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GIKI Admission Test Mathematics Syllabus
The Mathematics syllabus for GIKI Admission Test in full: 11 chapters and 44 topics, every one listed below, plus 50 Mathematics flashcards with answers.
Mathematics syllabus: all 11 chapters
Expand any chapter to see its topics and sub-topics.
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Number Systems, Sets & Functions
4 topics- Real and Complex Numbers
- Sets and Relations
- Functions
- Polynomials and Linear Equations
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Matrices & Determinants
4 topics- Matrix Operations
- Determinants and Their Properties
- Inverse of a Matrix
- Solution of Simultaneous Equations
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Quadratic Equations
4 topics- Solution of Quadratic Equations
- Nature of Roots
- Relationship Between Roots and Coefficients
- Equations Reducible to Quadratic Form
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Sequences & Series
4 topics- Arithmetic Progression
- Geometric Progression
- Harmonic Progression
- Sum to Infinity of Geometric Series
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Permutations, Combinations & Probability
4 topics- Permutations
- Combinations
- Basic Probability
- Elementary Statistics
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Mathematical Induction & Binomial Theorem
4 topics- Principle of Mathematical Induction
- Binomial Theorem for Positive Integral Index
- Binomial Series for Rational Index
- Applications of Binomial Expansion
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Trigonometry
5 topics- Angles and Their Measurement
- Trigonometric Functions and Identities
- Trigonometric Equations
- Inverse Trigonometric Functions
- Solution of Triangles and Applications
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Differential Calculus
4 topics- Limits and Continuity
- Differentiation Rules
- Derivatives of Trigonometric, Exponential and Logarithmic Functions
- Maxima, Minima and Rate of Change
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Integral Calculus
4 topics- Indefinite Integration
- Methods of Integration
- Definite Integrals
- Area Under Curves and Volumes
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Analytic Geometry
3 topics- Coordinate Geometry and Straight Lines
- Circle
- Conic Sections
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Vectors
4 topics- Vectors in Plane and Space
- Scalar (Dot) Product
- Vector (Cross) Product
- Scalar Triple Product
GIKI Admission Test Mathematics flashcards — 50 questions with answers
What is a complex number, and what are its real and imaginary parts in the form z = a + bi?
A complex number is a number of the form z = a + bi where a and b are real numbers and i = √(-1). Here 'a' is the real part Re(z) and 'b' is the imaginary part Im(z).
What is the conjugate of a complex number z = a + bi, and what is the product of z with its conjugate?
The conjugate is z̄ = a - bi. Their product z·z̄ = a² + b², which is a non-negative real number equal to |z|².
What is the modulus of a complex number z = a + bi?
The modulus is |z| = √(a² + b²), the distance of the point (a, b) from the origin in the Argand plane.
Classify the real number system: how do natural, whole, integer, rational, and irrational numbers relate?
Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational numbers. Rationals (expressible as p/q, q≠0) together with irrationals (non-terminating, non-repeating decimals like √2, π) form the real numbers.
What is the difference between a rational and 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 a fraction and has a non-terminating, non-repeating decimal.
Define the union and intersection of two sets A and B.
A ∪ B is the set of all elements in A or B (or both). A ∩ B is the set of elements common to both A and B.
What is the difference of sets A − B, and what is the complement of a set A?
A − B is the set of elements in A but not in B. The complement A' (or Aᶜ) is the set of all elements in the universal set U that are not in A.
State De Morgan's Laws for sets.
(A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.
For finite sets, what is the formula for the number of elements in A ∪ B (inclusion–exclusion)?
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
How many subsets does a set with n elements have, and how many of them are proper subsets?
A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets (excluding the set itself).
What is a relation from set A to set B?
A relation from A to B is any subset of the Cartesian product A × B; i.e., a set of ordered pairs (a, b) with a ∈ A and b ∈ B.
Define reflexive, symmetric, and transitive relations.
Reflexive: (a, a) ∈ R for all a. Symmetric: if (a, b) ∈ R then (b, a) ∈ R. Transitive: if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R. A relation that is all three is an equivalence relation.
What is a function, and how does it differ from a general relation?
A function from A to B is a relation that assigns to each element of A exactly one element of B. Unlike a general relation, no element of the domain may map to more than one image.
Define the domain, codomain, and range of a function.
Domain: the set of all input values (A). Codomain: the set B into which the function maps. Range: the set of actual output values, a subset of the codomain.
Distinguish between one-to-one (injective), onto (surjective), and bijective functions.
Injective: distinct inputs give distinct outputs. Surjective: every element of the codomain is an image of some input. Bijective: both injective and surjective (one-to-one correspondence).
What is the condition for a function f to have an inverse, and what does the inverse satisfy?
A function has an inverse if and only if it is bijective. The inverse f⁻¹ satisfies f⁻¹(f(x)) = x and f(f⁻¹(y)) = y, effectively swapping domain and range.
What is a polynomial of degree n in one variable?
A polynomial of degree n is an expression aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ where aₙ ≠ 0 and the exponents are non-negative integers.
State the Remainder Theorem.
When a polynomial P(x) is divided by (x − a), the remainder equals P(a).
State the Factor Theorem.
(x − a) is a factor of polynomial P(x) if and only if P(a) = 0.
What is the order (or dimension) of a matrix, and when is a matrix called square?
The order of a matrix is m × n where m is the number of rows and n the number of columns. A matrix is square when m = n (equal rows and columns).
Show the other 30 Mathematics flashcards
Under what condition can two matrices be multiplied, and what is the order of the product?
Matrices A (m × n) and B (p × q) can be multiplied only if n = p. The product AB has order m × q.
Is matrix multiplication commutative? State the key property.
No, matrix multiplication is generally not commutative: AB ≠ BA in general. However, it is associative (AB)C = A(BC) and distributive over addition.
What is the transpose of a matrix, and what is (AB)ᵀ?
The transpose Aᵀ is formed by interchanging rows and columns. For a product, (AB)ᵀ = BᵀAᵀ (reverse order).
Define a symmetric matrix and a skew-symmetric matrix.
A symmetric matrix satisfies Aᵀ = A. A skew-symmetric matrix satisfies Aᵀ = −A (its diagonal elements are all zero).
What is an identity matrix and what is its role in multiplication?
An identity matrix I is a square matrix with 1s on the main diagonal and 0s elsewhere. It acts as a multiplicative identity: AI = IA = A.
How is the determinant of a 2×2 matrix [[a, b],[c, d]] computed?
The determinant is ad − bc.
How is the determinant of a 3×3 matrix found by expansion along the first row?
For [[a,b,c],[d,e,f],[g,h,i]]: det = a(ei − fh) − b(di − fg) + c(dh − eg), expanding by cofactors along the first row with the +,−,+ sign pattern.
State two key properties of determinants relating to row operations.
Interchanging two rows (or columns) multiplies the determinant by −1. If two rows (or columns) are identical or proportional, the determinant is 0. Adding a multiple of one row to another leaves the determinant unchanged.
What is the determinant of a product of matrices, and of a transpose?
det(AB) = det(A)·det(B), and det(Aᵀ) = det(A).
When does a square matrix have an inverse (is non-singular)?
A square matrix A is invertible (non-singular) if and only if its determinant is non-zero, det(A) ≠ 0. If det(A) = 0 the matrix is singular and has no inverse.
What is the formula for the inverse of a matrix A using the adjoint?
A⁻¹ = (1/det(A)) · adj(A), where adj(A) is the adjoint (the transpose of the cofactor matrix), valid when det(A) ≠ 0.
How do you find the inverse of a 2×2 matrix [[a, b],[c, d]]?
A⁻¹ = (1/(ad − bc)) · [[d, −b],[−c, a]], provided ad − bc ≠ 0.
What is Cramer's Rule for solving simultaneous linear equations?
For a system AX = B with det(A) = D ≠ 0, each unknown xᵢ = Dᵢ / D, where Dᵢ is the determinant of A with its i-th column replaced by the constants column B.
In a system of linear equations AX = B, how does the determinant of A classify the solution?
If det(A) ≠ 0, there is a unique solution. If det(A) = 0, the system has either no solution (inconsistent) or infinitely many solutions (dependent).
What is the quadratic formula for ax² + bx + c = 0?
x = [−b ± √(b² − 4ac)] / (2a), where a ≠ 0.
What is the discriminant of a quadratic equation and what does it represent?
The discriminant is D = b² − 4ac. It determines the nature of the roots of ax² + bx + c = 0.
How does the discriminant determine the nature of the roots of ax² + bx + c = 0?
If D > 0: two distinct real roots. If D = 0: two equal (repeated) real roots. If D < 0: two complex conjugate roots. If D > 0 and is a perfect square (with rational coefficients): roots are rational.
For ax² + bx + c = 0 with roots α and β, what are the sum and product of the roots?
Sum of roots α + β = −b/a, and product of roots αβ = c/a.
How do you form a quadratic equation given the sum S and product P of its roots?
x² − Sx + P = 0, i.e., x² − (α+β)x + αβ = 0.
How are equations reducible to quadratic form, such as ax⁴ + bx² + c = 0, solved?
Substitute y = x² to convert it into ay² + by + c = 0, solve for y, then back-substitute and solve x² = y for x. The same substitution technique handles other equations expressible as a quadratic in some expression.
Define an arithmetic progression (AP) and its common difference.
An AP is a sequence in which each term differs from the previous by a constant called the common difference d, where d = aₙ − aₙ₋₁. Example: a, a+d, a+2d, ...
What is the formula for the nth term of an arithmetic progression?
aₙ = a + (n − 1)d, where a is the first term and d is the common difference.
What is 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 between two numbers a and b?
The arithmetic mean is (a + b)/2. It is the term inserted between a and b so the three form an AP.
Define a geometric progression (GP) and give the formula for its nth term.
A GP is a sequence where each term is the previous term multiplied by a constant ratio r (r = aₙ/aₙ₋₁). The nth term is aₙ = a·rⁿ⁻¹.
What is the sum of the first n terms of a geometric progression?
Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1 (equivalently a(1 − rⁿ)/(1 − r)). If r = 1, Sₙ = na.
What is the formula for the sum to infinity of a geometric series, and when does it apply?
S∞ = a/(1 − r), valid only when |r| < 1. If |r| ≥ 1 the infinite series diverges and has no finite sum.
What is the geometric mean between two positive numbers a and b?
The geometric mean is √(ab). It is the term inserted between a and b so the three form a GP.
Define a harmonic progression (HP) and how it relates to an AP.
A harmonic progression is a sequence whose reciprocals form an arithmetic progression. To find an HP term, take the corresponding AP of reciprocals, find its term, then take the reciprocal. The harmonic mean of a and b is 2ab/(a + b).
What is the formula for permutations of n distinct objects taken r at a time, and how do permutations differ from combinations?
ⁿPᵣ = n!/(n − r)! while ⁿCᵣ = n!/[r!(n − r)!]. Permutations count arrangements where order matters; combinations count selections where order does not matter, so ⁿPᵣ = ⁿCᵣ · r!.
GIKI Admission Test Mathematics FAQ
What is in the GIKI Admission Test Mathematics syllabus?
GIKI Admission Test Mathematics is split into 11 chapters: Number Systems, Sets & Functions, Matrices & Determinants, Quadratic Equations, Sequences & Series, Permutations, Combinations & Probability, Mathematical Induction & Binomial Theorem, Trigonometry, Differential Calculus, Integral Calculus, Analytic Geometry and Vectors. Together they contain 44 topics.
How many chapters are in GIKI Admission Test Mathematics?
11 chapters, containing 44 topics. The largest chapter is Trigonometry, with 5 topics.
Are there flashcards for GIKI 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.