RBSE SS Mathematics flashcards

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

Bijective function

A function that is both one-one and onto. Only bijections have inverses, because each codomain element then has exactly one preimage, so f⁻¹ can be defined unambiguously.

Equivalence relation

A relation that is reflexive, symmetric and transitive. Congruence modulo n is the standard example: aRb when a−b is divisible by n.

Number of relations

From a set with m elements to a set with n elements there are 2^(mn) relations, since each of the mn ordered pairs may be included or excluded independently.

Composition of functions

(g∘f)(x)=g(f(x)), defined only when the range of f lies inside the domain of g. Composition is generally not commutative, so g∘f need not equal f∘g.

Principal value of sin⁻¹

sin⁻¹x lies in [−π/2,π/2] for x in [−1,1]. This restricted range makes the inverse single-valued, so sin⁻¹(sin θ) equals θ only when θ is already in that interval.

Complementary inverse identity

For x in the domain, sin⁻¹x + cos⁻¹x = π/2, and tan⁻¹x + cot⁻¹x = π/2. These follow from the cofunction identities for complementary angles.

tan⁻¹ sum formula

tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)] when xy<1. If xy>1 with x,y>0, add π; if x,y<0, subtract π to stay in the principal range.

cos⁻¹ of negative argument

cos⁻¹(−x) = π − cos⁻¹x, not −cos⁻¹x, because the range of cos⁻¹ is [0,π]. The same pattern holds for cot⁻¹(−x) = π − cot⁻¹x.

Determinant of adjugate

For an n×n matrix A, |adj A| = |A|ⁿ⁻¹. Also A(adj A) = (adj A)A = |A|I, which gives A⁻¹ = adj A / |A| whenever |A|≠0.

Singular matrix condition

A square matrix is singular exactly when its determinant is zero, and then it has no inverse. A homogeneous system AX=0 has a non-trivial solution precisely in this case.

Skew-symmetric matrix

A square matrix satisfying Aᵀ = −A. All diagonal entries must be zero, and every square matrix decomposes into a symmetric part (A+Aᵀ)/2 plus a skew-symmetric part (A−Aᵀ)/2.

Scaling a determinant

For an n×n matrix A and scalar k, |kA| = kⁿ|A| because k multiplies each of the n rows. Also |AB| = |A||B| and |Aᵀ| = |A|.

Differentiability implies continuity

If f is differentiable at a point it is continuous there, but the converse fails: f(x)=|x| is continuous at 0 yet has no derivative there because left and right slopes differ.

Rolle's theorem

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. It guarantees a horizontal tangent between equal endpoint values.

Second derivative test

At a stationary point c with f′(c)=0: f″(c)<0 gives a local maximum, f″(c)>0 gives a local minimum. If f″(c)=0 the test fails and the first-derivative sign change must be checked.

Absolute extrema on closed interval

To find global maximum and minimum of f on [a,b], evaluate f at every critical point inside (a,b) and at both endpoints, then compare those values.

Integration by parts choice

∫u v′ dx = uv − ∫u′v dx. Choose u following ILATE: inverse trigonometric, logarithmic, algebraic, trigonometric, exponential - the earlier type in this list becomes u.

Area between two curves

If f(x)≥g(x) on [a,b], the enclosed area is ∫ₐᵇ [f(x)−g(x)]dx. Where the curves cross, split the interval and add the absolute areas of each piece.

Even and odd definite integrals

Over symmetric limits [−a,a], an odd integrand integrates to 0 and an even integrand gives 2∫₀ᵃ f(x)dx. This shortcut avoids lengthy antiderivative work.

Order and degree of ODE

Order is the highest derivative appearing; degree is the exponent of that highest derivative once the equation is polynomial in all derivatives. dy/dx + y² = 0 has order 1, degree 1.

Integrating factor for linear ODE

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

Homogeneous differential equation

An equation dy/dx = F(x,y) where F(tx,ty)=F(x,y). Substituting y=vx converts it to v + x dv/dx = F(1,v), which is separable in v and x.

Exponential growth model

If dN/dt = kN with initial value N₀, then N = N₀e^{kt}. Positive k models growth, negative k models decay such as radioactive disintegration or drug clearance.

Scalar triple product

[a b c] = a·(b×c), the volume of the parallelepiped with edges a, b, c. It equals zero exactly when the three vectors are coplanar.

Cross product magnitude

|a×b| = |a||b|sin θ, the area of the parallelogram with sides a and b. Half this value is the area of the triangle formed by the two vectors.

Distance from point to plane

For plane ax+by+cz=d and point (x₁,y₁,z₁), the perpendicular distance is |ax₁+by₁+cz₁−d| divided by √(a²+b²+c²), where (a,b,c) is the normal.

Corner point theorem

In linear programming, if the feasible region is bounded and non-empty, the optimal value of the objective function occurs at a vertex, or along a whole edge joining two optimal vertices.

Infeasible linear programme

A linear programming problem is infeasible when no point satisfies all constraints simultaneously, so the feasible region is empty and no optimal solution exists.

Conditional probability

P(A|B) = P(A∩B)/P(B) for P(B)>0. Events A and B are independent exactly when P(A|B)=P(A), equivalently P(A∩B)=P(A)P(B).

Binomial mean and variance

For X ~ B(n,p) with q=1−p, mean E(X)=np, variance=npq, and standard deviation √(npq). Each trial has two outcomes with constant probability p.

Bayes' theorem

P(Aᵢ|E) = P(Aᵢ)P(E|Aᵢ) / Σⱼ P(Aⱼ)P(E|Aⱼ), where the events Aⱼ partition the sample space. It reverses a conditional probability given evidence E.

Mathematics project structure

A project records aim, mathematical formulation, data or derivation, analysis, conclusion and bibliography. Marks depend on correct procedure, accurate computation, clear presentation and interpretation.