HEC USAT 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 HEC USAT, not a summary of it.

1. 1

   Algebra

   4 topics

   • Quadratic Functions and Equations
   • Matrices and Determinants
   • Sequences and Series
   • Binomial Theorem
2. 2

   Trigonometry

   3 topics

   • Trigonometric Ratios and Identities
   • Solutions of Triangles
   • Inverse Trigonometric Functions
3. 3

   Functions and Graphs

   3 topics

   • Types of Functions
   • Domain and Range
   • Graphing and Transformations
4. 4

   Calculus

   4 topics

   • Limits and Continuity
   • Differentiation
   • Applications of Derivatives
   • Integration
5. 5

   Coordinate Geometry

   3 topics

   • Straight Lines
   • Circles
   • Conic Sections
6. 6

   Vectors

   2 topics

   • Vector Operations
   • Dot and Cross Products

Mathematics flashcards for HEC USAT

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

1. What is the quadratic formula for the roots of ax² + bx + c = 0 (a ≠ 0)?

   x = [−b ± √(b² − 4ac)] / (2a)

2. For ax² + bx + c = 0, what does the discriminant b² − 4ac tell you about the roots?

   > 0: two distinct real roots; = 0: one repeated real root; < 0: two complex conjugate roots.

3. For ax² + bx + c = 0, what are the sum and product of the roots?

   Sum of roots = −b/a; Product of roots = c/a.

4. What is the vertex (turning point) of the parabola y = ax² + bx + c?

   x = −b/(2a), and y is found by substituting that x. It is a minimum if a > 0, a maximum if a < 0.

5. How do you form a quadratic equation given its roots α and β?

   x² − (α + β)x + αβ = 0, i.e. x² − (sum)x + (product) = 0.

6. In matrices, what is the condition for the product AB to be defined?

   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.

7. What is the formula for the determinant of a 2×2 matrix [[a, b], [c, d]]?

   det = ad − bc.

8. What is the formula for the inverse of a 2×2 matrix A = [[a, b], [c, d]]?

   A⁻¹ = (1/(ad − bc)) · [[d, −b], [−c, a]], valid when ad − bc ≠ 0.

9. What condition makes a square matrix singular (non-invertible)?

   Its determinant equals zero (det A = 0).

10. Define a symmetric matrix and a skew-symmetric matrix.

    Symmetric: Aᵀ = A (a_ij = a_ji). Skew-symmetric: Aᵀ = −A (a_ij = −a_ji, so diagonal entries are 0).

11. What is the transpose of a matrix, and what is (AB)ᵀ equal to?

    The transpose Aᵀ swaps rows and columns. (AB)ᵀ = BᵀAᵀ.

12. State Cramer's Rule for solving a system of linear equations.

    For AX = B with det A ≠ 0, each variable x_i = det(A_i)/det(A), where A_i is A with its i-th column replaced by B.

13. What is an identity matrix and what property defines it?

    A square matrix with 1s on the main diagonal and 0s elsewhere; AI = IA = A for any compatible matrix A.

14. What is the nth term of an arithmetic sequence with first term a and common difference d?

    a_n = a + (n − 1)d.

15. What is the sum of the first n terms of an arithmetic series?

    S_n = (n/2)[2a + (n − 1)d] = (n/2)(a + l), where l is the last term.

16. What is the nth term of a geometric sequence with first term a and common ratio r?

    a_n = a r^(n−1).

17. What is the sum of the first n terms of a geometric series (r ≠ 1)?

    S_n = a(1 − rⁿ)/(1 − r) = a(rⁿ − 1)/(r − 1).

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

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

19. What is the arithmetic mean A and geometric mean G between two numbers a and b?

    Arithmetic mean A = (a + b)/2; Geometric mean G = √(ab).

20. State the general (binomial) theorem expansion of (a + b)ⁿ for positive integer n.

    (a + b)ⁿ = Σ (from r=0 to n) C(n, r) a^(n−r) b^r.

21. What is the general term (the (r+1)th term) in the expansion of (a + b)ⁿ?

    T_(r+1) = C(n, r) a^(n−r) b^r.

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