COMSATS Admission 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 COMSATS Admission Test, not a summary of it.

1. 1

   Functions and Limits

   3 topics

   • Types of Functions
   • Limits and Continuity
   • Composition and Inverse Functions
2. 2

   Differentiation

   3 topics

   • Derivatives from First Principles
   • Rules of Differentiation
   • Applications of Derivatives
3. 3

   Integration

   3 topics

   • Indefinite Integrals
   • Definite Integrals
   • Area Under Curves
4. 4

   Trigonometry

   4 topics

   • Trigonometric Identities
   • Trigonometric Functions and Graphs
   • Solution of Triangles
   • Inverse Trigonometric Functions
5. 5

   Matrices and Determinants

   4 topics

   • Matrix Operations
   • Determinants
   • Inverse of a Matrix
   • Solving Systems of Equations
6. 6

   Vectors

   2 topics

   • Vector Operations
   • Scalar and Vector Products
7. 7

   Sequences and Series

   3 topics

   • Arithmetic Progression
   • Geometric Progression
   • Binomial Theorem
8. 8

   Analytic Geometry

   2 topics

   • Straight Lines
   • Conic Sections
9. 9

   Permutations, Combinations and Probability

   3 topics

   • Counting Principles
   • Permutations and Combinations
   • Basic Probability

Mathematics flashcards for COMSATS Admission Test

21 of 51 cards from the Mathematics deck — real questions with worked answers.

1. What is a function from set A to set B?

   A rule that assigns to each element of A (the domain) exactly one element of B (the codomain). No input may map to two different outputs.

2. Define an injective (one-to-one) function.

   A function where distinct inputs give distinct outputs: if f(x1)=f(x2) then x1=x2. No two domain elements share the same image.

3. Define a surjective (onto) function.

   A function whose range equals its codomain: every element of the codomain is the image of at least one element of the domain.

4. Define a bijective function and state why it matters.

   A function that is both injective and surjective. It matters because only bijective functions have an inverse.

5. What is an even function and what symmetry does its graph have?

   A function with f(-x)=f(x) for all x. Its graph is symmetric about the y-axis (e.g., x^2, cos x).

6. What is an odd function and what symmetry does its graph have?

   A function with f(-x)=-f(x) for all x. Its graph is symmetric about the origin (e.g., x^3, sin x).

7. How do you find the domain of a function involving a square root and a denominator?

   Require the expression under the square root to be greater than or equal to 0, and require the denominator to be non-zero; the domain is the intersection of these conditions.

8. What is a polynomial function and its general form?

   A function of the form f(x)=a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0 with non-negative integer exponents and real coefficients (a_n != 0).

9. Define the limit of a function: what does lim(x->a) f(x)=L mean?

   f(x) can be made arbitrarily close to L by taking x sufficiently close to a (but not equal to a). The value f(a) itself need not exist.

10. State the condition for a two-sided limit to exist at x=a.

    The left-hand limit and right-hand limit must both exist and be equal: lim(x->a-) f(x) = lim(x->a+) f(x).

11. What is the value of lim(x->0) (sin x)/x?

    1 (with x in radians). This is a standard limit used throughout calculus.

12. What is the value of lim(x->0) (1 - cos x)/x?

    0. (Related standard limit: lim(x->0) (1 - cos x)/x^2 = 1/2.)

13. State the three conditions for a function f to be continuous at x=a.

    1) f(a) is defined; 2) lim(x->a) f(x) exists; 3) lim(x->a) f(x) = f(a).

14. What is a removable discontinuity?

    A point where the limit exists but does not equal f(a) (or f(a) is undefined). It can be 'removed' by redefining the function at that single point.

15. What is the value of lim(x->infinity) (1 + 1/x)^x?

    e (approximately 2.718), the base of natural logarithms.

16. How do you evaluate the limit of a rational function as x approaches infinity?

    Divide numerator and denominator by the highest power of x. The limit depends on the ratio of leading coefficients (or is 0 or infinity if degrees differ).

17. What is the composition (f o g)(x)?

    (f o g)(x) = f(g(x)): apply g first, then apply f to the result. Its domain is the set of x in dom(g) such that g(x) is in dom(f).

18. Is function composition commutative? Give the general rule.

    No. In general (f o g)(x) is not equal to (g o f)(x); the order of composition matters.

19. How do you find the inverse f^{-1}(x) of a function algebraically?

    Write y=f(x), swap x and y, then solve for y. The result is f^{-1}(x). The function must be one-to-one for the inverse to exist.

20. What is the relationship between a function's graph and its inverse's graph?

    They are reflections of each other across the line y=x.

21. State the cancellation property of inverse functions.

    f(f^{-1}(x)) = x for all x in the domain of f^{-1}, and f^{-1}(f(x)) = x for all x in the domain of f.

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