_{n \ge 1}$ where $b_n = a_n - a_{n-1}, \forall n$ .

(ix) The number of partitions of 10 with no part larger than 5 is Q₁₀⁵.
(x) The number of integer solutions of the equation $x_1 + x_2 + \dots + x_{10} = 50$ in positive integers is $\binom{49}{9}$ .

2. (a) Consider the propositions ‘ $2 + 3 = 5$ ’ and 'The Sun rises in the West'.

i. Write the disjunction of the statements and give its truth value.

ii. Write the conjunction of the statements and give its truth value.

iii. Write the exclusive disjunction of the statements and give its truth value.

(b) If p and q are any statements, show that the statement

$$(p \rightarrow q) \land (\sim q) \rightarrow (\sim p)$

is a tautology.

(c) What is the probability of selecting 2 red and 3 green balls from a bag containing 4 red balls and 8 green balls?

(d) Write down the converse of each of the following statements:

i. If p is a prime number and a and b are any two natural numbers and if p divides a or b, then p divides ab.

ii. In a triangle $\triangle ABC$ , if $AB^2 + AC^2 = BC^2$ , then $\angle BAC = 90^{\circ}$ .

3. (a) If the solution of the recurrence relation $u_n + \alpha u_{n-1} + \beta u_{n-2} = f(n), (n \ge 2)$ is $u_n = 1 - 2n + 3.2^n$ , then determine the values of $\alpha, \beta$ and f(n).

(b) A bank pays you $4.5\%$ interest per year. In addition, you receive Rs. 100 as bonus at the end of the year (after the interest is paid). Find a recurrence for the amount of money after n years if you invest Rs. 2000.

(c) If 5 points are chosen in a square of side 2cm, show that there will always be two points at a distance of at most $\sqrt{2}$ cm.

4. (a) A die is rolled twice and the sum of the numbers that appear is observed. What is the probability that the sum is either a perfect square or a perfect cube?

(b) Write an algorithm that computes the sum of the squares of all the odd integers lying between two given numbers p and q, where $p \le q$ .

(c) For any natural numbers n and k, prove that $P_n^k = P_{n-1}^{k-1} + P_{n-k}^k$ .

5. (a) Write the expression $x_1 \lor x_2 \land x_3 \lor x_4$ in conjunction normal form and disjunctive normal form.

(b) Find the general form of the solution to a linear homogeneous recurrence relation with constant coefficients for which the characteristic roots are 4, -2 and 3 with multiplicities 2, 1 and 3, respectively. The relation also has a non-homogeneous part which is a linear combination of 3ⁿ and 4ⁿ.

6. (a) Use generating function to prove the identity

$$\sum_{k=0}^n \binom{r}{k} \binom{s}{n - k} = \binom{r + s}{n}.$

(b) Compute the time complexity of the merge sort algorithm.

7. (a) How many numbers form 0 to 999 (0 and 999 inclusive) are not divisible by 7 or 11?

(b) From a survey of 120 people, the following data was obtained:

90 owned a car, 35 owned a computer, 40 owned a house, 32 owned a car and a house, 21 owned a house and a computer, 26 owned a car and a computer, 17 owned all the three facilities.

i. How many people owned neither of the three.

ii. How many people owned only a car?

iii. How many people owned only a computer?

8. (a) Find the generating function of the recurrence $a_n = 6a_{n-1} - 5a_{n-2} + 1$ with initial conditions $a_0 = 2, a_1 = 5$ .

(b) Compute the Bell numbers B₆ and B₇.

(c) Draw the Ferrar graph of the partition $20 = 9 + 6 + 3 + 2$ . Also, write down the conjugate partition of this partition.

(d) A typical number plate for a four-wheeler in Delhi looks like DL3AXC4126, where the first two characters are fixed as DL, the third one is a digit, then the next three characters are letters, and finally the last four characters are digits. How many number plates in Delhi can be registered with this pattern?

BMTC 103 (January 2026 - July 2026) - HINDI

सत्रीय कार्य

पाठ्यक्रम कोड : BMTC-103
सत्रीय कार्य कोड : BMTC-103/TMA/2026
अधिकतम अंक : 100

1. जाँच कीजिए कि निम्नलिखित कथन सत्य हैं या असत्य। अपने उत्तरों की पुष्टि एक लघु उपपत्ति या प्रतिउदाहरण देकर कीजिए। (20)

i) "x² + y² - 3, 4 से विभाज्य नहीं है।" एक गणितीय कथन है।

ii) $\{1, 2, 3, 4, 5, 6\}$ से $\{a, b, c, d\}$ पर आच्छादित फलनों की संख्या 4! S₆⁴ है।

iii) किसी अनुक्रम का जनक फलन कभी भी एक बहुपद नहीं हो सकता।

iv) $n \log n \in O(n^2)$ है।

v) निकृष्टतम स्थिति में, वरणात्मक शाटन प्रविष्टि शाटन से तीव्र होता है।

vi) $a_n = a_{\frac{n}{2}} + n, a_1 = 0$ , जहाँ n, 2 का घात है, एक रैखिक पुनरावृत्ति संबंध है।

vii) अनुक्रम $\{1, 2, 3, \dots, n, \dots\}$ का जनक फलन (1 - z)^-2 है।

viii) यदि $g(x), \{a_n\}_{n \ge 1}$ का जनक फलन है, तो (1 - x)g(x) अनुक्रम $\{b_n\}_{n \ge 1}$ का जनक फलन है, जहाँ सभी $n \ge 1$ के लिए $b_n = a_n - 1$ है।

ix) 10 के उन विभाजनों की संख्या जिनमें कोई भी भाग 5 से बड़ा नहीं है, Q₁₀⁵ है।
x) समीकरण $x_1 + x_2 + \dots + x_{10} = 50$ के धन पूर्णांक हलों की संख्या $\binom{49}{9}$ है।

2. क) कथनों '2+3=5' और 'सूरज पश्चिम में उगता है।' पर विचार कीजिए।

i) कथनों का वियोजन लिखिए और इसका सत्य मान बताइए।

ii) कथनों का योग लिखिए और इसका सत्य मान बताइए।

iii) कथनों का अपवर्जी वियोजन लिखिए और इसका सत्य मान बताइए।

ख) यदि p और q कोई भी कथन हैं, तो दर्शाइए कि कथन $(p \rightarrow q) \wedge (\sim q) \rightarrow (\sim p)$ एक सर्वसत्य कथन है।

ग) 4 लाल और 8 हरी गेंदों वाले एक बस्ते से 2 हरी और 3 लाल गेंदें चुनने की प्रायिकता क्या है ?

घ) निम्नलिखित में से प्रत्येक कथन का विलोम लिखिए:

i) यदि p एक अभाज्य संख्या है, और a और b दो प्राकृतिक संख्याएँ हैं, और p, a या b को विभाजित करता है , तो p, ab को विभाजित करता है।

ii) एक त्रिभुज $\Delta ABC$ में, यदि $AB^2 + AC^2 = BC^2$ है, तो $\angle BAC = 90^\circ$ होगा।

3. क) यदि पुनरावृत्ति संबंध $u_n + \alpha u_{n-1} + \beta u_{n-2} = f(n), (n \ge 2)$ का हल $u_n = 1 - 2n + 3 \cdot 2^n$ है , तो $\alpha, \beta$ और f(n) के मान ज्ञात कीजिए।

ख) एक बैंक आपको 4.5% प्रति वर्ष ब्याज देता है। इसके अतिरिक्त, (ब्याज के भुगतान के बाद) आपको वर्ष के अंत में 100/- रुपए का बोनस मिलता है। यदि आपने 2000 रुपए निवेश किए हैं तो आपको n वर्ष के उपरांत मिलने वाली धनराशि के लिए एक पुनरावृत्ति संबंध बनाइए।

ग) यदि 2 सेमी. भुजा वाले एक वर्ग में 5 बिंदु चुने जाते हैं, तो दिखाइए कि इनमें से दो बिंदु ऐसे हमेशा होंगे जिनके बीच की दूरी अधिकतम $\sqrt{2}$ सेमी. होगी।

4. क) एक पासा दो बार फेंका जाता है, और उस पर आने वाली संख्याओं के योगफल का अवलोकन किया जाता है। इस बात की प्रायिकता क्या है कि यह योगफल या तो एक पूर्ण वर्ग है या एक पूर्ण घन है?

ख) एक ऐसी कलन-विधि लिखिए जो किन्हीं दो दी हुई संख्याओं p और q, जहाँ $p \le q$ , के बीच के सभी विषम पूर्णांकों के वर्गों का योगफल ज्ञात करती है।

ग) किन्हीं भी प्राकृतिक संख्याओं n और k के लिए, सिद्ध कीजिए कि $P_n^k = P_{n-1}^k + k P_{n-1}^{k-1}$ होता है।

5. क) व्यंजक $x_1 \vee x_2 \wedge x_3 \vee x_4$ को सर्वनिष्ठ प्रसामान्य समघात और सम्मिलन प्रसामान्य समघात के रूप में लिखिए।

ख) अचर गुणांकों वाले एक रैखिक समघात पुनरावृत्ति संबंध का व्यापक हल ज्ञात कीजिए जिसके अभिलक्षणिक मूल 4, -2 और 3 हैं, और बहुकताएँ क्रमशः 2, 1, और 3 हैं। संबंध का एक असमघात भाग भी है जो 3ⁿ और 4ⁿ का एकघात संचय है।

6. क) सर्वसमिका
$$\sum_{k=0}^{n} \binom{r}{k} \binom{s}{n-k} = \binom{r+s}{n}$
को जनक फलनों के प्रयोग से हल कीजिए।

ख) विलय शाटन कलन-विधि की काल जटिलता ज्ञात कीजिए।

7. क) 0 से 999 (0 और 999 दोनों को शामिल करके) तक की संख्याओं में से कितनी संख्याएँ 7 या 11 से विभाजित नहीं होती हैं? (4)

ख) 120 लोगों के एक सर्वेक्षण में निम्नलिखित आंकड़े प्राप्त हुए:

90 लोगों के पास कार है, 35 के पास कंप्यूटर है, 40 के पास घर है, 32 के पास घर और कार दोनों हैं, 21 के पास घर और कंप्यूटर दोनों हैं, 26 के पास कार और कंप्यूटर दोनों हैं, और 17 लोगों के पास ये तीनों सुविधाएँ हैं।

i) कितने लोगों के पास इन तीनों में से कोई भी सुविधा नहीं है?

ii) कितने लोगों के पास केवल कार है?

iii) कितने लोगों के पास केवल कंप्यूटर है?

8. क) प्रारम्भिक प्रतिबंधों $a_0 = 2, a_1 = 5$ के साथ, पुनरावृत्ति संबंध $a_n = 6a_{n-1} - 5a_{n-2} + 1$ का जनक फलन ज्ञात कीजिए।

ख) बैल संख्याएँ B₆ और B₇ ज्ञात कीजिए।

ग) विभाजन $20=9+6+3+2$ का फेरर ग्राफ बनाइए। साथ ही, इस विभाजन का संयुग्मी विभाजन लिखिए।

घ) दिल्ली की चार-पहिया गाड़ियों की संख्या पट्टिका सामान्यतया DL3AXC4126 जैसे दिखती है, जिसके प्रथम दो अक्षर DL नियत होते हैं, तीसरा अक्षर एक अंक होता है, और अगले तीन अक्षर अंग्रेजी के अक्षर हैं, और अंत के चार अक्षर अंक हैं। इस पैटर्न के साथ दिल्ली में कितनी संख्या पट्टिकाएँ पंजीकृत की जा सकती हैं?

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