Air University Entry Test Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in Air University Entry Test, not a summary of it.

1. 1

   Algebra

   5 topics

   • Quadratic equations
   • Functions and graphs
   • Sequences and series
   • Permutations and combinations
   • Binomial theorem
2. 2

   Matrices and Determinants

   3 topics

   • Matrix operations
   • Determinants
   • Solving systems of equations
3. 3

   Trigonometry

   3 topics

   • Trigonometric functions and identities
   • Solution of triangles
   • Inverse trigonometric functions
4. 4

   Analytic Geometry

   3 topics

   • Straight lines
   • Conic sections
   • Vectors
5. 5

   Calculus

   3 topics

   • Limits and continuity
   • Differentiation
   • Integration
6. 6

   Sets, Functions and Number Systems

   3 topics

   • Sets and operations
   • Complex numbers
   • Mathematical induction

Mathematics flashcards for Air University Entry Test

21 of 50 cards from the Mathematics deck — real questions with worked answers.

1. What is the standard form of a quadratic equation?

   ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.

2. State the quadratic formula for the roots of ax² + bx + c = 0.

   x = [−b ± √(b² − 4ac)] / (2a).

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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).

11. 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.

12. 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.

13. 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.

14. 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)).

15. What is the sum to infinity of a geometric series, and when does it exist?

    S∞ = a/(1 − r), valid only when |r| < 1.

16. 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).

17. What is the formula for the number of permutations of n distinct objects taken r at a time?

    P(n, r) = n! / (n − r)!.

18. 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)!].

19. 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).

20. 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.

21. 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.

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