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100+ Free BCS Written Mathematics Practice Questions

Prepare for the BCS Written Post-Related Mathematics, Subject Code 551, Total Marks 200 exam with instant access — no signup required.

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Key Facts: BCS Written Mathematics Exam

200 marks

Official written paper (Subject Code 551)

BPSC BCS written syllabus PDF

100 / 100

Part I / Part II split

BPSC BCS written syllabus PDF

4 hours

Duration used for BPSC 200-mark written papers on this site; confirm the current routine

Longstanding BPSC sitting practice

50% aggregate

Written-stage qualifying mark on 900 marks

BPSC BCS examination page

Tk. 200

50th BCS general application fee (Tk. 50 reduced categories); no separate written fee

50th BCS circular reporting citing PSC

Descriptive

Official item type; this bank is an English MCQ study adaptation

BPSC written syllabus structure

Subject Code 551 is a 200-mark descriptive written paper (Part I 100, Part II 100; about 4 hours). It sits inside the 900-mark BCS written stage, which requires 50% aggregate plus BPSC per-paper minima. This page offers 100 English MCQs as a study aid, not an official-paper simulation.

Sample BCS Written Mathematics Practice Questions

Try these sample questions to test your BCS Written Mathematics exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1For the positive real numbers 4 and 9, the difference (arithmetic mean) − (geometric mean) equals which value?
A.1
B.2
C.1/2
D.5/2
Explanation: The arithmetic mean is (4+9)/2 = 13/2. The geometric mean is √(4·9) = 6. Their difference is 13/2 − 6 = 1/2. AM–GM says this difference is nonnegative, and it is zero only when the two numbers are equal.
2If a1 ≤ a2 ≤ a3 and b1 ≤ b2 ≤ b3 are real numbers, Chebyshev's sum inequality asserts which comparison?
A.(a1b1 + a2b2 + a3b3)/3 ≥ ((a1+a2+a3)/3)((b1+b2+b3)/3)
B.(a1b1 + a2b2 + a3b3)/3 ≤ ((a1+a2+a3)/3)((b1+b2+b3)/3)
C.a1b3 + a2b2 + a3b1 ≥ a1b1 + a2b2 + a3b3
D.(a1+a2+a3)(b1+b2+b3) ≤ 3(a1b1 + a2b2 + a3b3) always fails for similarly ordered sequences
Explanation: Chebyshev's sum inequality for similarly ordered sequences says the mean of the products is at least the product of the means: (1/n) Σ ai bi ≥ ((1/n) Σ ai)((1/n) Σ bi). Equality holds when the sequences are proportional (or constant). The reverse inequality holds when the sequences are oppositely ordered.
3By De Moivre's theorem, (cos 15° + i sin 15°)^6 equals which complex number?
A.1
B.-1
C.i
D.-i
Explanation: De Moivre's theorem gives (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). Here nθ = 90°, so the power is cos 90° + i sin 90° = 0 + i·1 = i.
4Express 1+i in polar form and apply De Moivre's theorem. The value of (1+i)^8 is:
A.16
B.8i
C.-16
D.16i
Explanation: 1+i = √2 (cos(π/4) + i sin(π/4)). Then (1+i)^8 = (√2)^8 (cos(2π) + i sin(2π)) = 2^4 · 1 = 16. Equivalently (1+i)^2 = 2i, so (1+i)^8 = (2i)^4 = 16 i^4 = 16.
5If ω is a primitive cube root of unity, then 1 + ω + ω² equals:
A.1
B.ω
C.0
D.3
Explanation: The cube roots of unity are the roots of z³ − 1 = (z−1)(z²+z+1)=0. A primitive root satisfies ω ≠ 1 and ω³=1, so ω²+ω+1=0. Hence 1+ω+ω²=0. Also ω² + ω = −1 and (1+ω+ω²)(ω−1)=ω³−1=0.
6For |x|<1, the infinite sum Σ_{n=1}^∞ n x^n equals:
A.1/(1-x)
B.x/(1-x)^2
C.1/(1-x)^2
D.x/(1-x)
Explanation: Start from Σ_{n=0}^∞ x^n = 1/(1-x). Differentiate: Σ_{n=1}^∞ n x^{n-1} = 1/(1-x)^2. Multiply by x to obtain Σ n x^n = x/(1-x)^2. In particular, at x=1/2 the sum is 2.
7Using Horner's scheme (synthetic division) to evaluate p(x) = x³ + 4x² − 3x + 7 at x=2, the remainder (which equals p(2)) is:
A.17
B.21
C.25
D.29
Explanation: Synthetic division by x−2 on coefficients 1, 4, −3, 7 brings down 1, then 1·2+4=6, 6·2+(−3)=9, 9·2+7=25. The last number is the remainder, so p(2)=25. Directly, 8+16−6+7=25.
8By Descartes' rule of signs, the number of positive real roots of p(x) = x^4 + 3x² + 2, counted with multiplicity, is:
A.0
B.1
C.2
D.4
Explanation: Descartes' rule counts sign changes in the coefficient sequence. Here all nonzero coefficients are positive (1, 0, 3, 0, 2), so there are no sign changes and therefore no positive real roots. Replacing x by −x leaves the even function unchanged, so there are also no negative real roots; all four roots are nonreal.
9The product of the roots of the quadratic equation 2x² − 3x − 2 = 0 is:
A.3/2
B.-3/2
C.-1
D.1
Explanation: For ax² + bx + c = 0 the product of the roots is c/a. Here c/a = −2/2 = −1. The roots are 2 and −1/2, and 2·(−1/2)=−1, which matches.
10If α, β, γ are the roots of x³ − 4x² + 5x − 2 = 0, then α² + β² + γ² equals:
A.4
B.5
C.6
D.11
Explanation: For x³ − s1 x² + s2 x − s3 = 0 one has α+β+γ = s1 = 4 and αβ+βγ+γα = s2 = 5. Then α²+β²+γ² = (α+β+γ)² − 2(αβ+βγ+γα) = 16 − 10 = 6. (The roots are 1, 1, 2, and 1+1+4=6.)

About the BCS Written Mathematics Exam

BCS Written Post-Related Mathematics (Subject Code 551) is the 200-mark honours mathematics paper for technical/professional cadre and dual-cadre BCS applicants. Part I (100 marks) covers algebra, analytical geometry, linear algebra, and calculus with elementary real analysis. Part II (100 marks) covers mechanics and methods of applied mathematics, including ODE, Beta and Gamma functions, Legendre/Hermite/Laguerre/Bessel functions, and complex residues. The official paper is descriptive. This free bank is an English-language MCQ study adaptation of the assessable syllabus—not an official translation, format simulation, or substitute for writing full solutions.

Assessment

Part I (100 marks): Group A Algebra, Analytical Geometry, and Linear Algebra; Group B Differential and Integral Calculus including elementary real analysis. Part II (100 marks): Group A Mechanics; Group B Methods of Applied Mathematics (ODE, Beta/Gamma, special functions, complex analysis and residues). Official answers are written workings. This bank is an English-language MCQ study adaptation, not a simulation of the 200-mark descriptive paper.

Time Limit

4 hours (duration used for BPSC 200-mark written papers on this site and in longstanding BPSC sitting practice; confirm the current routine)

Passing Score

50% aggregate on the 900-mark written examination (BPSC page); per-paper minima also apply as set by BPSC

Exam Fee

Tk. 200 general (50th BCS); Tk. 50 for ethnic minority, persons with disabilities, and third gender; no separate written-stage fee. Confirm the current circular. (Bangladesh Public Service Commission (BPSC))

BCS Written Mathematics Exam Content Outline

Part I, 100 marks (with geometry and linear algebra)

Part I Group A: Algebra

Order properties of reals, mean inequalities and Chebyshev, complex numbers and De Moivre, finite series, polynomials, Horner's scheme, Descartes' rule of signs, and symmetric functions of roots.

Part I, 100 marks (with algebra and linear algebra)

Part I Group A: Analytical Geometry

Pairs of lines, coordinate transformations, the general second-degree equation and conics, 3-D planes and lines, shortest distance, and vector algebra.

Part I, 100 marks (with algebra and geometry)

Part I Group A: Linear Algebra

Matrices and linear systems, vector spaces over R, linear independence, basis and dimension, linear maps, rank-nullity, eigenvalues and eigenvectors.

Part I, 100 marks

Part I Group B: Calculus and Real Analysis

Suprema, completeness, sequences and series tests, continuity and IVT, Rolle/MVT/Taylor, maxima and minima, integration techniques, Riemann integral and FTC, improper integrals, areas and volumes.

Part II, 100 marks (with applied methods)

Part II Group A: Mechanics

Equilibrium, virtual work, stability, centre of gravity, rectilinear motion, SHM, central forces, rigid bodies, moments of inertia, d'Alembert, rotation about a fixed axis, Lagrange for holonomic systems.

Part II, 100 marks (with mechanics)

Part II Group B: Methods of Applied Mathematics

First- and second-order ODE, constant-coefficient linear equations, series solutions, Beta and Gamma, Legendre/Hermite/Laguerre/Bessel, analytic functions, Cauchy theorems, Laurent series, residues, and real integrals.

How to Pass the BCS Written Mathematics Exam

What You Need to Know

  • Passing score: 50% aggregate on the 900-mark written examination (BPSC page); per-paper minima also apply as set by BPSC
  • Assessment: Part I (100 marks): Group A Algebra, Analytical Geometry, and Linear Algebra; Group B Differential and Integral Calculus including elementary real analysis. Part II (100 marks): Group A Mechanics; Group B Methods of Applied Mathematics (ODE, Beta/Gamma, special functions, complex analysis and residues). Official answers are written workings. This bank is an English-language MCQ study adaptation, not a simulation of the 200-mark descriptive paper.
  • Time limit: 4 hours (duration used for BPSC 200-mark written papers on this site and in longstanding BPSC sitting practice; confirm the current routine)
  • Exam fee: Tk. 200 general (50th BCS); Tk. 50 for ethnic minority, persons with disabilities, and third gender; no separate written-stage fee. Confirm the current circular.

Keys to Passing

  • Work through all 100 available questions
  • Review every answer and explanation
  • Track weak areas and revisit them
  • Use our AI tutor for tough concepts

BCS Written Mathematics Study Tips from Top Performers

1Weight revision 50/50 to Part I and Part II; skipping mechanics or complex analysis leaves half the official paper untouched.
2Practise full workings for De Moivre, eigenpairs, series tests, Rolle/MVT, and integration—the official paper awards method, not a circled letter.
3Memorize Gamma/Beta identities, low-order Legendre/Hermite/Laguerre polynomials, and the residue at a pole of order m before contour problems.
4For mechanics, keep virtual work, SHM period, parallel-axis MOI, and Lagrange's equation for a simple pendulum as instant-recall templates.
5Use this MCQ bank to lock results and common traps, then write timed descriptive answers from the BPSC syllabus headings.

Frequently Asked Questions

What is BCS Written Post-Related Mathematics?

It is the 200-mark post-related written paper (Subject Code 551) in the BCS written stage, used by technical/professional cadre and dual-cadre applicants. The official BPSC syllabus splits it into Part I (algebra, geometry, linear algebra, calculus/real analysis) and Part II (mechanics and methods of applied mathematics), 100 marks each.

Is the official paper multiple choice?

No. BPSC examines this subject as a 200-mark descriptive written paper. This bank is an English-language MCQ study adaptation of the syllabus, not an official translation, format simulation, or substitute for writing full solutions under timed conditions.

How long is the paper?

Four hours is the duration used for BPSC 200-mark written papers on this site and in longstanding BPSC sitting practice. Always confirm the sitting routine published for your BCS cycle.

What is the pass mark?

The BPSC page requires 50% aggregate on the 900-mark written examination. Per-paper minima also apply as set by BPSC. There is no published MCQ pass score because the official paper is not MCQ.

What is the exam fee?

There is no separate fee for the Mathematics written paper. For the 50th BCS the application fee is Tk. 200 general and Tk. 50 for ethnic minority, persons with disabilities, and third gender. Confirm the live circular before paying.

Who sits this paper?

Technical/professional cadre candidates sit a 200-mark post-related subject instead of the extra general-cadre papers. Dual-cadre applicants sit this extra 200-mark paper as well. General-cadre-only candidates do not sit Subject Code 551.

How does the viva relate to this paper?

After written qualification, candidates face a viva voce. The official BPSC page still lists 200 marks with a 50% pass. News coverage from the 47th BCS onward has reported a 100-mark viva, and PSC has discussed reducing it further. Confirm the current circular.

In what language is this practice bank?

This bank is an original English-language MCQ study adaptation of the official BPSC Mathematics syllabus. It is not an official BPSC translation or a simulation of the language or written format of the real paper.