About This Solved Paper
| Prepared by | Unnati Education, IGNOU-experienced academic content writer |
|---|---|
| Qualification | Graduate with specialization in Mathematics and Computer Applications |
| Programme | IGNOU Bachelor of Computer Applications (BCA), Post Graduate Diploma in Computer Applications (PGDCA and PGDCA NEW), Master of Computer Applications (MCA) |
| Institution Reference | IGNOU Term-End Examination December 2025 |
BCS-012 is a core paper for IGNOU's BCA program and also relevant to PGDCA and MCA students who need a strong mathematical foundation. The subject covers algebra, matrices, calculus, progressions, vectors, and linear programming. It is one of those papers where concept clarity directly decides your marks. The December 2025 Term-End Examination tests both proof-based and numerical questions. Unnati Education's solved BCS 012 Question Paper December 2025 gives you step-by-step working for every question, written precisely to IGNOU's marking expectations.
What is BCS-012 Question Paper December 2025?
The BCS 012 December 2025 question paper is the official IGNOU Term-End Examination for Basic Mathematics under BCA, PGDCA, and MCA programs, carrying 100 marks with Question 1 compulsory and any three from the remaining four to be attempted.
This paper was conducted under the IGNOU December 2025 Term-End Examination schedule. Question 1 carries 40 marks and contains eight sub-parts covering determinants, matrix algebra, mathematical induction, arithmetic progressions, complex numbers, polynomial roots, integration, and coordinate geometry. Questions 2 to 5 carry 20 marks each and go deeper into harmonic progressions, calculus, vectors, complex number operations, Cramer's rule, matrix inverse, and linear programming.
About BCS-012 — Basic Mathematics
BCS-012 covers the mathematical concepts that underpin computer science and applications. Matrices and determinants are central — you learn to compute determinants, show matrix properties, and find inverses. Algebra includes arithmetic progressions, harmonic progressions, geometric series, complex numbers, and solving polynomial equations. Mathematical induction is tested as a proof technique. Calculus covers differentiation and integration, including finding derivatives of composite functions and evaluating definite and indefinite integrals. Vectors and three-dimensional geometry appear through direction cosines, cross products, and scalar triple products. Linear programming rounds out the syllabus with maximization problems under given constraints. Most students notice that the breadth of topics makes strategic question selection very important in this exam.
BCS 012 Exam Pattern
The paper runs for three hours and carries 100 marks. Question 1 is compulsory and carries 40 marks across eight sub-parts, each worth 5 marks. You must then attempt any three questions from Questions 2 to 5, each carrying 20 marks and divided into four sub-parts of 5 marks each. Here's the important part — since all optional questions follow the same structure, your choice of three questions should be based on which topics you are strongest in. Choosing poorly can cost you significantly even if you know the subject.
All Questions — BCS 012 Question Paper December 2025 IGNOU
(a) Show that: (5) |-a² ab ac| |ba -b² bc| = 4a²b²c² |ca cb -c²|
(b) If A = [(-1, -2), (3, 6)], show that A² = 5A. (5)
(c) Use principle of mathematical induction to show that: (5) 1 + 3 + ......... + (2n - 1) = n², for all n belonging to N (where N is a set of natural numbers).
(d) If m is not equal to n and m times the mth term of an Arithmetic Progression (A.P.) is equal to n times the nth term of the A.P., then show that (m + n)th term of the A.P. is zero. (5)
(e) If z belongs to C and |z - i| = |z + i|, show that Im(z) = 0, where z is a complex number and C is a set of complex numbers. (5)
(f) Find the roots of the equation: (5) x³ - 13x² + 15x + 189 = 0 given that one root exceeds the other by 2 and the roots are integers.
(g) Evaluate integral (I) given below: (5) I = integral of x³ divided by (x + 1)² dx
(h) Show that the diagonals of a rhombus are at right angles. (5)
(a) Find the 10th term of the Harmonic Progression (H.P.): (5) 1/5, 1/11, 1/17, 1/23, ..........
(b) If x = a + b, y = a·omega + b·omega² and z = a·omega² + b·omega, show that xyz = a³ + b³. [Here omega is not equal to 1 is a cube root of unity.] (5)
(c) Find the direction cosines of the line joining (0, 1, -1) and (3, 2, 1). (5)
(d) If y = (e^x + e^(-x)) divided by (e^x - e^(-x)), find dy/dx. (5)
(a) If y = ln[e^x multiplied by ((x - 2)/(x + 2))^(3/4)], show that dy/dx = (x² - 1)/(x² - 4). (5)
(b) Find the intervals in which the function f(x) = 3x^(5/2) - 5x^(3/2), x > 0 is (i) increasing, and (ii) decreasing. (5)
(c) Evaluate: (5) Integral of (3^x + 2^x) divided by 6^x dx
(d) If vector a = 4i + j + 3k and vector b = -2i + j - 2k, find a unit vector perpendicular to both vector a and vector b. (5)
(a) If x belongs to the set of real numbers, solve the inequality: (5) 9 divided by (x - 3) is less than 5 (where the set of real numbers R is given)
(b) If a, b, x, y belong to the set of real numbers, a is not equal to 0 and b is not equal to 0: (5) (a - ib)(x + iy) = (a² + b²)i, find x and y.
(c) Use Cramer's rule to solve the system of equations: (5) 2x - y + z = 4 3x - y = 5 2y - z = 1
(d) If A = [(2, 2, -4), (-4, 2, -4), (2, -1, 5)] and B = [(1, -1, 0), (2, 3, 4), (0, 1, 2)], show that AB = BA = 6I₃. Find A⁻¹. (5)
(a) Use mathematical induction to show that: (5) 1 + 1/2 + 1/2² + ..... + 1/2ⁿ < 2, for all n belonging to N.
(b) Find quadratic equation with real coefficients, one of its roots is -2 + 3i. (5)
(c) If vector a = -i + j + 2k, vector b = 2i - j + k and vector c = 3i - j + 2k, find (vector a cross vector b) dot vector c. (5)
(d) Maximize: (5) P = 5x + 2y subject to: 10x + 2y is greater than or equal to 2100 x + (1/2)y is less than or equal to 600 y is less than or equal to 800 x is greater than or equal to 0, y is greater than or equal to 0.
BCS 012 Syllabus Topics Covered in Question Paper
The December 2025 paper covers the complete BCS-012 curriculum systematically. Matrices and determinants form a significant section — including determinant proofs, matrix multiplication, showing A² properties, and finding the inverse of a matrix. Mathematical induction appears twice across the paper. Arithmetic and harmonic progressions test your sequence knowledge. Complex numbers appear in multiple forms — modulus conditions, finding roots with real coefficients, and algebraic operations involving complex sets. Calculus is tested through differentiation of logarithmic and hyperbolic functions, finding increasing and decreasing intervals, and evaluating definite and indefinite integrals. Vectors cover direction cosines, cross products, and scalar triple products. Cramer's rule, inequalities, and linear programming with constraints complete the full picture.
Sample Answer Preview — BCS 012 Basic Mathematics Solved Question Paper
Here is how our solved paper handles Question 1(f) on finding roots of the cubic equation x³ - 13x² + 15x + 189 = 0. The solution begins by using the given condition that one root exceeds another by 2, so if one root is r, another is r + 2. Using Vieta's formulas, the three roots sum to 13 and their product equals -189. Substituting and solving gives the three integer roots. The answer shows every algebraic step clearly, including verification by substitution. For a 5-mark question, this approach demonstrates full method marks. To access the complete working, get the BCS-012 Solved Question Paper from Unnati Education.
How to Write High-Scoring Answers in BCS 012
Mathematics answers in IGNOU are marked on method, not just final results. This is the single most important thing to understand about this paper. For proof-based questions like mathematical induction or showing determinant results, write every step — base case, inductive hypothesis, inductive step, and conclusion. For numerical problems involving matrices or Cramer's rule, show your cofactor expansions and intermediate calculations even when they seem obvious. For integration, write the substitution clearly before evaluating. A 5-mark question typically needs four to five visible steps plus a conclusion. Skipping steps, even correct ones, will cost you marks.
Who Should Use This BCS-012 Solved Question Paper?
This solved paper is for IGNOU BCA students in the first semester, PGDCA students who need the mathematics bridge course, and MCA students appearing for BCS-012 as part of their curriculum. It is particularly valuable if your school-level mathematics is rusty and you are finding university-level proofs and calculus challenging. Repeat candidates who could not follow the working in free resources will find the step-by-step format much clearer. Students appearing in the December 2025 exam who want model solutions to revise from will get the most direct benefit.
Why This is Better Than Free BCS-012 PDFs and Telegram Files
Free BCS-012 material found online is rarely verified for mathematical accuracy. Wrong intermediate steps, incomplete proofs, and missing working are common problems. For a mathematics paper, an incorrect step invalidates everything that follows — and that directly affects your score. Our BCS-012 December 2025 solved paper is prepared by subject specialists who have verified every calculation and proof. Matrix operations are checked numerically. Integration answers are validated. Proofs are written in the standard format IGNOU expects. You are not studying from guesswork — you are studying from verified solutions.
Student Reviews — BCS-012 Solved Question Paper by Unnati Education
"The determinant proof in Q1(a) was explained so clearly. I had been trying to work it out for hours and the solved paper showed me exactly where I was going wrong." — Pooja R., BCA Semester 1, Delhi
"Cramer's rule solution in Q4(c) was perfect — all steps shown, final answers verified. This is the quality I needed for a maths paper." — Manish S., PGDCA, Bhopal
"The linear programming maximization in Q5(d) had the graphical explanation clearly described. Worth every rupee for this paper." — Nidhi V., BCA, Pune
How to Get the BCS-012 Solved Question Paper — Step by Step
The process takes just a few minutes. Step one: contact Unnati Education through WhatsApp or the contact form on this page. Step two: confirm your program name — BCA, PGDCA, or MCA — and specify that you need the BCS-012 Basic Mathematics December 2025 solved paper. Step three: receive the complete solved paper on WhatsApp or email, usually within a few hours of your request being confirmed.
Frequently Asked Questions
What is BCS-012?
BCS-012 is the Basic Mathematics paper under IGNOU's BCA Revised program and is also used in PGDCA and MCA curricula. It covers matrices, determinants, mathematical induction, arithmetic and harmonic progressions, complex numbers, calculus including differentiation and integration, vectors, coordinate geometry, linear programming, and inequalities. The December 2025 Term-End Examination carries 100 marks with Question 1 compulsory at 40 marks.
What is the exam pattern for BCS-012 December 2025?
The exam is three hours long and totals 100 marks. Question 1 is compulsory and carries 40 marks across eight sub-parts of 5 marks each. Students must attempt any three questions from Questions 2 to 5, each carrying 20 marks and divided into four sub-parts of 5 marks. Every sub-part is an independent question, so partial preparation across topics can still earn good marks.
Which topics are most important in BCS-012 December 2025?
Based on the actual December 2025 paper, key topics include determinants, matrix multiplication and inverse, mathematical induction, arithmetic and harmonic progressions, complex numbers including modulus conditions and quadratic roots, cubic equation roots, integration and differentiation, direction cosines, cross product and scalar triple product of vectors, Cramer's rule, solving inequalities, linear programming with constraints, and showing AB equals BA type matrix problems.
How can I access the IGNOU BCS-012 December 2025 question paper?
The full BCS-012 December 2025 question paper with all questions and sub-parts is available on this page, reproduced exactly as printed in the official IGNOU paper. For completely solved answers with step-by-step mathematical working verified by subject specialists, contact Unnati Education on WhatsApp or through the contact form and receive the solved paper within hours.
About Unnati Education — IGNOU Study Material Experts
Unnati Education supports IGNOU students across India in preparing for their Term-End Examinations with materials that are built for the actual exam, not generic study guides. Our team has subject specialists in mathematics, computer science, commerce, economics, and humanities. For technical papers like BCS-012, we ensure that every calculation is verified and every proof follows the correct format. We serve BCA, PGDCA, MCA, BCOMG, BBA, BA, MA students and update all materials each exam cycle so you always get the most relevant content.
Explore More IGNOU BCS-012 Solved Papers and Study Material
Unnati Education also offers solved papers for BCS-011, BCS-040, BCS-041, BCSL series lab papers, and other BCA semester papers. PGDCA students can find solved papers for their complete program as well. You can also access IGNOU assignment solutions and guess papers for the June 2026 session. All materials match the same quality standard as this BCS-012 solved paper, with mathematical accuracy as a non-negotiable requirement.
Get Your BCS 012 Question Paper December 2025 Solved Paper Now
You have seen the complete question paper and you know exactly what the December 2025 exam covers. Mathematics rewards preparation — and the right kind of preparation means studying from solutions that show every step correctly. The BCS-012 Solved Question Paper December 2025 from Unnati Education gives you verified, step-by-step solutions for every question in the paper. Contact us on WhatsApp or through the form below and get your solved paper today.