How many formulas are on the Shunya formula sheet?

468 formulas in 83 groups, drawn from all ten Shunya labs and grouped by class and topic.

Can I filter the formula sheet by class?

Yes. Pick All, or class 9, 10, 11 or 12, and only the matching groups stay on screen. You can also search by topic or formula name.

Does each formula link to a lab?

Yes. Every group has a link to the Shunya lab that lets you try that formula on a live example.

Maths Formula Sheet: Class 9–12 (83 Groups, 468 Formulas)

Function & Graph Explorer Open the lab →

Coordinate geometry and the graph of a line Class 9

Equation of a line: $y = mx + c$
Slope (rise ÷ run): $m = \dfrac{y_2 - y_1}{x_2 - x_1}$
y-intercept (put x = 0): $(0,\ c)$
x-intercept (put y = 0): $x = -\dfrac{c}{m}$
Angle the line makes with the x-axis: $\theta = \tan^{-1} m$

Pair of linear equations and zeros of a quadratic Class 10

Pair of linear equations: $a_1x + b_1y = c_1,\qquad a_2x + b_2y = c_2$
Determinants (Cramer's rule): $D = a_1b_2 - a_2b_1,\quad D_x = c_1b_2 - c_2b_1,\quad D_y = a_1c_2 - a_2c_1$
The one solution when D ≠ 0: $x = \dfrac{D_x}{D},\qquad y = \dfrac{D_y}{D}$
One solution (lines cross): $\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$
No solution (parallel lines): $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$
Infinitely many (same line): $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$
Quadratic: $y = ax^2 + bx + c$
Discriminant: $D = b^2 - 4ac$
Zeros (D > 0): $x = \dfrac{-b \pm \sqrt{D}}{2a}$
Repeated zero (D = 0): $x = -\dfrac{b}{2a}$
Complex roots (D < 0): $x = \dfrac{-b \pm i\sqrt{-D}}{2a}$
Sum and product of the zeros: $\alpha + \beta = -\dfrac{b}{a},\qquad \alpha\beta = \dfrac{c}{a}$
Vertex form: $y = a(x - h)^2 + k,\quad h = -\dfrac{b}{2a},\ k = c - \dfrac{b^2}{4a}$
Factored form: $y = a(x - \alpha)(x - \beta)$

Polynomials: zeros and the shape of the graph Class 10

Cubic: $y = ax^3 + bx^2 + cx + d$
Sum of the zeros: $\alpha + \beta + \gamma = -\dfrac{b}{a}$
Sum of products of pairs: $\alpha\beta + \beta\gamma + \gamma\alpha = \dfrac{c}{a}$
Product of the zeros: $\alpha\beta\gamma = -\dfrac{d}{a}$
Polynomial built from its zeros: $y = a(x - \alpha)(x - \beta)(x - \gamma)\cdots$
Number of real zeros of a degree-n polynomial: $\le n$

Functions, symmetry and coordinate geometry Class 11

Even function (symmetric about the y-axis): $f(-x) = f(x)$
Odd function (half-turn symmetry about the origin): $f(-x) = -f(x)$
Stretched sine wave: $y = a\sin(bx) + c$
Amplitude, period, vertical lift: $|a|,\qquad \dfrac{2\pi}{|b|},\qquad c$
Where a function is undefined: $\dfrac{1}{x}:\ x = 0, \qquad \tan x:\ x = \dfrac{\pi}{2} + n\pi$
Distance between two points: $AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint: $M = \left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)$
Section formula (internal, m : n): $P = \left(\dfrac{mx_2 + nx_1}{m + n},\ \dfrac{my_2 + ny_1}{m + n}\right)$
Area of a triangle: $\tfrac{1}{2}\,\bigl|\,x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\,\bigr|$

One-one functions, inverses and a first look at slope Class 12

Inverse function: $f^{-1}\bigl(f(x)\bigr) = x$
Reflection in y = x: $(a,\ b) \ \longleftrightarrow\ (b,\ a)$
One-one test: $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$
Derivative (slope of the tangent): $f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$
Tangent line at x₀: $y - f(x_0) = f'(x_0)\,(x - x_0)$
Power rule: $\dfrac{d}{dx}\,x^n = n x^{n-1}$
Sum, product and quotient rules: $(u \pm v)' = u' \pm v',\quad (uv)' = u'v + uv',\quad \left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$
Chain rule: $\dfrac{d}{dx}\,f\bigl(g(x)\bigr) = f'\bigl(g(x)\bigr)\cdot g'(x)$
Standard derivatives: $(\sin x)' = \cos x,\ (\cos x)' = -\sin x,\ (\tan x)' = \sec^2 x,\ (e^x)' = e^x,\ (\ln x)' = \dfrac{1}{x}$

Trigonometry & Height-Distance Open the lab →

Trigonometric ratios of an acute angle Class 10

The three ratios: $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$
Their reciprocals: $\operatorname{cosec}\theta = \dfrac{1}{\sin\theta},\quad \sec\theta = \dfrac{1}{\cos\theta},\quad \cot\theta = \dfrac{1}{\tan\theta}$
Tan as a quotient: $\tan\theta = \dfrac{\sin\theta}{\cos\theta},\qquad \cot\theta = \dfrac{\cos\theta}{\sin\theta}$
Pythagoras theorem: $\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2$
Sides from the hypotenuse h: $\text{opposite} = h\sin\theta,\qquad \text{adjacent} = h\cos\theta$
Complementary angles: $\sin(90^\circ - \theta) = \cos\theta,\quad \cos(90^\circ - \theta) = \sin\theta,\quad \tan(90^\circ - \theta) = \cot\theta$
Values at 0°, 30°, 45°, 60°, 90°: $\sin:\ 0,\ \tfrac12,\ \tfrac{1}{\sqrt2},\ \tfrac{\sqrt3}{2},\ 1 \qquad \cos:\ 1,\ \tfrac{\sqrt3}{2},\ \tfrac{1}{\sqrt2},\ \tfrac12,\ 0 \qquad \tan:\ 0,\ \tfrac{1}{\sqrt3},\ 1,\ \sqrt3,\ \text{not defined}$

Trigonometric identities Class 10

Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$
Tan and sec: $1 + \tan^2\theta = \sec^2\theta$
Cot and cosec: $1 + \cot^2\theta = \operatorname{cosec}^2\theta$
Tan as a quotient: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
Complementary angles: $\sin(90^\circ - \theta) = \cos\theta,\qquad \tan(90^\circ - \theta) = \cot\theta$

Heights and distances Class 10

Angle from the sides: $\tan\theta = \dfrac{H}{d}\ \Rightarrow\ \theta = \tan^{-1}\dfrac{H}{d}$
Height from angle and distance (e = eye height, 0 at ground level): $H = d\tan\theta + e$
Distance from height and angle: $d = \dfrac{H}{\tan\theta}$
When a string or a ladder is involved: $\sin\theta = \dfrac{\text{height}}{\text{length}},\quad \cos\theta = \dfrac{\text{distance}}{\text{length}}$
Length from the height and angle: $\text{length} = \dfrac{\text{height}}{\sin\theta}$
Check with Pythagoras: $\text{length} = \sqrt{d^2 + H^2}$
Angle of depression = angle of elevation (alternate angles): $\theta_{\text{depression}} = \theta_{\text{elevation}}$
Two observation points, distance s apart: $H = \dfrac{s\,\tan\theta_1\tan\theta_2}{\tan\theta_2 - \tan\theta_1}$
Tower above a building: $\text{tower} = H_{\text{building}} + d\tan(\text{elevation})$
Handy values for 30°, 45°, 60°: $\tan 30^\circ = \tfrac{1}{\sqrt3},\quad \tan 45^\circ = 1,\quad \tan 60^\circ = \sqrt3$

The unit circle, radians and signs Class 11

A point on the unit circle: $P = (\cos\theta,\ \sin\theta),\qquad x^2 + y^2 = 1$
Radians and degrees: $\pi\ \text{rad} = 180^\circ,\qquad \theta_{\text{rad}} = \theta_{\text{deg}} \times \dfrac{\pi}{180}$
Tan from the coordinates: $\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{y}{x}$
Reference angle, by quadrant: $\text{I: } \theta \qquad \text{II: } 180^\circ - \theta \qquad \text{III: } \theta - 180^\circ \qquad \text{IV: } 360^\circ - \theta$
Positive ratios in quadrants I and II: $\text{I: all},\qquad \text{II: } \sin$
Positive ratios in quadrants III and IV: $\text{III: } \tan,\qquad \text{IV: } \cos$
Size from the reference angle: $|\sin\theta| = \sin(\text{ref. angle}),\quad |\cos\theta| = \cos(\text{ref. angle})$
Repeating every turn: $\sin(\theta + 360^\circ) = \sin\theta,\qquad \tan(\theta + 180^\circ) = \tan\theta$

Graphs of sin, cos and tan Class 11

The general transformed wave: $y = A\,f\bigl(B(x - C)\bigr) + D$
Amplitude: $|A|$
Period of sin and cos: $\dfrac{2\pi}{|B|}\quad\left(\dfrac{360^\circ}{|B|}\right)$
Period of tan: $\dfrac{\pi}{|B|}\quad\left(\dfrac{180^\circ}{|B|}\right)$
Phase shift and midline: $\text{shift right by } C,\qquad \text{midline } y = D$
Range of sin and cos: $-1 \le \sin\theta \le 1,\qquad -1 \le \cos\theta \le 1$
Where tan has asymptotes: $\cos\theta = 0\ \Rightarrow\ \theta = 90^\circ + 180^\circ n$
Cos is sin shifted: $\cos\theta = \sin(\theta + 90^\circ)$

Statistics & Probability Open the lab →

Experimental probability Class 9

Experimental probability: $P(E) = \dfrac{\text{trials in which } E \text{ happened}}{\text{total number of trials}}$
Expected number of times in n trials: $n \times P(E)$
Difference from the theoretical value: $\bigl|\,\text{experimental } P - \text{theoretical } P\,\bigr|$
Law of large numbers: $\text{experimental } P(E) \longrightarrow \text{theoretical } P(E) \text{ as } n \text{ grows}$

Probability of an event Class 10

Equally likely outcomes: $P(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$
Range of a probability: $0 \le P(E) \le 1,\qquad P(\text{impossible}) = 0,\qquad P(\text{certain}) = 1$
Complement: $P(\text{not } E) = 1 - P(E)$
All outcomes together: $\sum P(\text{outcome}) = 1$
Two dice, sum s (out of 36 pairs): $P(\text{sum} = s) = \dfrac{6 - |s - 7|}{36},\qquad 2 \le s \le 12$
Two tosses of a coin with P(H) = p: $P(0\text{ H}) = q^2,\quad P(1\text{ H}) = 2pq,\quad P(2\text{ H}) = p^2,\quad q = 1 - p$

Mean, median and mode Class 9

Mean: $\bar{x} = \dfrac{\sum x}{n}$
Median, n odd (the middle value of the sorted list): $\left(\dfrac{n + 1}{2}\right)\text{th value}$
Median, n even: $\dfrac{\left(\frac{n}{2}\right)\text{th value} + \left(\frac{n}{2} + 1\right)\text{th value}}{2}$
Mode (there can be more than one, or none): $\text{most frequent value}$
Range: $\text{largest} - \text{smallest}$

Grouped data Class 10

Class mark and class width: $x = \dfrac{l + u}{2},\qquad h = u - l$
Mean (direct method): $\bar{x} = \dfrac{\sum f x}{\sum f}$
Median: $\text{Median} = l + \dfrac{\frac{n}{2} - cf}{f} \times h$
Mode: $\text{Mode} = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$
Empirical relation: $\text{Mode} \approx 3\,\text{Median} - 2\,\text{Mean}$
Median class (the first class with cumulative frequency at least n/2): $cf \ge \dfrac{n}{2}$

Spread: quartiles, variance and standard deviation Class 11

First quartile (median of the lower half): $Q_1 = \text{median}(\text{lower half})$
Third quartile (median of the upper half): $Q_3 = \text{median}(\text{upper half})$
Interquartile range: $\text{IQR} = Q_3 - Q_1$
Variance: $\sigma^2 = \dfrac{\sum (x - \bar{x})^2}{n}$
Standard deviation: $\sigma = \sqrt{\sigma^2}$

Conditional probability and independent events Class 12

Multiplication rule: $P(A \cap B) = P(A)\,P(B \mid A)$
Conditional probability: $P(B \mid A) = \dfrac{P(A \cap B)}{P(A)}$
Independent events: $P(A \cap B) = P(A)\,P(B)$
Two draws with replacement: $P(\text{both red}) = \dfrac{r}{N}\times\dfrac{r}{N}$
Two draws without replacement: $P(\text{both red}) = \dfrac{r}{N}\times\dfrac{r - 1}{N - 1}$
Same colour, and its complement: $P(\text{same}) = P(RR) + P(BB) + P(GG),\qquad P(\text{different}) = 1 - P(\text{same})$

Binomial distribution Class 12

Probability of k successes in n trials: $P(X = k) = \binom{n}{k} p^k q^{n-k},\qquad q = 1 - p$
Binomial coefficient: $\binom{n}{k} = \dfrac{n!}{k!\,(n - k)!}$
At most k successes: $P(X \le k) = \sum_{i=0}^{k} P(X = i)$
At least k successes: $P(X \ge k) = 1 - P(X \le k - 1)$
Mean, variance and standard deviation: $\mu = np,\qquad \sigma^2 = npq,\qquad \sigma = \sqrt{npq}$
Most likely value: $\lfloor (n + 1)p \rfloor$
Normal curve fits well when: $np \ge 5 \ \text{ and } \ nq \ge 5$

Normal distribution Class 12

Density of the normal curve: $f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2\sigma^2}}$
Standardising (z-score): $z = \dfrac{x - \mu}{\sigma}$
Probability between a and b: $P(a < X < b) = \Phi(z_b) - \Phi(z_a)$
The 68 – 95 – 99.7 rule: $P(|X - \mu| < \sigma) \approx 0.6827,\quad P(|X - \mu| < 2\sigma) \approx 0.9545,\quad P(|X - \mu| < 3\sigma) \approx 0.9973$
Total area under the curve: $\int_{-\infty}^{\infty} f(x)\,dx = 1$

Geometry & Constructions Open the lab →

Lines and angles Class 9

Linear pair: $\angle 1 + \angle 2 = 180^\circ$
Vertically opposite angles: $\angle 1 = \angle 3$
Corresponding angles (l ∥ m): $\angle 1 = \angle 5$
Alternate interior angles (l ∥ m): $\angle 3 = \angle 5$
Co-interior angles (l ∥ m): $\angle 3 + \angle 6 = 180^\circ$
Converse: $\text{equal corresponding angles} \Rightarrow l \parallel m$

Triangles Class 9

Angle sum: $\angle A + \angle B + \angle C = 180^\circ$
Exterior angle: $\angle ACD = \angle A + \angle B$
Distance between two points: $AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Angle from the sides (law of cosines): $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$
Area from the coordinates: $\tfrac12\,\bigl|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\bigr|$
Heron's formula: $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)},\qquad s = \dfrac{a + b + c}{2}$
Triangle inequality: $a + b > c$
Larger angle, longer side: $\angle A > \angle B \iff a > b$

Congruence of triangles Class 9

SSS: $AB = PQ,\ BC = QR,\ CA = RP$
SAS: $AB = PQ,\ \angle A = \angle P,\ AC = PR$
ASA: $\angle A = \angle P,\ AB = PQ,\ \angle B = \angle Q$
AAS: $\angle A = \angle P,\ \angle B = \angle Q,\ BC = QR$
RHS: $\angle C = \angle R = 90^\circ,\ AB = PQ,\ BC = QR$
Not a rule: $\text{SSA can fit two different triangles}$
Corresponding parts (CPCT): $\triangle ABC \cong \triangle PQR \Rightarrow \text{all six matching parts are equal}$

Quadrilaterals Class 9

Angle sum of a quadrilateral: $\angle A + \angle B + \angle C + \angle D = 360^\circ$
Parallelogram: $AB \parallel DC,\ BC \parallel AD \ \Rightarrow\ AB = DC,\ BC = AD$
Diagonals of a parallelogram: $\text{they bisect each other}$
Rhombus (a parallelogram with all sides equal): $AC \perp BD$
Rectangle (a parallelogram with a right angle): $AC = BD$
Midpoint theorem: $DE \parallel BC,\quad DE = \tfrac12 BC$
Midpoint quadrilateral (always a parallelogram): $\text{area} = \tfrac12\,[ABCD]$

Similar triangles Class 10

Basic proportionality theorem: $DE \parallel BC \ \Rightarrow\ \dfrac{AD}{DB} = \dfrac{AE}{EC}$
Ratio of the sides: $\dfrac{AD}{AB} = \dfrac{AE}{AC} = \dfrac{DE}{BC} = k$
Perimeters: $\dfrac{\text{perimeter of } ADE}{\text{perimeter of } ABC} = k$
Areas: $\dfrac{\text{area of } ADE}{\text{area of } ABC} = k^2$
AA similarity: $\angle A = \angle P,\ \angle B = \angle Q \ \Rightarrow\ \triangle ABC \sim \triangle PQR$

Pythagoras theorem Class 10

Pythagoras: $a^2 + b^2 = c^2$
Converse: $a^2 + b^2 = c^2 \ \Rightarrow\ \angle C = 90^\circ$
Acute triangle: $a^2 + b^2 > c^2$
Obtuse triangle: $a^2 + b^2 < c^2$
Pythagorean triples: $3\text{-}4\text{-}5,\quad 5\text{-}12\text{-}13,\quad 8\text{-}15\text{-}17$

Triangle centres Class 10

Centroid: $G = \left(\dfrac{x_1 + x_2 + x_3}{3},\ \dfrac{y_1 + y_2 + y_3}{3}\right)$
Centroid divides a median: $AG : GD = 2 : 1$
Circumradius: $R = \dfrac{abc}{4\,\text{Area}}$
Inradius: $r = \dfrac{\text{Area}}{s}$
Incentre: $I = \dfrac{aA + bB + cC}{a + b + c}$
Orthocentre: $H = A + B + C - 2O$
Euler line: $O,\ G,\ H \text{ are collinear},\qquad OG : GH = 1 : 2$

Constructions and tangents Class 10

Perpendicular bisector: $PA = PB \ \text{for every point } P \text{ on it}$
Angle bisector (by SSS): $\triangle VDF \cong \triangle VEF \Rightarrow \angle DVF = \angle EVF$
Dividing a segment: $\dfrac{AP}{PB} = \dfrac{m}{n}$
Tangent and radius: $OT \perp PT$
Tangent length: $PT^2 = OP^2 - r^2$
Two tangents from a point: $PT_1 = PT_2$
Angle in a semicircle: $\angle OTP = 90^\circ \ \text{when } OP \text{ is a diameter}$

Circles & Conic Sections Open the lab →

Chords Class 9

Length of a chord at distance d from the centre: $AB = 2\sqrt{r^2 - d^2}$
Perpendicular from the centre: $OM \perp AB \ \Rightarrow\ AM = MB$
Equal chords: $AB = CD \iff OM = ON$
Longer chord, nearer the centre: $AB > CD \iff OM < ON$
The diameter is the longest chord: $d = 0 \ \Rightarrow\ AB = 2r$

Angles in a circle Class 9

Angle at the centre: $\angle AOB = 2\,\angle APB$
Same segment: $\angle APB = \angle AQB$
Opposite segments: $\angle APB + \angle AQB = 180^\circ$
Angle in a semicircle: $\angle APB = 90^\circ$
Converse (P then lies on the circle with diameter AB): $\angle APB = 90^\circ$

Cyclic quadrilaterals Class 9

Opposite angles: $\angle A + \angle C = 180^\circ,\quad \angle B + \angle D = 180^\circ$
Exterior angle: $\text{exterior angle at } C = \angle A$
Converse: $\angle A + \angle C = 180^\circ \Rightarrow ABCD \text{ is cyclic}$
Angle sum: $\angle A + \angle B + \angle C + \angle D = 360^\circ$

Tangents Class 10

Tangent and radius: $OT \perp PT$
Tangent length: $PT^2 = OP^2 - r^2$
Two tangents from a point: $PT_1 = PT_2$
Angle between the tangents: $\angle T_1PT_2 + \angle T_1OT_2 = 180^\circ$
Intersecting chords: $PA \cdot PA' = PB \cdot PB'$
Power of a point: $PA \cdot PA' = |OP^2 - r^2|$

Areas related to circles Class 10

Circumference and area: $C = 2\pi r,\qquad A = \pi r^2$
Arc length: $l = \dfrac{\theta}{360^\circ} \times 2\pi r$
Area of a sector: $\dfrac{\theta}{360^\circ} \times \pi r^2$
Area of a segment: $\text{sector} - \text{triangle} = \dfrac{r^2}{2}\,(\theta - \sin\theta)$
Chord of angle θ: $AB = 2r\sin\dfrac{\theta}{2}$
Perimeter of a sector: $l + 2r$

Circle and line Class 11

Standard form: $(x - h)^2 + (y - k)^2 = r^2$
General form: $x^2 + y^2 + 2gx + 2fy + c = 0$
Centre and radius: $(-g,\ -f),\qquad r = \sqrt{g^2 + f^2 - c}$
Distance from the centre to y = mx + c: $d = \dfrac{|mh - k + c|}{\sqrt{1 + m^2}}$
Secant, tangent, outside: $d < r,\quad d = r,\quad d > r$
Tangent condition: $(c + mh - k)^2 = r^2(1 + m^2)$
Two circles: $d > r_1 + r_2,\quad d = r_1 + r_2,\quad |r_1 - r_2| < d < r_1 + r_2,\ \dots$

Parabola Class 11

Standard forms: $y^2 = 4ax,\quad y^2 = -4ax,\quad x^2 = 4ay,\quad x^2 = -4ay$
Focus and directrix of y² = 4ax: $F = (a,\ 0),\qquad x = -a$
Definition: $PF = PM$
Latus rectum: $4a$
Eccentricity: $e = 1$
Tangent at (x₁, y₁): $y\,y_1 = 2a\,(x + x_1)$

Ellipse Class 11

Standard form: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1,\qquad a > b$
Distance to the foci: $c^2 = a^2 - b^2$
Eccentricity: $e = \dfrac{c}{a} < 1$
Foci and directrices: $(\pm ae,\ 0),\qquad x = \pm\dfrac{a}{e}$
Definition: $PF_1 + PF_2 = 2a$
Latus rectum: $\dfrac{2b^2}{a}$
Area: $\pi a b$

Hyperbola Class 11

Standard form: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$
Distance to the foci: $c^2 = a^2 + b^2$
Eccentricity: $e = \dfrac{c}{a} > 1$
Foci and directrices: $(\pm ae,\ 0),\qquad x = \pm\dfrac{a}{e}$
Definition: $|PF_1 - PF_2| = 2a$
Asymptotes: $y = \pm\dfrac{b}{a}\,x$
Latus rectum: $\dfrac{2b^2}{a}$

Eccentricity and the family Class 11

Focus–directrix definition: $PF = e \cdot PM$
Polar form (focus at the origin): $r = \dfrac{\ell}{1 + e\cos\theta}$
Kind of conic: $e = 0:\ \text{circle},\quad e < 1:\ \text{ellipse},\quad e = 1:\ \text{parabola},\quad e > 1:\ \text{hyperbola}$
Semi-major axis of an ellipse: $a = \dfrac{\ell}{1 - e^2}$
Semi-transverse axis of a hyperbola: $a = \dfrac{\ell}{e^2 - 1}$
Relation between a, b, e: $b^2 = a^2(1 - e^2)\ \text{(ellipse)},\qquad b^2 = a^2(e^2 - 1)\ \text{(hyperbola)}$

Calculus Lab Open the lab →

Limits Class 11

Definition: $\lim_{x \to a} f(x) = L$
The limit exists when: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$
Algebra of limits: $\lim (f \pm g) = \lim f \pm \lim g,\qquad \lim (fg) = \lim f \cdot \lim g$
Standard limits: $\lim_{x \to 0}\dfrac{\sin x}{x} = 1,\qquad \lim_{x \to 0}\dfrac{e^x - 1}{x} = 1$
More standard limits: $\lim_{x \to 0}\dfrac{\ln(1 + x)}{x} = 1,\qquad \lim_{x \to a}\dfrac{x^n - a^n}{x - a} = na^{n-1}$
The number e: $\lim_{n \to \infty}\left(1 + \dfrac{1}{n}\right)^n = e$

Derivative Class 11

First principles: $f'(x) = \lim_{h \to 0}\dfrac{f(x + h) - f(x)}{h}$
Sum and product: $(u \pm v)' = u' \pm v',\qquad (uv)' = u'v + uv'$
Quotient: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$
Chain rule: $\dfrac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)$
Power and exponential: $(x^n)' = nx^{n-1},\qquad (e^x)' = e^x,\qquad (\ln x)' = \dfrac{1}{x}$
Trigonometric: $(\sin x)' = \cos x,\quad (\cos x)' = -\sin x,\quad (\tan x)' = \sec^2 x$

Continuity and differentiability Class 12

Continuous at a: $\lim_{x \to a} f(x) = f(a)$
Differentiable at a: $\lim_{h \to 0^-}\dfrac{f(a + h) - f(a)}{h} = \lim_{h \to 0^+}\dfrac{f(a + h) - f(a)}{h}$
Differentiable implies continuous: $f'(a) \text{ exists} \Rightarrow f \text{ is continuous at } a$
The reverse is false (|x| is continuous at 0 but not differentiable): $f(x) = |x|$

Tangent and normal Class 12

Slope of the tangent: $m = f'(x_0)$
Tangent line: $y - y_0 = f'(x_0)\,(x - x_0)$
Normal line: $y - y_0 = -\dfrac{1}{f'(x_0)}\,(x - x_0)$
Rate of change of y with respect to x: $\dfrac{dy}{dx}$

Increasing, decreasing, maxima and minima Class 12

Increasing and decreasing: $f'(x) > 0 \Rightarrow \text{increasing},\qquad f'(x) < 0 \Rightarrow \text{decreasing}$
Stationary points: $f'(x) = 0$
First derivative test: $f' \text{ changes } + \to - \Rightarrow \max,\qquad - \to + \Rightarrow \min$
Second derivative test: $f'(c) = 0,\ f''(c) < 0 \Rightarrow \max;\quad f''(c) > 0 \Rightarrow \min$
Concavity and inflection: $f'' > 0:\ \text{concave up},\quad f'' < 0:\ \text{concave down}$
Absolute extrema on [a, b]: compare the ends and every stationary point c: $f(a),\ f(b),\ f(c)$

Optimisation Class 12

Fixed perimeter P: $A(x) = x\left(\dfrac{P}{2} - x\right),\qquad x = \dfrac{P}{4}$
Pen against a wall, fencing L: $A(x) = x(L - 2x),\qquad x = \dfrac{L}{4}$
Open box from a square sheet s: $V(x) = x(s - 2x)^2,\qquad x = \dfrac{s}{6}$
Can of volume V, least metal: $S(r) = 2\pi r^2 + \dfrac{2V}{r},\qquad r = \sqrt[3]{\dfrac{V}{2\pi}},\ h = 2r$
Rectangle in a semicircle of radius R: $A(x) = 2x\sqrt{R^2 - x^2},\qquad x = \dfrac{R}{\sqrt{2}}$
The four steps: $\text{write } f(x);\ f'(x) = 0;\ \text{check } f''(x);\ \text{state the answer}$

Integrals Class 12

Riemann sum: $\sum_{i=1}^{n} f(x_i)\,\Delta x,\qquad \Delta x = \dfrac{b - a}{n}$
Definite integral: $\int_a^b f(x)\,dx = \lim_{n \to \infty}\sum_{i=1}^{n} f(x_i)\,\Delta x$
Fundamental theorem: $\int_a^b f(x)\,dx = F(b) - F(a),\qquad F'(x) = f(x)$
Standard integrals: $\int x^n dx = \dfrac{x^{n+1}}{n + 1},\quad \int e^x dx = e^x,\quad \int \dfrac{dx}{x} = \ln x$
Trigonometric: $\int \sin x\,dx = -\cos x,\qquad \int \cos x\,dx = \sin x$
Properties: $\int_a^b f = -\int_b^a f,\qquad \int_a^b f = \int_a^c f + \int_c^b f$
Trapezoid rule: $\dfrac{\Delta x}{2}\,\bigl[f(x_0) + 2f(x_1) + \dots + f(x_n)\bigr]$

Differential equations Class 12

Order and degree: $\text{order} = \text{highest derivative},\qquad \text{degree} = \text{its power}$
Variable separable: $\dfrac{dy}{dx} = f(x)g(y) \Rightarrow \int \dfrac{dy}{g(y)} = \int f(x)\,dx$
Growth and decay: $\dfrac{dy}{dx} = ky \Rightarrow y = Ce^{kx}$
Linear first order: $\dfrac{dy}{dx} + Py = Q,\qquad \text{integrating factor } e^{\int P\,dx}$
Euler's method: $y_{n+1} = y_n + h\,f(x_n, y_n)$
Particular solution: $\text{use } y(x_0) = y_0 \text{ to find the constant } C$

Matrices & Transformations Open the lab →

Matrices: order and types Class 12

Order of a matrix: $m \times n$
Identity matrix: $AI = IA = A$
Symmetric: $A^T = A$
Skew-symmetric: $A^T = -A$

Matrix arithmetic Class 12

Sum (same order): $(A + B)_{ij} = a_{ij} + b_{ij}$
Scalar multiple: $(kA)_{ij} = k\,a_{ij}$
Product entry: $(AB)_{ij} = \sum_k a_{ik}b_{kj}$
Order of a product: $(m \times n)(n \times p) = m \times p$
Not commutative: $AB \ne BA \text{ in general}$

Transpose Class 12

Definition: $(A^T)_{ij} = a_{ji}$
Rules: $(A^T)^T = A,\qquad (A + B)^T = A^T + B^T$
Transpose of a product: $(AB)^T = B^TA^T$
Transpose of a scalar multiple: $(kA)^T = kA^T$

Determinant Class 12

2 × 2: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$
Expansion along row i: $\det A = \sum_j a_{ij}C_{ij}$
Cofactor: $C_{ij} = (-1)^{i+j} M_{ij}$
Product rule: $\det(AB) = \det A \cdot \det B$
Scalar multiple (n × n): $\det(kA) = k^n \det A$
Transpose: $\det(A^T) = \det A$

Row operations and the determinant Class 12

Swap two rows: $\det \to -\det$
Multiply a row by k: $\det \to k\det$
Add a multiple of a row to another: $\det \text{ unchanged}$
Two equal rows, or a row of zeros: $\det = 0$

Adjoint and inverse Class 12

Adjoint: $\operatorname{adj} A = (C_{ij})^T$
Key identity: $A\,(\operatorname{adj} A) = (\det A)\,I$
Inverse: $A^{-1} = \dfrac{\operatorname{adj} A}{\det A},\qquad \det A \ne 0$
2 × 2 inverse: $\dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
Inverse of a product: $(AB)^{-1} = B^{-1}A^{-1}$

Solving linear systems Class 12

Matrix form: $AX = B$
Matrix method (det A ≠ 0): $X = A^{-1}B$
Cramer's rule: $x = \dfrac{D_x}{D},\quad y = \dfrac{D_y}{D},\quad z = \dfrac{D_z}{D}$
One solution: $\text{rank } A = \text{rank } [A|B] = n$
No solution: $\text{rank } A \lt \text{rank } [A|B]$
Infinitely many: $\text{rank } A = \text{rank } [A|B] \lt n$

Matrices as transformations (beyond the core syllabus) Class 12

Image of a point: $\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$
Rotation by θ: $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
Area scale factor: $|\det M|$
Eigenvalues: $\lambda^2 - (\operatorname{tr} M)\lambda + \det M = 0$
Composition (B after A): $M = BA$

Maths Explainers Open the lab →

Number systems Class 9

A rational number: $\dfrac{p}{q},\qquad p, q \text{ integers},\ q \ne 0$
Terminating decimal: $q = 2^m \cdot 5^n \ \Rightarrow\ \dfrac{p}{q} \text{ terminates}$
Repeating decimal to a fraction: $0.\overline{ab} = \dfrac{ab}{99},\qquad 0.\overline{a} = \dfrac{a}{9}$
Rationalising: $\dfrac{1}{a + b\sqrt{c}} = \dfrac{a - b\sqrt{c}}{a^2 - b^2 c}$
Laws of surds: $\sqrt{a}\sqrt{b} = \sqrt{ab},\qquad (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$

Arithmetic progressions Class 10

nth term: $a_n = a + (n - 1)d$
Sum of n terms: $S_n = \dfrac{n}{2}\,[2a + (n - 1)d]$
Sum from the last term l: $S_n = \dfrac{n}{2}\,(a + l)$
A term from the sums: $a_n = S_n - S_{n-1}$
Three numbers in AP: $2b = a + c$

Complex numbers Class 11

The imaginary unit: $i^2 = -1,\qquad i^4 = 1$
Modulus and conjugate: $|z| = \sqrt{a^2 + b^2},\qquad z\bar{z} = |z|^2$
Polar form: $z = r(\cos\theta + i\sin\theta),\qquad r = |z|$
Product and quotient: $|z_1z_2| = |z_1||z_2|,\qquad \arg(z_1z_2) = \arg z_1 + \arg z_2$
Quotient by the conjugate: $\dfrac{z_1}{z_2} = \dfrac{z_1\bar{z_2}}{|z_2|^2}$
Triangle inequality: $|z_1 + z_2| \le |z_1| + |z_2|$

Roots of complex numbers Class 11

De Moivre's theorem: $[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$
The nth roots of z (k = 0, 1, …, n − 1): modulus and angle: $r^{1/n},\quad \dfrac{\theta + 2\pi k}{n}$
Roots of unity: $\omega = e^{2\pi i/n},\qquad 1 + \omega + \dots + \omega^{n-1} = 0$
Cube roots of unity: $\omega^3 = 1,\qquad 1 + \omega + \omega^2 = 0$

Counting Class 11

Multiplication principle: $m \times n \times p \text{ ways}$
Permutations: ${}^nP_r = \dfrac{n!}{(n - r)!}$
Combinations: ${}^nC_r = \dfrac{n!}{r!\,(n - r)!}$
Link between them: ${}^nP_r = {}^nC_r \cdot r!,\qquad {}^nC_r = {}^nC_{n-r}$
Pascal's rule: ${}^nC_r = {}^{n-1}C_{r-1} + {}^{n-1}C_r$
Arrangements with repeats: $\dfrac{n!}{p!\,q!\,r!}$
In a circle: $(n - 1)!$

Binomial theorem Class 11

Expansion: $(a + b)^n = \sum_{r=0}^{n} {}^nC_r\, a^{n-r}b^r$
General term: $T_{r+1} = {}^nC_r\, a^{n-r}b^r$
Number of terms: $n + 1$
Sum of the coefficients: ${}^nC_0 + {}^nC_1 + \dots + {}^nC_n = 2^n$
Alternating sum: ${}^nC_0 - {}^nC_1 + {}^nC_2 - \dots = 0$

Geometric progressions and means Class 11

nth term: $a_n = ar^{n-1}$
Sum of n terms: $S_n = \dfrac{a(1 - r^n)}{1 - r},\qquad r \ne 1$
Sum to infinity: $S_\infty = \dfrac{a}{1 - r},\qquad |r| \lt 1$
The three means: $AM = \dfrac{a + b}{2},\quad GM = \sqrt{ab},\quad HM = \dfrac{2ab}{a + b}$
Order and link: $AM \ge GM \ge HM,\qquad GM^2 = AM \cdot HM$

Standard sums Class 11

Natural numbers: $1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}$
Squares: $1^2 + 2^2 + \dots + n^2 = \dfrac{n(n + 1)(2n + 1)}{6}$
Cubes: $1^3 + 2^3 + \dots + n^3 = \left[\dfrac{n(n + 1)}{2}\right]^2$
Fibonacci numbers: $F_n = F_{n-1} + F_{n-2},\qquad F_1 + \dots + F_n = F_{n+2} - 1$

Sets Class 11

Two sets: $|A \cup B| = |A| + |B| - |A \cap B|$
Three sets: $|A \cup B \cup C| = \sum |A| - \sum |A \cap B| + |A \cap B \cap C|$
De Morgan's laws: $(A \cup B)' = A' \cap B',\qquad (A \cap B)' = A' \cup B'$
Difference and complement: $A - B = A \cap B',\qquad A' = U - A$
Number of subsets: $|P(A)| = 2^n$

Linear programming Class 12

Objective function: $Z = ax + by$
Constraints: $a_1x + b_1y \le c_1,\quad a_2x + b_2y \ge c_2,\quad x, y \ge 0$
Corner point theorem: the best Z is at: $\text{a corner point}$
Two optimal corners: the optimum is: $\text{the whole edge}$
Unbounded region: check that: $Z \gt M \text{ meets it}$

Mensuration & 3D Shapes Open the lab →

Area of a triangle Class 9

Base and height: $A = \tfrac{1}{2}\,b\,h$
Semi-perimeter: $s = \dfrac{a + b + c}{2}$
Heron's formula: $A = \sqrt{s(s - a)(s - b)(s - c)}$
Equilateral triangle of side a: $A = \dfrac{\sqrt{3}}{4}\,a^2$
Isosceles (equal sides a, base b): $A = \dfrac{b}{4}\sqrt{4a^2 - b^2}$

Quadrilaterals Class 9

Rectangle and square: $A = lb,\qquad A = a^2$
Parallelogram: $A = b \times h$
Trapezium: $A = \tfrac{1}{2}(a + b)\,h$
Rhombus: $A = \tfrac{1}{2}\,d_1 d_2,\qquad \text{side} = \sqrt{\left(\tfrac{d_1}{2}\right)^2 + \left(\tfrac{d_2}{2}\right)^2}$
Regular polygon: $A = \tfrac{1}{2} \times \text{perimeter} \times \text{apothem}$

Cuboid and cube Class 9

Volume of a cuboid: $V = l \times b \times h$
Lateral surface area: $2h(l + b)$
Total surface area: $2(lb + bh + hl)$
Space diagonal: $\sqrt{l^2 + b^2 + h^2}$
Cube of edge a: $V = a^3,\quad \text{TSA} = 6a^2,\quad d = a\sqrt{3}$

Cylinder Class 9

Volume: $V = \pi r^2 h$
Curved surface area: $\text{CSA} = 2\pi r h$
Total surface area: $\text{TSA} = 2\pi r(r + h)$
Hollow cylinder: $V = \pi h\,(R^2 - r^2)$

Cone Class 9

Slant height: $l = \sqrt{r^2 + h^2}$
Volume: $V = \tfrac{1}{3}\pi r^2 h$
Curved surface area: $\text{CSA} = \pi r l$
Total surface area: $\text{TSA} = \pi r(r + l)$
The unrolled sector: $\theta = \dfrac{360^\circ \times r}{l}$

Sphere and hemisphere Class 9

Sphere: $V = \tfrac{4}{3}\pi r^3,\qquad S = 4\pi r^2$
Hemisphere volume: $V = \tfrac{2}{3}\pi r^3$
Hemisphere surface areas: $\text{CSA} = 2\pi r^2,\qquad \text{TSA} = 3\pi r^2$
Spherical shell: $V = \tfrac{4}{3}\pi(R^3 - r^3)$
Cone : sphere : cylinder (r, 2r): $1 : 2 : 3$

Combinations of solids Class 10

Volume: $V = V_1 + V_2 + \dots$
Surface area: $\text{visible surfaces only}$
Ice-cream cone (cone + hemisphere): $V = \tfrac{1}{3}\pi r^2 h + \tfrac{2}{3}\pi r^3,\quad S = \pi r l + 2\pi r^2$
Capsule (cylinder + 2 hemispheres): $V = \pi r^2 h + \tfrac{4}{3}\pi r^3,\quad S = 2\pi r h + 4\pi r^2$

Converting one solid into another Class 10

Volume is unchanged: $V_{\text{old}} = V_{\text{new}}$
Number of small solids: $n = \dfrac{V_{\text{big}}}{V_{\text{small}}}$
Sphere into spheres: $n = \left(\dfrac{R}{r}\right)^3$
Surface area: $\text{changes: } n \text{ small solids have more}$

Frustum of a cone Class 10

Slant height: $l = \sqrt{h^2 + (R - r)^2}$
Volume: $V = \tfrac{1}{3}\pi h\,(R^2 + Rr + r^2)$
Curved surface area: $\text{CSA} = \pi(R + r)\,l$
Total surface area: $\text{TSA} = \pi(R + r)\,l + \pi R^2 + \pi r^2$
Big cone minus small cone: $V = \tfrac{1}{3}\pi R^2 H - \tfrac{1}{3}\pi r^2 (H - h)$

Vectors & 3D Geometry Open the lab →

Points in three dimensions Class 11

Distance between A and B (Δx = x₂ − x₁, and so on): $AB = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$
Distance from the origin: $OP = \sqrt{x^2 + y^2 + z^2}$
Midpoint (the x-coordinate; y and z are the same): $x = \dfrac{x_1 + x_2}{2}$
Internal section m : n (the x-coordinate): $x = \dfrac{mx_2 + nx_1}{m + n}$
External section m : n: $\dfrac{m\,\vec{b} - n\,\vec{a}}{m - n}$

Vectors Class 12

Magnitude: $|\vec{a}| = \sqrt{x^2 + y^2 + z^2}$
Unit vector: $\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}$
Direction cosines: $l = \dfrac{x}{|\vec{a}|},\quad m = \dfrac{y}{|\vec{a}|},\quad n = \dfrac{z}{|\vec{a}|}$
The identity: $l^2 + m^2 + n^2 = 1$
Vector from A to B: $\overrightarrow{AB} = \vec{b} - \vec{a}$
Triangle inequality: $|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|$

Dot product Class 12

Definition: $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$
By components: $\vec{a}\cdot\vec{b} = x_1x_2 + y_1y_2 + z_1z_2$
Angle between vectors: $\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$
Projection of a on b: $\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$
Perpendicular: $\vec{a}\cdot\vec{b} = 0$

Cross product Class 12

Magnitude: $|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$
By components: $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$
Area of a parallelogram: $|\vec{a}\times\vec{b}|$
Area of a triangle: $\tfrac{1}{2}\,|\vec{a}\times\vec{b}|$
Order matters: $\vec{a}\times\vec{b} = -\,\vec{b}\times\vec{a}$
Parallel vectors: $\vec{a}\times\vec{b} = \vec{0}$

Scalar triple product Class 12

Definition: $[\vec{a}\ \vec{b}\ \vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c})$
Cyclic: $[\vec{a}\ \vec{b}\ \vec{c}] = [\vec{b}\ \vec{c}\ \vec{a}]$
Volume of a parallelepiped: $V = |[\vec{a}\ \vec{b}\ \vec{c}]|$
Volume of a tetrahedron: $V = \tfrac{1}{6}\,|[\vec{a}\ \vec{b}\ \vec{c}]|$
Coplanar vectors: $[\vec{a}\ \vec{b}\ \vec{c}] = 0$

Lines in space Class 12

Vector form: $\vec{r} = \vec{a} + \lambda\vec{b}$
Cartesian form: $\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}$
Through two points: $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$
Angle between two lines: $\cos\theta = \dfrac{|\vec{b}_1\cdot\vec{b}_2|}{|\vec{b}_1||\vec{b}_2|}$
Perpendicular and parallel: $\vec{b}_1\cdot\vec{b}_2 = 0,\qquad \vec{b}_1\times\vec{b}_2 = \vec{0}$

Shortest distance between lines Class 12

Skew lines: $d = \dfrac{|(\vec{a}_2 - \vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2)|}{|\vec{b}_1\times\vec{b}_2|}$
Parallel lines: $d = \dfrac{|(\vec{a}_2 - \vec{a}_1)\times\vec{b}|}{|\vec{b}|}$
Lines that meet: $(\vec{a}_2 - \vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2) = 0$
Parallel directions: $\vec{b}_1\times\vec{b}_2 = \vec{0}$

Planes Class 12

Vector form: $\vec{r}\cdot\vec{n} = d$
Cartesian form: $ax + by + cz = d$
Through three points: $\vec{n} = (\vec{b} - \vec{a})\times(\vec{c} - \vec{a})$
Intercept form: $\dfrac{x}{p} + \dfrac{y}{q} + \dfrac{z}{r} = 1$
Distance of a point from a plane: $\dfrac{|\vec{n}\cdot\vec{Q} - d|}{|\vec{n}|}$

Angles with planes Class 12

Line and plane: $\sin\theta = \dfrac{|\vec{n}\cdot\vec{b}|}{|\vec{n}||\vec{b}|}$
Two planes: $\cos\theta = \dfrac{|\vec{n}_1\cdot\vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$
Line of intersection: $\text{direction } \vec{n}_1\times\vec{n}_2$
Perpendicular planes: $\vec{n}_1\cdot\vec{n}_2 = 0$

Vedic Math Shortcuts Open the lab →

Multiplication shortcuts

Multiply by 11 (2-digit): $\overline{a\,b} \times 11 = \overline{a\ (a+b)\ b}$
Same leading digit, units add to 10: $(10t+u)\bigl(10t+(10-u)\bigr) = 100\,t(t+1) + u(10-u)$
Nikhilam: two numbers near a base B: $(B+d_1)(B+d_2) = B(B+d_1+d_2) + d_1d_2$
Multiply by 9, 99, 999, …: $n \times (10^k - 1) = n\times 10^k - n$
Multiply by 5, 25, 50, 125: $n\times 5 = \dfrac{n}{2}\times 10,\qquad n\times 25 = \dfrac{n}{4}\times 100,\qquad n \times 125 = \dfrac{n}{8}\times 1000$
Vertically and crosswise (2-digit × 2-digit): $(10a+b)(10c+d) = 100ac + 10(ad+bc) + bd$

Squaring shortcuts

Square a number ending in 5: $(10a+5)^2 = 100\,a(a+1) + 25$
Square any number (base method): $n^2 = (n+d)(n-d) + d^2$

Divisibility checks

Divisible by 3 or 9: $n \bmod 3,\ n \bmod 9 \;=\; (\text{digit sum}) \bmod 3,\ (\text{digit sum}) \bmod 9$
Divisible by 11: $n \bmod 11 = (\text{alternating digit sum}) \bmod 11$

Maths formula sheet

Function & Graph Explorer Open the lab →

Coordinate geometry and the graph of a line Class 9

Pair of linear equations and zeros of a quadratic Class 10

Polynomials: zeros and the shape of the graph Class 10

Functions, symmetry and coordinate geometry Class 11

One-one functions, inverses and a first look at slope Class 12

Trigonometry & Height-Distance Open the lab →

Trigonometric ratios of an acute angle Class 10

Trigonometric identities Class 10

Heights and distances Class 10

The unit circle, radians and signs Class 11

Graphs of sin, cos and tan Class 11

Statistics & Probability Open the lab →

Experimental probability Class 9

Probability of an event Class 10

Mean, median and mode Class 9

Grouped data Class 10

Spread: quartiles, variance and standard deviation Class 11

Conditional probability and independent events Class 12

Binomial distribution Class 12

Normal distribution Class 12

Geometry & Constructions Open the lab →

Lines and angles Class 9

Triangles Class 9

Congruence of triangles Class 9

Quadrilaterals Class 9

Similar triangles Class 10

Pythagoras theorem Class 10

Triangle centres Class 10

Constructions and tangents Class 10

Circles & Conic Sections Open the lab →

Chords Class 9

Angles in a circle Class 9

Cyclic quadrilaterals Class 9

Tangents Class 10

Areas related to circles Class 10

Circle and line Class 11

Parabola Class 11

Ellipse Class 11

Hyperbola Class 11

Eccentricity and the family Class 11

Calculus Lab Open the lab →

Limits Class 11

Derivative Class 11

Continuity and differentiability Class 12

Tangent and normal Class 12

Increasing, decreasing, maxima and minima Class 12

Optimisation Class 12

Integrals Class 12

Differential equations Class 12

Matrices & Transformations Open the lab →

Matrices: order and types Class 12

Matrix arithmetic Class 12

Transpose Class 12

Determinant Class 12

Row operations and the determinant Class 12

Adjoint and inverse Class 12

Solving linear systems Class 12

Matrices as transformations (beyond the core syllabus) Class 12

Maths Explainers Open the lab →

Number systems Class 9

Arithmetic progressions Class 10

Complex numbers Class 11

Roots of complex numbers Class 11

Counting Class 11

Binomial theorem Class 11

Geometric progressions and means Class 11

Standard sums Class 11

Sets Class 11

Linear programming Class 12

Mensuration & 3D Shapes Open the lab →

Area of a triangle Class 9

Quadrilaterals Class 9

Cuboid and cube Class 9

Cylinder Class 9

Cone Class 9

Sphere and hemisphere Class 9

Combinations of solids Class 10

Converting one solid into another Class 10