Numbers, Quantification and Numerical Applications
Key Points to Remember
- →Modular arithmetic: a mod m = remainder when a is divided by m. Properties: (a+b) mod m = [(a mod m) + (b mod m)] mod m.
- →Congruence: a ≡ b (mod m) means m divides (a−b). Useful for checking divisibility patterns.
- →Successive discounts: equivalent discount for two discounts d₁ and d₂ = d₁ + d₂ − (d₁ × d₂)/100.
- →Partnership: profit sharing ∝ (Capital × Time). Direct ratio for investment duration.
- →Boats and Streams: Downstream speed = B + S, Upstream speed = B − S. Distance = Speed × Time.
- →Mixture problems: alligation method — draw cross diagram, find ratio of ingredients to achieve target concentration.
Exam Tips
Successive discount formula is the most tested topic from this chapter — know it cold.
For clocks: angle moved by hour hand = 0.5° per minute. Minute hand = 6° per minute.
Partnership with unequal time: multiply each partner's capital by their time period.
Algebra — Matrices and Determinants
Key Points to Remember
- →Matrix addition requires same order. Matrix multiplication: A (m×n) × B (n×p) = C (m×p).
- →Transpose: (A')' = A, (AB)' = B'A', (A+B)' = A'+B'.
- →Determinant of 2×2: |a b; c d| = ad − bc. Determinant of 3×3 by expansion along any row/column.
- →Properties: det(AB) = det(A)·det(B). det(kA) = kⁿ det(A) for n×n matrix.
- →Inverse: A⁻¹ = (1/|A|) × Adj(A). Exists only if |A| ≠ 0 (non-singular).
- →System of equations: AX = B → X = A⁻¹B (matrix method). Or use Cramer's rule: x = D_x/D, y = D_y/D.
Exam Tips
Inverse matrix: find cofactor matrix → take transpose (adjugate) → divide by determinant.
Cramer's rule: D = |coefficient matrix|, D_x = replace x-column with constants, D_y = replace y-column.
Singular matrix (|A|=0): no unique solution — either no solution or infinite solutions.
Calculus — Applications
Key Points to Remember
- →Limits: lim(x→a) [xⁿ − aⁿ]/(x−a) = naⁿ⁻¹. lim(x→0) sin x/x = 1. lim(x→0) (eˣ−1)/x = 1.
- →Continuity at x = a: f(a) exists, left limit = right limit = f(a).
- →Differentiation: d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, product rule, quotient rule, chain rule.
- →Marginal cost MC = dC/dx. Marginal revenue MR = dR/dx. Profit π = R − C. Maximise: dπ/dx = 0 and d²π/dx² < 0.
- →Elasticity of demand: e = (dq/dp) × (p/q). Elastic: |e| > 1. Inelastic: |e| < 1. Unitary: |e| = 1.
- →Maxima/minima: f'(x) = 0 at critical points. f''(x) < 0 → maximum; f''(x) > 0 → minimum.
Exam Tips
Business application: always define what x represents. Write the formula clearly before differentiating.
MR = MC at profit maximisation — can also be asked without a specific function.
Elasticity of demand < 1: price increase → revenue increase. |e| > 1: price increase → revenue decrease.
Probability Distributions
Key Points to Remember
- →Random variable X: discrete (finite/countable values) or continuous (range of values).
- →Probability distribution: P(X = xᵢ) ≥ 0 for all i; ΣP(xᵢ) = 1.
- →Expected value (mean): E(X) = Σ[xᵢ · P(xᵢ)]. Variance: Var(X) = E(X²) − [E(X)]².
- →Binomial distribution: n trials, probability p of success each trial. P(X = r) = nCr · pʳ · (1−p)ⁿ⁻ʳ.
- →Binomial mean = np. Variance = npq where q = 1−p. Standard deviation = √(npq).
- →Normal distribution: symmetric bell curve. Mean = median = mode. 68-95-99.7 rule.
Exam Tips
Verify E(X) and Var(X) using the formula in tabular form — clear and gets full marks.
Binomial: n and p given or deducible from context. Compute nCr carefully.
Standard normal z = (X − μ)/σ — use z-table when probability question involves normal distribution.
Index Numbers and Time-Based Data
Key Points to Remember
- →Simple Index = (Current price / Base price) × 100.
- →Weighted index: Laspeyre's = Σ(p₁q₀)/Σ(p₀q₀) × 100 (base year quantities).
- →Paasche's = Σ(p₁q₁)/Σ(p₀q₁) × 100 (current year quantities).
- →Fisher's Ideal = √(Laspeyre's × Paasche's). Satisfies time-reversal and factor-reversal tests.
- →Consumer Price Index (CPI): measures cost of living change. Used to adjust wages and salaries.
- →Time series components: Trend (T), Seasonal (S), Cyclical (C), Irregular (I).
- →Moving averages: 3-year MA = average of 3 consecutive values. Smoothens out fluctuations to reveal trend.
Exam Tips
Always set up a table with columns: Commodity, p₀, p₁, q₀, q₁, p₁q₀, p₀q₀, p₁q₁, p₀q₁.
Fisher's index: geometric mean of Laspeyre's and Paasche's — reduces bias.
Time reversal test: P₀₁ × P₁₀ = 1. Only Fisher's satisfies both reversal tests.
Financial Mathematics
Key Points to Remember
- →Simple Interest: SI = PRT/100. Compound Interest: A = P(1 + r/n)^(nt).
- →Effective rate of interest: i_eff = (1 + r/n)^n − 1.
- →Present Value: PV = FV / (1 + r)ⁿ. Future Value: FV = PV × (1 + r)ⁿ.
- →Annuity: equal payments at regular intervals. PV of annuity = PMT × [1 − (1+r)⁻ⁿ] / r.
- →EMI = P × r(1+r)ⁿ / [(1+r)ⁿ − 1] where r = monthly rate, n = number of months.
- →Depreciation (SLM) = (Cost − Salvage) / Useful life years.
- →Depreciation (WDV): Book value at end = Cost × (1 − d)ⁿ where d = depreciation rate.
- →Sinking fund: regular deposits to accumulate a future amount. A = PMT × [(1+r)ⁿ − 1] / r.
Exam Tips
EMI: convert annual rate to monthly (divide by 12). Convert years to months (multiply by 12).
WDV depreciation: multiply each year by (1 − d) — value decreases each year.
Distinguish annuity due (payment at start) from ordinary annuity (payment at end) — exam specifies.
Linear Programming
Key Points to Remember
- →LPP components: decision variables (x, y), objective function (maximise/minimise), constraints (inequalities), non-negativity (x ≥ 0, y ≥ 0).
- →Graphical method: plot each constraint as a line. Determine feasible region (satisfies all constraints and non-negativity).
- →Corner point theorem: optimal value of objective function occurs at one of the corner (vertex) points of feasible region.
- →Evaluate objective function at each vertex → optimal value is maximum or minimum as required.
- →Unbounded solution: feasible region has no bound in the direction of optimisation.
- →No feasible solution: constraints are contradictory — no region satisfies all simultaneously.
Exam Tips
Practise identifying corner points from intersection of constraint lines — solve systems of 2 equations.
Always check whether the problem is maximise or minimise — easy to swap under exam pressure.
Write out all constraint inequalities from the problem statement before graphing.
Inferential Statistics
Key Points to Remember
- →Population: entire group of interest. Sample: subset of population used for study.
- →Types of sampling: Simple Random (each has equal chance), Stratified (population divided into strata), Systematic (every kth item), Cluster (random selection of groups).
- →Sampling distribution of mean: mean = μ, standard error = σ/√n.
- →Central Limit Theorem: for large n (≥30), distribution of sample means is approximately normal.
- →Confidence interval: estimate ± (z × SE). 95% CI: z = 1.96. 99% CI: z = 2.576.
- →Hypothesis testing: H₀ (null hypothesis), H₁ (alternative). z-test for large samples. t-test for small samples (n<30).
- →p-value: probability of observing test statistic as extreme as calculated. Reject H₀ if p < significance level (0.05 or 0.01).
Exam Tips
CI formula: x̄ ± z(σ/√n). Calculate z × σ/√n first, then add and subtract from x̄.
Distinguish between sampling error (random) and non-sampling error (systematic bias).
Type I error: rejecting true H₀. Type II error: accepting false H₀. Significance level = P(Type I error).