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NTS NAT-IE Mathematics Flashcards
51 question-and-answer cards covering Mathematics as it is examined in NTS NAT-IE. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Mathematics deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
Give the integrals of sin x and cos x.
Integral of sin x dx = -cos x + C; Integral of cos x dx = sin x + C.
Give the integrals of e^x and 1/x.
Integral of e^x dx = e^x + C; Integral of (1/x) dx = ln|x| + C.
Give the integral of sec^2 x and of 1/(1 + x^2).
Integral of sec^2 x dx = tan x + C; Integral of 1/(1 + x^2) dx = arctan x + C.
State the integration by parts formula.
Integral of u dv = u*v - Integral of v du. Choose u (to differentiate) and dv (to integrate); the LIATE rule helps pick u.
What is the method of integration by substitution (u-substitution)?
Let u = g(x) so du = g'(x) dx, converting Integral of f(g(x))g'(x) dx into Integral of f(u) du. It reverses the chain rule.
State the Fundamental Theorem of Calculus (evaluation part).
If F is an antiderivative of f on [a, b], then the definite integral from a to b of f(x) dx = F(b) - F(a).
What does a definite integral represent geometrically?
The net signed area between the curve y = f(x) and the x-axis from x = a to x = b: area above the axis counts positive, area below counts negative.
State three key properties of definite integrals.
1) Integral from a to a = 0; 2) Integral a to b = -(Integral b to a); 3) Integral a to b = Integral a to c + Integral c to b (additivity over intervals).
How do you find the area between a curve y = f(x) and the x-axis when the curve crosses the axis on [a, b]?
Split the interval at the x-intercepts, integrate over each piece, and add the absolute values: Area = integral of |f(x)| dx so areas below the axis count as positive.
How do you find the area between two curves y = f(x) (upper) and y = g(x) (lower) on [a, b]?
Area = Integral from a to b of [f(x) - g(x)] dx, with f the upper curve. Find intersection points to determine the limits.
State the three Pythagorean trigonometric identities.
sin^2 x + cos^2 x = 1; 1 + tan^2 x = sec^2 x; 1 + cot^2 x = cosec^2 x.
State the sine and cosine addition formulas for (A + B).
sin(A + B) = sin A cos B + cos A sin B; cos(A + B) = cos A cos B - sin A sin B.
State the double-angle formulas for sin 2x and cos 2x.
sin 2x = 2 sin x cos x; cos 2x = cos^2 x - sin^2 x = 2cos^2 x - 1 = 1 - 2sin^2 x.
State the tangent addition formula tan(A + B).
tan(A + B) = (tan A + tan B) / (1 - tan A tan B).
State the half-angle identities for sin^2 x and cos^2 x (power-reduction).
sin^2 x = (1 - cos 2x)/2; cos^2 x = (1 + cos 2x)/2.
What is the general solution of sin x = 0, and of cos x = 0?
sin x = 0 gives x = n*pi; cos x = 0 gives x = (2n + 1)*pi/2, where n is any integer.
What is the general solution of sin x = sin a?
x = n*pi + (-1)^n * a, where n is any integer (a is the principal value).
What is the general solution of cos x = cos a, and of tan x = tan a?
cos x = cos a: x = 2n*pi ± a. tan x = tan a: x = n*pi + a, where n is any integer.
State the Law of Sines for a triangle with sides a, b, c opposite angles A, B, C.
a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. Used when given two angles and a side (AAS/ASA) or two sides and a non-included angle (SSA).
State the Law of Cosines.
c^2 = a^2 + b^2 - 2ab*cos C (and cyclic permutations). Used for SAS (two sides and included angle) or SSS (three sides).
Give two formulas for the area of a triangle used in solution of triangles.
Area = (1/2)ab sin C (two sides and included angle); and Heron's formula: Area = sqrt[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2.
For an arithmetic progression with first term a and common difference d, give the nth term and the sum of n terms.
nth term: a_n = a + (n - 1)d. Sum: S_n = (n/2)[2a + (n - 1)d] = (n/2)(a + l), where l is the last term.
For a geometric progression with first term a and common ratio r, give the nth term and the sum of n terms.
nth term: a_n = a*r^(n-1). Sum of n terms: S_n = a(1 - r^n)/(1 - r) for r not equal to 1.
What is the sum of an infinite geometric series, and when does it converge?
S(infinity) = a/(1 - r), valid only when |r| < 1. If |r| >= 1 the series diverges. Also: arithmetic mean = (a+b)/2; geometric mean = sqrt(ab).
What this deck covers
The Mathematics deck follows the NTS NAT-IE Mathematics syllabus — 9 chapters and 23 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 5.7 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 102 characters, which is long enough to carry the reasoning and short enough to say out loud.
A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.
Mathematics flashcards FAQ
How many Mathematics flashcards are in this NTS NAT-IE deck?
51 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these NTS NAT-IE flashcards free?
Yes. The preview here is free to read with no signup, and the full 51-card deck is free inside the Examius app.
What do the Mathematics cards cover?
They follow the NTS NAT-IE Mathematics syllabus — 9 chapters and 23 topics — so the questions track what is actually examinable.
How should I use these flashcards?
Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.