CBSE Class 12 Applied Mathematics

Chapters: 8

1. Numbers, Quantification and Numerical Applications

Modulo arithmetic and congruence · Alligation and mixture · Numerical problems · Numerical inequalities

Modular Arithmetic: Remainders and Congruences – Euclidean division writes any integer a as a = n × q + r with 0 ≤ r < n. The remainder r is "a mod n". Two numbers are congruent modulo n (a ≡ b mod n) when they leave the same remainder, which means n divides a − b. Congruences can be added, subtracted, multiplied and raised to powers, so we can find remainders of huge numbers using small ones. Remainders of powers repeat in cycles.
Alligation and Mixture – When two ingredients with prices (or strengths) c and d are mixed, the mixture's mean price m lies between them. The rule of alligation says the quantities must be in the ratio (d − m) : (m − c), cheaper to dearer. It is a short cut for the weighted average m = (q₁c + q₂d) ÷ (q₁ + q₂). For repeated replacement, if x units are drawn from a vessel of V units and replaced with water n times, the original liquid left is V(1 − x/V)ⁿ.
Speed, Distance and Time (with Time and Work) – Speed = distance ÷ time. To change km/h into m/s multiply by 5/18. Average speed = total distance ÷ total time. Two bodies moving towards each other close the gap at the sum of their speeds; in the same direction, at the difference. A train must cover its own length (plus the platform or other train). Downstream speed = boat + stream; upstream = boat − stream. Work and pipes use the same idea: add rates (per hour), subtract leaks.

2. Algebra

Matrices · Determinants and inverse · Solving simultaneous equations

Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
Determinants: Minors, Cofactors, Adjoint, Inverse and Linear Systems – A determinant is one number made from a square matrix, written |A|. For a 2 × 2 matrix it is ad − bc, and it equals the (signed) area made by the columns. For a 3 × 3 matrix we expand along a row using minors and cofactors. |A| = 0 means A is singular and has no inverse. Half of a determinant gives the area of a triangle. The adjoint (transpose of the cofactor matrix) gives A⁻¹ = (adj A)/|A|, and then a system AX = B is solved by X = A⁻¹B. The value of |A| and (adj A)B tell us if a system is consistent.

3. Calculus

Derivatives and applications · Integration and applications · Differential equations and modelling

Application of Derivatives – The derivative measures how fast one quantity changes compared with another. Its sign tells whether a function goes up (f′ > 0, increasing) or down (f′ < 0, decreasing). Where f′ = 0 the tangent is flat: these critical points may be a local maximum or minimum, checked by the first derivative test (sign change) or the second derivative test (sign of f″). This lets us solve real problems like the biggest box or the cheapest tank.
Integrals – Integration undoes differentiation: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. We integrate with standard formulas, substitution (replace an inside part by u), partial fractions (split a fraction) and by parts (∫u dv = uv − ∫v du). A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve from a to b, and the fundamental theorem says it equals F(b) − F(a). Its properties make many hard integrals easy.
Differential Equations – A differential equation connects a function y with its derivatives. Its order is the highest derivative present and its degree is the power of that derivative (when the equation is a polynomial in derivatives). A general solution has arbitrary constants; a condition like y(0) = 1 fixes them to give a particular solution. Class 12 solves first-order equations of three kinds: variables separable, homogeneous (put y = vx) and linear dy/dx + Py = Q (multiply by the integrating factor e^∫P dx).

4. Probability Distributions

Random variables and expectation · Binomial, Poisson and normal distributions

Random Variables and Probability Distributions – A random variable X gives a number to every outcome of a chance experiment. A discrete random variable takes separate values; its probability distribution lists each value x with its probability P(X = x), and these add up to 1. The mean (expected value) is E(X) = Σ x·P(x). The variance Var(X) = E(X²) − [E(X)]² measures spread, and the standard deviation is its square root.
Probability Distributions of Discrete Random Variables – A random variable X turns each outcome of an experiment into a number. Its probability distribution lists every value x with its probability P(X = x); each P is between 0 and 1 and they add to 1. The mean E(X) = Σx·P(x) is the long-run average (balance point). The variance Var(X) = Σ(x − μ)²P(x) = E(X²) − μ² measures spread; σ = √Var. Special models: uniform, binomial B(n, p) with mean np and variance np(1 − p), and Poisson with mean = variance = λ.

5. Inferential Statistics

Population and sample · Parameter, statistic and inference · t-test

Sampling: Learning About a Population from a Sample – A population is the whole group we want to know about; a sample is a smaller part we actually check. A good sample is chosen at random so that it represents the population. Simple random sampling gives everyone an equal chance; stratified sampling takes the right share from each group; systematic sampling takes every k-th item. Different samples give slightly different answers (sampling variation), but bigger samples wobble less (law of large numbers). A biased sample gives a wrong answer however big it is.
Statistical Inference – Statistical inference means using a sample to say something about a whole population. A number that describes the population (like the true proportion p or the mean μ) is a parameter. A number worked out from a sample (like p̂ or x̄) is a statistic, and we use it as a point estimate. Different random samples give different answers: this is sampling variability. If we took many samples, their statistics would form the sampling distribution, centred on the true value, with spread called the standard error: SE = √(p(1−p)/n) for a proportion and σ/√n for a mean. Bigger samples give smaller spread. A 95% confidence interval is estimate ± 1.96 × SE; about 95 of every 100 such intervals catch the true value. Simulation helps us check whether a claimed model fits the data. Good inference needs random sampling, and an association in data does not prove cause.
Hypothesis Testing – A hypothesis test checks a claim about a population using a sample. Start with the null hypothesis H₀ (no change, e.g. p = 0.5) and the alternative H₁ (what we suspect, e.g. p > 0.5). Choose a significance level such as 5%. Work out how likely the sample result (or more extreme) is if H₀ were true: the p-value. If the p-value is below the level, or the result falls in the critical region, reject H₀. Otherwise there is not enough evidence to reject it. Type I error = rejecting a true H₀; Type II error = not rejecting a false H₀.

6. Time-based Data

Time series

Time Series: Moving Averages, Trend and Seasonal Variation – A time series is a set of data recorded at equal time intervals, such as sales every quarter or rainfall every month. We plot it as a line graph with time on the horizontal axis. Real series go up and down with the seasons (seasonal variation), in longer waves (cyclic variation) and with random noise. To see the long-term direction (the trend) we smooth the data with moving averages: the mean of each group of consecutive values, for example 4 values for quarterly data. A line of best fit through the moving averages is the trend line. Seasonal variation = actual value − trend value; the mean seasonal variation for each season, added to the trend, gives a forecast.

7. Financial Mathematics

Perpetuity and sinking funds · Valuation of bonds · EMI · CAGR and depreciation

Annuities: Equal Payments, Future Value and Present Value – An annuity is a series of equal payments made at equal time gaps, like a monthly saving or a loan instalment. Each payment earns compound interest, so the payments form a geometric series. Future value of an ordinary annuity (payments at the end of each period): FV = P[(1+i)ⁿ − 1]/i. Present value (what the payments are worth today): PV = P[1 − (1+i)⁻ⁿ]/i. Loans use PV: the loan amount equals the present value of all instalments.
Valuation of Bonds – A bond is a loan to a government or company. It pays a fixed coupon (coupon rate × face value) every period and returns the face value at maturity. Its fair price today is the present value of all these payments, discounted at the market yield r: P = C × [1 − (1 + r)⁻ⁿ] ÷ r + F ÷ (1 + r)ⁿ. When the market yield is below the coupon rate the bond sells at a premium; equal, at par; above, at a discount. Price and yield always move in opposite directions.
EMI: Equated Monthly Instalments, Flat Rate and Reducing Balance – An EMI (equated monthly instalment) is the same amount paid every month to clear a loan with interest. In the flat-rate method interest is worked out on the full loan for the whole time: EMI = (P + P×R×T/100) ÷ n. In the reducing-balance method interest is charged only on the amount still owed, so EMI = P·r·(1+r)^n ÷ ((1+r)^n − 1), where r is the monthly rate as a decimal and n the number of months. Each EMI pays some interest and some principal; the interest part falls and the principal part grows until the balance is zero.

8. Linear Programming

Linear programming

Linear Programming (Class 12): find the best answer with a graph – Linear programming finds the biggest profit or the smallest cost when you must obey some rules. The rules are straight-line inequalities (constraints). Together they cut out a region of allowed points (the feasible region). The goal, Z = ax + by (the objective function), is always best at a corner of that region. So: draw the lines, shade, find the corners, put each corner in Z, pick the largest or smallest. If the region is open (unbounded), check once more that the answer really holds.