COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS) Syllabus

Mathematics (NAT-IE / NAT-ICS) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics (NAT-IE / NAT-ICS) in COMSATS NTS-based Test, not a summary of it.

1. 1

   Algebra and Functions

   4 topics

   • Sets and Functions
   • Sequences and Series
   • Permutations and Combinations
   • Binomial Theorem
2. 2

   Trigonometry

   3 topics

   • Trigonometric Ratios and Identities
   • Trigonometric Equations
   • Inverse Trigonometric Functions
3. 3

   Calculus

   3 topics

   • Limits and Continuity
   • Differentiation
   • Integration
4. 4

   Analytic Geometry

   3 topics

   • Straight Lines
   • Conic Sections
   • Vectors
5. 5

   Matrices and Determinants

   3 topics

   • Matrix Operations
   • Determinants
   • Solving Systems of Equations

Mathematics (NAT-IE / NAT-ICS) flashcards for COMSATS NTS-based Test

24 of 50 cards from the Mathematics (NAT-IE / NAT-ICS) deck — real questions with worked answers.

1. Define a function from set A to set B.

   A relation that assigns to each element of A exactly one element of B. A is the domain; B is the codomain.

2. State De Morgan's laws for sets A and B.

   (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.

3. What is the number of subsets and the number of proper subsets of a set with n elements?

   Subsets = 2^n; proper subsets = 2^n − 1.

4. Distinguish a one-to-one (injective) function from an onto (surjective) function.

   Injective: distinct inputs give distinct outputs (no two x map to same y). Surjective: every element of the codomain is an image of some input (range = codomain).

5. State the formula for the nth term of an arithmetic progression (AP).

   a_n = a + (n − 1)d, where a is the first term and d the common difference.

6. State the sum of the first n terms of an arithmetic progression.

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

7. State the nth term and the sum of n terms of a geometric progression (GP).

   a_n = a r^{n−1}; S_n = a(r^n − 1)/(r − 1) for r ≠ 1.

8. State the sum to infinity of a geometric series and its condition.

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

9. What is the relationship between the arithmetic mean (A) and geometric mean (G) of two positive numbers?

   A ≥ G, with equality if and only if the two numbers are equal (AM ≥ GM).

10. Insert the formula for a single geometric mean between two positive numbers a and b.

    G = √(ab).

11. State the formula for the number of permutations of n distinct objects taken r at a time.

    nPr = n! / (n − r)!.

12. State the formula for the number of combinations of n distinct objects taken r at a time.

    nCr = n! / [r!(n − r)!].

13. How many distinct arrangements are there of n objects where some are alike (p alike, q alike, etc.)?

    n! / (p! q! r! ...), dividing by the factorials of the counts of each group of identical objects.

14. How many circular permutations of n distinct objects are there?

    (n − 1)! (or (n − 1)!/2 if clockwise and anticlockwise are considered identical, e.g. a necklace).

15. State the symmetry property of combinations.

    nCr = nC(n−r).

16. State the Binomial Theorem for (a + b)^n where n is a positive integer.

    (a + b)^n = Σ_{r=0}^{n} nCr · a^{n−r} · b^r.

17. Write the general (r+1)th term in the expansion of (a + b)^n.

    T_{r+1} = nCr · a^{n−r} · b^r.

18. How do you find the term independent of x (constant term) in a binomial expansion?

    Write the general term, simplify the power of x, set the exponent of x equal to zero, solve for r, then substitute back.

19. State the binomial series for (1 + x)^n when n is not a positive integer, and its validity condition.

    (1 + x)^n = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + ..., valid for |x| < 1.

20. Give the exact values of sin, cos, and tan of 30°.

    sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3.

21. State the three fundamental Pythagorean trigonometric identities.

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

22. State the sine and cosine of a sum: sin(A+B) and cos(A+B).

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

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

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

24. State the formula for tan(A + B).

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

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