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

1. 1

   Functions and Limits

   2 topics

   • Types of Functions
   • Limits and Continuity
2. 2

   Differentiation

   2 topics

   • Derivatives and Rules
   • Applications of Derivatives
3. 3

   Integration

   3 topics

   • Indefinite Integrals
   • Definite Integrals
   • Area Under a Curve
4. 4

   Trigonometry

   3 topics

   • Trigonometric Identities
   • Trigonometric Equations
   • Solution of Triangles
5. 5

   Sequences and Series

   3 topics

   • Arithmetic Progression
   • Geometric Progression
   • Binomial Theorem
6. 6

   Matrices and Determinants

   3 topics

   • Matrix Operations
   • Determinants and Inverse
   • Systems of Linear Equations
7. 7

   Analytic Geometry

   2 topics

   • Straight Lines
   • Conic Sections
8. 8

   Vectors

   2 topics

   • Vector Algebra
   • Scalar and Vector Products
9. 9

   Permutation, Combination and Probability

   3 topics

   • Counting Principles
   • Permutations and Combinations
   • Basic Probability

Mathematics flashcards for NTS NAT-IE

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

1. What is the definition of an even function, and what symmetry does its graph have?

   A function f is even if f(-x) = f(x) for all x in its domain. Its graph is symmetric about the y-axis. Example: f(x) = x^2, cos x.

2. What is the definition of an odd function, and what symmetry does its graph have?

   A function f is odd if f(-x) = -f(x) for all x in its domain. Its graph is symmetric about the origin. Example: f(x) = x^3, sin x.

3. What conditions make a function one-to-one (injective)?

   A function is one-to-one if distinct inputs give distinct outputs: f(a) = f(b) implies a = b. Graphically, it passes the horizontal line test.

4. What is an onto (surjective) function?

   A function f: A to B is onto if every element of the codomain B is the image of at least one element of A, i.e. the range equals the codomain.

5. What is the condition for a function to have an inverse, and how do the graphs of f and f-inverse relate?

   A function has an inverse if and only if it is bijective (one-to-one and onto). The graphs of f and f^(-1) are reflections of each other across the line y = x.

6. How do you find the composition (f o g)(x), and is composition commutative?

   (f o g)(x) = f(g(x)): apply g first, then f. Composition is generally NOT commutative; f o g need not equal g o f.

7. State the formal (epsilon-delta) idea of the limit of f(x) as x approaches a.

   lim(x->a) f(x) = L means for every epsilon > 0 there exists delta > 0 such that 0 < |x - a| < delta implies |f(x) - L| < epsilon.

8. What are 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). All three must hold.

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

   lim(x->0) sin(x)/x = 1 (x in radians). A fundamental trigonometric limit.

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

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

11. What limit defines the number e involving (1 + 1/n)^n?

    e = lim(n->infinity) (1 + 1/n)^n. Equivalently, lim(x->0) (1 + x)^(1/x) = e, approximately 2.71828.

12. What is a removable discontinuity versus a jump discontinuity?

    Removable: the limit exists but does not equal f(a) (or f(a) is undefined) - the 'hole' can be patched. Jump: left-hand and right-hand limits exist but are unequal.

13. State the limit definition of the derivative f'(x).

    f'(x) = lim(h->0) [f(x + h) - f(x)] / h, provided the limit exists. It gives the instantaneous rate of change / slope of the tangent line.

14. State the power rule for differentiation.

    d/dx [x^n] = n*x^(n-1) for any real n.

15. State the product rule for derivatives.

    d/dx [u*v] = u'v + uv' (the derivative of the first times the second plus the first times the derivative of the second).

16. State the quotient rule for derivatives.

    d/dx [u/v] = (u'v - uv') / v^2, where v is not zero.

17. State the chain rule for derivatives.

    d/dx [f(g(x))] = f'(g(x)) * g'(x): the derivative of the outer function evaluated at the inner, times the derivative of the inner.

18. Give the derivatives of sin x, cos x, and tan x.

    d/dx[sin x] = cos x; d/dx[cos x] = -sin x; d/dx[tan x] = sec^2 x.

19. Give the derivatives of e^x and ln x.

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

20. Give the derivatives of sec x, cosec x, and cot x.

    d/dx[sec x] = sec x tan x; d/dx[cosec x] = -cosec x cot x; d/dx[cot x] = -cosec^2 x.

21. What does the sign of the first derivative f'(x) tell you about a function?

    If f'(x) > 0 the function is increasing; if f'(x) < 0 it is decreasing; if f'(x) = 0 there is a critical point (possible max, min, or inflection).

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