Examius
🇵🇰 FBISE Examinations · flashcards
FBISE Examinations Mathematics (HSSC) Flashcards
50 question-and-answer cards covering Mathematics (HSSC) as it is examined in FBISE Examinations. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Mathematics (HSSC) deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
State the First Derivative Test for a maximum and a minimum.
At a critical point x=c: if f' changes from + to - as x increases through c, it is a local maximum; if f' changes from - to +, it is a local minimum; no sign change means neither.
State the Second Derivative Test for local extrema.
If f'(c)=0: when f''(c)<0, f has a local maximum at c; when f''(c)>0, a local minimum at c; when f''(c)=0 the test is inconclusive.
What are the steps to find the absolute maximum/minimum of a continuous function on a closed interval [a,b]?
Find critical points in (a,b) where f'(x)=0 or is undefined, evaluate f at these points and at the endpoints a and b, then compare values: the largest is the absolute max and the smallest the absolute min.
What is an indefinite integral (antiderivative)?
∫f(x)dx = F(x) + C, where F'(x)=f(x). It represents the family of all antiderivatives of f, and C is the arbitrary constant of integration.
State the power rule for indefinite integration.
∫x^n dx = x^(n+1)/(n+1) + C, valid for n ≠ -1. For n=-1, ∫(1/x)dx = ln|x| + C.
Give the integrals of sin x, cos x, and sec^2 x.
∫sin x dx = -cos x + C; ∫cos x dx = sin x + C; ∫sec^2 x dx = tan x + C.
Give the integrals of e^x, a^x, and 1/x.
∫e^x dx = e^x + C; ∫a^x dx = a^x/ln a + C; ∫(1/x)dx = ln|x| + C.
State the integration by parts formula.
∫u dv = uv - ∫v du. Used to integrate products; choose u so that du simplifies (often by the LIATE priority).
What is the substitution method (u-substitution) used for in integration?
To simplify ∫f(g(x))g'(x)dx by letting u=g(x), so du=g'(x)dx, transforming the integral into ∫f(u)du, which is easier to evaluate.
State the Fundamental Theorem of Calculus for evaluating a definite integral.
If F is an antiderivative of f on [a,b], then ∫(a→b) f(x)dx = F(b) - F(a).
How does a definite integral differ from an indefinite integral?
A definite integral ∫(a→b)f(x)dx has limits a and b and yields a definite number (with no constant C). An indefinite integral ∫f(x)dx yields a family of functions F(x)+C.
State two key properties: the value of ∫(a→a)f(x)dx and the effect of swapping limits.
∫(a→a)f(x)dx = 0; and ∫(a→b)f(x)dx = -∫(b→a)f(x)dx (swapping limits changes the sign).
State the additivity property of definite integrals over an interval.
∫(a→b)f(x)dx = ∫(a→c)f(x)dx + ∫(c→b)f(x)dx, for any c (typically a≤c≤b).
How is the area under a curve y=f(x) above the x-axis from x=a to x=b found?
Area = ∫(a→b) f(x) dx, provided f(x) ≥ 0 on [a,b].
How do you handle the area when the curve f(x) lies below the x-axis on [a,b]?
The definite integral is negative, so the area is |∫(a→b)f(x)dx| = -∫(a→b)f(x)dx. If the curve crosses the axis, split the interval at the crossing points and add the absolute values of each piece.
What is the formula for the area bounded between two curves y=f(x) (upper) and y=g(x) (lower) from a to b?
Area = ∫(a→b) [f(x) - g(x)] dx, where f(x) ≥ g(x) on [a,b].
State the sum and difference formulae for sine.
sin(A+B) = sin A cos B + cos A sin B; sin(A-B) = sin A cos B - cos A sin B.
State the sum and difference formulae for cosine.
cos(A+B) = cos A cos B - sin A sin B; cos(A-B) = cos A cos B + sin A sin B.
State the sum and difference formulae for tangent.
tan(A+B) = (tan A + tan B)/(1 - tan A tan B); tan(A-B) = (tan A - tan B)/(1 + tan A tan B).
State the double angle formula for sin 2A.
sin 2A = 2 sin A cos A.
State the three forms of the double angle formula for cos 2A.
cos 2A = cos^2 A - sin^2 A = 2 cos^2 A - 1 = 1 - 2 sin^2 A.
State the double angle formula for tan 2A.
tan 2A = (2 tan A)/(1 - tan^2 A).
State the half-angle formulas for sin(A/2) and cos(A/2).
sin(A/2) = ±√[(1 - cos A)/2]; cos(A/2) = ±√[(1 + cos A)/2]. The sign depends on the quadrant of A/2.
Express sin^2 A and cos^2 A in terms of cos 2A (power-reduction).
sin^2 A = (1 - cos 2A)/2; cos^2 A = (1 + cos 2A)/2.
What this deck covers
The Mathematics (HSSC) deck follows the FBISE Examinations Mathematics (HSSC) syllabus — 8 chapters and 25 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 6.3 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 97 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 (HSSC) flashcards FAQ
How many Mathematics (HSSC) flashcards are in this FBISE Examinations 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 FBISE Examinations 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 (HSSC) cards cover?
They follow the FBISE Examinations Mathematics (HSSC) syllabus — 8 chapters and 25 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.