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🇵🇰 COMSATS NTS-based Test · flashcards
COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS) Flashcards
50 question-and-answer cards covering Mathematics (NAT-IE / NAT-ICS) as it is examined in COMSATS NTS-based Test. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Mathematics (NAT-IE / NAT-ICS) deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
Give the general solution of sinθ = sinα.
θ = nπ + (−1)^n α, where n is any integer.
Give the general solution of cosθ = cosα.
θ = 2nπ ± α, where n is any integer.
Give the general solution of tanθ = tanα.
θ = nπ + α, where n is any integer.
State the domain and range of the principal-value function y = sin⁻¹x.
Domain: [−1, 1]; Range: [−π/2, π/2].
State the principal-value range of y = cos⁻¹x and y = tan⁻¹x.
cos⁻¹x: [0, π]; tan⁻¹x: (−π/2, π/2).
State the addition formula for inverse tangents: tan⁻¹x + tan⁻¹y.
tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)], when xy < 1.
State the formal definition that the limit of f(x) as x→a equals L.
For every ε > 0 there exists δ > 0 such that |f(x) − L| < ε whenever 0 < |x − a| < δ.
State the standard limit of sin x / x as x → 0.
lim_{x→0} sin x / x = 1 (x in radians).
State the limit defining e: lim (1 + 1/n)^n as n → ∞.
lim_{n→∞} (1 + 1/n)^n = e ≈ 2.71828.
State the three conditions for f(x) 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).
State the limit definition of the derivative of f(x).
f'(x) = lim_{h→0} [f(x + h) − f(x)] / h.
State the power rule and the derivative of sin x and cos x.
d/dx(x^n) = n x^{n−1}; d/dx(sin x) = cos x; d/dx(cos x) = −sin x.
State the product rule and quotient rule of differentiation.
Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².
State the chain rule for y = f(g(x)).
dy/dx = f'(g(x)) · g'(x), i.e. dy/dx = (dy/du)(du/dx).
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 a local maximum; if f''(x) = 0 the test is inconclusive.
Give derivatives of e^x, ln x, and tan x.
d/dx(e^x) = e^x; d/dx(ln x) = 1/x; d/dx(tan x) = sec²x.
State the power rule for integration of x^n.
∫ x^n dx = x^{n+1}/(n+1) + C, for n ≠ −1; for n = −1, ∫ (1/x) dx = ln|x| + C.
State the Fundamental Theorem of Calculus for evaluating a definite integral.
∫_a^b f(x) dx = F(b) − F(a), where F is an antiderivative of f.
State the integration by parts formula.
∫ u dv = uv − ∫ v du.
Give ∫ sin x dx and ∫ cos x dx.
∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C.
State the distance formula between two points (x₁,y₁) and (x₂,y₂) and the slope formula.
Distance = √[(x₂−x₁)² + (y₂−y₁)²]; slope m = (y₂−y₁)/(x₂−x₁).
State the condition for two lines with slopes m₁ and m₂ to be parallel and to be perpendicular.
Parallel: m₁ = m₂. Perpendicular: m₁·m₂ = −1.
State the standard equation of a circle and of an ellipse centered at the origin.
Circle: x² + y² = r². Ellipse: x²/a² + y²/b² = 1.
State the standard equations of a parabola (vertex at origin, opening right) and a hyperbola centered at the origin.
Parabola: y² = 4ax. Hyperbola: x²/a² − y²/b² = 1.
What this deck covers
The Mathematics (NAT-IE / NAT-ICS) deck follows the COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS) syllabus — 5 chapters and 16 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 10.0 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 53 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 (NAT-IE / NAT-ICS) flashcards FAQ
How many Mathematics (NAT-IE / NAT-ICS) flashcards are in this COMSATS NTS-based Test deck?
50 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these COMSATS NTS-based Test flashcards free?
Yes. The preview here is free to read with no signup, and the full 50-card deck is free inside the Examius app.
What do the Mathematics (NAT-IE / NAT-ICS) cards cover?
They follow the COMSATS NTS-based Test Mathematics (NAT-IE / NAT-ICS) syllabus — 5 chapters and 16 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.