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PIEAS Admission Test Mathematics Syllabus
The Mathematics syllabus for PIEAS Admission Test in full: 10 chapters and 34 topics, every one listed below, plus 50 Mathematics flashcards with answers.
Mathematics syllabus: all 10 chapters
Expand any chapter to see its topics and sub-topics.
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Algebra and Functions
4 topics- Number Systems and Complex Numbers
- Sets, Functions and Groups
- Partial Fractions
- Binomial Theorem and Mathematical Induction
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Quadratic Equations
3 topics- Nature of Roots
- Solving Quadratic and Reducible Equations
- Systems of Equations
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Sequences and Series
4 topics- Arithmetic Progression
- Geometric Progression
- Harmonic Progression and Means
- Infinite Series and Sum to Infinity
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Trigonometry
4 topics- Trigonometric Functions and Identities
- Trigonometric Equations
- Solution of Triangles
- Inverse Trigonometric Functions
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Coordinate Geometry
3 topics- Straight Lines
- Circles
- Conic Sections
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Vectors
3 topics- Vector Algebra and Components
- Scalar (Dot) Product
- Vector (Cross) Product
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Matrices and Determinants
4 topics- Matrix Operations
- Determinants and Properties
- Inverse of a Matrix
- Solving Linear Systems (Cramer's Rule)
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Differential Calculus
3 topics- Limits and Continuity
- Derivatives and Rules of Differentiation
- Applications of Derivatives
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Integral Calculus
3 topics- Indefinite Integration and Techniques
- Definite Integrals
- Area Under a Curve
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Probability and Statistics
3 topics- Permutations and Combinations
- Basic Probability
- Measures of Central Tendency
PIEAS Admission Test Mathematics flashcards — 50 questions with answers
What is the standard form of a complex number, and what do the real and imaginary parts represent?
A complex number is written z = a + bi, where a is the real part Re(z), b is the imaginary part Im(z), and i = √(-1) with i² = -1.
How do you compute the modulus and conjugate of a complex number z = a + bi?
Modulus |z| = √(a² + b²); conjugate z̄ = a − bi. Also z·z̄ = a² + b² = |z|².
What are the four powers of the imaginary unit i in cyclic form?
i¹ = i, i² = −1, i³ = −i, i⁴ = 1; the pattern repeats every 4 powers.
How is a complex number a + bi expressed in polar (trigonometric) form?
z = r(cos θ + i sin θ), where r = |z| = √(a²+b²) and θ = arg(z) = tan⁻¹(b/a).
State De Moivre's Theorem for a complex number in polar form.
[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ) for any integer n.
Classify the real number system into its main subsets.
Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational numbers; together with Irrational numbers they form the Real numbers. Reals ⊂ Complex numbers.
What distinguishes a rational number from an irrational number?
A rational number can be written as p/q with integers p, q (q≠0) and has a terminating or repeating decimal; an irrational number cannot be expressed as such and has a non-terminating, non-repeating decimal (e.g., √2, π).
Define a function in terms of a relation between two sets.
A function f: A → B is a relation that assigns to each element of A exactly one element of B (no input has two outputs).
Distinguish between an injective (one-to-one), surjective (onto), and bijective function.
Injective: distinct inputs give distinct outputs. Surjective: every element of the codomain is mapped to. Bijective: both injective and surjective, hence invertible.
State De Morgan's Laws for sets.
(A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.
For a set with n elements, how many subsets and how many proper subsets does it have?
It has 2ⁿ subsets in total and 2ⁿ − 1 proper subsets (excluding the set itself).
What four properties must a set with a binary operation satisfy to be a group?
Closure, Associativity, Existence of an Identity element, and Existence of an Inverse for every element. If it is also commutative, it is an Abelian group.
What is the condition on a rational fraction P(x)/Q(x) before it can be split into partial fractions?
It must be a proper fraction (degree of numerator < degree of denominator). If improper, first divide to get a polynomial plus a proper fraction.
What form of partial fraction corresponds to a non-repeated linear factor (ax + b) in the denominator?
A single term of the form A/(ax + b), where A is a constant to be determined.
What partial fraction form corresponds to a repeated linear factor (ax + b)² in the denominator?
A/(ax + b) + B/(ax + b)² — one term for each power up to the multiplicity.
What partial fraction form corresponds to a non-repeated irreducible quadratic factor (ax² + bx + c)?
A term of the form (Ax + B)/(ax² + bx + c), with a linear numerator.
State the Binomial Theorem for (a + b)ⁿ where n is a positive integer.
(a + b)ⁿ = Σ (from r=0 to n) C(n,r) aⁿ⁻ʳ bʳ, where C(n,r) = n!/[r!(n−r)!].
What is the general (r+1)th term in the expansion of (a + b)ⁿ?
T₍ᵣ₊₁₎ = C(n, r) aⁿ⁻ʳ bʳ.
How many terms are in the expansion of (a + b)ⁿ, and what is the sum of the binomial coefficients?
There are (n + 1) terms; the sum of all binomial coefficients C(n,0)+C(n,1)+...+C(n,n) = 2ⁿ.
State the two steps of the Principle of Mathematical Induction.
1) Base step: prove the statement true for n = 1 (or the starting value). 2) Inductive step: assume true for n = k, then prove it true for n = k + 1.
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What does the discriminant of a quadratic ax² + bx + c = 0 equal, and what does it determine?
Discriminant = b² − 4ac; it determines the nature (type) of the roots.
Describe the nature of the roots of a quadratic based on the discriminant b² − 4ac.
If >0: two real distinct roots (rational if a perfect square, otherwise irrational). If =0: two real equal (repeated) roots. If <0: two complex conjugate roots.
For a quadratic ax² + bx + c = 0 with roots α and β, what are the sum and product of the roots?
Sum α + β = −b/a; Product αβ = c/a.
How do you form a quadratic equation given its roots α and β?
x² − (α + β)x + αβ = 0, i.e., x² − (sum of roots)x + (product of roots) = 0.
State the quadratic formula for the roots of ax² + bx + c = 0.
x = [−b ± √(b² − 4ac)] / (2a).
What is a reducible (equation reducible to quadratic) equation, with an example method?
An equation not quadratic in x but quadratic in some expression; substitute that expression (e.g., let y = x²) to convert it into a quadratic, solve, then back-substitute. Example: x⁴ − 5x² + 4 = 0 with y = x².
What does it mean for a system of linear equations to be consistent versus inconsistent?
Consistent: has at least one solution. Inconsistent: has no solution. A consistent system may have a unique solution or infinitely many.
Using determinants, what is Cramer's Rule for solving a system of linear equations?
For AX = B, each unknown xᵢ = Dᵢ / D, where D is the determinant of the coefficient matrix and Dᵢ replaces the i-th column of A with B (requires D ≠ 0).
What does the value of the coefficient determinant D tell you about a system's solutions?
If D ≠ 0: unique solution. If D = 0: either no solution (inconsistent) or infinitely many solutions (dependent).
What is the general (nth) term of an arithmetic progression with first term a and common difference d?
aₙ = a + (n − 1)d.
State the formula for the sum of the first n terms of an arithmetic progression.
Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l), where l is the last term.
What is the arithmetic mean (A.M.) between two numbers a and b?
A.M. = (a + b)/2.
What is the general (nth) term of a geometric progression with first term a and common ratio r?
aₙ = a·rⁿ⁻¹.
State the formula for the sum of the first n terms of a geometric progression (r ≠ 1).
Sₙ = a(1 − rⁿ)/(1 − r) for r < 1, or a(rⁿ − 1)/(r − 1) for r > 1.
What is the geometric mean (G.M.) between two positive numbers a and b?
G.M. = √(ab).
What is a harmonic progression, and how is its nth term found?
A sequence whose reciprocals form an arithmetic progression. To find the nth term, take the reciprocal of the corresponding A.P. term: 1/[a + (n−1)d].
What is the harmonic mean (H.M.) between two numbers a and b?
H.M. = 2ab/(a + b).
State the relationship (and inequality) between A.M., G.M., and H.M. of two positive numbers.
G.M.² = A.M. × H.M., and A.M. ≥ G.M. ≥ H.M. (equality when the numbers are equal).
Under what condition does an infinite geometric series converge, and what is its sum to infinity?
It converges when |r| < 1; then S∞ = a/(1 − r).
What happens to the sum of an infinite geometric series when |r| ≥ 1?
The series diverges — it has no finite sum to infinity.
Define the six trigonometric functions of an angle in a right triangle.
sin = opp/hyp, cos = adj/hyp, tan = opp/adj, csc = hyp/opp, sec = hyp/adj, cot = adj/opp.
State the three Pythagorean trigonometric identities.
sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.
State the sine and cosine addition formulas.
sin(A + B) = sinA cosB + cosA sinB; cos(A + B) = cosA cosB − sinA sinB.
State the double-angle formula for sin 2θ and the three forms for cos 2θ.
sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.
State the tangent addition formula tan(A + B).
tan(A + B) = (tanA + tanB)/(1 − tanA tanB).
What are the periods of sin θ, cos θ, and tan θ?
sin and cos have period 2π (360°); tan and cot have period π (180°).
How do you convert between degrees and radians?
π radians = 180°. To convert degrees to radians multiply by π/180; radians to degrees multiply by 180/π.
State the Law of Sines and the Law of Cosines for a triangle.
Law of Sines: a/sinA = b/sinB = c/sinC. Law of Cosines: c² = a² + b² − 2ab cosC.
State the half-angle formulas for sin(θ/2) and cos(θ/2).
sin(θ/2) = ±√[(1 − cosθ)/2]; cos(θ/2) = ±√[(1 + cosθ)/2], sign depending on the quadrant of θ/2.
In which quadrants are sin, cos, and tan positive (the 'All, Sin, Tan, Cos' rule)?
Quadrant I: all positive. Quadrant II: only sin (and csc) positive. Quadrant III: only tan (and cot) positive. Quadrant IV: only cos (and sec) positive.
PIEAS Admission Test Mathematics FAQ
What is in the PIEAS Admission Test Mathematics syllabus?
PIEAS Admission Test Mathematics is split into 10 chapters: Algebra and Functions, Quadratic Equations, Sequences and Series, Trigonometry, Coordinate Geometry, Vectors, Matrices and Determinants, Differential Calculus, Integral Calculus and Probability and Statistics. Together they contain 34 topics.
How many chapters are in PIEAS Admission Test Mathematics?
10 chapters, containing 34 topics. The largest chapter is Algebra and Functions, with 4 topics.
Are there flashcards for PIEAS Admission Test Mathematics?
Yes — a 50-card Mathematics deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.