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NUST NET Mathematics Syllabus

The Mathematics syllabus for NUST NET in full: 12 chapters and 50 topics, every one listed below, plus 50 Mathematics flashcards with answers.

12Chapters
50Topics
50Flashcards

Mathematics syllabus: all 12 chapters

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

  1. Number Systems and Complex Numbers

     4 topics
    • Real Numbers and Properties
    • Complex Numbers and Operations
    • Argand Diagram and Modulus
    • De Moivre's Theorem

  2. Matrices and Determinants

     4 topics
    • Types of Matrices and Operations
    • Determinants and Properties
    • Inverse of a Matrix
    • Systems of Linear Equations

  3. Quadratic Equations

     4 topics
    • Methods of Solving Quadratics
    • Nature of Roots and Discriminant
    • Sum and Product of Roots
    • Equations Reducible to Quadratic Form

  4. Sequences and Series

     4 topics
    • Arithmetic Progression
    • Geometric Progression
    • Harmonic Progression
    • Means and Sum Formulas

  5. Permutations, Combinations and Probability

     4 topics
    • Fundamental Counting Principle
    • Permutations (nPr)
    • Combinations (nCr)
    • Probability Theorems

  6. Mathematical Induction and Binomial Theorem

     4 topics
    • Principle of Mathematical Induction
    • Binomial Expansion
    • General and Middle Term
    • Binomial Series for Rational Index

  7. Trigonometry

     5 topics
    • Trigonometric Functions and Identities
    • Sum, Difference and Multiple Angle Formulas
    • Trigonometric Equations
    • Inverse Trigonometric Functions
    • Solution of Triangles

  8. Functions and Graphs

     4 topics
    • Domain and Range
    • Types of Functions
    • Composition and Inverse Functions
    • Graphical Transformations

  9. Differential Calculus

     5 topics
    • Limits and Continuity
    • Differentiation Rules
    • Derivatives of Trigonometric, Logarithmic and Exponential Functions
    • Applications: Maxima and Minima
    • Rate of Change and Tangents

  10. Integral Calculus

     4 topics
    • Indefinite Integrals
    • Techniques of Integration
    • Definite Integrals
    • Area Under Curves

  11. Analytic Geometry

     3 topics
    • Straight Lines
    • Circles
    • Conic Sections

  12. Vectors

     5 topics
    • Vectors in 2D and 3D
    • Addition and Subtraction of Vectors
    • Scalar (Dot) Product
    • Vector (Cross) Product
    • Geometric Applications of Vectors

NUST NET Mathematics flashcards — 50 questions with answers

  1. What is the definition of a rational number?

    A number that can be expressed as p/q where p and q are integers and q ≠ 0. Its decimal expansion is either terminating or recurring.

  2. What is an irrational number, and give two examples.

    A real number that cannot be written as p/q (q ≠ 0); its decimal is non-terminating and non-recurring. Examples: √2 and π.

  3. State the closure, commutative, associative, and distributive properties of real numbers under addition/multiplication.

    Closure: a+b and a·b are real. Commutative: a+b=b+a, ab=ba. Associative: (a+b)+c=a+(b+c), (ab)c=a(bc). Distributive: a(b+c)=ab+ac.

  4. What are the additive identity, multiplicative identity, additive inverse, and multiplicative inverse for real numbers?

    Additive identity: 0. Multiplicative identity: 1. Additive inverse of a: −a. Multiplicative inverse of a (a≠0): 1/a.

  5. State the trichotomy property of real numbers.

    For any two real numbers a and b, exactly one of these holds: a < b, a = b, or a > b.

  6. Define the imaginary unit i and give the values of i², i³, and i⁴.

    i = √(−1). Then i² = −1, i³ = −i, and i⁴ = 1. Powers of i cycle with period 4.

  7. What is the standard (rectangular) form of a complex number, and what are its real and imaginary parts?

    z = a + bi, where a = Re(z) is the real part and b = Im(z) is the imaginary part (a, b real).

  8. How do you add and subtract two complex numbers (a+bi) and (c+di)?

    Add/subtract real and imaginary parts separately: (a+bi)±(c+di) = (a±c) + (b±d)i.

  9. How do you multiply two complex numbers (a+bi)(c+di)?

    (a+bi)(c+di) = (ac − bd) + (ad + bc)i, using i² = −1.

  10. What is the complex conjugate of z = a + bi, and what is z·z̄?

    The conjugate is z̄ = a − bi. Their product z·z̄ = a² + b² = |z|², a real non-negative number.

  11. How do you divide complex numbers (a+bi)/(c+di)?

    Multiply numerator and denominator by the conjugate of the denominator: (a+bi)(c−di) / (c²+d²).

  12. On the Argand diagram, how is a complex number z = a + bi represented?

    As the point (a, b) in the plane, with the horizontal axis as the real axis and the vertical axis as the imaginary axis.

  13. What is the modulus of a complex number z = a + bi?

    |z| = √(a² + b²), the distance from the origin to the point (a, b) on the Argand diagram.

  14. What is the argument (amplitude) of a complex number z = a + bi?

    arg(z) = θ where tan θ = b/a; it is the angle the line from origin to (a,b) makes with the positive real axis (adjusting for the correct quadrant).

  15. What is the polar (trigonometric) form of a complex number?

    z = r(cos θ + i sin θ), where r = |z| is the modulus and θ = arg(z) is the argument.

  16. State the property of the modulus of a product and a quotient of complex numbers.

    |z₁z₂| = |z₁||z₂| and |z₁/z₂| = |z₁|/|z₂| (z₂ ≠ 0).

  17. State De Moivre's Theorem.

    For any integer n: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).

  18. Using De Moivre's Theorem, what is the formula for the n-th roots of a complex number r(cos θ + i sin θ)?

    r^(1/n)[cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)] for k = 0, 1, …, n−1, giving n distinct roots.

  19. What are the cube roots of unity, and what is their key property?

    1, ω, ω², where ω = (−1+i√3)/2. They satisfy 1 + ω + ω² = 0 and ω³ = 1.

  20. How is De Moivre's Theorem used to express cos(nθ) and sin(nθ)?

    Expand (cos θ + i sin θ)ⁿ via the binomial theorem and equate real and imaginary parts to (cos nθ + i sin nθ).

Show the other 30 Mathematics flashcards

  1. Define a row matrix and a column matrix.

    A row matrix has a single row (order 1×n); a column matrix has a single column (order m×1).

  2. What is a square matrix and what is its order?

    A matrix with an equal number of rows and columns, of order n×n.

  3. Define a diagonal matrix, a scalar matrix, and an identity matrix.

    Diagonal: square matrix with all off-diagonal entries 0. Scalar: diagonal matrix with equal diagonal entries. Identity: scalar matrix with all diagonal entries 1.

  4. What is a symmetric matrix and a skew-symmetric matrix?

    Symmetric: Aᵀ = A (aᵢⱼ = aⱼᵢ). Skew-symmetric: Aᵀ = −A (aᵢⱼ = −aⱼᵢ, with zero diagonal).

  5. Under what condition can two matrices be added, and how is matrix addition performed?

    They must have the same order. Add corresponding entries: (A+B)ᵢⱼ = aᵢⱼ + bᵢⱼ.

  6. Under what condition can two matrices A and B be multiplied, and what is the order of the product?

    Columns of A must equal rows of B. If A is m×n and B is n×p, then AB is m×p.

  7. Is matrix multiplication commutative? State the correct property.

    No; in general AB ≠ BA. Matrix multiplication is associative and distributive but not commutative.

  8. 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.

  9. How do you compute the determinant of a 2×2 matrix [[a, b],[c, d]]?

    det = ad − bc.

  10. How do you compute the determinant of a 3×3 matrix by cofactor expansion along the first row?

    det = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁).

  11. What happens to a determinant if two rows (or columns) are interchanged?

    The determinant changes sign (is multiplied by −1).

  12. What is the value of a determinant if two rows or columns are identical or proportional?

    The determinant is zero.

  13. State the relationship between det(AB), det(A), and det(B), and det(Aᵀ).

    det(AB) = det(A)·det(B), and det(Aᵀ) = det(A).

  14. What is a singular matrix and a non-singular matrix?

    Singular: det(A) = 0 (no inverse). Non-singular: det(A) ≠ 0 (invertible).

  15. What is the formula for the inverse of a matrix A in terms of its adjoint?

    A⁻¹ = (1/det A) · adj(A), valid when det A ≠ 0.

  16. 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.

  17. What is the adjoint (adjugate) of a matrix?

    The transpose of the matrix of cofactors of A.

  18. State the property (AB)⁻¹ and (A⁻¹)⁻¹.

    (AB)⁻¹ = B⁻¹A⁻¹ and (A⁻¹)⁻¹ = A.

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

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

  20. In the matrix method, how is the solution of AX = B found when A is non-singular?

    X = A⁻¹B, computed by multiplying the inverse of the coefficient matrix by the constant matrix.

  21. When does a system of linear equations have a unique solution, no solution, or infinitely many solutions (in terms of the determinant)?

    If det A ≠ 0: unique solution. If det A = 0: either no solution (inconsistent) or infinitely many solutions (dependent).

  22. What is the quadratic formula for ax² + bx + c = 0?

    x = [−b ± √(b² − 4ac)] / (2a).

  23. Name the four standard methods of solving a quadratic equation.

    Factorization, completing the square, the quadratic formula, and graphical method.

  24. What is the discriminant of a quadratic equation, and what symbol denotes it?

    D = b² − 4ac (the expression under the square root in the quadratic formula).

  25. How does the discriminant determine the nature of the roots of ax² + bx + c = 0?

    D > 0: real and distinct roots. D = 0: real and equal roots. D < 0: complex conjugate (imaginary) roots.

  26. When are the roots of a quadratic real and rational versus real and irrational?

    If D = b²−4ac is a perfect square (and a,b,c rational): rational roots. If D > 0 but not a perfect square: irrational (conjugate surd) roots.

  27. 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.

  28. 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.

  29. How is an equation of the form ax⁴ + bx² + c = 0 reduced to a quadratic?

    Substitute y = x², giving ay² + by + c = 0; solve for y, then back-substitute x = ±√y.

  30. How can an equation like x + √x − 6 = 0 be reduced to quadratic form?

    Let y = √x (y ≥ 0), giving y² + y − 6 = 0; solve for y, then x = y² (rejecting negative y).

NUST NET Mathematics FAQ

What is in the NUST NET Mathematics syllabus?

NUST NET Mathematics is split into 12 chapters: Number Systems and Complex Numbers, Matrices and Determinants, Quadratic Equations, Sequences and Series, Permutations, Combinations and Probability, Mathematical Induction and Binomial Theorem, Trigonometry, Functions and Graphs, Differential Calculus, Integral Calculus, Analytic Geometry and Vectors. Together they contain 50 topics.

How many chapters are in NUST NET Mathematics?

12 chapters, containing 50 topics. The largest chapter is Trigonometry, with 5 topics.

Are there flashcards for NUST NET 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.