📐 CBSE 2026–27 · 15 Chapters · 80 Marks

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Unit I Number System 1 ch · Ch 1 · 7 marks · 12 periods
Ch 1Number System
Rational & irrational numbers, √(2) irrationality, square root spiral
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1.1Introduction to rational numbers
  • A rational number is any number of the form p/q where p,q are integers and q≠0.
  • Integers, fractions, and terminating/repeating decimals are all rational.
  • ℕ ⊂ ℤ ⊂ ℚ.
  • Rationals are closed under +,-,x, and division by non-zero.
Formulas & Key Relations
Rational formp/q, q≠0, p,q ∈ ℤ
Closurea/b +/- c/d = (ad+/-bc)/(bd)
Reverse - testCan the number be written as p/q? Yes=rational
1.2Representation on the number line
  • Every rational maps to exactly one point on the number line.
  • To place p/q: divide unit into q equal parts, count p from 0.
  • Successive magnification: zoom into intervals of 1/10n to pin down any decimal.
  • Negative rationals lie left of 0; positive right.
Formulas & Key Relations
Position of p/qDivide unit into q parts; count p parts from 0
Successive magnificationZoom into interval of size 1/10n at each step
1.3Density of rational numbers
  • Between any two distinct rationals there is always another rational.
  • The mean (a+b)/2 is rational and always lies strictly between a and b.
  • Infinitely many rationals exist in any interval.
  • Despite being dense, rationals leave gaps filled by irrationals.
Formulas & Key Relations
Rational between a and bm = (a+b)/2
n rationals between a and bak = a + k(b-a)/(n+1) for k=1..n
Reverse - find bound from meana = 2m-b or b = 2m-a— given mean m and one bound
1.4Finding rationals between two rationals
  • Average method: (a+b)/2 always lies between a and b.
  • For n rationals: divide the interval into (n+1) equal parts.
  • Common denominator method: integers between numerators give intermediate rationals.
Formulas & Key Relations
Average methodm = (a+b)/2
n equally spaced valuesak = a + k(b-a)/(n+1)
Reverse - find spacinggap = (b-a)/(n+1)
1.5Decimal representation of rationals
  • Every rational has a terminating OR non-terminating repeating decimal.
  • Terminates iff denominator (lowest terms) has only 2s and 5s as prime factors.
  • Repeating decimals can always be converted back to p/q.
  • Bar notation: 0.3̅ means 0.333...
Formulas & Key Relations
Terminating conditionq = 2a * 5b (lowest terms)
Repeating to fraction0.d̅ = d/9; 0.ab̅ = (ab-a)/90
Reverse - decimal from fractionPerform long division; pattern reveals period
1.6Introduction to irrational numbers
  • Irrationals cannot be written as p/q for any integers p,q.
  • Decimal expansion: non-terminating AND non-repeating.
  • Famous irrationals: √(2), √(3), π, e.
  • Together with rationals they form the real numbers R.
Formulas & Key Relations
Irrational definitionx ∉ ℚ
Decimal testNon-terminating AND non-repeating → irrational
Real number lineℝ = ℚ ∪ {irrationals}
1.7Proof of irrationality of √(2) and √(3)
  • Proof by contradiction: assume √(2) = p/q in lowest terms with gcd(p,q)=1.
  • Then p2 = 2q2; p2 even → p even; write p=2k.
  • Substituting: 4k2=2q2 → q2=2k2 → q even. Contradicts gcd=1.
  • Same structure proves √(3) using divisibility by 3.
Formulas & Key Relations
Key lemmaIf p2 divisible by prime n → p divisible by n
Contradiction √(2)Assume √(2)=p/q (gcd=1) → p2=2q2 → 2|p and 2|q → contradiction
For √(3)p2=3q2 → 3|p → p=3k → 3|q → contradiction
1.8The square root spiral
  • Start: right triangle with both legs=1; hypotenuse=√(2).
  • Add new triangle on hypotenuse with new short leg=1; new hypotenuse=√(3).
  • Continuing produces √(2),√(3),√(4),√(5),...
  • Also called the Wheel of Theodorus.
Formulas & Key Relations
Hypotenuse at step nhn = √(n+1)— step 1 gives √(2)
Triangle at step nLegs: √(n) and 1; Hypotenuse: √(n+1)
Reverse - find stepn = (target)2 - 1
Unit II Algebra 4 ch · Ch 2–5 · 20 marks · 66 periods
Ch 2Introduction to Polynomials
Degree, linear polynomials, slope & y-intercept, graphing
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2.1Algebraic expressions
  • Combines constants, variables, and operations (+,-,x,/).
  • Like terms share same variable(s) and exponent(s) and can be combined.
  • Monomial:1 term; Binomial:2; Trinomial:3.
2.2Definition of a polynomial
  • All exponents of the variable must be non-negative integers (0,1,2,...).
  • Expressions like x(-1) or x(1/2) are NOT polynomials.
  • Every real number is a constant (degree-0) polynomial.
Formulas & Key Relations
Standard forman*xn + ... + a1*x + a0
Zero of p(x)p(x)=0 → solve for x
Reverse - verify rootSubstitute x=c; if p(c)=0 then c is a zero
2.3Degree of a polynomial
  • Degree = highest power with non-zero coefficient.
  • Degree 0→constant; 1→linear; 2→quadratic; 3→cubic.
  • Degree determines max number of roots and graph shape.
Formulas & Key Relations
Degreemax exponent with non-zero coefficient
Linear° 1: ax+b
Quadratic° 2: ax2+bx+c
Cubic° 3: ax3+bx2+cx+d
2.4Linear polynomials and applications
  • f(x)=ax+b (a≠0) has exactly one zero.
  • Models constant-rate change: taxi fare = fixed + rate x km.
  • Graph is always a straight line.
Formulas & Key Relations
Linear polynomialf(x) = ax+b, a≠0
Zerox = -b/a
Reverse - find a,b from zero zAny a≠0; b = -az
2.5Exploring linear patterns
  • A linear pattern has a constant difference between consecutive terms.
  • Graphs of linear patterns are straight lines.
  • Check: are all consecutive differences equal?
Formulas & Key Relations
nth termT(n) = a+(n-1)d
Common differenced = T(n+1)-T(n)
Reverse - find positionn = (T(n)-a)/d + 1
2.6Modelling linear growth and decay
  • Growth: fixed increase per unit time. Decay: fixed decrease.
  • Model: y=mx+c with m>0 (growth) or m<0 (decay).
  • Initial value at x=0 is c; rate of change is m.
Formulas & Key Relations
Growthy = mx+c, m>0
Decayy = mx+c, m<0
Reverse - find x from yx = (y-c)/m
Reverse - find c from pointc = y-mx
2.7Linear relationships and visualising
  • Two variables are linearly related when one changes at a constant rate.
  • Plot three points; if collinear, the relationship is linear.
  • Check equal y-differences for equal x-steps.
Formulas & Key Relations
Two-point form(y-y1)/(y2-y1) = (x-x1)/(x2-x1)
Slope from two pointsm = (y2-y1)/(x2-x1)
Reverse - find missing coordy = y1+m(x-x1)
2.8Slope and y-intercept of y=ax+b
  • Slope m: positive→up, negative→down, zero→horizontal.
  • y-intercept b: where the line crosses the y-axis (set x=0).
  • Parallel lines have equal slopes. Perpendicular: m1*m2=-1.
Formulas & Key Relations
Slopem = (y2-y1)/(x2-x1) = rise/run
y-interceptSet x=0 → y=b
x-interceptSet y=0 → x=-b/m
Reverse - slope from interceptsm = -(y-intercept)/(x-intercept)
Reverse - find b from point and mb = y-mx
Ch 3Sequences and Progressions
AP, GP, nth term, fractals, Tower of Hanoi
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3.1Introduction to sequences
  • An ordered list of numbers following a rule: a1,a2,a3,...
  • Each number is a term at position n.
  • Examples: perfect squares 1,4,9,16...; primes 2,3,5,7,11...
Formulas & Key Relations
Sequence notationa1, a2, ..., an, ...
General terman = f(n)
3.2Explicit (general) rule
  • Gives the nth term directly from n - no prior terms needed.
  • Efficient: compute any term instantly.
  • Identify by finding a function pattern in (n, an) pairs.
Formulas & Key Relations
Explicit formulaan = f(n)
Examplean = 2n+1 → 3,5,7,9,...
Reverse - find n from ann = finverse(an)
3.3Recursive rule
  • Defines each term from previous term(s) plus an initial condition.
  • Requires computing all earlier terms to reach a specific one.
  • Fibonacci is the most famous recursive sequence.
Formulas & Key Relations
General recursivea1=c; an=g(an-1)
AP recursivea1=a; an=an-1+d
FibonacciF1=1,F2=1; Fn=Fn-1+Fn-2
3.4Arithmetic Progressions (AP): nth term
  • AP: constant difference d between consecutive terms.
  • General form: a, a+d, a+2d, ...
  • Check: all consecutive differences equal → AP.
Formulas & Key Relations
nth terman = a+(n-1)d
Common differenced = an+1-an
Reverse - find nn = (an-a)/d+1
Reverse - find aa = an-(n-1)d
Reverse - find dd = (an-a)/(n-1)
Middle termIf a,b,c in AP → b=(a+c)/2
3.5Sum of AP and practical contexts
  • Sum formula: pair first and last terms (Gauss trick).
  • Plot (n, an): points on straight line - slope=d.
  • Practical: staircase steps, equal monthly savings.
Formulas & Key Relations
Sum SnSn = n/2 * (2a+(n-1)d)
Alt formSn = n*(a+an)/2
Reverse - find n from Sndn2/2 + n(a-d/2) - Sn=0 (quadratic in n)
Reverse - find a from Sn,d,na = Sn/n - (n-1)d/2
3.6Sum of first n natural numbers
  • Gauss trick: pair (1+n),(2+n-1),... each=n+1; n/2 pairs.
  • Special AP with a=1,d=1.
  • Fundamental in counting, algebra, algorithm analysis.
Formulas & Key Relations
Sum formula1+2+...+n = n(n+1)/2
Reverse - find n given sum Sn = (√(8S+1)-1)/2
Sum of first n odd numbersn2
Sum of first n even numbersn(n+1)
3.7Geometric Progressions (GP): nth term
  • GP: each term multiplied by constant ratio r to get the next.
  • r>1→grows; 0<r<1→shrinks; r<0→alternates sign.
  • Examples: 2,6,18,54 (r=3); 81,27,9,3 (r=1/3).
Formulas & Key Relations
nth terman = a*r(n-1)
Common ratior = an+1/an
Reverse - find nn = log(an/a)/log(r) + 1
Reverse - find aa = an/r(n-1)
Reverse - find rr = (an/a)(1/(n-1))
Geometric meana,b,c in GP → b2=ac → b=√(ac)
3.8Visualising GP and practical contexts
  • Plot (n,an): exponential curve for r>1.
  • Compound interest, population growth, radioactive decay (r<1).
  • Doubling sequence (r=2): 1,2,4,8,... in binary and computing.
Formulas & Key Relations
Sum (r≠1)Sn = a(rn-1)/(r-1)
Infinite sum (|r|<1)S∞ = a/(1-r)
Reverse - find a from S∞a = S∞*(1-r)
Reverse - find r from S∞r = 1 - a/S∞
3.9Applications of GP in fractals
  • Fractals show self-similarity at every scale.
  • Sierpinski triangle: remaining area forms GP with r=3/4.
  • Koch snowflake: perimeter grows as GP with r=4/3 (infinite perimeter!).
  • Model coastlines, clouds, ferns, blood vessels.
Formulas & Key Relations
Sierpinski area after n stepsAn = A0*(3/4)n
Koch perimeter after n stepsPn = P0*(4/3)n
Reverse - steps for given arean = log(An/A0)/log(3/4)
3.10Tower of Hanoi puzzle
  • Move n discs from peg A to C; larger never on smaller.
  • Recursive: n-1 to B, disc n to C, n-1 from B to C.
  • Minimum moves = 2n-1 — exponential growth.
  • For 64 discs at 1 move/sec: ~585 billion years!
Formulas & Key Relations
Minimum movesT(n) = 2n - 1
Recursive formT(1)=1; T(n)=2T(n-1)+1
Reverse - find n from movesn = log2(T+1)
64 discs264-1 ≈ 1.8x1019 moves
Ch 4Exploring Algebraic Identities
Identities, geometric proofs, factorisation, rational expressions
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4.1Revisiting algebraic identities
  • An identity is true for ALL values of the variables.
  • Used to expand, factorise, and simplify quickly.
  • Memorise all standard forms and reverses for exam speed.
Formulas & Key Relations
(a+b)2a2+2ab+b2
(a-b)2a2-2ab+b2
(a+b)(a-b)a2-b2
(a+b+c)2a2+b2+c2+2ab+2bc+2ca
(a+b)3a3+3a2*b+3ab2+b3
(a-b)3a3-3a2*b+3ab2-b3
Reverse - find ab given sumsab = [(a+b)2-(a2+b2)]/2
Reverse - find a2+b2a2+b2 = (a+b)2-2ab
4.2Visualising identities geometrically
  • (a+b)2: square of side (a+b) = a2 + 2ab + b2 (four regions).
  • (a+b)(a-b): L-shaped region = rectangle.
  • These proofs without words make algebra tangible.
Formulas & Key Relations
Area proof (a+b)2Square side (a+b) = a2+2(ab)+b2
Area proof a2-b2L-shape area = (a+b)(a-b)
4.3Factorisation using identities
  • Always take out HCF first, then apply identities.
  • Recognise: x2-25=(x+5)(x-5); x2+6x+9=(x+3)2.
  • Factorisation is the reverse of expansion.
Formulas & Key Relations
Difference of squaresa2-b2 = (a+b)(a-b)
Perfect squarea2+2ab+b2 = (a+b)2
Sum of cubesa3+b3 = (a+b)(a2-ab+b2)
Difference of cubesa3-b3 = (a-b)(a2+ab+b2)
4.4More identities and applications
  • (x+a)(x+b) useful for fast mental multiplication.
  • 103x97=(100+3)(100-3)=10000-9=9991.
  • Brahmagupta and Bhaskara developed systematic algebraic methods.
Formulas & Key Relations
(x+a)(x+b)x2+(a+b)x+ab
Reverse - sum & product of rootsSum=-(a+b); Product=ab
Mental multiplication103x97=(100)2-32=9991
4.5Factorisation through algebra tiles
  • Algebra tiles: x2=big square, x=rectangle, 1=unit square.
  • Arrange tiles into a rectangle; dimensions are the factors.
  • If tiles cannot form rectangle → no integer factors.
Formulas & Key Relations
Area modelx2+5x+6=(x+2)(x+3)
Verify(x+2)(x+3)=x2+5x+6
4.6Finding new identities
  • New identities derived by combining existing ones.
  • This discovery spirit is central to mathematical thinking.
  • All identities verified by expanding both sides.
Formulas & Key Relations
a2+b2 from sum/producta2+b2 = (a+b)2-2ab
2ab2ab = (a+b)2-(a2+b2)
(a-b)2(a-b)2 = (a+b)2-4ab
Reverse - find a+ba+b = √(a2+b2+2ab)
4.7Simplifying rational expressions
  • A rational expression: fraction with polynomials top and bottom.
  • Factorise both, then cancel common factors.
  • Always note domain restrictions from cancelled factors.
Formulas & Key Relations
SimplificationFactor both; cancel common factors
Example(x2-4)/(x+2) = (x-2) for x≠-2
Domain restrictionSet denominator ≠ 0
Ch 5Linear Equations in Two Variables
Graphical & algebraic solutions, substitution & elimination
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5.1Introduction through practical examples
  • One equation in two unknowns has infinitely many solutions.
  • Two unknowns need two equations for a unique solution.
  • Real-life: cost of items, mixing problems, speed-time.
Formulas & Key Relations
Standard formax+by+c=0, a,b not both 0
Solutions (one eq)Infinitely many - a whole line
5.2Graphical representation
  • Each solution (x,y) is a point; all solutions form a straight line.
  • To draw: find x-intercept (set y=0) and y-intercept (set x=0); join.
  • Every point on the line satisfies the equation.
Formulas & Key Relations
x-interceptSet y=0, solve for x
y-interceptSet x=0, solve for y
Verify a pointSubstitute (x0,y0); check equality
5.3Slope-intercept form
  • Rearrange ax+by=c to y=mx+k for immediate graphing.
  • Slope tells direction and steepness; y-intercept is the starting value.
Formulas & Key Relations
Slope-intercepty = mx+k (m=slope, k=y-intercept)
Convert from standardy = (-a/b)x+(c/b)
Reverse - standard formmx-y+k=0
5.4Graphs over restricted domains
  • Variables sometimes restricted to whole numbers (e.g. number of items).
  • Discrete graphs show dots, not a continuous line.
  • Recognising discrete vs continuous matters for real-world modelling.
Formulas & Key Relations
Discrete domainx ∈ {0,1,2,...} → plot only those points
5.5Pair of linear equations
  • Geometrically two lines; solution is their intersection.
  • Three possible cases: unique, none, infinitely many.
  • The ratio test tells you which case without drawing.
Formulas & Key Relations
Systema1x+b1y=c1 and a2x+b2y=c2
Unique solutiona1/a2 ≠ b1/b2
No solutiona1/a2 = b1/b2 ≠ c1/c2
Infinite solutionsa1/a2 = b1/b2 = c1/c2
5.6Graphical method
  • Draw both lines; intersection point = solution.
  • Useful to visualise the nature of the solution.
  • Exact algebraic methods preferred for precise answers.
Formulas & Key Relations
StepsFind 2 points on each line; draw; read intersection
LimitationApproximate for non-integer solutions
5.7Nature of solutions
  • Consistent: at least one solution. Inconsistent: none.
  • Dependent (coincident): infinitely many solutions.
  • Test using the ratio of coefficients.
Formulas & Key Relations
ConsistentLines intersect: a1/a2 ≠ b1/b2
InconsistentParallel: a1/a2=b1/b2≠c1/c2
DependentCoincident: a1/a2=b1/b2=c1/c2
5.8Substitution method
  • Express one variable from one equation; substitute into the other.
  • Best when a coefficient is 1 or -1.
  • Always verify by substituting back into both equations.
Formulas & Key Relations
StepsExpress one var; substitute; solve; back-substitute
Reverse - verifySubstitute (x,y) into both original equations
5.9Elimination method
  • Multiply equations so one variables coefficients match; add/subtract.
  • More efficient than substitution for large coefficients.
  • Determinant (a1b2-a2b1)=0 when lines are parallel.
Formulas & Key Relations
Solution formulax=(b1c2-b2c1)/(a1b2-a2b1); y=(a2c1-a1c2)/(a1b2-a2b1)
Determinant DD = a1b2-a2b1
Reverse - D=0 meansLines parallel or coincident
Unit III Coordinate Geometry 1 ch · Ch 6 · 4 marks · 6 periods
Ch 6Coordinate Geometry
2-D Cartesian system, distance, midpoint, collinearity
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6.1Brief history
  • Rene Descartes (1637) linked geometry and algebra - analytic geometry.
  • Pierre de Fermat independently developed similar ideas.
  • Foundation of calculus and modern mathematics.
6.2The 2-D Cartesian coordinate system
  • Two perpendicular axes cross at origin O(0,0).
  • x-axis horizontal; y-axis vertical.
  • Four quadrants: I(+,+), II(-,+), III(-,-), IV(+,-).
Formulas & Key Relations
Point notationP(x,y); x=abscissa, y=ordinate
Reflect across x-axis(x,y) → (x,-y)
Reflect across y-axis(x,y) → (-x,y)
Reflect across origin(x,y) → (-x,-y)
6.3Distance between two points
  • Derived from Pythagoras: horizontal and vertical differences are the legs.
  • Distance is always ≥ 0.
  • Key use: verify equilateral, isosceles, or right triangles from coordinates.
Formulas & Key Relations
Distance formulad = √[(x2-x1)2+(y2-y1)2]
From origind = √(x2+y2)
Reverse - find missing coordSet up d2 equation and solve
6.4Midpoint of a line segment
  • Average the x-coordinates and the y-coordinates.
  • Divides the segment in ratio 1:1.
  • Used to find centroids, circumcentres, and 4th vertices of parallelograms.
Formulas & Key Relations
MidpointM = ((x1+x2)/2, (y1+y2)/2)
Reverse - find endpointx2=2Mx-x1; y2=2My-y1
Section formula (m:n)P = ((mx2+nx1)/(m+n), (my2+ny1)/(m+n))
6.5Checking collinearity
  • Three methods: area, slope, or distance.
  • Area=0 means the three points form a degenerate (flat) triangle.
  • Slope equality is the quickest check.
Formulas & Key Relations
Area method (0→collinear)A = (1/2)|x1(y2-y3)+x2(y3-y1)+x3(y1-y2)|
Slope methodslope(AB) = slope(BC) → collinear
Reverse - find unknown for collinearitySet area=0 and solve for missing coord
6.6Checking right-angled triangles
  • Compute all three distances using the distance formula.
  • Check if a2+b2=c2 where c is the longest.
  • Right angle is at the vertex opposite the longest side.
Formulas & Key Relations
Pythagoras checka2+b2 = c2 (c=longest side)
Dot product (right angle at A)(x2-x1)(x3-x1)+(y2-y1)(y3-y1)=0
Reverse - find vertex for right angleSet dot product=0 and solve
Unit IV Geometry 5 ch · Ch 7–11 · 25 marks · 69 periods
Ch 7Introduction to Euclids Geometry
History, Sulbasutras, axioms, 5 postulates, parallelism
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7.1History of geometry
  • Geometry (geo=earth, metron=measure) began in Egypt/Mesopotamia.
  • Baudhayana Sulbasutras (800-600 BCE): earliest formal geometric rules from India.
  • Euclid (~300 BCE) compiled "Elements" - systematic axiomatic geometry.
7.2Baudhayana Sulbasutras
  • Sulbasutras: rope-and-peg altar construction rules.
  • Diagonal of a square produces double the area - early Pythagorean theorem.
  • Constructions required exact right angles for religious altar precision.
Formulas & Key Relations
Baudhayana theoremd2=a2+b2 (diagonal of rectangle)
Square side from areas=√(A)
Reverse - area from sideA=s2
7.3Euclid definitions
  • Point: "that which has no part"; Line: "breadthless length".
  • These are intuitive primitives - accepted without proof.
  • Modern maths treats them as undefined.
7.4Axioms of measurement
  • Axioms are universal logical truths assumed without proof.
  • They apply beyond geometry.
  • All theorems ultimately derive from axioms + postulates.
Formulas & Key Relations
Axiom 1Things equal to the same thing are equal to each other
Axiom 2Equals added to equals give equals
Axiom 3Equals subtracted from equals give equals
Axiom 4Things that coincide are equal
Axiom 5The whole is greater than the part
7.55 postulates of Euclidean geometry
  • P5 is the most complex - its denial leads to non-Euclidean geometry.
  • All of Euclidean geometry is derived from just these 5 postulates + 5 axioms.
Formulas & Key Relations
P1A line segment between any two points
P2Any segment can be extended indefinitely
P3Circle with any centre and radius
P4All right angles are equal
P5 (Parallel)Interior angles < 180° on one side → lines meet on that side
Playfair equivalentThrough a point, exactly one parallel to a given line
7.6Definition of parallelism
  • Two coplanar lines are parallel if they never meet however far extended.
  • Through any external point there is exactly one parallel (Playfair).
  • Parallel lines are always equidistant.
Formulas & Key Relations
NotationAB ∥ CD
Reverse - prove parallelShow alternate angles equal OR co-interior angles supplementary
Ch 8Lines and Angles
Rays, angle types, intersecting lines, parallel lines theorems
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8.1Rays and angles
  • Ray: one endpoint, extends infinitely in one direction.
  • ∠: two rays sharing a vertex.
  • Measured in degrees; complete turn = 360°.
Formulas & Key Relations
∠ notation∠ ABC (B=vertex; BA,BC=arms)
Full revolution360°
Straight angle180°
8.2Measures of angles
  • Five types: acute, right, obtuse, straight, reflex.
  • Protractor measures; compass constructs.
Formulas & Key Relations
Acute0<θ<90°
Rightθ=90°
Obtuse90<θ<180°
Straightθ=180°
Reflex180<θ<360°
Reverse - reflex from partnerReflex = 360-θ
8.3Intersecting lines and angles
  • Two intersecting lines form 2 pairs of vertically opposite angles.
  • Adjacent angles at intersection are supplementary (linear pair).
  • All four angles at intersection sum to 360°.
Formulas & Key Relations
Sum at intersectionAll 4 angles = 360°
Linear pairAdjacent angles on straight line = 180°
Vertically opposite∠ A = ∠ C and ∠ B = ∠ D
8.4Pairs of angles
  • Complementary: sum 90°. Supplementary: sum 180°.
  • Linear pair: adjacent supplementary forming a straight line.
  • Vertically opposite: always equal.
Formulas & Key Relations
ComplementaryA+B=90°
SupplementaryA+B=180°
Reverse - complementComplement=90-θ
Reverse - supplementSupplement=180-θ
8.5Linear pair theorem and converse
  • Theorem: ray on a line makes two angles summing to 180°.
  • Converse: two adjacent angles summing to 180° → outer arms are collinear.
Formulas & Key Relations
Theorem∠ AOC + ∠ BOC = 180°
ConverseSum=180°, adjacent → outer arms form a line
8.6Proof: vertically opposite angles are equal
  • Uses two linear pairs at intersection, then subtracts the common angle.
Formulas & Key Relations
Step 1∠ AOC+∠ AOD=180°
Step 2∠ BOD+∠ AOD=180°
Result∠ AOC=∠ BOD and ∠ AOD=∠ BOC
8.7Corresponding, alternate, and co-interior angles
  • A transversal crossing two lines creates 8 angles.
  • Corresponding/alternate equal <→ lines are parallel.
  • Co-interior supplementary <→ lines are parallel.
Formulas & Key Relations
Corresponding (F-shape)Equal when lines parallel
Alternate interior (Z-shape)Equal when parallel
Co-interior (C-shape)Supplementary when parallel: sum=180°
Reverse - prove parallelShow corresponding OR alternate angles equal
8.8Transitivity of parallelism
  • If l ∥ m and m ∥ n then l ∥ n.
Formulas & Key Relations
Transitivityl ∥ m and m ∥ n → l ∥ n
8.9Theorems on parallel lines
  • ∠ sum of triangle=180° proved by drawing parallel through one vertex.
  • Exterior angle = sum of two opposite interior angles.
  • Triangle must have at least two acute angles.
Formulas & Key Relations
∠ sum of triangleA+B+C=180°
Exterior angleExt = sum of two non-adjacent interior angles
Reverse - third angleC=180-A-B
Reverse - interior from exteriorSum of two interior = Exterior
Ch 9Triangles - Congruence Theorems
SAS, SSS, ASA, AAS, RHS; isosceles; converses
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9.1Rigidity and applications
  • Triangle is the only rigid polygon - shape fixed once side lengths are fixed.
  • Used in bridges, cranes, roof trusses.
  • A quadrilateral is NOT rigid.
9.2Congruence of two triangles
  • Congruent: same shape AND size.
  • CPCT: once congruence proved, all corresponding parts are equal.
  • Order of vertices in notation specifies correspondence.
Formulas & Key Relations
NotationΔ ABC = Δ PQR (A<→P, B<→Q, C<→R)
CPCTCorresponding Parts of Congruent Triangles are equal
9.3SAS congruence axiom
  • SAS is an axiom - accepted without proof.
  • The ∠ MUST be between (included by) the two sides.
Formulas & Key Relations
SAS2 sides + included angle equal → triangles congruent
Reverse (CPCT)Prove SAS → deduce unknown sides/angles
9.4SSS congruence condition
  • All three sides match → triangle shape completely determined.
  • Proved using construction and SAS axiom.
Formulas & Key Relations
SSS3 sides equal → triangles congruent
9.5ASA congruence condition
  • The side MUST be between (included by) the two angles.
Formulas & Key Relations
ASA2 angles + included side equal → triangles congruent
Included sideSide between the two given angles
9.6AAS congruence condition
  • Two equal angles force the third to be equal (angle sum=180).
Formulas & Key Relations
AAS2 angles + any non-included side equal → congruent
Why2 equal angles → 3rd also equal → reduces to ASA
9.7RHS congruence condition
  • Only for right-angled triangles.
  • Third side determined by Pythagoras → uniqueness.
Formulas & Key Relations
RHSRight angle + Hypotenuse + one Leg equal → congruent
Missing legleg = √(hyp2 - otherleg2)
9.8Properties of isosceles triangles
  • Isosceles: two equal sides (legs); third side is the base.
  • Base angles are always equal - proved using SAS.
  • Altitude from apex bisects base perpendicularly.
Formulas & Key Relations
Base angles theoremAB=AC → ∠ B = ∠ C
Converse∠ B=∠ C → AB=AC
Apex altitudeAltitude from apex = angle bisector = ⊥ bisector of base
9.9Propositions and their converses
  • Not all converses are true - always check with counter-example.
  • Contrapositive is always logically equivalent to the original.
Formulas & Key Relations
PropositionIf P, then Q
ConverseIf Q, then P
ContrapositiveIf not Q, then not P (equivalent to original)
9.10Why SSA is not valid
  • SSA in general allows two different triangles - the ambiguous case.
  • Exception: angle=90° → RHS (valid). ∠ obtuse → valid.
Formulas & Key Relations
Ambiguous caseTwo different triangles can share same two sides and non-included angle
Exception 1∠=90° → becomes RHS
Exception 2∠ obtuse → unique triangle
Ch 104-gons (Quadrilaterals)
Parallelograms, midpoint theorem, medians, symmetry, tiling
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10.1Definition and properties
  • Quadrilateral: 4 sides, 4 angles, 4 vertices.
  • Interior angle sum = 360°.
  • Types: parallelogram, rectangle, rhombus, square, trapezium, kite.
Formulas & Key Relations
∠ sumSum of interior angles = 360°
ProofDivide into 2 triangles: 2x180=360
10.2Properties of a parallelogram
  • Both pairs of opposite sides parallel AND equal.
  • Both pairs of opposite angles equal.
  • Diagonals bisect each other.
Formulas & Key Relations
Opposite sidesAB=CD; BC=AD
Opposite angles∠ A=∠ C; ∠ B=∠ D
Consecutive angles∠ A+∠ B=180°
DiagonalsAO=OC; BO=OD (O=intersection)
Reverse - find unknownUse opposite=equal or consecutive=supplementary
10.3Midpoint theorem
  • Segment joining midpoints of two sides || third side and = half of it.
  • Proved by constructing a parallelogram using the midpoints.
Formulas & Key Relations
Midpoint theoremM,N midpoints of AB,AC → MN ∥ BC and MN=BC/2
Reverse - find BCBC=2*MN
Reverse - find MNMN=BC/2
ConverseThrough midpoint of AB, ∥ BC → bisects AC
10.4Converse of midpoint theorem
  • Line through midpoint of one side, parallel to another, bisects the third.
Formulas & Key Relations
ConverseThrough midpoint M of AB, draw MN ∥ BC → N bisects AC
10.5Medians and centroid
  • All three medians concurrent at centroid G.
  • Centroid divides each median 2:1 from vertex.
  • Centroid = centre of gravity of uniform triangular lamina.
Formulas & Key Relations
Centroid divides medianAG:GM = 2:1 (A=vertex, M=foot)
Centroid coordinatesG = ((x1+x2+x3)/3, (y1+y2+y3)/3)
Reverse - find vertex from Gx1 = 3Gx-x2-x3
10.6Varignon theorem
  • Joining midpoints of any quadrilaterals sides → parallelogram.
  • Each side of inner shape = half a diagonal of outer shape.
  • Area of inner parallelogram = half area of outer quadrilateral.
Formulas & Key Relations
Varignon theoremMidpoints of sides of any 4-gon form a parallelogram
Side lengthEach side of inner gram = half a diagonal of outer 4-gon
AreaArea(Varignon) = (1/2)*Area(original)
10.7Coordinates of midpoint and fourth vertex
  • In a parallelogram, midpoints of diagonals are the same.
  • Use this to find 4th vertex given the other three.
Formulas & Key Relations
Midpoints equalMidpoint(AC) = Midpoint(BD)
Fourth vertexD = (Ax+Cx-Bx, Ay+Cy-By)
Reverse - verifyCheck midpoints of both diagonals are equal
10.8Symmetries of quadrilaterals
  • Parallelogram: only 180° rotational symmetry; no line symmetry.
  • Rectangle: 2 lines + 180° rotation.
  • Square: 4 lines + 90,180,270° rotation.
Formulas & Key Relations
ParallelogramRotational 180° only
Rectangle2 lines + 180° rotation
Rhombus2 lines (diagonals) + 180° rotation
Square4 lines + 90,180,270,360° rotation
10.9How any 4-gon tiles the plane
  • Any quadrilateral tiles the plane - angles sum to 360°.
  • Rotate copies 180° around edge midpoints to fill plane.
Formulas & Key Relations
Key factAngles of any 4-gon sum to 360°
Tiling4 copies with all 4 angles at one point → 360° exactly
Ch 11Circles
Chords, arcs, subtended angles, cyclic quadrilaterals
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11.1Definitions
  • Circle: all points at distance r from centre O.
  • Chord: segment joining two points on circle; diameter is longest chord.
  • Arc: continuous piece; Segment: between chord and arc; Sector: pie slice.
Formulas & Key Relations
Circle equation(x-h)2+(y-k)2=r2
Diameterd=2r
Reverse - find rr=d/2
Arc lengthl=(θ/360)*2*π*r
Sector areaA=(θ/360)*π*r2
11.2Unique circle through three non-collinear points
  • Exactly one circle passes through any three non-collinear points.
  • Circumcentre is equidistant from all three vertices.
  • Location: inside (acute), on hyp (right), outside (obtuse) triangle.
Formulas & Key Relations
CircumradiusR=abc/(4*Area)
CircumcentreIntersection of ⊥ bisectors of any two sides
11.3Circumcircle and circumcentre
  • Circumcircle passes through all three vertices.
  • For right triangle: R = hypotenuse/2.
Formulas & Key Relations
CircumradiusR=abc/(4*Area) = a/(2*sinA)
Reverse - find sidea=2R*sinA
Right triangleR = hypotenuse/2
11.4Equal chords subtend equal angles
  • Equal chords subtend equal central angles.
  • Proved using SSS congruence.
  • Converse also true.
Formulas & Key Relations
TheoremAB=CD → ∠ AOB = ∠ COD
Converse∠ AOB=∠ COD → AB=CD
ProofΔ OAB = Δ OCD (SSS)
11.5Perpendicular from centre bisects chord
  • The perpendicular from centre to any chord bisects it.
  • Proved using RHS congruence.
  • Use to find chord length from distance, or vice versa.
Formulas & Key Relations
TheoremOM ⊥ AB → AM=MB
Half-chordAM=√(r2-d2) (d=distance centre to chord)
Reverse - chord lengthAB=2*√(r2-d2)
Reverse - find distanced=√(r2-(AB/2)2)
11.6Chord length vs distance from centre
  • Longer chord → closer to centre.
  • Diameter (d=0) is the longest chord.
  • Equal chords are equidistant from centre.
Formulas & Key Relations
Relationshipr2=d2+(AB/2)2
Equal chords equidistantAB=CD <→ d(O,AB)=d(O,CD)
ReverseLarger chord is closer to centre
11.7Central angle = 2 x inscribed angle
  • Central angle = double any inscribed angle subtending the same arc.
  • All inscribed angles on the same arc are equal.
  • Special case (Thales): inscribed angle in semicircle = 90°.
Formulas & Key Relations
Inscribed angle theorem∠ AOB = 2 * ∠ APB (P on remaining arc)
Semicircle∠ APB=90° when AB is diameter
Reverse - central from inscribed∠ AOB = 2*∠ APB
Reverse - inscribed from central∠ APB = ∠ AOB/2
11.8Angles in the same segment are equal
  • All inscribed angles subtending same chord from same side are equal.
  • Follows from central angle theorem.
  • Used to prove four points are concyclic.
Formulas & Key Relations
Theorem∠ APB=∠ AQB (P,Q on same side of AB)
Reverse - prove concyclicEqual angles from same side → points are concyclic
11.9∠ in semicircle = 90° (Thales)
  • Any angle inscribed in a semicircle equals 90°.
  • Proof: central angle of semicircle=180; inscribed=half=90.
  • Converse: angle=90 in triangle → hypotenuse is circumdiameter.
Formulas & Key Relations
Thales theoremAB=diameter → ∠ APB=90°
Converse∠ APB=90° → AB is diameter
Circumradius of right triangleR=hypotenuse/2
11.10Cyclic quadrilateral: opposite angles supplementary
  • Opposite angles of cyclic quadrilateral sum to 180°.
  • Converse: supplementary opposite angles → quadrilateral is cyclic.
Formulas & Key Relations
TheoremABCD cyclic → A+C=180° and B+D=180°
ConverseA+C=180° → ABCD is cyclic
Reverse - find angleC=180-A
Unit V Mensuration 2 ch · Ch 12–13 · 14 marks · 27 periods
Ch 12Area and Perimeter
π, arc length, Herons formula, Brahmagupta, sector area
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12.1Perimeter of shapes
  • Perimeter = total length of the boundary.
  • Measured in linear units (cm, m, km).
  • Composite shapes: add all outer edge lengths.
Formulas & Key Relations
TriangleP=a+b+c
RectangleP=2(l+w)
SquareP=4s
Regular polygon (n sides)P=n*s
Reverse - find l from P,wl=P/2-w
Reverse - find s from Ps=P/n
12.2Circumference and introduction to π
  • C/d = π is constant for every circle.
  • π ≈ 3.14159... - non-terminating, non-repeating.
  • Historical: Archimedes 22/7; Aryabhata 62832/20000; Zu Chongzhi 355/113.
Formulas & Key Relations
CircumferenceC=2*π*r=π*d
Reverse - find r from Cr=C/(2*π)
Reverse - find d from Cd=C/π
12.3Irrationality of π
  • π is irrational (Lambert, 1768) and transcendental (Lindemann, 1882).
  • Madhava (~1400 CE) found the first exact infinite series for π.
  • We can approximate π to any precision but never express it exactly as p/q.
Formulas & Key Relations
Madhava seriesπ/4=1-1/3+1/5-1/7+...
Approximation 22/7Error ≈ 0.04%
Better: 355/113Accurate to 6 decimal places
12.4Length of an arc
  • Arc length is a fraction (θ/360) of the full circumference.
  • Radian measure: θ=l/r - why radians are natural.
  • Used for curved roads, belt length, bridge arcs.
Formulas & Key Relations
Arc length (degrees)l=(θ/360)*2*π*r
Arc length (radians)l=r*θ
Reverse - find r from l,θr=l*360/(2*π*θ)
Reverse - find θ from l,rθ=l*360/(2*π*r) degrees
12.5Areas: rectangles, parallelograms, triangles
  • Parallelogram: base x perpendicular height (NOT slant side).
  • Triangle area = half parallelogram with same base and height.
  • A median divides a triangle into two equal-area triangles.
Formulas & Key Relations
RectangleA=l*w
SquareA=s2
ParallelogramA=base*h (h ⊥ base)
TriangleA=(1/2)*base*h
Reverse - find h from A (triangle)h=2A/base
Reverse - find base from A (triangle)base=2A/h
12.6Heron formula
  • Computes area from three sides alone - no height needed.
  • Named after Heron of Alexandria (~60 CE).
  • Useful for triangular plots of land with known boundary lengths.
Formulas & Key Relations
Semi-perimeters=(a+b+c)/2
Heron formulaArea=√[s(s-a)(s-b)(s-c)]
Reverse - find heighth=2*Area/base (use Heron for Area)
Verify with base-heightAlso equals (1/2)*base*height
12.7Squaring a rectangle (Baudhayana)
  • "Squaring" a shape: construct square of equal area.
  • Requires constructing geometric mean √(l*w).
  • Used in Vedic altar construction.
Formulas & Key Relations
Square of equal areaSide=√(l*w) (geometric mean)
Reverse - rectangle from squareAny l,w with l*w=s2 works
12.8Area of a circle
  • Derived by cutting circle into thin sectors rearranged into a rectangle.
  • Archimedes proved this using the method of exhaustion.
Formulas & Key Relations
AreaA=π*r2
Reverse - find r from Ar=√(A/π)
Reverse - find d from Ad=2*√(A/π)
Area in terms of CA=C2/(4*π)=C*r/2
12.9Area of a sector
  • Sector area is proportional to the central angle.
  • Segment = sector - triangle formed by two radii and chord.
  • Applications: pie charts, sprinkler coverage.
Formulas & Key Relations
Sector area (degrees)A=(θ/360)*π*r2
Sector area (radians)A=r2*θ/2
Reverse - find r from A,θr=√(360A/(π*θ))
Reverse - find θ from A,rθ=360A/(π*r2) degrees
Segment area=Sector area - Triangle area
12.10Brahmaguptas formula
  • Brahmagupta (7th century CE) generalised Herons to cyclic quadrilaterals.
  • Requires the quadrilateral to be inscribed in a circle.
  • Herons is the special case when the fourth side = 0.
Formulas & Key Relations
Brahmaguptas formulaArea=√[(s-a)(s-b)(s-c)(s-d)]
Semi-perimeters=(a+b+c+d)/2
Valid forCyclic quadrilaterals only
Special case d=0→ Herons formula
12.11Herons as special case of Brahmagupta's
  • Set d=0 - the quadrilateral collapses to a triangle.
  • The (s-d) term reduces to s, giving exactly Herons formula.
  • Demonstrates 'special case' and 'generalisation' - key mathematical concepts.
Formulas & Key Relations
Set d=0s=(a+b+c)/2; (s-d)=s → Area=√[s(s-a)(s-b)(s-c)]
GeneralisationBrahmagupta → Heron: 4-gon degenerates to triangle
Ch 13Surface Area and Volume
Cuboid, cylinder, cone, pyramid, sphere, hemisphere
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13.1Cuboid and cube
  • Cube is a special cuboid with l=w=h.
  • TSA includes all 6 faces; LSA excludes top and bottom.
  • Diagonal uses 3D Pythagoras.
Formulas & Key Relations
Cuboid TSA2(lw+wh+lh)
Cuboid LSA2h(l+w)
Cuboid VolumeV=l*w*h
Cube TSA6*s2
Cube VolumeV=s3
Reverse - find s from Vs=∛(V)
Reverse - find l from V,w,hl=V/(w*h)
Diagonal (cuboid)d=√(l2+w2+h2)
Diagonal (cube)d=s*√(3)
13.2Right circular cylinder
  • Curved SA = unrolled lateral surface → rectangle 2*π*r x h.
  • TSA adds two circular ends.
  • Real-world: tins, pipes, tanks.
Formulas & Key Relations
Curved SACSA=2*π*r*h
Total SATSA=2*π*r*(r+h)
VolumeV=π*r2*h
Reverse - find r from V,hr=√(V/(π*h))
Reverse - find h from V,rh=V/(π*r2)
Reverse - find h from CSA,rh=CSA/(2*π*r)
13.3Cone
  • Slant height l links r, h and curved surface.
  • Volume = (1/3) x cylinder volume with same base and height.
  • Three cones fill one cylinder.
Formulas & Key Relations
Slant heightl=√(r2+h2)
Curved SACSA=π*r*l
Total SATSA=π*r*(l+r)
VolumeV=(1/3)*π*r2*h
Reverse - find r from V,hr=√(3V/(π*h))
Reverse - find h from V,rh=3V/(π*r2)
Reverse - find r from l,hr=√(l2-h2)
13.4Pyramid
  • The (1/3) factor applies to ALL pyramids (Cavalieri principle).
  • Slant height of square pyramid: l=√(h2+(a/2)2).
  • Used for Great Pyramid calculations.
Formulas & Key Relations
Volume (any pyramid)V=(1/3)*Base Area*h
Square pyramid LSA2al (a=base side, l=slant height)
Square pyramid TSAa2+2al
Reverse - find h from Vh=3V/Base Area
Slant heightl=√(h2+(a/2)2)
13.5Sphere
  • SA=4*π*r2 equals lateral SA of circumscribed cylinder (Archimedes).
  • Ratio sphere:cylinder:cone volumes = 2:3:1.
  • Why bubbles are spherical: sphere maximises V for given SA.
Formulas & Key Relations
Surface areaSA=4*π*r2
VolumeV=(4/3)*π*r3
Reverse - find r from SAr=√(SA/(4*π))
Reverse - find r from Vr=∛(3V/(4*π))
Sphere:cylinder:cone2:3:1 (same r, cylinder h=2r)
13.6Hemisphere
  • Hemisphere = half sphere; curved SA=2*π*r2.
  • TSA adds flat base: 2*π*r2+π*r2=3*π*r2.
  • Applications: bowls, domes, satellite dishes.
Formulas & Key Relations
Curved SACSA=2*π*r2
Total SATSA=3*π*r2
VolumeV=(2/3)*π*r3
Reverse - find r from TSAr=√(TSA/(3*π))
Reverse - find r from Vr=∛(3V/(2*π))
13.7Comparing 3-D shapes by varying dimensions
  • Doubling radius quadruples SA but octuples volume.
  • Sphere encloses maximum volume for given SA.
  • Small cells have high SA/V ratio - efficient for nutrient exchange.
Formulas & Key Relations
SA scales asr2 (double r → 4x SA)
V scales asr3 (double r → 8x V)
SA/V ratio (sphere)3/r (smaller r → larger ratio)
Isoperimetric inequalitySphere maximises V for given SA
Unit VI Statistics & Probability 2 ch · Ch 14–15 · 10 marks · 24 periods
Ch 14Statistics
Mean, median, mode, weighted average, graphs
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14.1Collecting and organising data
  • Statistical cycle: question→collect→organise→analyse→interpret→communicate.
  • Primary data: collected directly. Secondary: from existing sources.
  • Frequency tables and tally charts organise raw data.
14.2Graphical representation
  • Bar graph: discrete categories. Histogram: continuous data (no gaps).
  • Frequency polygon: connect tops of histogram bars.
  • Ogive: cumulative frequency curve.
14.3Mean
  • Mean uses ALL values; sensitive to outliers.
  • Best for symmetric distributions.
  • Minimises sum of squared deviations.
Formulas & Key Relations
Mean (ungrouped)x̅ = Σ(xi)/n
Mean (grouped)x̅ = Σ(fi*xi)/Σ(fi) (xi=class midpoint)
Reverse - find missing valuemissing = n*x̅ - Σ(other values)
Reverse - find nn = Σ(xi)/x̅
Effect of shift by kNew mean = old mean + k
Effect of scale by kNew mean = k * old mean
14.4Median
  • Middle value when sorted; NOT affected by outliers.
  • Best for skewed distributions (incomes, house prices).
  • For grouped data use interpolation within the median class.
Formulas & Key Relations
Odd nMedian = ((n+1)/2)th value (sorted)
Even nMedian = average of (n/2)th and (n/2+1)th values
Grouped medianM = l + [(n/2-cf)/f]*h
Wherel=lower bound, cf=cum freq before, f=class freq, h=width
Reverse - find classFind class where cum freq first reaches n/2
14.5Mode
  • Most frequent value; can be multiple modes.
  • Only measure usable for categorical data.
  • Not affected by extreme values.
Formulas & Key Relations
Ungrouped modeValue with highest frequency
Grouped mode estimateMode = l+[(f1-f0)/(2f1-f0-f2)]*h
Wheref1=modal freq, f0=prev, f2=next, h=width, l=lower bound
14.6Weighted average
  • Different values given different weights.
  • GPA: credits are weights. Price index: quantities are weights.
  • Ordinary mean is a special case with all weights=1.
Formulas & Key Relations
Weighted meanx̅w = Σ(wi*xi)/Σ(wi)
GPA exampleGPA = Σ(credits*gradepoints)/totalcredits
Reverse - find missing weightSet up equation and solve for unknown wi
14.7Stacked and 100% stacked bar graphs
  • Stacked bar: shows totals AND sub-group composition.
  • 100% stacked: shows proportions, not absolute values.
  • Limitation: middle segments hard to compare.
Ch 15Introduction to Probability
Randomness, empirical & theoretical probability, tree diagrams
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15.1Concept of probability and randomness
  • Random experiment: outcome unpredictable in advance.
  • Examples: coin toss, die roll, card draw.
  • Probability was formalised in 17th century by Pascal and Fermat.
Formulas & Key Relations
Probability range0 ≤ P(A) ≤ 1
Certain eventP(S) = 1
Impossible eventP(empty set) = 0
15.2The probability scale
  • P always between 0 and 1.
  • P=0: impossible. P=1: certain. P=0.5: equally likely to happen or not.
  • Expressed as fraction, decimal, or percentage.
Formulas & Key Relations
Scale0 <- unlikely → 0.5 <- likely → 1
P(A)=0Impossible (rolling 7 on standard die)
P(A)=1Certain (rolling ≤ 6 on standard die)
15.3Empirical (experimental) probability
  • Based on actual data - not theoretical reasoning.
  • More trials → empirical probability approaches theoretical (Law of Large Numbers).
  • Used when exact outcomes are unknown.
Formulas & Key Relations
Empirical PP(A) = (number of times A occurred)/(total trials)
Law of Large NumbersEmpirical P → Theoretical P as n → ∞
Reverse - estimate trials neededn ≈ (required events)/P(A)
15.4Theoretical probability: sample space and events
  • Sample space S = all possible outcomes.
  • Event A = subset of S.
  • Assumes equally likely outcomes.
Formulas & Key Relations
Theoretical PP(A) = n(A)/n(S) (equally likely outcomes)
ComplementP(A') = 1-P(A)
Mutually exclusiveP(A or B) = P(A)+P(B)
Reverse - find P(A) from complementP(A) = 1-P(A')
15.5Probability using tree diagrams
  • Tree diagrams show all outcomes of multi-stage experiments.
  • Multiply along branches (AND); add across branches (OR).
  • Essential when probabilities differ at each stage.
Formulas & Key Relations
Multiplication rule (independent)P(A and B) = P(A)*P(B)
Read treeMultiply along branches; add branches for desired event
Conditional probabilityP(B|A) = P(A and B)/P(A)
15.6Probability using tables
  • Two-way table displays all combined outcomes.
  • Count cells matching the event; divide by total cells.
  • Efficient and visual - no outcomes missed.
Formulas & Key Relations
Two dice sample space6x6=36 equally likely outcomes
P(sum=7)6/36=1/6 (six favourable outcomes)
P(sum=2)1/36 (only (1,1))
Reverse - find n from Pn = P * total outcomes
Class 9 Mathematics · CBSE Syllabus 2026–27 · GEMS New Millennium School Dubai
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