NTS NAT-ICS 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 NTS NAT-ICS, not a summary of it.

1. 1

   Algebra and Functions

   4 topics

   • Functions and Graphs
   • Sequences and Series
   • Binomial Theorem
   • Matrices and Determinants
2. 2

   Trigonometry

   3 topics

   • Trigonometric Ratios and Identities
   • Solution of Triangles
   • Trigonometric Equations
3. 3

   Calculus

   3 topics

   • Limits and Continuity
   • Differentiation
   • Integration Basics
4. 4

   Coordinate Geometry

   2 topics

   • Straight Lines
   • Conic Sections
5. 5

   Sets, Logic and Permutations

   3 topics

   • Sets and Venn Diagrams
   • Permutations and Combinations
   • Probability

Mathematics flashcards for NTS NAT-ICS

21 of 59 cards from the Mathematics deck — real questions with worked answers.

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

   A rule that assigns to each element of A exactly one element of B. No element of A may map to two different elements of B.

2. How do you determine the domain of a real-valued function f(x)?

   The set of all real x for which f(x) is defined (real). Exclude values making a denominator zero or producing a negative number under an even root or a non-positive argument of a logarithm.

3. What is the graphical test to check if a curve represents a function?

   The vertical line test: if any vertical line cuts the graph in more than one point, it is not a function.

4. When is a function called one-to-one (injective), and what is its graphical test?

   When different inputs give different outputs (f(a)=f(b) implies a=b). Graphical test: the horizontal line test — no horizontal line meets the graph more than once.

5. How is the composite function (f o g)(x) defined?

   (f o g)(x) = f(g(x)); apply g first, then f. Defined only where g(x) lies in the domain of f.

6. What condition must a function satisfy to have an inverse, and how is the inverse graph related to the original?

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

7. Define even and odd functions and state their graph symmetries.

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

8. What is the general term of an arithmetic sequence with first term a and common difference d?

   a_n = a + (n - 1)d.

9. 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.

10. What is the general term of a geometric sequence with first term a and common ratio r?

    a_n = a r^(n-1).

11. What is the sum of the first n terms of a geometric series (r not equal to 1)?

    S_n = a(1 - r^n)/(1 - r) = a(r^n - 1)/(r - 1).

12. What is the sum to infinity of a geometric series, and when does it exist?

    S_infinity = a/(1 - r), valid only when |r| < 1.

13. What is the arithmetic mean (A) and geometric mean (G) between two numbers a and b?

    A = (a + b)/2 and G = sqrt(ab). Also A >= G for positive a, b.

14. State the formulas for the sum of the first n natural numbers, their squares, and their cubes.

    Sum n = n(n+1)/2; Sum n^2 = n(n+1)(2n+1)/6; Sum n^3 = [n(n+1)/2]^2.

15. State the Binomial Theorem for a positive integer n.

    (a + b)^n = sum_{r=0}^{n} C(n,r) a^(n-r) b^r, where C(n,r) = n!/[r!(n-r)!].

16. What is the general (r+1)th term in the expansion of (a + b)^n?

    T_{r+1} = C(n, r) a^(n-r) b^r.

17. How many terms are there in the expansion of (a + b)^n, and how do you find the middle term(s)?

    There are n + 1 terms. If n is even, the single middle term is the (n/2 + 1)th term; if n is odd, there are two middle terms, the ((n+1)/2)th and ((n+3)/2)th.

18. 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! x^2 + n(n-1)(n-2)/3! x^3 + ..., valid for |x| < 1.

19. What does Pascal's triangle give in a binomial expansion?

    The binomial coefficients C(n, r); each entry is the sum of the two entries directly above it.

20. What is the order of a matrix, and when are two matrices conformable for multiplication?

    Order is m x n (rows x columns). A (m x n) and B (p x q) can be multiplied as AB only if n = p; the product is m x q.

21. Define a symmetric and a skew-symmetric matrix.

    Symmetric: A^T = A. Skew-symmetric: A^T = -A (its diagonal entries are all zero).

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