NEB Class 12 Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the roots are x = [−b ± √(b² − 4ac)]/(2a). The discriminant b² − 4ac decides whether the roots are real, equal, or complex.

Sum and product of roots

For ax² + bx + c = 0, the sum of roots is −b/a and the product is c/a. A quadratic with given roots α and β can be written x² − (α + β)x + αβ = 0.

Complex modulus and conjugate

For z = a + bi, the modulus is |z| = √(a² + b²) and the conjugate is z̄ = a − bi. Because z·z̄ = |z|², division by a complex number uses its conjugate.

Arithmetic sequence formulas

The nth term of an arithmetic sequence is a + (n − 1)d, and the sum of n terms is n/2[2a + (n − 1)d], where a is the first term and d is the common difference.

Geometric sequence sum

For a geometric sequence, the nth term is ar^(n−1). The sum of n terms is a(r^n − 1)/(r − 1) for r ≠ 1, and the infinite sum is a/(1 − r) when |r| < 1.

Combination formula

The number of selections of r objects from n distinct objects is nCr = n!/[r!(n − r)!]. Also nCr = nC(n − r), and ordered arrangements satisfy nPr = r!·nCr.

Pythagorean identities

The Pythagorean identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ. They are used to simplify trigonometric expressions and prove identities.

Sine and cosine addition

The addition formulas are sin(A ± B) = sinA cosB ± cosA sinB and cos(A ± B) = cosA cosB ∓ sinA sinB; the sign in the cosine formula is opposite to the angle sign.

Double-angle formulas

Double-angle formulas include sin2θ = 2sinθcosθ and cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; also tan2θ = 2tanθ/(1 − tan²θ).

General trigonometric solutions

General solutions are sinθ = sinα ⇒ θ = nπ + (−1)^n α; cosθ = cosα ⇒ θ = 2nπ ± α; tanθ = tanα ⇒ θ = nπ + α, where n is any integer.

Sine and cosine rules

In any triangle ABC, a/sinA = b/sinB = c/sinC and a² = b² + c² − 2bc cosA. The area is ½bc sinA = ½ab sinC.

Distance and midpoint

The distance between (x1, y1) and (x2, y2) is √[(x2 − x1)² + (y2 − y1)²]. Their midpoint is ((x1 + x2)/2, (y1 + y2)/2).

Slope and line equations

The slope of a line through two points is m = (y2 − y1)/(x2 − x1). Point-slope form is y − y1 = m(x − x1); slope-intercept form is y = mx + c.

Perpendicular and parallel lines

For slopes m1 and m2, lines are parallel when m1 = m2 and perpendicular when m1m2 = −1. The acute angle θ between them satisfies tanθ = |(m1 − m2)/(1 + m1m2)|.

Circle standard equation

A circle with centre (h,k) and radius r is (x−h)²+(y−k)²=r². For x²+y²+2gx+2fy+c=0, the centre is (−g,−f) and radius is √(g²+f²−c).

Parabola y²=4ax

The parabola y² = 4ax has vertex (0,0), focus (a,0), directrix x = −a, and latus rectum length 4a. For x² = 4ay, the focus is (0,a).

Dot product formula

For vectors a and b, a·b = |a||b|cosθ = a1b1 + a2b2 + a3b3. Perpendicular vectors have a·b = 0; the scalar projection of a on b is (a·b)/|b|.

Cross product formula

The cross product a × b has magnitude |a||b|sinθ and direction given by the right-hand rule. Its magnitude equals the area of the parallelogram; triangle area is ½|a × b|.

Scalar triple product

The scalar triple product [a b c] = a·(b × c) gives the volume of the parallelepiped formed by a, b, c. Three vectors are coplanar exactly when this product is zero.

Vector equation of line

The vector equation r = a + λb represents a line through the point with position vector a, parallel to vector b; λ is a real parameter that locates each point.

Direction cosines

For a = a1i + a2j + a3k, the direction cosines are l = a1/|a|, m = a2/|a|, n = a3/|a|. They satisfy l² + m² + n² = 1.

Limit sin x over x

Standard limit results: lim x→0 (sin x)/x = 1, lim x→0 (1 − cos x)/x² = 1/2, and lim x→0 (tan x)/x = 1.

Product and quotient rules

The product rule is (uv)′ = u′v + uv′; the quotient rule is (u/v)′ = (u′v − uv′)/v². The chain rule states dy/dx = dy/du · du/dx.

Derivatives of exponentials and logs

Derivatives: d/dx(e^x) = e^x, d/dx(a^x) = a^x ln a, d/dx(ln x) = 1/x, and d/dx(log_a x) = 1/(x ln a).

Maxima and minima test

At a critical point, f′(x) = 0 or f′(x) is undefined. If f″(x) < 0, the point is a local maximum; if f″(x) > 0, it is a local minimum.

Integration by parts

Integration by parts uses ∫u dv = uv − ∫v du. The LIATE order suggests choosing u as logarithmic, inverse trigonometric, algebraic, trigonometric, or exponential, in that priority.

Definite integral area

The area between y = f(x) and the x-axis from x = a to x = b is ∫_a^b |f(x)| dx. Between two curves, integrate upper minus lower over the intersection interval.

Mean of grouped data

For grouped data, the mean is x̄ = Σfx/Σf, where x denotes each class midpoint and f its corresponding frequency.

Variance and standard deviation

Variance is σ² = Σf(x − x̄)²/Σf and standard deviation is σ = √σ². The coefficient of variation is (σ/x̄) × 100%, useful for comparing relative variability.

Correlation coefficient

Pearson's correlation coefficient is r = Σ(x − x̄)(y − ȳ)/√[Σ(x − x̄)² Σ(y − ȳ)²]. It ranges from −1 to 1 and measures the strength and direction of linear association.

Conditional probability

Conditional probability is P(A|B) = P(A∩B)/P(B) when P(B) > 0. Events A and B are independent exactly when P(A∩B) = P(A)P(B), so one event does not change the other's probability.

Binomial distribution

For n independent trials with success probability p, the binomial probability is P(X = r) = nCr p^r(1 − p)^(n−r). The mean is np and variance is np(1 − p).