If your BEY-019 exam is scheduled for 2026 or a later session, the BEY 019 Question Paper December 2025 is one of the sharpest revision tools you can keep open on your desk. It tells you exactly how IGNOU frames questions on real analysis, sets, relations, and calculus, which areas repeat across sessions, and how to use three hours without running out of time. Unnati Education has built this solved paper for students who want honest clarity before stepping into the actual term end exam.
About This Solved Paper
| Prepared by | Unnati Education, IGNOU-experienced academic content writer |
|---|---|
| Subject expert qualification | Graduate team with specialisation in Mathematics and Applied Science |
| Programme | IGNOU Bachelor of Science Applied Science Energy (BSCAEY) |
| Institution Reference | IGNOU December 2025 Term-End Examination |
You can download the BEY 019 Solved Question Paper December 2025 directly from this page as a clean PDF file. The file opens smoothly on any phone, tablet, or laptop and remains with you through every revision round. Students who prepare with real previous year papers almost always score better than those who stay stuck inside textbook exercises. That's why beginning your revision with a proper solved paper changes the final weeks of preparation for the better.
What is BEY 019 Question Paper December 2025 Definition
The BEY 019 December 2025 question paper is the official IGNOU Term End Examination paper for Real Analysis and Discrete Mathematics, conducted in December 2025 for BSCAEY students. It carries 70 marks, runs for three hours, and asks you to attempt any seven questions of equal marks, with a non programmable scientific calculator allowed.
The paper tests how well you handle logic, sets, relations, functions, coordinate geometry, progressions, simple interest, difference equations, limits, derivatives, and integration. You are expected to write clearly, justify each step, and reach the final answer with proper working.
About IGNOU BEY-019 Real Analysis and Discrete Mathematics
BEY-019 is a core mathematical paper in the Bachelor of Science (Applied Science Energy) programme offered by IGNOU. It builds the logical and analytical thinking every energy science graduate needs. The syllabus covers logic, sets, relations, functions, coordinate geometry, arithmetic progressions, simple interest, difference equations, limits, derivatives, and integration. In practice, this is the paper that connects basic reasoning with real mathematical application. Most students handle progressions easily but slow down on difference equations, and that is exactly where a solved paper becomes useful.
BEY 019 Question Paper December 2025 Exam Pattern and Marks Breakdown (BEY-019 December 2025 Question Paper)
The paper runs for three hours with a total of 70 marks. Question 1 is compulsory and carries 10 marks split across five true or false statements, each worth two marks. You then attempt any six questions from Questions 2 to 9, each worth 10 marks, usually split into two subparts. Here's the important part, the compulsory question rewards clear reasoning. A wrong guess without justification loses the mark, so always back your answer with a short proof or counter example.
All Questions BEY 019 Question Paper December 2025 IGNOU (Complete List)
Below is the full paper exactly as it appeared in the IGNOU Term End Examination December 2025. The attached PDF shows the original paper with full mathematical formatting.
Note: (i) Question No. 1 is compulsory. (ii) Answer any six questions from the remaining question nos. 2 to 9. (iii) Use of scientific (non programmable) calculator is allowed. (iv) Symbols have their usual meanings.
Question 1 (Compulsory, 10 marks)
State whether the following statements are True or False? Justify your answer with a short proof or a counter example:
(a) Every continuous function is differentiable.
(b) The contrapositive statement for p → q is ~p → ~q.
(c) y2 – 4y – 2x = 0 represents a parabola.
(d) The relation (≤) on the set of integers is an anti symmetric relation.
(e) For the function f(x) = 7 when x < 2, ax + b when x > 2, a + 5 when x = 2, the values of 'a' and 'b' are 2 and 5 respectively, provided f(x) is continuous at x = 2.
Question 2 (10 marks)
(a) Out of 50 people in a group, 24 speak English, 15 speak Hindi, 18 speak Sanskrit, 6 speak English and Hindi, 8 speak English and Sanskrit, 5 Hindi and Sanskrit and 10 people do not speak any of the three languages. Find, how many speak: (i) All the three languages, (ii) Hindi but not Sanskrit, and (iii) English and Sanskrit but not Hindi.
(b) Let f : ℕ → Y be a function defined as f(x) = 4x + 3, where Y = {y ∈ ℕ : y = 4x + 3 for some x ∈ ℕ}. Show that f is invertible. Find the inverse.
Question 3 (10 marks)
(a) Find the domain and range of the real function f(x) = √(9 – x2).
(b) Let R be a relation from ℕ to ℕ defined by R = {(a, b) : a, b ∈ ℕ and a = b2}. Are the following true? (i) (a, b) ∈ R implies (b, a) ∈ R (ii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R Justify your answer in each case.
Question 4 (10 marks)
(a) Find the co-ordinates of the point which divides the line joining the points (4, 7) and (3, – 5) in the ratio 3 : 4.
(b) Show that the equation x2 + y2 – 2x + 6y – 6 = 0 is a circle. Find its centre and radius.
Question 5 (10 marks)
(a) Write the first five terms of an Arithmetic Progression whose third and eleventh terms are 21 and 85 respectively.
(b) A man deposited ₹ 10,000 in a bank at the rate of 5% simple interest annually. Find the amount in the 15th year since he deposited the amount and also calculate the total amount after 20 years.
Question 6 (10 marks)
(a) Solve the following difference equation: y(t+2) – 4yt = 5.
(b) What do you understand by a phase diagram? What is it used for?
Question 7 (10 marks)
(a) Evaluate: lim (x→3) [1/(x – 3) – 3/(x2 – 3x)].
(b) Find: dy/dx if x = 2 + 4t2, y = 9t2 + 3t + 1.
Question 8 (10 marks)
(a) Evaluate: ∫ from 1 to 4 of |x – 2| dx.
(b) Evaluate: ∫ x3 · ax dx.
Question 9 (10 marks)
(a) Using integration, find the area bounded by the line y = 3x + 2, the x axis and the ordinates x = 0 and x = 1.
(b) Find dy/dx, where y = u2, u = 3v, v = x/(x + 1).
BEY 019 Question Paper December 2025 Syllabus Topics Covered
The December 2025 paper pulls from almost every important unit of the BEY-019 syllabus. You will notice questions on logic and contrapositives, continuity, conic sections, antisymmetric relations, Venn diagrams, invertible functions, domain and range, coordinate geometry, circles, arithmetic progressions, simple interest, difference equations, phase diagrams, limits, parametric differentiation, definite integrals with modulus, integration by parts, area under curves, and chain rule. The balance between discrete mathematics and real analysis stays steady across recent BSCAEY sessions.
Sample Answer Preview BEY-019 Real Analysis and Discrete Mathematics Solved Question Paper Explanation
Take Question 1(a) as an example. The statement is false. Continuity does not guarantee differentiability. The classic counter example is f(x) equals modulus of x, which is continuous at x equals zero but fails the left hand and right hand derivative test at that point. Our solved paper shows the proof with both one sided derivatives worked out, followed by a short graph description. Every true or false answer follows this same structure, one line verdict, formal reason, and where possible a counter example.
How to Write High-Scoring Answers in BEY 019 Question Paper December 2025
For true or false questions, always start with a clear verdict in one line, then add a short proof or counter example underneath. For set theory questions with three languages, draw a proper three circle Venn diagram before writing numbers. For integration, show the substitution, the new limits, and the simplified answer on separate lines. Most students lose marks by writing long paragraphs without visible structure. Keep answers tight, underline final results, and label every diagram clearly so the evaluator does not have to guess what each region means.
Who Should Use This BEY-019 Solved Question Paper December 2025?
This solved paper is built for IGNOU BSCAEY students appearing in June 2026, December 2026, or any later session. It suits first time exam takers who want direction on how IGNOU frames discrete mathematics questions, repeaters aiming to close earlier weak spots, and working students who can spare only a few hours each week. If sets and relations feel easy but calculus slows you down, the worked solutions bring both halves of the syllabus into balance without overwhelming you.
Why This is Better Than Free BEY-019 December 2025 Question Paper PDFs and Telegram Files
Free files circulating on Telegram often arrive without proper mathematical symbols, so integrals look like plain text and fractions break across lines. Some PDFs carry wrong answers copied from unrelated years. A few skip the compulsory question entirely or show only partial working. Unnati Education's solved paper is checked by writers who have handled IGNOU mathematical papers for years. You get clean notation, correct proofs, accurate numerical answers, and solutions that match IGNOU's marking style closely.
Student Reviews BEY-019 Solved Question Paper by Unnati Education
Neha from Nagpur said the Venn diagram solution finally cleared up how to handle three set problems without confusion. Siddharth, a BSCAEY student from Guwahati, wrote that the difference equation answer was the first explanation that actually made sense to him. Pooja from Indore appreciated how the parametric differentiation question was shown step by step with proper substitution. These short notes reach our team regularly and keep us focused on careful, checked solutions rather than rushed answer keys.
How to Get the BEY-019 Solved Question Paper December 2025 Step by Step
First, click the download or buy button visible on this product page. Second, pay through UPI, debit card, credit card, or net banking, whichever option you prefer on your device. Third, your PDF link arrives in your email within minutes, and the same file stays accessible on your account page for repeat downloads anytime. No subscription trap. No monthly charge. No waiting for a support agent to share the file through WhatsApp late at night before your exam.
Frequently Asked Questions BEY 019 Question Paper December 2025 IGNOU
Is this the official IGNOU BEY 019 December 2025 paper?
Yes, this is the actual Term End Examination paper conducted by IGNOU in December 2025 for BEY 019 under the Bachelor of Science Applied Science Energy BSCAEY programme. Every question on this page is reproduced exactly as it appeared in the official paper, which makes it a reliable source for your 2026 exam preparation and for any future Term End attempt you may plan later in your studies.
Will this paper help me for my 2026 exam?
It will help you strongly. IGNOU often repeats question themes in BEY 019, especially on logic, sets, relations, Venn diagrams, progressions, simple interest, difference equations, limits, differentiation and integration. By practicing the December 2025 paper with solved answers, you understand the examiner style, the expected length for a 10 mark question, and how to structure both theory and calculation based answers for full marks in your Term End exam.
Are all true or false questions answered with proper justification?
Yes. Every part of Question 1 in our solved paper carries a clear true or false verdict along with a short proof or counter example. Our team never gives one word answers because IGNOU awards marks only when you justify your reasoning. You see the logic written the same way a university level examiner would expect.
Does it cover every unit of the syllabus?
The December 2025 paper covers a wide cross section of the syllabus, including logic, sets, relations, functions, coordinate geometry, progressions, difference equations, limits, differentiation, and integration. Our solutions treat each unit with the same level of detail. By the time you complete the full solved paper, you have effectively revised most of the BEY-019 course in one focused, practical sitting.
Is the download instant after payment?
Yes, delivery is instant. Once your payment is confirmed, the BEY 019 solved question paper PDF is sent to your registered email and also saved inside your Unnati Education account dashboard. You can open it on mobile, tablet or laptop, and there is no limit on how many times you download, print or revise the paper during your BSCAEY preparation journey in any upcoming Term End exam cycle across years.
About Unnati Education IGNOU Study Material Experts
Unnati Education is an academic resource platform built for IGNOU students across India. Our team includes graduates, postgraduates, engineers and subject mentors who have personally handled IGNOU coursework. We prepare solved question papers, handwritten assignments and revision notes in student friendly language. Our focus is simple, we want you to stop depending on unreliable free files and start preparing with material you can actually trust during your exam cycle.
Explore More IGNOU BEY-019 Solved Papers and Study Material
You can also explore BEY 019 solved assignments, previous year question papers from June and December cycles, important questions lists, and short revision notes on our website. Together, these resources give you a complete preparation package for Real Analysis and Discrete Mathematics, from the first reading of the course to your final revision the night before your IGNOU BSCAEY Term End examination in any upcoming session.