Maharashtra HSC Mathematics & Statistics - Commerce (MSBSHSE) flashcards

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

Statement (proposition)

A declarative sentence that is either true or false but not both. Questions, commands and exclamations are not statements because no truth value can be assigned.

Conjunction p ∧ q

The compound statement 'p and q'. It is true only when both component statements are true and false in all other three cases of the truth table.

Implication p → q

The conditional 'if p then q'. It is false only when p is true and q is false; it is true whenever the antecedent p is false.

Contrapositive equivalence

The statement ~q → ~p is logically equivalent to p → q. The converse q → p and the inverse ~p → ~q are not equivalent to the original implication.

De Morgan's laws

~(p ∧ q) is equivalent to ~p ∨ ~q, and ~(p ∨ q) is equivalent to ~p ∧ ~q. These laws allow negation to be pushed inside brackets.

Number of elements in union

n(A ∪ B) = n(A) + n(B) − n(A ∩ B). For three sets, subtract the three pairwise intersections and add back the triple intersection.

Matrix multiplication condition

Product AB exists only when the number of columns of A equals the number of rows of B. The resulting matrix has order (rows of A) × (columns of B).

Determinant of 2×2 matrix

For matrix with first row a, b and second row c, d, the determinant is ad − bc. A zero determinant means the matrix is singular and has no inverse.

Inverse of 2×2 matrix

A⁻¹ equals the adjoint divided by the determinant, giving (1/(ad − bc)) times the matrix with rows d, −b and −c, a.

Cramer's rule

For two linear equations, each variable equals a determinant ratio: replace the corresponding coefficient column by the constants column, then divide by Δ.

Chain rule

If y depends on u and u depends on x, then dy/dx = (dy/du)(du/dx). It handles composite functions such as powers of expressions.

Quotient rule

Derivative of u divided by v is (u′v − uv′)/v², valid where v is not zero. The numerator order matters and cannot be reversed.

Elasticity of demand

η = −(p/x)(dx/dp), usually reported as a positive number. Demand is elastic when η > 1, inelastic when η < 1, unit elastic when η = 1.

Marginal cost and marginal revenue

Marginal cost is the derivative of total cost with respect to output; marginal revenue is the derivative of total revenue. Both approximate the change from one extra unit.

Profit maximisation condition

Profit is maximised where marginal revenue equals marginal cost, MR = MC, and the second derivative of the profit function is negative at that output.

Minimum average cost

Average cost C(x)/x reaches its minimum where average cost equals marginal cost. At that output the cost per unit is lowest.

Break-even point

The output at which total revenue equals total cost, so profit P(x) = R(x) − C(x) is zero. Below it the firm makes a loss.

Fundamental theorem of calculus

The definite integral from a to b of f(x) dx equals F(b) − F(a), where F is any antiderivative of f. It links areas to antiderivatives.

Integration by parts

The rule ∫u v′ dx = uv − ∫u′v dx. Choose u in the order inverse, logarithmic, algebraic, trigonometric, exponential to simplify the remaining integral.

Consumer's surplus

The area between the demand curve and the price line, computed as the integral of demand price from zero to x₀ minus p₀x₀. It measures gain to consumers.

Area between two curves

If f(x) lies above g(x) on the interval from a to b, the bounded area equals the definite integral of f(x) − g(x) over that interval.

Total cost by integration

Total cost equals the integral of marginal cost from zero to x plus fixed cost. The constant of integration is the fixed cost.

Order of differential equation

The order is the order of the highest derivative appearing in the equation. The general solution contains exactly that many arbitrary constants.

Linear differential equation

An equation dy/dx + Py = Q has integrating factor e to the power integral of P dx, and solution y times that factor equals the integral of Q times the factor.

Exponential growth model

The equation dy/dt = ky has solution y = y₀e^{kt}. Positive k gives growth, negative k gives decay, and y₀ is the initial value.

Binomial distribution

P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ for r from 0 to n, with q = 1 − p. Mean is np and variance is npq.

Poisson distribution

P(X = r) = e^{−m}mʳ/r! for r = 0, 1, 2, …, where m is the mean count. Mean and variance both equal m.

Poisson approximation to binomial

When n is large and p is small, the binomial with mean np is approximated by a Poisson distribution with m = np. This simplifies computation of rare-event counts.

Variance of random variable

Var(X) = E(X²) − [E(X)]², where E(X²) is the sum of x² times p(x). Standard deviation is the positive square root of variance.

Laspeyres price index

A weighted aggregative index using base-year quantities: (Σp₁q₀ / Σp₀q₀) × 100. It overstates price rises because the basket is not updated.

Fisher's ideal index

The geometric mean of the Laspeyres and Paasche indices, written as the square root of their product. It satisfies the time-reversal and factor-reversal tests.

Moving average

A smoothing technique that replaces each value by the average of a fixed number of neighbouring values. For quarterly data, a centred four-period moving average removes seasonal variation.