Karnataka II PUC Mathematics (KSEAB) flashcards

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

Injective function

A function f is injective (one-one) when distinct inputs always give distinct outputs, i.e. f(a)=f(b) implies a=b. Equivalently, every horizontal line meets the graph at most once.

Bijective function

A function that is both one-one and onto its codomain. Only bijections possess an inverse function defined on the whole codomain, and the inverse then satisfies f⁻¹(f(x))=x.

Equivalence relation conditions

A relation on a set that is reflexive (aRa), symmetric (aRb implies bRa) and transitive (aRb and bRc imply aRc). Such a relation splits the set into disjoint equivalence classes.

Principal value of sin⁻¹

sin⁻¹x is defined for x in [−1,1] and takes its value in [−π/2,π/2]. So sin⁻¹(−1/2)=−π/6, and sin⁻¹(1)=π/2, never values outside that interval.

Complementary inverse trig identity

For |x|≤1, sin⁻¹x+cos⁻¹x=π/2. Similarly tan⁻¹x+cot⁻¹x=π/2 and sec⁻¹x+cosec⁻¹x=π/2 wherever both terms are defined.

tan⁻¹ sum formula

If xy<1, tan⁻¹x+tan⁻¹y=tan⁻¹((x+y)/(1−xy)). When xy>1 and both x,y are positive, the value lies in the second quadrant, so add π to the principal result.

Determinant of product

For square matrices of the same order, |AB|=|A||B|. Consequences include |Aᵀ|=|A|, |A⁻¹|=1/|A| and |kA|=kⁿ|A| for an n×n matrix A.

Adjoint inverse formula

A square matrix A is invertible exactly when |A|≠0, and then A⁻¹=(1/|A|)adj A. This also gives A(adj A)=(adj A)A=|A|I.

Skew-symmetric matrix property

A matrix satisfying Aᵀ=−A. Every diagonal entry must be zero, and the determinant of a skew-symmetric matrix of odd order is always zero.

Continuity at a point

f is continuous at x=a when the left limit, the right limit and f(a) all exist and are equal. If either one-sided limit differs, the function has a jump or removable discontinuity there.

Rolle's theorem hypotheses

If f is continuous on [a,b], differentiable on (a,b) and f(a)=f(b), then some c in (a,b) satisfies f′(c)=0. All three hypotheses are needed for the conclusion.

Derivative of aˣ

For a fixed base a>0, d/dx(aˣ)=aˣ ln a. The special case a=e gives d/dx(eˣ)=eˣ because ln e=1, which is why the exponential is its own derivative.

Second derivative test

At a stationary point where f′(c)=0: if f″(c)<0 the point is a local maximum; if f″(c)>0 it is a local minimum; if f″(c)=0 the test gives no conclusion.

Absolute maximum on closed interval

For f continuous on [a,b], compute f at every critical point inside (a,b) and at both endpoints. Compare all these values; the largest is the absolute maximum, the smallest the absolute minimum.

Tangent and normal slopes

At a point on y=f(x), the tangent slope is f′(x) and the normal slope is −1/f′(x), provided f′(x)≠0. A horizontal tangent means a vertical normal.

Integration by parts choice

Using ∫uv dx=u∫v dx−∫u′∫v dx, pick u by ILATE order: inverse trigonometric, logarithmic, algebraic, trigonometric, exponential. The first function present becomes u.

Even and odd definite integrals

On a symmetric interval [−a,a], ∫f(x)dx equals 2∫₀ᵃf(x)dx when f is even and equals 0 when f is odd. Verify f(−x)=f(x) or f(−x)=−f(x) first.

Area between two curves

If f(x)≥g(x) on [a,b], the area enclosed is ∫ₐᵇ(f(x)−g(x))dx. Find intersection points to get a and b, and split the interval wherever the upper curve changes.

Order of a differential equation

The order is the order of the highest derivative appearing in the equation. Degree is the power of that highest derivative once the equation is polynomial in all derivatives.

Integrating factor for linear ODE

For dy/dx+Py=Q with P,Q functions of x, the integrating factor is e^∫P dx. Multiplying through makes the left side the derivative of y times that factor.

Homogeneous ODE substitution

For dy/dx=F(x,y) with F homogeneous of degree zero, put y=vx. Then dy/dx=v+x dv/dx, producing a separable equation in v and x that can be integrated directly.

Vector dot product

a·b=a₁b₁+a₂b₂+a₃b₃=|a||b|cos θ. It is zero exactly when the vectors are perpendicular, and the projection of a on b has length |a·b|/|b|.

Scalar triple product meaning

[a b c]=a·(b×c) equals the signed volume of the parallelepiped with edges a, b, c. It vanishes exactly when the three vectors are coplanar, and it changes sign if two are swapped.

Distance from point to plane

For plane ax+by+cz+d=0 and point (x₁,y₁,z₁), the perpendicular distance is |ax₁+by₁+cz₁+d| divided by √(a²+b²+c²). Use the plane in this standard form first.

Corner point theorem

When a linear programming feasible region is bounded, the maximum or minimum of the objective occurs at a corner point. Evaluate Z at every vertex and compare.

Unbounded feasible region check

If the region is unbounded and Z has a corner maximum, check whether Z can grow without limit in the open half-plane. If it can, no maximum exists; otherwise the corner value stands.

Conditional probability formula

P(A|B)=P(A∩B)/P(B) for P(B)>0. Rearranging gives the multiplication rule P(A∩B)=P(A)P(B|A), which extends to sequences of events.

Binomial mean and variance

For X~B(n,p) with q=1−p: P(X=r)=ⁿCᵣpʳq^(n−r), mean E(X)=np, variance=npq, standard deviation=√(npq). The distribution is symmetric when p=1/2.

Bayes' theorem

P(Aᵢ|B)=P(Aᵢ)P(B|Aᵢ) divided by ΣP(Aⱼ)P(B|Aⱼ). It reverses a conditional probability using prior probabilities of a partition of the sample space.

Number of relations and functions

From an m-element set to an n-element set there are 2^(mn) relations and n^m functions. If m=n, exactly n! of those functions are bijections.

Combination formula

ⁿCᵣ=n!/(r!(n−r)!) counts r-element subsets of an n-element set, so order is ignored. Key properties: ⁿCᵣ=ⁿC₍ₙ₋ᵣ₎ and ⁿCᵣ+ⁿC₍ᵣ₋₁₎=⁽ⁿ⁺¹⁾Cᵣ.

Sum of geometric progression

For first term a and ratio r≠1, Sₙ=a(rⁿ−1)/(r−1). If |r|<1 the infinite sum converges to a/(1−r); if |r|≥1 the infinite series diverges.