CBSE Class 11 Applied Mathematics
Chapters: 7
1. Numbers, Quantification and Numerical Applications
Binary numbers · Indices, logarithms and antilogarithms · Bhartiya system of numeration · Clocks and calendar · Time and work; speed, distance and time · Seating arrangement
- Number Systems and Encoding – A number system is a way to write numbers using a set of digits and a base. Decimal (base 10) uses 0–9, binary (base 2) uses 0 and 1, octal (base 8) uses 0–7, and hexadecimal (base 16) uses 0–9 and A–F. To go from decimal to any base, divide repeatedly by the base and read remainders bottom to top; for fractions, multiply by the base and read the integer parts top to bottom. To go to decimal, multiply each digit by its place value and add. Binary ↔ octal uses groups of 3 bits, binary ↔ hex groups of 4. Text is stored with encoding schemes: ASCII (7-bit, 128 characters), ISCII (8-bit, Indian scripts) and Unicode (every script), stored as UTF-8 (1–4 bytes) or UTF-32 (4 bytes).
- Logarithms – A logarithm tells you the power you need. log_b x = y means b^y = x. So log₂ 8 = 3 because 2³ = 8. Logs turn multiplying into adding: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a. Base 10 is the common log, base e is the natural log (ln). Any base can be changed: log_b x = log x ÷ log b.
- The Indian System of Numeration – In the Indian system, the first comma comes after 3 digits from the right and then after every 2 digits: ones, thousands, lakhs, crores. 1 lakh = 1,00,000 = 100 thousand; 1 crore = 1,00,00,000 = 10 million. The international system puts a comma after every 3 digits: thousands, millions, billions. India also gave the world zero and the decimal place-value idea, with old names for every power of ten.
- Clock and Calendar: Telling Time by the Sky and by Numbers – Our time units come from the sky. One spin of the Earth relative to the Sun is a solar day (24 h); relative to the stars it is a sidereal day (about 23 h 56 min). The Moon's phases repeat every 29.53 days (a month). The Earth goes round the Sun in about 365.2422 days (a year). Calendars are lunar, solar or lunisolar. The Gregorian leap year rule keeps the calendar in step with the seasons. With numbers we can then solve clock problems (angle = |30H − 5.5M|) and calendar problems (odd days).
- 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.
- Logical Reasoning – Logical reasoning means getting from facts (premises) to a conclusion in a way we can check. A statement is either true or false. Statements are joined with NOT, AND, OR, IF…THEN and IF AND ONLY IF, and truth tables show when the result is true. In deduction, if the premises are true and the form is valid, the conclusion must be true (modus ponens, modus tollens, syllogisms). Venn diagrams test syllogisms with 'all', 'no' and 'some'. Induction and analogy go from examples to a general idea: the conclusion is only probable. A direct proof goes step by step from what is known; an indirect proof assumes the opposite and reaches a contradiction. Fallacies are tempting but faulty arguments.
2. Algebra
Sets · Relations · Mathematical logic · Sequences and series
- Sets: Representation, Types, Subsets, Venn Diagrams and Operations – A set is a well-defined collection of different objects. We write it in roster form {2, 4, 6} or set-builder form {x : x is even}. Sets can be empty, finite, infinite or equal. If every element of B is in A, B is a subset of A (B ⊂ A); a set with n elements has 2ⁿ subsets. Intervals like (a, b) and [a, b] are subsets of real numbers. The universal set U holds everything under study. With Venn diagrams we see union A ∪ B, intersection A ∩ B, difference A − B and complement A′ = U − A.
- Mathematical Reasoning: Puzzles You Can Solve with Rules – Reasoning questions test whether you can spot a rule and use it. Odd one out: find the rule most items share. Syllogism: draw circles for "all", "some" and "no" and accept only what MUST be true. Blood relations: draw a family tree. Coding-decoding: find how letters or numbers shift. Cryptarithms: digits hide behind letters; use place value, carries and odd/even (parity) to find them.
- Sequences and Series: AP, GP, AM and GM – A sequence is a list of numbers in a definite order: a₁, a₂, a₃, … A series is what we get when we add the terms. In an arithmetic progression (AP) we add the same number d each time; aₙ = a + (n − 1)d and Sₙ = n/2[2a + (n − 1)d]. The arithmetic mean (AM) of a and b is (a + b)/2; n AMs between a and b use d = (b − a)/(n + 1). In a geometric progression (GP) we multiply by the same number r each time; aₙ = arⁿ⁻¹ and Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1. If |r| < 1 the terms shrink and the infinite sum is S∞ = a/(1 − r). The geometric mean (GM) of two positive numbers a and b is √(ab), and n GMs between them use r = (b/a)^(1/(n+1)). For positive a and b, AM ≥ GM, with equality only when a = b.
3. Calculus
Functions and graphs · Limits and continuity · Differentiation
- Limits (Class 11): What a Function Gets Close To – A limit tells us which number f(x) gets close to when x gets close to a point. We walk towards the point from the left and from the right. If both sides reach the same number, that number is the limit. You will learn limits of polynomial, rational, trigonometric, exponential and log functions, with the standard results you must know.
4. Combinatorics and Probability
Permutations and combinations · Probability
- Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.
5. Descriptive Statistics
Measures of dispersion and percentiles · Correlation · Regression
- Measures of Dispersion: Range, Mean Deviation, Variance and SD – Dispersion means spread: how far the values sit from the centre. Range = largest − smallest. Mean deviation = average distance from the mean (or median). Variance = average of squared distances from the mean. Standard deviation = √variance. The same ideas work for grouped data when every term is multiplied by its frequency.
- Correlation: Scatter Diagram, Karl Pearson's Coefficient and Spearman's Rank Correlation – Correlation tells how two variables move together. It is positive when both rise together, negative when one rises as the other falls, and zero when there is no straight-line pattern. A scatter diagram shows it as a picture. Karl Pearson's coefficient r measures its direction and strength and always lies between −1 and +1. Spearman's rank correlation R uses ranks and works for qualities like beauty or honesty; tied ranks need a small correction.
- Linear Regression and the Least Squares Line – Linear regression finds the straight line ŷ = a + bx that best follows paired data (x, y). A residual is the gap between a real point and the line: e = y − ŷ. The least squares line makes the sum of squared residuals as small as possible. Its slope is b = Sxy ÷ Sxx and it always passes through the mean point (x̄, ȳ). We use it to predict y from x, but only inside the data range, and a strong link does not prove that x causes y.
6. Basics of Financial Mathematics
Interest and interest rates · Annuities · Taxes and utility bills
- Compound Interest – Interest is money paid for using money. Simple interest is paid only on the starting amount (the principal), so it grows by the same step every year. Compound interest is paid on the principal plus all the interest earned so far, so the steps get bigger: interest earns interest. The amount after n years is A = P(1 + r/100)ⁿ. If interest is added k times a year, use rate r/k and k × n periods: A = P(1 + r/(100k))^(kn). Adding it every instant gives continuous compounding, A = Pe^(rt). The same idea run backwards gives depreciation: A = P(1 − r/100)ⁿ. Compound growth is exponential growth.
- 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.
- Taxation: Types of Taxes and How to Calculate Them – A tax is money people and businesses must pay to the government. The government uses it for public goods (roads, schools, hospitals, defence), to help poorer people and to manage the economy. Direct taxes are paid straight from income or wealth (income tax, corporate tax, property tax). Indirect taxes are added to the price of goods and services (GST, VAT, excise, customs). A tax is progressive if richer people pay a bigger share of income, proportional (flat) if all pay the same share, and regressive if poorer people pay a bigger share. Income tax often uses slabs: each slab of income has its own rate. Effective rate = total tax ÷ income × 100.
7. Coordinate Geometry
Straight lines · Circles and parabola
- Straight Lines (Class 11): Slope, Forms of the Equation and Distance – The slope of a line is rise ÷ run = (y₂ − y₁)/(x₂ − x₁) = tan θ. Parallel lines have equal slopes; perpendicular lines have m₁m₂ = −1. The angle between two lines is given by tan θ = |(m₂ − m₁)/(1 + m₁m₂)|. A line can be written as y = b or x = a (parallel to an axis), y − y₁ = m(x − x₁) (point-slope), y = mx + c (slope-intercept), the two-point form, or x/a + y/b = 1 (intercept form). The distance of (x₁, y₁) from Ax + By + C = 0 is |Ax₁ + By₁ + C|/√(A² + B²).
- Conic Sections (Class 11): Circle, Parabola, Ellipse and Hyperbola – Cutting a double cone with a plane gives a circle, an ellipse, a parabola or a hyperbola; a cut through the vertex gives a point, a line or a pair of lines (degenerate conics). Circle: (x − h)² + (y − k)² = r². Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a, e = 1. Ellipse x²/a² + y²/b² = 1 (a > b): c² = a² − b², foci (±c, 0), e = c/a < 1, latus rectum 2b²/a, PF₁ + PF₂ = 2a. Hyperbola x²/a² − y²/b² = 1: c² = a² + b², e = c/a > 1, latus rectum 2b²/a, |PF₁ − PF₂| = 2a.