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Air University Entry Test Mathematics Syllabus
The Mathematics syllabus for Air University Entry Test in full: 6 chapters and 20 topics, every one listed below, plus 50 Mathematics flashcards with answers.
Mathematics syllabus: all 6 chapters
Expand any chapter to see its topics and sub-topics.
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Algebra
5 topics- Quadratic equations
- Functions and graphs
- Sequences and series
- Permutations and combinations
- Binomial theorem
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Matrices and Determinants
3 topics- Matrix operations
- Determinants
- Solving systems of equations
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Trigonometry
3 topics- Trigonometric functions and identities
- Solution of triangles
- Inverse trigonometric functions
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Analytic Geometry
3 topics- Straight lines
- Conic sections
- Vectors
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Calculus
3 topics- Limits and continuity
- Differentiation
- Integration
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Sets, Functions and Number Systems
3 topics- Sets and operations
- Complex numbers
- Mathematical induction
Air University Entry Test Mathematics flashcards — 50 questions with answers
What is the standard form of a quadratic equation?
ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
State the quadratic formula for the roots of ax² + bx + c = 0.
x = [−b ± √(b² − 4ac)] / (2a).
What is the discriminant of a quadratic equation, and what does it determine?
D = b² − 4ac. It determines the nature of the roots: D > 0 → two distinct real roots; D = 0 → one repeated real root; D < 0 → two complex conjugate roots.
For 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.
What is a function, and what are its domain and range?
A function is a relation that assigns each input exactly one output. The domain is the set of all valid inputs; the range is the set of all resulting outputs.
How do you test whether a graph represents a function?
The vertical line test: if any vertical line intersects the graph at more than one point, it is not a function.
Define an even function and an odd function (with their symmetry).
Even: f(−x) = f(x), symmetric about the y-axis. Odd: f(−x) = −f(x), symmetric about the origin.
What condition must a function satisfy to have an inverse, and what is the graphical relationship between f and f⁻¹?
It must be one-to-one (bijective / passes the horizontal line test). The graph of f⁻¹ is the reflection of f across the line y = x.
Describe the graph transformations for y = f(x) + k, y = f(x − h), and y = a·f(x).
f(x) + k shifts vertically by k; f(x − h) shifts horizontally right by h; a·f(x) stretches vertically by factor a (and reflects in x-axis if a < 0).
What is the general term (nth term) of an arithmetic sequence?
aₙ = a + (n − 1)d, where a is the first term and d is the common difference.
State the formula for the sum of the first n terms of an arithmetic series.
Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
What is the nth term of a geometric sequence?
aₙ = a·r^(n−1), where a is the first term and r is the common ratio.
State the sum of the first n terms of a geometric series.
Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1 (equivalently a(rⁿ − 1)/(r − 1)).
What is the sum to infinity of a geometric series, and when does it exist?
S∞ = a/(1 − r), valid only when |r| < 1.
What is the arithmetic mean between two numbers a and b, and the geometric mean?
Arithmetic mean = (a + b)/2; Geometric mean = √(ab) (for positive a, b).
What is the formula for the number of permutations of n distinct objects taken r at a time?
P(n, r) = n! / (n − r)!.
What is the formula for the number of combinations of n distinct objects taken r at a time?
C(n, r) = n! / [r!(n − r)!].
What is the key conceptual difference between a permutation and a combination?
In a permutation order matters (arrangements); in a combination order does not matter (selections).
How many distinct arrangements of n objects exist when there are repeated objects (p alike of one kind, q of another)?
n! / (p! q! …), dividing the total factorial by the factorials of the counts of each repeated group.
Show the other 30 Mathematics flashcards
State the properties C(n, r) = C(n, n − r) and C(n, 0) = C(n, n) = ?
C(n, r) = C(n, n − r) (symmetry); C(n, 0) = C(n, n) = 1.
State the Binomial Theorem expansion for (a + b)ⁿ.
(a + b)ⁿ = Σ (from r=0 to n) C(n, r) a^(n−r) bʳ.
What is the general (r+1)th term in the expansion of (a + b)ⁿ?
Tᵣ₊₁ = C(n, r) a^(n−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 the binomial coefficients C(n,0)+C(n,1)+…+C(n,n) = 2ⁿ.
What is the binomial series for (1 + x)ⁿ when n is not a positive integer, and its condition of validity?
(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … , valid for |x| < 1.
How are two matrices added, and what condition must they satisfy?
They must have the same order (dimensions); add corresponding entries: (A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ.
State the condition for matrix multiplication AB and the order of the resulting matrix.
The number of columns of A must equal the number of rows of B. If A is m×n and B is n×p, then AB is m×p.
Is matrix multiplication commutative? State the key property.
No; in general AB ≠ BA. Matrix multiplication is associative and distributive but not commutative.
What is the identity matrix, and what is its effect under multiplication?
The identity matrix I is a square matrix with 1's on the main diagonal and 0's elsewhere. For any compatible matrix A: AI = IA = A.
What is the transpose of a matrix, and what is (AB)ᵀ?
The transpose Aᵀ is formed by interchanging rows and columns. (AB)ᵀ = BᵀAᵀ.
Define a symmetric matrix and a skew-symmetric matrix.
Symmetric: Aᵀ = A. Skew-symmetric: Aᵀ = −A (its diagonal entries are all zero).
How do you compute the determinant of a 2×2 matrix [[a, b], [c, d]]?
det = ad − bc.
What does it mean when det(A) = 0?
The matrix is singular: it has no inverse, and a related system of equations has either no unique solution (no solution or infinitely many).
State the formula for the inverse of a 2×2 matrix A = [[a, b], [c, d]].
A⁻¹ = (1/det A) [[d, −b], [−c, a]], provided det A = ad − bc ≠ 0.
State the property det(AB) and det(Aᵀ) in terms of determinants.
det(AB) = det(A)·det(B); det(Aᵀ) = det(A).
What is the cofactor expansion method for a 3×3 determinant?
Expand along a row or column: det = Σ aᵢⱼ·Cᵢⱼ, where Cᵢⱼ = (−1)^(i+j)·Mᵢⱼ and Mᵢⱼ is the minor (determinant of the matrix with row i and column j deleted).
State Cramer's Rule for solving a system Ax = b.
Each variable xᵢ = det(Aᵢ)/det(A), where Aᵢ is A with its ith column replaced by b, provided det(A) ≠ 0.
In the matrix method, how is the solution to AX = B found?
X = A⁻¹B, provided A is invertible (det A ≠ 0).
Classify the consistency of a linear system by its number of solutions.
Consistent with a unique solution; consistent with infinitely many solutions; or inconsistent (no solution).
What are the three elementary row operations used in Gaussian elimination?
1) Swap two rows; 2) Multiply a row by a nonzero scalar; 3) Add a multiple of one row to another row.
Define the six trigonometric ratios in a right triangle for angle θ.
sin θ = opp/hyp; cos θ = adj/hyp; tan θ = opp/adj; csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ.
State the three Pythagorean identities.
sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.
State the sine and cosine addition formulas.
sin(A + B) = sin A cos B + cos A sin B; cos(A + B) = cos A cos B − sin A sin B.
State the double angle formulas for sin 2θ and cos 2θ.
sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.
State the Law of Sines for a triangle with sides a, b, c opposite angles A, B, C.
a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.
State the Law of Cosines for side a.
a² = b² + c² − 2bc·cos A.
Give two formulas for the area of a triangle using its sides/angles.
Area = ½·ab·sin C (two sides and included angle); or Heron's formula: √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2.
State the domain and range of the inverse function y = sin⁻¹x (arcsin).
Domain: −1 ≤ x ≤ 1; Range (principal values): −π/2 ≤ y ≤ π/2.
State the slope formula and the slope-intercept form of a straight line.
Slope m = (y₂ − y₁)/(x₂ − x₁); slope-intercept form: y = mx + c, where c is the y-intercept.
State the point-slope form of a line and the conditions for two lines to be parallel or perpendicular.
Point-slope: y − y₁ = m(x − x₁). Parallel lines have equal slopes (m₁ = m₂); perpendicular lines satisfy m₁·m₂ = −1.
Air University Entry Test Mathematics FAQ
What is in the Air University Entry Test Mathematics syllabus?
Air University Entry Test Mathematics is split into 6 chapters: Algebra, Matrices and Determinants, Trigonometry, Analytic Geometry, Calculus and Sets, Functions and Number Systems. Together they contain 20 topics.
How many chapters are in Air University Entry Test Mathematics?
6 chapters, containing 20 topics. The largest chapter is Algebra, with 5 topics.
Are there flashcards for Air University Entry 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.