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🇵🇰 NTS NAT-ICS · flashcards
NTS NAT-ICS Mathematics Flashcards
59 question-and-answer cards covering Mathematics as it is examined in NTS NAT-ICS. 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.
What is the general solution of sin x = sin a?
x = n*pi + (-1)^n a, for integer n.
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.
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).
What is the value of lim x->0 (sin x)/x?
1 (with x in radians).
What is the value of lim x->0 (1 + x)^(1/x) and lim n->infinity (1 + 1/n)^n?
Both equal e.
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).
State the limit formula for lim x->0 (a^x - 1)/x.
ln a (and for a = e it equals 1).
What is the limit definition of the derivative f'(x)?
f'(x) = lim h->0 [f(x + h) - f(x)]/h.
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.
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).
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.
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.
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.
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.
State the formula for integration by parts.
Integral of u dv = uv - Integral of v du.
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).
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).
How is the slope of a line through points (x1, y1) and (x2, y2) computed?
m = (y2 - y1)/(x2 - x1).
What are the slope conditions for two lines to be parallel and perpendicular?
Parallel: m1 = m2. Perpendicular: m1 * m2 = -1 (slopes are negative reciprocals).
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).
What is the standard equation of a circle with centre (h, k) and radius r?
(x - h)^2 + (y - k)^2 = r^2.
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.
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.
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.
What this deck covers
The Mathematics deck follows the NTS NAT-ICS Mathematics syllabus — 5 chapters and 15 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 11.8 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 65 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-ICS deck?
59 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-ICS flashcards free?
Yes. The preview here is free to read with no signup, and the full 59-card deck is free inside the Examius app.
What do the Mathematics cards cover?
They follow the NTS NAT-ICS Mathematics syllabus — 5 chapters and 15 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.