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NTS NAT-ICS Mathematics Syllabus

The Mathematics syllabus for NTS NAT-ICS in full: 5 chapters and 15 topics, every one listed below, plus 59 Mathematics flashcards with answers.

5Chapters
15Topics
59Flashcards

Mathematics syllabus: all 5 chapters

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

  1. Algebra and Functions

     4 topics
    • Functions and Graphs
    • Sequences and Series
    • Binomial Theorem
    • Matrices and Determinants

  2. Trigonometry

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

  3. Calculus

     3 topics
    • Limits and Continuity
    • Differentiation
    • Integration Basics

  4. Coordinate Geometry

     2 topics
    • Straight Lines
    • Conic Sections

  5. Sets, Logic and Permutations

     3 topics
    • Sets and Venn Diagrams
    • Permutations and Combinations
    • Probability

NTS NAT-ICS Mathematics flashcards — 59 questions with 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.

Show the other 39 Mathematics flashcards

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

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

  2. What is the value of the determinant of a 2x2 matrix [[a, b],[c, d]]?

    ad - bc.

  3. What is the formula for the inverse of a 2x2 matrix A = [[a, b],[c, d]]?

    A^(-1) = 1/(ad - bc) * [[d, -b],[-c, a]], provided ad - bc is not zero.

  4. When is a square matrix called singular versus non-singular?

    Singular if its determinant is 0 (no inverse exists); non-singular (invertible) if its determinant is not 0.

  5. What is the adjoint of a matrix and how is it used to find the inverse?

    The adjoint (adj A) is the transpose of the cofactor matrix. A^(-1) = (1/det A) adj A.

  6. State Cramer's rule for solving a system of linear equations.

    For AX = B, each variable x_i = det(A_i)/det(A), where A_i is A with its i-th column replaced by B; requires det(A) not equal to 0.

  7. List the values of sin, cos and tan for 0, 30, 45, 60 and 90 degrees.

    sin: 0, 1/2, 1/sqrt2, sqrt3/2, 1. cos: 1, sqrt3/2, 1/sqrt2, 1/2, 0. tan: 0, 1/sqrt3, 1, sqrt3, undefined.

  8. State the three Pythagorean trigonometric identities.

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

  9. State the sine and cosine addition formulas.

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

  10. State the double-angle formulas for sin 2A and cos 2A.

    sin 2A = 2 sinA cosA; cos 2A = cos^2 A - sin^2 A = 1 - 2 sin^2 A = 2 cos^2 A - 1.

  11. State the formula for tan(A + B) and tan 2A.

    tan(A+B) = (tanA + tanB)/(1 - tanA tanB); tan 2A = 2 tanA/(1 - tan^2 A).

  12. State the Law of Sines for a triangle with sides a, b, c opposite angles A, B, C.

    a/sinA = b/sinB = c/sinC = 2R, where R is the circumradius.

  13. State the Law of Cosines.

    a^2 = b^2 + c^2 - 2bc cosA (and cyclic forms). Equivalently cosA = (b^2 + c^2 - a^2)/(2bc).

  14. State two formulas for the area of a triangle in terms of its sides/angles.

    Area = (1/2) ab sinC (two sides and included angle); or Heron's formula Area = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2.

  15. What is the general solution of sin x = 0 and cos x = 0?

    sin x = 0 gives x = n*pi; cos x = 0 gives x = (2n + 1)*pi/2, for integer n.

  16. What is the general solution of sin x = sin a?

    x = n*pi + (-1)^n a, for integer n.

  17. What is the general solution of cos x = cos a and of tan x = tan a?

    cos x = cos a gives x = 2n*pi +/- a; tan x = tan a gives x = n*pi + a, for integer n.

  18. What does it mean for the limit of f(x) as x approaches a to exist?

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

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

    1 (with x in radians).

  20. What is the value of lim x->0 (1 + x)^(1/x) and lim n->infinity (1 + 1/n)^n?

    Both equal e.

  21. State the three conditions for f(x) to be continuous at x = a.

    f(a) is defined; lim x->a f(x) exists; and lim x->a f(x) = f(a).

  22. State the limit formula for lim x->0 (a^x - 1)/x.

    ln a (and for a = e it equals 1).

  23. What is the limit definition of the derivative f'(x)?

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

  24. State the power rule, product rule and quotient rule for differentiation.

    d/dx(x^n) = n x^(n-1); (uv)' = u'v + uv'; (u/v)' = (u'v - uv')/v^2.

  25. State the chain rule for differentiation.

    If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x); i.e. dy/dx = (dy/du)(du/dx).

  26. Give the derivatives of sin x, cos x, tan x, e^x and ln x.

    d/dx sin x = cos x; cos x -> -sin x; tan x -> sec^2 x; e^x -> e^x; ln x -> 1/x.

  27. How do you classify a stationary point using the second derivative test?

    At f'(x)=0: if f''(x) > 0 it is a local minimum; if f''(x) < 0 it is a local maximum; if f''(x) = 0 the test is inconclusive.

  28. State the power rule for integration of x^n.

    Integral of x^n dx = x^(n+1)/(n+1) + C, for n not equal to -1; for n = -1, Integral of (1/x) dx = ln|x| + C.

  29. Give the integrals of cos x, sin x, e^x and sec^2 x.

    Integral cos x dx = sin x + C; Integral sin x dx = -cos x + C; Integral e^x dx = e^x + C; Integral sec^2 x dx = tan x + C.

  30. State the formula for integration by parts.

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

  31. What does the Fundamental Theorem of Calculus state for a definite integral?

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

  32. What is the slope-intercept form and the point-slope form of a straight line?

    Slope-intercept: y = mx + c. Point-slope: y - y1 = m(x - x1).

  33. How is the slope of a line through points (x1, y1) and (x2, y2) computed?

    m = (y2 - y1)/(x2 - x1).

  34. What are the slope conditions for two lines to be parallel and perpendicular?

    Parallel: m1 = m2. Perpendicular: m1 * m2 = -1 (slopes are negative reciprocals).

  35. What is the distance from a point (x1, y1) to the line ax + by + c = 0?

    |a*x1 + b*y1 + c| / sqrt(a^2 + b^2).

  36. What is the standard equation of a circle with centre (h, k) and radius r?

    (x - h)^2 + (y - k)^2 = r^2.

  37. What is the standard equation of a parabola opening to the right, and its focus and directrix?

    y^2 = 4ax; focus (a, 0), directrix x = -a, vertex at origin.

  38. What is the standard equation of an ellipse, and the relationship among a, b and c?

    x^2/a^2 + y^2/b^2 = 1 (a > b); foci at (+/-c, 0) with c^2 = a^2 - b^2; eccentricity e = c/a < 1.

  39. What is the standard equation of a hyperbola, the c relationship, and its asymptotes?

    x^2/a^2 - y^2/b^2 = 1; c^2 = a^2 + b^2; eccentricity e = c/a > 1; asymptotes y = +/- (b/a) x.

NTS NAT-ICS Mathematics FAQ

What is in the NTS NAT-ICS Mathematics syllabus?

NTS NAT-ICS Mathematics is split into 5 chapters: Algebra and Functions, Trigonometry, Calculus, Coordinate Geometry and Sets, Logic and Permutations. Together they contain 15 topics.

How many chapters are in NTS NAT-ICS Mathematics?

5 chapters, containing 15 topics. The largest chapter is Algebra and Functions, with 4 topics.

Are there flashcards for NTS NAT-ICS Mathematics?

Yes — a 59-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.