BCS 042 key facts: June 2026
- Course code: BCS-042 (BCS 042)
- Subject: Introduction to Algorithm Design
- Programme: Bachelor of Computer Applications, BCA (Revised) Semester IV; also BCA Online (BCAOL)
- Blocks and units: 2 blocks, 6 units
- Maximum marks: 50 (term-end exam); assignment weightage 30%
- Duration: 2 hours
- Questions in the paper: 5 questions split into 12 parts; Q1 (20 marks) compulsory, any three of Q2 to Q5 (10 marks each)
- Samples shown on this page: 2 (Q1(a) and Q3(b)); all 12 parts in the paid file
BCS 042 Solved Question Paper June 2026 gives you complete, exam-length answers to the IGNOU Introduction to Algorithm Design term-end paper, from the sum of squares induction proof in Q1(a) to the DFS traversal from node 1 in Q5(b). Two answers are open below so you can check our standard, and the full file with all 12 parts comes on WhatsApp 9899436384.
1. What the June 2026 File Covers
BCS 042 Introduction to Algorithm Design is the fourth-semester IGNOU Bachelor of Computer Applications course that teaches you how to design an algorithm and measure how fast it runs.
The June 2026 paper has five questions split into 12 parts. Question 1 is compulsory and tests four ideas: an induction proof, a recursion tree, why analysis matters and asymptotic bounds. Questions 2 to 5 are mostly worked problems: bubble sort, GCD(595, 252), polynomial evaluation, a seven-object knapsack, Kruskal and Prim on a six-vertex graph, divide and conquer, adjacency representation and DFS.
If you are collecting every BCA Question Paper for Semester IV, this one deserves extra attention, because 11 of its 12 parts cover topics that already appeared in the four earlier official BCS 042 papers we checked.
Our file answers each part at the length its marks demand. A 2-mark definition gets a tight paragraph, while the 8-mark spanning tree question gets two full edge-by-edge tables, one for each algorithm, plus the final cost.
2. Two Answers Solved in Full, Ten Held Back
Two full answers are shown here, Q1(a) on mathematical induction and Q3(b) on the knapsack problem; the other 10 parts are answered only in the paid file.
We picked these two because induction on the sum of squares has now been set three times in recent papers, and the knapsack table is exactly the kind of numerical answer where marks are won or lost on layout.
Induction proof for the sum of squares
Question 1 (a): Prove the following proposition using induction. P(n): 12 + 22 + 32 + ... + n2 = n(n + 1)(2n + 1) / 6 (5)
We prove P(n) for every positive integer n using the principle of mathematical induction, which needs a basis step, an inductive hypothesis and an inductive step.
Step 1: basis step (n = 1)
LHS = 12 = 1. RHS = 1(1 + 1)(2 × 1 + 1) / 6 = (1 × 2 × 3) / 6 = 1. Since LHS = RHS, P(1) is true.
Step 2: inductive hypothesis
Assume P(k) is true for some positive integer k, that is, 12 + 22 + 32 + ... + k2 = k(k + 1)(2k + 1) / 6.
Step 3: inductive step
We must show that P(k + 1) is true, that is, 12 + 22 + ... + k2 + (k + 1)2 = (k + 1)(k + 2)(2k + 3) / 6.
- LHS = [12 + 22 + ... + k2] + (k + 1)2
- = k(k + 1)(2k + 1) / 6 + (k + 1)2 (by the inductive hypothesis)
- = (k + 1) [k(2k + 1) + 6(k + 1)] / 6 (taking (k + 1) / 6 common)
- = (k + 1)(2k2 + k + 6k + 6) / 6
- = (k + 1)(2k2 + 7k + 6) / 6
- = (k + 1)(k + 2)(2k + 3) / 6 (since 2k2 + 7k + 6 = (k + 2)(2k + 3))
- = (k + 1)[(k + 1) + 1][2(k + 1) + 1] / 6 = RHS
So P(k + 1) is true whenever P(k) is true.
Conclusion. P(1) is true, and P(k) implies P(k + 1). By the principle of mathematical induction, 12 + 22 + 32 + ... + n2 = n(n + 1)(2n + 1) / 6 for all positive integers n.
Quick check. For n = 3, LHS = 1 + 4 + 9 = 14 and RHS = (3 × 4 × 7) / 6 = 84 / 6 = 14. Writing one numerical check like this shows the examiner you understand the formula, not just the algebra.
Fractional knapsack by the greedy ratio method
Question 3 (b): Find the optimal solution for the Knapsack instance n = 7 and m = 15. (5)
Profits (P1 to P7) = (10, 5, 15, 7, 6, 18, 3)
Weights (W1 to W7) = (2, 3, 5, 7, 1, 4, 1)
This is the continuous (fractional) knapsack problem from the greedy technique unit. Objects may be split, so the greedy rule that gives the optimal solution is: compute profit per unit weight (pi/wi) for every object, then fill the knapsack in decreasing order of that ratio, taking a fraction of the last object if it does not fit whole.
Step 1: profit by weight ratio
| Object i | Profit pi | Weight wi | pi/wi |
|---|---|---|---|
| 1 | 10 | 2 | 5.00 |
| 2 | 5 | 3 | 1.67 |
| 3 | 15 | 5 | 3.00 |
| 4 | 7 | 7 | 1.00 |
| 5 | 6 | 1 | 6.00 |
| 6 | 18 | 4 | 4.50 |
| 7 | 3 | 1 | 3.00 |
Step 2: order by decreasing ratio
Object 5 (6.00), object 1 (5.00), object 6 (4.50), object 3 (3.00), object 7 (3.00), object 2 (1.67), object 4 (1.00). Objects 3 and 7 tie at 3.00; either order gives the same result because both fit whole.
Step 3: fill the knapsack (m = 15)
| Object taken | Fraction xi | Weight added | Capacity left | Profit added | Total profit |
|---|---|---|---|---|---|
| 5 | 1 | 1 | 14 | 6 | 6 |
| 1 | 1 | 2 | 12 | 10 | 16 |
| 6 | 1 | 4 | 8 | 18 | 34 |
| 3 | 1 | 5 | 3 | 15 | 49 |
| 7 | 1 | 1 | 2 | 3 | 52 |
| 2 | 2/3 | 2 | 0 | 5 × 2/3 = 3.33 | 55.33 |
| 4 | 0 | 0 | 0 | 0 | 55.33 |
Step 4: optimal solution
Solution vector (x1, x2, ..., x7) = (1, 2/3, 1, 0, 1, 1, 1).
Check: total weight = 2 + 2 + 5 + 0 + 1 + 4 + 1 = 15 = m. Maximum profit = 10 + 3.33 + 15 + 0 + 6 + 18 + 3 = 55.33 (exactly 166/3).
Complexity. Computing the ratios takes O(n) and sorting them takes O(n log n), so the greedy knapsack runs in O(n log n) time for n objects.
3. Questions Answered Only in the Full File
The remaining 10 parts of the June 2026 paper are listed below exactly as printed. Their answers, workings and diagrams are in the paid file only.
Question 1 (b): Solve the recurrence relation T(n) = T(n/3) + T(2n/3) + n using the recursion tree method. (5)
Question 1 (c): Why is analysis of an algorithm important? Justify your answer with an example. (5)
Question 1 (d): Explain asymptotic notations. Also discuss what upper and lower bounds are in asymptotic notations. (5)
Question 2 (a): Write an algorithm for bubble sort and write its worst case and best case analysis. (6)
Question 2 (b): Find GCD(595, 252) by Euclid's algorithm. (4)
Question 3 (a): Write a basic algorithm to evaluate a polynomial and find its complexity. (5)
Question 4 (a): Apply Kruskal's and Prim's algorithm to find the minimum cost of a spanning tree of the graph given in the paper (weighted graph on vertices A to F). (8)
Question 4 (b): Define the divide and conquer approach. (2)
Question 5 (a): What is an Adjacency Matrix? Write the adjacency matrix and adjacency list for the graph given in the paper (vertices 1 to 5). (5)
Question 5 (b): Write the Depth-First Search (DFS) algorithm. Also traverse the graph given in the paper using DFS from node 1. (5)
Q4(a) alone asks for two full spanning tree walk-throughs on the same graph, and Q1(b) needs a recursion tree with unequal branches, which many students leave half-drawn. Both are worked completely in the file, with the graphs redrawn.
4. Syllabus by Block and Unit
BCS 042 is organised into 2 blocks and 6 units: Block 1 covers how to analyse an algorithm, and Block 2 covers greedy, divide and conquer and graph algorithms.
| Block | Unit | Key topics | June 2026 parts from this unit |
|---|---|---|---|
| Block 1: Introduction to Algorithm | Unit 1: Basics of an Algorithm | Characteristics of an algorithm, Tower of Hanoi problem formulation, substitution, iteration and master methods for recurrences | Q1(a) induction (Sample 1), Q1(b) recursion tree |
| Block 1: Introduction to Algorithm | Unit 2: Asymptotic Bounds | Big O, Omega and Theta notations; comparing linear, quadratic, polynomial and exponential growth | Q1(c) why analysis matters, Q1(d) upper and lower bound |
| Block 1: Introduction to Algorithm | Unit 3: Analysis of Simple Algorithms | Euclid's algorithm for GCD, Horner's rule for polynomial evaluation, n × n matrix multiplication, exponent evaluation | Q2(a) bubble sort, Q2(b) GCD(595, 252), Q3(a) polynomial |
| Block 2: Design Techniques | Unit 1: Greedy Technique | Elements of greedy strategy, activity selection, continuous knapsack, coin changing, minimum cost spanning trees | Q3(b) knapsack (Sample 2), Q4(a) Kruskal and Prim |
| Block 2: Design Techniques | Unit 2: Divide and Conquer Approach | Binary search, merge sort, quick sort, integer multiplication | Q4(b) definition of divide and conquer |
| Block 2: Design Techniques | Unit 3: Graph Algorithms | Adjacency matrix, adjacency list, depth first search, breadth first search | Q5(a) adjacency matrix and list, Q5(b) DFS from node 1 |
Block 1, Unit 3 carries three of the twelve parts on its own, so it pays to revise Euclid and Horner before anything else. Pair this table with our BCS 042 Study Material if you want unit notes alongside the solved paper.
5. Question Paper Pattern and Marks Split
The BCS 042 Question Paper for June 2026 carries 50 marks in 2 hours: Question 1 (20 marks) is compulsory, and you answer any three of Questions 2 to 5 (10 marks each).
| Question | Parts and marks | What it tests | Status |
|---|---|---|---|
| Q1 | (a) 5, (b) 5, (c) 5, (d) 5 = 20 | Sum of squares induction, unequal-split recursion tree, need for analysis, asymptotic bounds | Compulsory |
| Q2 | (a) 6, (b) 4 = 10 | Bubble sort with best and worst case, Euclid's GCD of 595 and 252 | Any three of Q2 to Q5 |
| Q3 | (a) 5, (b) 5 = 10 | Polynomial evaluation cost, seven-object knapsack with m = 15 | Any three of Q2 to Q5 |
| Q4 | (a) 8, (b) 2 = 10 | Kruskal and Prim on a weighted graph A to F, divide and conquer definition | Any three of Q2 to Q5 |
| Q5 | (a) 5, (b) 5 = 10 | Adjacency matrix and list for vertices 1 to 5, DFS algorithm with traversal from node 1 | Any three of Q2 to Q5 |
The paper prints 60 marks of questions, but you attempt 50. The term-end result counts for 70% of your course score, with the BCS 042 assignment making up the other 30%.
This layout matches the June 2024 and June 2025 papers, which also print the 20-mark note on Question 1. You can compare them yourself on our IGNOU Previous Year Question Paper page.
6. Topics IGNOU Keeps Repeating in BCS 042
Across the four official BCS 042 papers we could find (June 2023, December 2023, June 2024 and June 2025), 11 of the 12 June 2026 parts had been asked before in some form.
| Topic (June 2026 part) | Jun 2023 | Dec 2023 | Jun 2024 | Jun 2025 | Times seen (of 4) |
|---|---|---|---|---|---|
| DFS algorithm and traversal (Q5(b)) | Q2(a) | Q4(b) | Q2(b) | Q4(a) | 4 |
| Minimum cost spanning tree, Kruskal or Prim (Q4(a)) | Q3(a) Kruskal | Q5(b) Prim | Q5(a) Kruskal | Q4(b) Prim | 4 |
| Asymptotic notations and bounds (Q1(d)) | Q3(b), Q4(b)(i) | Q1(a) | Q1(a), Q1(b), Q2(a)(iii) | Q2(b) | 4 |
| Recurrence relations (Q1(b)) | Q2(b) recurrence tree | Q1(b) | Q1(c) Fibonacci | Q2(a) same recurrence | 4 |
| Mathematical induction (Q1(a), Sample 1) | Q1(c) sum of squares | Q5(a) sum of n | Q3(b) sum of squares | — | 3 |
| Adjacency matrix or list (Q5(a)) | Q1(d) matrix | Q1(d) list | Q1(e) both | — | 3 |
| Algorithm definition and why analysis matters (Q1(c)) | Q1(a) | — | Q4(a) | Q1(a) | 3 |
| Divide and conquer, including quick sort (Q4(b)) | — | Q1(e) quick sort | Q5(b) | Q5(a), Q5(b) | 3 |
| Fractional knapsack (Q3(b), Sample 2) | Q4(a) | — | — | Q1(c)(ii) | 2 |
| Bubble sort (Q2(a)) | Q5(a) | — | — | Q3(a) | 2 |
| Euclid's algorithm for GCD (Q2(b)) | — | Q2(b) | — | Q1(c)(iv) | 2 |
| Polynomial evaluation (Q3(a)) | — | — | — | — | 0 |
December 2024 and December 2025 papers were not available on IGNOU's official question paper pages when we checked, so the counts above cover four sessions only. Polynomial evaluation is the one genuinely new topic in June 2026, which makes Q3(a) worth reading carefully in the full file.
Use the pattern, not a single paper, to plan revision: DFS, spanning trees, asymptotic bounds and recurrences have come up every time. For a wider shortlist across courses, see our IGNOU Solved Guess Paper collection.
Reading the June 2026 paper next to June 2025 is revealing: the recurrence T(n/3) + T(2n/3) + n and the bubble sort best and worst case appear in both, almost unchanged.
7. How We Write and Check Every Answer
Every BCS 042 answer is written to the marks it carries, worked by hand, and reviewed before it goes into the file.
Each answer is written in the order the paper prints it. Algorithms are given as numbered steps in the pseudocode style the IGNOU blocks use, and every complexity claim is shown, not just stated, so an examiner can follow the reasoning line by line.
Method and notation follow the official course material on eGyanKosh, for example the continuous knapsack rule and the recurrence methods of Block 1. Where the blocks use a particular name, such as Minimum Cost Spanning Tree (MCST), we keep it.
Numerical parts are solved twice by different people and the totals compared. The knapsack profit, the GCD remainders and the spanning tree cost must agree before the answer is approved, and every table carries clear column headings so it reads well in a handwritten copy.
8. Solved File Index and the BCSL-045 Lab Link
BCS 042 is a theory paper, so this index maps all 12 answer parts to their unit; the coding practice for this course happens separately in BCSL-045.
| Part | Marks | Topic | Unit | Answer available |
|---|---|---|---|---|
| Q1(a) | 5 | Induction proof for the sum of squares | Block 1, Unit 1 | Sample 1 above |
| Q1(b) | 5 | Recursion tree for T(n/3) + T(2n/3) + n | Block 1, Unit 1 | Full file |
| Q1(c) | 5 | Why algorithm analysis matters, with example | Block 1, Unit 2 | Full file |
| Q1(d) | 5 | Asymptotic notations, upper and lower bound | Block 1, Unit 2 | Full file |
| Q2(a) | 6 | Bubble sort algorithm, best and worst case | Block 1, Unit 3 | Full file |
| Q2(b) | 4 | Euclid's algorithm for GCD(595, 252) | Block 1, Unit 3 | Full file |
| Q3(a) | 5 | Polynomial evaluation algorithm and complexity | Block 1, Unit 3 | Full file |
| Q3(b) | 5 | Knapsack n = 7, m = 15 by greedy ratio | Block 2, Unit 1 | Sample 2 above |
| Q4(a) | 8 | Kruskal and Prim on the A to F graph | Block 2, Unit 1 | Full file |
| Q4(b) | 2 | Definition of divide and conquer | Block 2, Unit 2 | Full file |
| Q5(a) | 5 | Adjacency matrix and list for a 5-vertex graph | Block 2, Unit 3 | Full file |
| Q5(b) | 5 | DFS algorithm and traversal from node 1 | Block 2, Unit 3 | Full file |
The Semester IV lab course BCSL-045 Introduction to Algorithm Design Lab is where you code these same ideas in C, such as sorting and graph traversal. That lab has its own assignment and practical exam, so its record is not part of this solved paper.
9. Programmes That Study BCS 042
BCS 042 Introduction to Algorithm Design is studied in Semester IV of the IGNOU BCA (revised syllabus) and by learners on BCA Online (BCAOL).
In the Bachelor of Computer Applications, BCS 042 sits alongside BCS-040 Statistical Techniques, MCS-024, BCS-041 Fundamentals of Computer Networks and the labs MCSL-016, BCSL-043, BCSL-044 and BCSL-045. The algorithm course is usually the one students find hardest to write answers for, because marks depend on method as well as the final figure.
Both BCA and BCAOL learners sit the same term-end paper and submit the same assignment, so one solved file serves both groups.
10. How to Get the Full Solved Question Paper
Send "BCS 042 June 2026" on WhatsApp 9899436384 to receive the complete file with all 12 answer parts.
The file holds the 2 answers you have read here plus the 10 parts listed above, including both spanning tree walk-throughs, the recursion tree, the bubble sort passes and the DFS order from node 1.
Our team replies with the price and payment details, and the PDF is shared in the same chat once payment is confirmed. If you are sitting the exam soon, mention your exam date so we can prioritise your message.
11. BCS 042 Algorithm Design FAQs
What does the Solved Question Paper June 2026 include?
It includes complete answers to all 12 parts of the IGNOU June 2026 Introduction to Algorithm Design paper, set out in the printed order from Q1(a) to Q5(b). Proofs, recurrence trees, sorting passes, spanning tree steps and DFS traversals are worked in full, with tables where examiners expect them. Two answers, Q1(a) and Q3(b), are shown on this page so you can judge the quality first.
How do I get the full file on WhatsApp?
Message WhatsApp 9899436384 with the text "BCS 042 June 2026 solved paper" and our team will reply with the price and payment details. Once payment is confirmed, the PDF arrives in the same chat. It carries all 12 answer parts: the 2 shown on this page plus the 10 parts listed here as question text only. Keep it on your phone for quick revision before the exam.
Which BCS 042 topics repeat most in IGNOU papers?
DFS traversal, minimum cost spanning trees, asymptotic notations and recurrence relations appeared in all four official papers we checked: June 2023, December 2023, June 2024 and June 2025. Mathematical induction, adjacency representation and divide and conquer appeared in three of them. The June 2026 recurrence T(n/3) + T(2n/3) + n was also set in June 2025 with the same recursion tree method.
Is BCS 042 a theory or practical course?
BCS 042 is a theory course with a two-hour, 50-mark term-end exam. The hands-on side of algorithm design sits in a separate lab course, BCSL-045 Introduction to Algorithm Design Lab, in the same fourth semester of the BCA. This file therefore covers written answers only, and its index maps each question to the block and unit it comes from rather than to lab exercises.
How should I attempt the June 2026 paper in two hours?
Question 1 is compulsory and carries 20 marks across four 5-mark parts, so give it about 45 minutes. Then pick any three of Questions 2 to 5, each worth 10 marks, and spend around 25 minutes on each. Choose numerical questions such as knapsack, GCD or spanning trees if you have practised them, because a complete worked table earns marks quickly and is easy to check.
Who writes and checks the BCS 042 answers?
Answers are written by Prateek Talwar and the Unnati Educations academic team, then reviewed by Sheetal Kirola before release. Every numerical answer is recomputed by hand, and each proof is checked step by step against the method taught in the official IGNOU study material on eGyanKosh. We also compare answer length and layout with what earlier official BCS 042 papers expect for each mark value.
Which programmes include the BCS 042 exam?
BCS 042 is a fourth-semester course in the IGNOU Bachelor of Computer Applications (BCA, revised syllabus) and is also studied by learners on the BCA Online programme (BCAOL). Both sit the same Introduction to Algorithm Design term-end paper, so this solved file works for either. Students re-appearing after an earlier attempt can use it too, since the paper pattern has stayed stable across recent sessions.
12. Who Wrote and Checked the BCS 042 Answers
The BCS 042 answers were written by Prateek Talwar and the Unnati Educations academic team, and reviewed by Sheetal Kirola.
Prateek Talwar prepares IGNOU computer-science solved papers from the course blocks and the official term-end paper. Sheetal Kirola, Academic Head (M.Ed.), recomputes every numerical by hand and checks each proof step by step against the method taught in the official study material.