Examius

🇵🇰 COMSATS Admission Test · subject

COMSATS Admission Test Mathematics Syllabus

The Mathematics syllabus for COMSATS Admission Test in full: 9 chapters and 27 topics, every one listed below, plus 51 Mathematics flashcards with answers.

9Chapters
27Topics
51Flashcards

Mathematics syllabus: all 9 chapters

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

  1. Functions and Limits

     3 topics
    • Types of Functions
    • Limits and Continuity
    • Composition and Inverse Functions

  2. Differentiation

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

  3. Integration

     3 topics
    • Indefinite Integrals
    • Definite Integrals
    • Area Under Curves

  4. Trigonometry

     4 topics
    • Trigonometric Identities
    • Trigonometric Functions and Graphs
    • Solution of Triangles
    • Inverse Trigonometric Functions

  5. Matrices and Determinants

     4 topics
    • Matrix Operations
    • Determinants
    • Inverse of a Matrix
    • Solving Systems of Equations

  6. Vectors

     2 topics
    • Vector Operations
    • Scalar and Vector Products

  7. Sequences and Series

     3 topics
    • Arithmetic Progression
    • Geometric Progression
    • Binomial Theorem

  8. Analytic Geometry

     2 topics
    • Straight Lines
    • Conic Sections

  9. Permutations, Combinations and Probability

     3 topics
    • Counting Principles
    • Permutations and Combinations
    • Basic Probability

COMSATS Admission Test Mathematics flashcards — 51 questions with 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.

Show the other 31 Mathematics flashcards

  1. 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.

  2. Give the first-principles (limit) definition of the derivative f'(x).

    f'(x) = lim(h->0) [f(x+h) - f(x)] / h, provided the limit exists.

  3. Using first principles, what is the derivative of f(x)=x^2?

    f'(x) = lim(h->0) [(x+h)^2 - x^2]/h = lim(h->0)(2x + h) = 2x.

  4. Using first principles, what is the derivative of f(x)=sqrt(x)?

    f'(x) = 1/(2 sqrt(x)), obtained by rationalizing the difference quotient with the conjugate.

  5. State the power rule for differentiation.

    d/dx (x^n) = n x^{n-1} for any real number n.

  6. State the product rule for differentiation.

    d/dx [u v] = u'v + u v'. (Derivative of first times second, plus first times derivative of second.)

  7. State the quotient rule for differentiation.

    d/dx [u/v] = (u'v - u v') / v^2, where v != 0.

  8. State the chain rule for differentiation.

    d/dx f(g(x)) = f'(g(x)) * g'(x). Differentiate the outer function, keep the inner, then multiply by the derivative of the inner.

  9. What are the derivatives of sin x and cos x?

    d/dx sin x = cos x; d/dx cos x = -sin x.

  10. What are the derivatives of e^x and ln x?

    d/dx e^x = e^x; d/dx ln x = 1/x (for x>0).

  11. What is the derivative of tan x?

    d/dx tan x = sec^2 x.

  12. What does the first derivative tell you about increasing/decreasing behavior?

    Where f'(x) > 0 the function is increasing; where f'(x) < 0 it is decreasing. f'(x)=0 indicates a critical point.

  13. State the First Derivative Test for local extrema.

    At a critical point, if f' changes from + to -, it is a local maximum; if f' changes from - to +, it is a local minimum; if no sign change, it is neither.

  14. What does the second derivative tell you about concavity and inflection points?

    f''(x) > 0 means concave up; f''(x) < 0 means concave down. An inflection point is where concavity changes (f''=0 and sign changes).

  15. State the Second Derivative Test for a critical point where f'(c)=0.

    If f''(c) > 0, f has a local minimum at c; if f''(c) < 0, a local maximum; if f''(c)=0, the test is inconclusive.

  16. What is the geometric meaning of the derivative at a point?

    f'(a) is the slope of the tangent line to the curve y=f(x) at the point (a, f(a)); it also gives the instantaneous rate of change.

  17. What is an indefinite integral and why is +C included?

    The indefinite integral of f(x) is the family of all antiderivatives F(x)+C, where F'(x)=f(x). +C accounts for the fact that constants vanish under differentiation.

  18. State the power rule for indefinite integration.

    Integral of x^n dx = x^{n+1}/(n+1) + C, for n != -1.

  19. What is the integral of 1/x dx?

    Integral of 1/x dx = ln|x| + C (this covers the n=-1 case excluded by the power rule).

  20. What are the integrals of sin x and cos x?

    Integral of sin x dx = -cos x + C; Integral of cos x dx = sin x + C.

  21. State the integration-by-parts formula.

    Integral of u dv = u v - Integral of v du.

  22. State the Fundamental Theorem of Calculus (evaluation part).

    If F is an antiderivative of f on [a,b], then the integral from a to b of f(x) dx = F(b) - F(a).

  23. What property results from swapping the limits of a definite integral?

    Integral from a to b of f dx = - Integral from b to a of f dx. Swapping the limits negates the value.

  24. How do you change the limits when using substitution in a definite integral?

    Replace the x-limits with the corresponding u-values from u=g(x), so no need to convert back to x. Evaluate the new integrand between the new limits.

  25. How do you compute the area under a curve y=f(x) above the x-axis from a to b?

    Area = Integral from a to b of f(x) dx, provided f(x) >= 0 on [a,b].

  26. How do you find the area between two curves f(x) and g(x) on [a,b]?

    Area = Integral from a to b of [f(x) - g(x)] dx, where f is the upper curve and g is the lower curve over the interval.

  27. State the Pythagorean trigonometric identities.

    sin^2(x) + cos^2(x) = 1; 1 + tan^2(x) = sec^2(x); 1 + cot^2(x) = csc^2(x).

  28. State the sine and cosine sum identities.

    sin(A+B) = sin A cos B + cos A sin B; cos(A+B) = cos A cos B - sin A sin B.

  29. State the double-angle identities for sine and cosine.

    sin(2x) = 2 sin x cos x; cos(2x) = cos^2 x - sin^2 x = 2cos^2 x - 1 = 1 - 2sin^2 x.

  30. State the period, amplitude, and range of y = a sin(bx).

    Amplitude = |a|; period = 2π/|b|; range = [-|a|, |a|]. The graph oscillates b times faster than sin x.

  31. State the Law of Sines and the Law of Cosines for solving triangles.

    Law of Sines: a/sin A = b/sin B = c/sin C. Law of Cosines: c^2 = a^2 + b^2 - 2ab cos C (used for SAS or SSS cases).

COMSATS Admission Test Mathematics FAQ

What is in the COMSATS Admission Test Mathematics syllabus?

COMSATS Admission Test Mathematics is split into 9 chapters: Functions and Limits, Differentiation, Integration, Trigonometry, Matrices and Determinants, Vectors, Sequences and Series, Analytic Geometry and Permutations, Combinations and Probability. Together they contain 27 topics.

How many chapters are in COMSATS Admission Test Mathematics?

9 chapters, containing 27 topics. The largest chapter is Trigonometry, with 4 topics.

Are there flashcards for COMSATS Admission Test Mathematics?

Yes — a 51-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.