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HEC USAT Mathematics Syllabus

The Mathematics syllabus for HEC USAT in full: 6 chapters and 19 topics, every one listed below, plus 50 Mathematics flashcards with answers.

6Chapters
19Topics
50Flashcards

Mathematics syllabus: all 6 chapters

Expand any chapter to see its topics and sub-topics.

  1. Algebra

     4 topics
    • Quadratic Functions and Equations
    • Matrices and Determinants
    • Sequences and Series
    • Binomial Theorem

  2. Trigonometry

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

  3. Functions and Graphs

     3 topics
    • Types of Functions
    • Domain and Range
    • Graphing and Transformations

  4. Calculus

     4 topics
    • Limits and Continuity
    • Differentiation
    • Applications of Derivatives
    • Integration

  5. Coordinate Geometry

     3 topics
    • Straight Lines
    • Circles
    • Conic Sections

  6. Vectors

     2 topics
    • Vector Operations
    • Dot and Cross Products

HEC USAT Mathematics flashcards — 50 questions with 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.

Show the other 30 Mathematics flashcards

  1. 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.

  2. 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,n) = 2ⁿ.

  3. What is the formula for the binomial coefficient C(n, r)?

    C(n, r) = n! / [r!(n − r)!].

  4. State the values of sin, cos, and tan for 30°, 45°, and 60°.

    sin: 1/2, √2/2, √3/2. cos: √3/2, √2/2, 1/2. tan: 1/√3, 1, √3 (for 30°, 45°, 60°).

  5. State the three Pythagorean trigonometric identities.

    sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.

  6. State the sine and cosine addition formulas for (A + B).

    sin(A + B) = sinA cosB + cosA sinB; cos(A + B) = cosA cosB − sinA sinB.

  7. State the double-angle formulas for sin 2θ and cos 2θ.

    sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

  8. What is the formula for tan(A + B)?

    tan(A + B) = (tanA + tanB) / (1 − tanA tanB).

  9. State the Law of Sines for a triangle with sides a, b, c opposite angles A, B, C.

    a/sinA = b/sinB = c/sinC = 2R, where R is the circumradius.

  10. State the Law of Cosines for side a of a triangle.

    a² = b² + c² − 2bc cosA (and cyclic versions for b² and c²).

  11. Give two formulas for the area of a triangle used in solution of triangles.

    Area = (1/2) ab sinC (two sides and included angle); or Heron's formula √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2.

  12. In solving triangles, which case can produce two possible triangles (the ambiguous case)?

    The SSA case (two sides and a non-included angle) when using the Law of Sines.

  13. What is the domain and range of y = sin⁻¹ x (arcsin)?

    Domain: −1 ≤ x ≤ 1; Range: −π/2 ≤ y ≤ π/2.

  14. What is the domain and range of y = cos⁻¹ x (arccos)?

    Domain: −1 ≤ x ≤ 1; Range: 0 ≤ y ≤ π.

  15. What is the domain and range of y = tan⁻¹ x (arctan)?

    Domain: all real numbers; Range: −π/2 < y < π/2.

  16. What is the derivative of sin⁻¹ x and of tan⁻¹ x?

    d/dx (sin⁻¹ x) = 1/√(1 − x²); d/dx (tan⁻¹ x) = 1/(1 + x²).

  17. Define an injective (one-to-one), surjective (onto), and bijective function.

    Injective: distinct inputs give distinct outputs. Surjective: every element of the codomain is an image. Bijective: both injective and surjective.

  18. What is the difference between an even function and an odd function?

    Even: f(−x) = f(x), symmetric about the y-axis. Odd: f(−x) = −f(x), symmetric about the origin.

  19. What condition must a function satisfy to have an inverse, and how are their graphs related?

    It must be one-to-one (bijective on its range). The graph of f⁻¹ is the reflection of f across the line y = x.

  20. How do you find the domain of a function?

    Find all real x for which the function is defined: exclude values making a denominator zero, a square-root argument negative, or a logarithm argument ≤ 0.

  21. What is the range of f(x) = x² and of f(x) = √x (principal root)?

    For x²: range is [0, ∞). For √x: domain [0, ∞) and range [0, ∞).

  22. How does y = f(x) + k and y = f(x − h) transform the graph of f?

    f(x) + k shifts the graph up by k (down if k < 0); f(x − h) shifts it right by h (left if h < 0).

  23. How do y = −f(x) and y = f(−x) transform a graph?

    −f(x) reflects the graph in the x-axis; f(−x) reflects it in the y-axis.

  24. How does y = a·f(x) (a > 1) versus y = f(bx) (b > 1) transform a graph?

    a·f(x) stretches vertically by factor a; f(bx) compresses horizontally by factor 1/b.

  25. State the formal definition of a function being continuous at x = a.

    f is continuous at a if: f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x) = f(a).

  26. What is the standard limit lim(x→0) sin x / x?

    It equals 1 (with x in radians).

  27. State the limit definition (first principle) of the derivative f'(x).

    f'(x) = lim(h→0) [f(x + h) − f(x)] / h.

  28. State the power rule, product rule, and quotient rule for differentiation.

    Power: d/dx xⁿ = n xⁿ⁻¹. Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².

  29. State the chain rule for differentiating a composite function y = f(g(x)).

    dy/dx = f'(g(x)) · g'(x), i.e. dy/dx = (dy/du)(du/dx).

  30. State the derivatives of sin x, cos x, e^x, and ln x.

    d/dx sin x = cos x; d/dx cos x = −sin x; d/dx e^x = e^x; d/dx ln x = 1/x.

HEC USAT Mathematics FAQ

What is in the HEC USAT Mathematics syllabus?

HEC USAT Mathematics is split into 6 chapters: Algebra, Trigonometry, Functions and Graphs, Calculus, Coordinate Geometry and Vectors. Together they contain 19 topics.

How many chapters are in HEC USAT Mathematics?

6 chapters, containing 19 topics. The largest chapter is Algebra, with 4 topics.

Are there flashcards for HEC USAT 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.