Shunya / Formula Sheet

Maths formula sheet

Every formula from every Shunya lab, on one page: 468 formulas in 83 groups. Filter by class, search by name or topic, and jump into the lab to see a formula in use.

Function & Graph Explorer Open the lab →

Coordinate geometry and the graph of a line Class 9

Equation of a line
y=mx+cy = mx + c
Slope (rise ÷ run)
m=y2−y1x2−x1m = \dfrac{y_2 - y_1}{x_2 - x_1}
y-intercept (put x = 0)
(0, c)(0,\ c)
x-intercept (put y = 0)
x=−cmx = -\dfrac{c}{m}
Angle the line makes with the x-axis
θ=tan⁡−1m\theta = \tan^{-1} m

Pair of linear equations and zeros of a quadratic Class 10

Pair of linear equations
a1x+b1y=c1,a2x+b2y=c2a_1x + b_1y = c_1,\qquad a_2x + b_2y = c_2
Determinants (Cramer's rule)
D=a1b2−a2b1,Dx=c1b2−c2b1,Dy=a1c2−a2c1D = 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=DxD,y=DyDx = \dfrac{D_x}{D},\qquad y = \dfrac{D_y}{D}
One solution (lines cross)
a1a2≠b1b2\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}
No solution (parallel lines)
a1a2=b1b2≠c1c2\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}
Infinitely many (same line)
a1a2=b1b2=c1c2\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}
Quadratic
y=ax2+bx+cy = ax^2 + bx + c
Discriminant
D=b2−4acD = b^2 - 4ac
Zeros (D > 0)
x=−b±D2ax = \dfrac{-b \pm \sqrt{D}}{2a}
Repeated zero (D = 0)
x=−b2ax = -\dfrac{b}{2a}
Complex roots (D < 0)
x=−b±i−D2ax = \dfrac{-b \pm i\sqrt{-D}}{2a}
Sum and product of the zeros
α+β=−ba,αβ=ca\alpha + \beta = -\dfrac{b}{a},\qquad \alpha\beta = \dfrac{c}{a}
Vertex form
y=a(x−h)2+k,h=−b2a, k=c−b24ay = a(x - h)^2 + k,\quad h = -\dfrac{b}{2a},\ k = c - \dfrac{b^2}{4a}
Factored form
y=a(x−α)(x−β)y = a(x - \alpha)(x - \beta)

Polynomials: zeros and the shape of the graph Class 10

Cubic
y=ax3+bx2+cx+dy = ax^3 + bx^2 + cx + d
Sum of the zeros
α+β+γ=−ba\alpha + \beta + \gamma = -\dfrac{b}{a}
Sum of products of pairs
αβ+βγ+γα=ca\alpha\beta + \beta\gamma + \gamma\alpha = \dfrac{c}{a}
Product of the zeros
αβγ=−da\alpha\beta\gamma = -\dfrac{d}{a}
Polynomial built from its zeros
y=a(x−α)(x−β)(x−γ)⋯y = a(x - \alpha)(x - \beta)(x - \gamma)\cdots
Number of real zeros of a degree-n polynomial
≤n\le n

Functions, symmetry and coordinate geometry Class 11

Even function (symmetric about the y-axis)
f(−x)=f(x)f(-x) = f(x)
Odd function (half-turn symmetry about the origin)
f(−x)=−f(x)f(-x) = -f(x)
Stretched sine wave
y=asin⁡(bx)+cy = a\sin(bx) + c
Amplitude, period, vertical lift
∣a∣,2π∣b∣,c|a|,\qquad \dfrac{2\pi}{|b|},\qquad c
Where a function is undefined
1x: x=0,tan⁡x: x=π2+nπ\dfrac{1}{x}:\ x = 0, \qquad \tan x:\ x = \dfrac{\pi}{2} + n\pi
Distance between two points
AB=(x2−x1)2+(y2−y1)2AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Midpoint
M=(x1+x22, y1+y22)M = \left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)
Section formula (internal, m : n)
P=(mx2+nx1m+n, my2+ny1m+n)P = \left(\dfrac{mx_2 + nx_1}{m + n},\ \dfrac{my_2 + ny_1}{m + n}\right)
Area of a triangle
12 ∣ x1(y2−y3)+x2(y3−y1)+x3(y1−y2) ∣\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(f(x))=xf^{-1}\bigl(f(x)\bigr) = x
Reflection in y = x
(a, b) ⟷ (b, a)(a,\ b) \ \longleftrightarrow\ (b,\ a)
One-one test
f(x1)=f(x2)⇒x1=x2f(x_1) = f(x_2) \Rightarrow x_1 = x_2
Derivative (slope of the tangent)
f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}
Tangent line at x₀
y−f(x0)=f′(x0) (x−x0)y - f(x_0) = f'(x_0)\,(x - x_0)
Power rule
ddx xn=nxn−1\dfrac{d}{dx}\,x^n = n x^{n-1}
Sum, product and quotient rules
(u±v)′=u′±v′,(uv)′=u′v+uv′,(uv)′=u′v−uv′v2(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
ddx f(g(x))=f′(g(x))⋅g′(x)\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⁡2x, (ex)′=ex, (ln⁡x)′=1x(\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⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\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
cosec⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\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⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta},\qquad \cot\theta = \dfrac{\cos\theta}{\sin\theta}
Pythagoras theorem
opposite2+adjacent2=hypotenuse2\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2
Sides from the hypotenuse h
opposite=hsin⁡θ,adjacent=hcos⁡θ\text{opposite} = h\sin\theta,\qquad \text{adjacent} = h\cos\theta
Complementary angles
sin⁡(90∘−θ)=cos⁡θ,cos⁡(90∘−θ)=sin⁡θ,tan⁡(90∘−θ)=cot⁡θ\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, 12, 12, 32, 1cos⁡: 1, 32, 12, 12, 0tan⁡: 0, 13, 1, 3, not defined\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⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
Tan and sec
1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta
Cot and cosec
1+cot⁡2θ=cosec⁡2θ1 + \cot^2\theta = \operatorname{cosec}^2\theta
Tan as a quotient
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta}
Complementary angles
sin⁡(90∘−θ)=cos⁡θ,tan⁡(90∘−θ)=cot⁡θ\sin(90^\circ - \theta) = \cos\theta,\qquad \tan(90^\circ - \theta) = \cot\theta

Heights and distances Class 10

Angle from the sides
tan⁡θ=Hd ⇒ θ=tan⁡−1Hd\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=dtan⁡θ+eH = d\tan\theta + e
Distance from height and angle
d=Htan⁡θd = \dfrac{H}{\tan\theta}
When a string or a ladder is involved
sin⁡θ=heightlength,cos⁡θ=distancelength\sin\theta = \dfrac{\text{height}}{\text{length}},\quad \cos\theta = \dfrac{\text{distance}}{\text{length}}
Length from the height and angle
length=heightsin⁡θ\text{length} = \dfrac{\text{height}}{\sin\theta}
Check with Pythagoras
length=d2+H2\text{length} = \sqrt{d^2 + H^2}
Angle of depression = angle of elevation (alternate angles)
θdepression=θelevation\theta_{\text{depression}} = \theta_{\text{elevation}}
Two observation points, distance s apart
H=s tan⁡θ1tan⁡θ2tan⁡θ2−tan⁡θ1H = \dfrac{s\,\tan\theta_1\tan\theta_2}{\tan\theta_2 - \tan\theta_1}
Tower above a building
tower=Hbuilding+dtan⁡(elevation)\text{tower} = H_{\text{building}} + d\tan(\text{elevation})
Handy values for 30°, 45°, 60°
tan⁡30∘=13,tan⁡45∘=1,tan⁡60∘=3\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⁡θ, sin⁡θ),x2+y2=1P = (\cos\theta,\ \sin\theta),\qquad x^2 + y^2 = 1
Radians and degrees
π rad=180∘,θrad=θdeg×π180\pi\ \text{rad} = 180^\circ,\qquad \theta_{\text{rad}} = \theta_{\text{deg}} \times \dfrac{\pi}{180}
Tan from the coordinates
tan⁡θ=sin⁡θcos⁡θ=yx\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{y}{x}
Reference angle, by quadrant
I: θII: 180∘−θIII: θ−180∘IV: 360∘−θ\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
I: all,II: sin⁡\text{I: all},\qquad \text{II: } \sin
Positive ratios in quadrants III and IV
III: tan⁡,IV: cos⁡\text{III: } \tan,\qquad \text{IV: } \cos
Size from the reference angle
∣sin⁡θ∣=sin⁡(ref. angle),∣cos⁡θ∣=cos⁡(ref. angle)|\sin\theta| = \sin(\text{ref. angle}),\quad |\cos\theta| = \cos(\text{ref. angle})
Repeating every turn
sin⁡(θ+360∘)=sin⁡θ,tan⁡(θ+180∘)=tan⁡θ\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(B(x−C))+Dy = A\,f\bigl(B(x - C)\bigr) + D
Amplitude
∣A∣|A|
Period of sin and cos
2π∣B∣(360∘∣B∣)\dfrac{2\pi}{|B|}\quad\left(\dfrac{360^\circ}{|B|}\right)
Period of tan
π∣B∣(180∘∣B∣)\dfrac{\pi}{|B|}\quad\left(\dfrac{180^\circ}{|B|}\right)
Phase shift and midline
shift right by C,midline y=D\text{shift right by } C,\qquad \text{midline } y = D
Range of sin and cos
−1≤sin⁡θ≤1,−1≤cos⁡θ≤1-1 \le \sin\theta \le 1,\qquad -1 \le \cos\theta \le 1
Where tan has asymptotes
cos⁡θ=0 ⇒ θ=90∘+180∘n\cos\theta = 0\ \Rightarrow\ \theta = 90^\circ + 180^\circ n
Cos is sin shifted
cos⁡θ=sin⁡(θ+90∘)\cos\theta = \sin(\theta + 90^\circ)

Statistics & Probability Open the lab →

Experimental probability Class 9

Experimental probability
P(E)=trials in which E happenedtotal number of trialsP(E) = \dfrac{\text{trials in which } E \text{ happened}}{\text{total number of trials}}
Expected number of times in n trials
n×P(E)n \times P(E)
Difference from the theoretical value
∣ experimental P−theoretical P ∣\bigl|\,\text{experimental } P - \text{theoretical } P\,\bigr|
Law of large numbers
experimental P(E)⟶theoretical P(E) as n grows\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)=number of favourable outcomestotal number of outcomesP(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}
Range of a probability
0≤P(E)≤1,P(impossible)=0,P(certain)=10 \le P(E) \le 1,\qquad P(\text{impossible}) = 0,\qquad P(\text{certain}) = 1
Complement
P(not E)=1−P(E)P(\text{not } E) = 1 - P(E)
All outcomes together
∑P(outcome)=1\sum P(\text{outcome}) = 1
Two dice, sum s (out of 36 pairs)
P(sum=s)=6−∣s−7∣36,2≤s≤12P(\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 H)=q2,P(1 H)=2pq,P(2 H)=p2,q=1−pP(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
xˉ=∑xn\bar{x} = \dfrac{\sum x}{n}
Median, n odd (the middle value of the sorted list)
(n+12)th value\left(\dfrac{n + 1}{2}\right)\text{th value}
Median, n even
(n2)th value+(n2+1)th value2\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)
most frequent value\text{most frequent value}
Range
largest−smallest\text{largest} - \text{smallest}

Grouped data Class 10

Class mark and class width
x=l+u2,h=u−lx = \dfrac{l + u}{2},\qquad h = u - l
Mean (direct method)
xˉ=∑fx∑f\bar{x} = \dfrac{\sum f x}{\sum f}
Median
Median=l+n2−cff×h\text{Median} = l + \dfrac{\frac{n}{2} - cf}{f} \times h
Mode
Mode=l+f1−f02f1−f0−f2×h\text{Mode} = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h
Empirical relation
Mode≈3 Median−2 Mean\text{Mode} \approx 3\,\text{Median} - 2\,\text{Mean}
Median class (the first class with cumulative frequency at least n/2)
cf≥n2cf \ge \dfrac{n}{2}

Spread: quartiles, variance and standard deviation Class 11

First quartile (median of the lower half)
Q1=median(lower half)Q_1 = \text{median}(\text{lower half})
Third quartile (median of the upper half)
Q3=median(upper half)Q_3 = \text{median}(\text{upper half})
Interquartile range
IQR=Q3−Q1\text{IQR} = Q_3 - Q_1
Variance
σ2=∑(x−xˉ)2n\sigma^2 = \dfrac{\sum (x - \bar{x})^2}{n}
Standard deviation
σ=σ2\sigma = \sqrt{\sigma^2}

Conditional probability and independent events Class 12

Multiplication rule
P(A∩B)=P(A) P(B∣A)P(A \cap B) = P(A)\,P(B \mid A)
Conditional probability
P(B∣A)=P(A∩B)P(A)P(B \mid A) = \dfrac{P(A \cap B)}{P(A)}
Independent events
P(A∩B)=P(A) P(B)P(A \cap B) = P(A)\,P(B)
Two draws with replacement
P(both red)=rN×rNP(\text{both red}) = \dfrac{r}{N}\times\dfrac{r}{N}
Two draws without replacement
P(both red)=rN×r−1N−1P(\text{both red}) = \dfrac{r}{N}\times\dfrac{r - 1}{N - 1}
Same colour, and its complement
P(same)=P(RR)+P(BB)+P(GG),P(different)=1−P(same)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)=(nk)pkqn−k,q=1−pP(X = k) = \binom{n}{k} p^k q^{n-k},\qquad q = 1 - p
Binomial coefficient
(nk)=n!k! (n−k)!\binom{n}{k} = \dfrac{n!}{k!\,(n - k)!}
At most k successes
P(X≤k)=∑i=0kP(X=i)P(X \le k) = \sum_{i=0}^{k} P(X = i)
At least k successes
P(X≥k)=1−P(X≤k−1)P(X \ge k) = 1 - P(X \le k - 1)
Mean, variance and standard deviation
μ=np,σ2=npq,σ=npq\mu = np,\qquad \sigma^2 = npq,\qquad \sigma = \sqrt{npq}
Most likely value
⌊(n+1)p⌋\lfloor (n + 1)p \rfloor
Normal curve fits well when
np≥5  and  nq≥5np \ge 5 \ \text{ and } \ nq \ge 5

Normal distribution Class 12

Density of the normal curve
f(x)=1σ2π e−(x−μ)22σ2f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2\sigma^2}}
Standardising (z-score)
z=x−μσz = \dfrac{x - \mu}{\sigma}
Probability between a and b
P(a<X<b)=Φ(zb)−Φ(za)P(a < X < b) = \Phi(z_b) - \Phi(z_a)
The 68 – 95 – 99.7 rule
P(∣X−μ∣<σ)≈0.6827,P(∣X−μ∣<2σ)≈0.9545,P(∣X−μ∣<3σ)≈0.9973P(|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
∫−∞∞f(x) dx=1\int_{-\infty}^{\infty} f(x)\,dx = 1

Geometry & Constructions Open the lab →

Lines and angles Class 9

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

Triangles Class 9

Angle sum
∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^\circ
Exterior angle
∠ACD=∠A+∠B\angle ACD = \angle A + \angle B
Distance between two points
AB=(x2−x1)2+(y2−y1)2AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Angle from the sides (law of cosines)
cos⁡A=b2+c2−a22bc\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}
Area from the coordinates
12 ∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\tfrac12\,\bigl|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\bigr|
Heron's formula
Area=s(s−a)(s−b)(s−c),s=a+b+c2\text{Area} = \sqrt{s(s - a)(s - b)(s - c)},\qquad s = \dfrac{a + b + c}{2}
Triangle inequality
a+b>ca + b > c
Larger angle, longer side
∠A>∠B  ⟺  a>b\angle A > \angle B \iff a > b

Congruence of triangles Class 9

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

Quadrilaterals Class 9

Angle sum of a quadrilateral
∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^\circ
Parallelogram
AB∥DC, BC∥AD ⇒ AB=DC, BC=ADAB \parallel DC,\ BC \parallel AD \ \Rightarrow\ AB = DC,\ BC = AD
Diagonals of a parallelogram
they bisect each other\text{they bisect each other}
Rhombus (a parallelogram with all sides equal)
AC⊥BDAC \perp BD
Rectangle (a parallelogram with a right angle)
AC=BDAC = BD
Midpoint theorem
DE∥BC,DE=12BCDE \parallel BC,\quad DE = \tfrac12 BC
Midpoint quadrilateral (always a parallelogram)
area=12 [ABCD]\text{area} = \tfrac12\,[ABCD]

Similar triangles Class 10

Basic proportionality theorem
DE∥BC ⇒ ADDB=AEECDE \parallel BC \ \Rightarrow\ \dfrac{AD}{DB} = \dfrac{AE}{EC}
Ratio of the sides
ADAB=AEAC=DEBC=k\dfrac{AD}{AB} = \dfrac{AE}{AC} = \dfrac{DE}{BC} = k
Perimeters
perimeter of ADEperimeter of ABC=k\dfrac{\text{perimeter of } ADE}{\text{perimeter of } ABC} = k
Areas
area of ADEarea of ABC=k2\dfrac{\text{area of } ADE}{\text{area of } ABC} = k^2
AA similarity
∠A=∠P, ∠B=∠Q ⇒ △ABC∼△PQR\angle A = \angle P,\ \angle B = \angle Q \ \Rightarrow\ \triangle ABC \sim \triangle PQR

Pythagoras theorem Class 10

Pythagoras
a2+b2=c2a^2 + b^2 = c^2
Converse
a2+b2=c2 ⇒ ∠C=90∘a^2 + b^2 = c^2 \ \Rightarrow\ \angle C = 90^\circ
Acute triangle
a2+b2>c2a^2 + b^2 > c^2
Obtuse triangle
a2+b2<c2a^2 + b^2 < c^2
Pythagorean triples
3-4-5,5-12-13,8-15-173\text{-}4\text{-}5,\quad 5\text{-}12\text{-}13,\quad 8\text{-}15\text{-}17

Triangle centres Class 10

Centroid
G=(x1+x2+x33, y1+y2+y33)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=2:1AG : GD = 2 : 1
Circumradius
R=abc4 AreaR = \dfrac{abc}{4\,\text{Area}}
Inradius
r=Areasr = \dfrac{\text{Area}}{s}
Incentre
I=aA+bB+cCa+b+cI = \dfrac{aA + bB + cC}{a + b + c}
Orthocentre
H=A+B+C−2OH = A + B + C - 2O
Euler line
O, G, H are collinear,OG:GH=1:2O,\ G,\ H \text{ are collinear},\qquad OG : GH = 1 : 2

Constructions and tangents Class 10

Perpendicular bisector
PA=PB for every point P on itPA = PB \ \text{for every point } P \text{ on it}
Angle bisector (by SSS)
△VDF≅△VEF⇒∠DVF=∠EVF\triangle VDF \cong \triangle VEF \Rightarrow \angle DVF = \angle EVF
Dividing a segment
APPB=mn\dfrac{AP}{PB} = \dfrac{m}{n}
Tangent and radius
OT⊥PTOT \perp PT
Tangent length
PT2=OP2−r2PT^2 = OP^2 - r^2
Two tangents from a point
PT1=PT2PT_1 = PT_2
Angle in a semicircle
∠OTP=90∘ when OP is a diameter\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=2r2−d2AB = 2\sqrt{r^2 - d^2}
Perpendicular from the centre
OM⊥AB ⇒ AM=MBOM \perp AB \ \Rightarrow\ AM = MB
Equal chords
AB=CD  ⟺  OM=ONAB = CD \iff OM = ON
Longer chord, nearer the centre
AB>CD  ⟺  OM<ONAB > CD \iff OM < ON
The diameter is the longest chord
d=0 ⇒ AB=2rd = 0 \ \Rightarrow\ AB = 2r

Angles in a circle Class 9

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

Cyclic quadrilaterals Class 9

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

Tangents Class 10

Tangent and radius
OT⊥PTOT \perp PT
Tangent length
PT2=OP2−r2PT^2 = OP^2 - r^2
Two tangents from a point
PT1=PT2PT_1 = PT_2
Angle between the tangents
∠T1PT2+∠T1OT2=180∘\angle T_1PT_2 + \angle T_1OT_2 = 180^\circ
Intersecting chords
PA⋅PA′=PB⋅PB′PA \cdot PA' = PB \cdot PB'
Power of a point
PA⋅PA′=∣OP2−r2∣PA \cdot PA' = |OP^2 - r^2|

Areas related to circles Class 10

Circumference and area
C=2πr,A=πr2C = 2\pi r,\qquad A = \pi r^2
Arc length
l=θ360∘×2πrl = \dfrac{\theta}{360^\circ} \times 2\pi r
Area of a sector
θ360∘×πr2\dfrac{\theta}{360^\circ} \times \pi r^2
Area of a segment
sector−triangle=r22 (θ−sin⁡θ)\text{sector} - \text{triangle} = \dfrac{r^2}{2}\,(\theta - \sin\theta)
Chord of angle θ
AB=2rsin⁡θ2AB = 2r\sin\dfrac{\theta}{2}
Perimeter of a sector
l+2rl + 2r

Circle and line Class 11

Standard form
(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2
General form
x2+y2+2gx+2fy+c=0x^2 + y^2 + 2gx + 2fy + c = 0
Centre and radius
(−g, −f),r=g2+f2−c(-g,\ -f),\qquad r = \sqrt{g^2 + f^2 - c}
Distance from the centre to y = mx + c
d=∣mh−k+c∣1+m2d = \dfrac{|mh - k + c|}{\sqrt{1 + m^2}}
Secant, tangent, outside
d<r,d=r,d>rd < r,\quad d = r,\quad d > r
Tangent condition
(c+mh−k)2=r2(1+m2)(c + mh - k)^2 = r^2(1 + m^2)
Two circles
d>r1+r2,d=r1+r2,∣r1−r2∣<d<r1+r2, …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
y2=4ax,y2=−4ax,x2=4ay,x2=−4ayy^2 = 4ax,\quad y^2 = -4ax,\quad x^2 = 4ay,\quad x^2 = -4ay
Focus and directrix of y² = 4ax
F=(a, 0),x=−aF = (a,\ 0),\qquad x = -a
Definition
PF=PMPF = PM
Latus rectum
4a4a
Eccentricity
e=1e = 1
Tangent at (x₁, y₁)
y y1=2a (x+x1)y\,y_1 = 2a\,(x + x_1)

Ellipse Class 11

Standard form
x2a2+y2b2=1,a>b\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1,\qquad a > b
Distance to the foci
c2=a2−b2c^2 = a^2 - b^2
Eccentricity
e=ca<1e = \dfrac{c}{a} < 1
Foci and directrices
(±ae, 0),x=±ae(\pm ae,\ 0),\qquad x = \pm\dfrac{a}{e}
Definition
PF1+PF2=2aPF_1 + PF_2 = 2a
Latus rectum
2b2a\dfrac{2b^2}{a}
Area
πab\pi a b

Hyperbola Class 11

Standard form
x2a2−y2b2=1\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1
Distance to the foci
c2=a2+b2c^2 = a^2 + b^2
Eccentricity
e=ca>1e = \dfrac{c}{a} > 1
Foci and directrices
(±ae, 0),x=±ae(\pm ae,\ 0),\qquad x = \pm\dfrac{a}{e}
Definition
∣PF1−PF2∣=2a|PF_1 - PF_2| = 2a
Asymptotes
y=±ba xy = \pm\dfrac{b}{a}\,x
Latus rectum
2b2a\dfrac{2b^2}{a}

Eccentricity and the family Class 11

Focus–directrix definition
PF=e⋅PMPF = e \cdot PM
Polar form (focus at the origin)
r=ℓ1+ecos⁡θr = \dfrac{\ell}{1 + e\cos\theta}
Kind of conic
e=0: circle,e<1: ellipse,e=1: parabola,e>1: hyperbolae = 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=ℓ1−e2a = \dfrac{\ell}{1 - e^2}
Semi-transverse axis of a hyperbola
a=ℓe2−1a = \dfrac{\ell}{e^2 - 1}
Relation between a, b, e
b2=a2(1−e2) (ellipse),b2=a2(e2−1) (hyperbola)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→af(x)=L\lim_{x \to a} f(x) = L
The limit exists when
lim⁡x→a−f(x)=lim⁡x→a+f(x)\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)
Algebra of limits
lim⁡(f±g)=lim⁡f±lim⁡g,lim⁡(fg)=lim⁡f⋅lim⁡g\lim (f \pm g) = \lim f \pm \lim g,\qquad \lim (fg) = \lim f \cdot \lim g
Standard limits
lim⁡x→0sin⁡xx=1,lim⁡x→0ex−1x=1\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→0ln⁡(1+x)x=1,lim⁡x→axn−anx−a=nan−1\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→∞(1+1n)n=e\lim_{n \to \infty}\left(1 + \dfrac{1}{n}\right)^n = e

Derivative Class 11

First principles
f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0}\dfrac{f(x + h) - f(x)}{h}
Sum and product
(u±v)′=u′±v′,(uv)′=u′v+uv′(u \pm v)' = u' \pm v',\qquad (uv)' = u'v + uv'
Quotient
(uv)′=u′v−uv′v2\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}
Chain rule
ddxf(g(x))=f′(g(x)) g′(x)\dfrac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)
Power and exponential
(xn)′=nxn−1,(ex)′=ex,(ln⁡x)′=1x(x^n)' = nx^{n-1},\qquad (e^x)' = e^x,\qquad (\ln x)' = \dfrac{1}{x}
Trigonometric
(sin⁡x)′=cos⁡x,(cos⁡x)′=−sin⁡x,(tan⁡x)′=sec⁡2x(\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→af(x)=f(a)\lim_{x \to a} f(x) = f(a)
Differentiable at a
lim⁡h→0−f(a+h)−f(a)h=lim⁡h→0+f(a+h)−f(a)h\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) exists⇒f is continuous at af'(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∣f(x) = |x|

Tangent and normal Class 12

Slope of the tangent
m=f′(x0)m = f'(x_0)
Tangent line
y−y0=f′(x0) (x−x0)y - y_0 = f'(x_0)\,(x - x_0)
Normal line
y−y0=−1f′(x0) (x−x0)y - y_0 = -\dfrac{1}{f'(x_0)}\,(x - x_0)
Rate of change of y with respect to x
dydx\dfrac{dy}{dx}

Increasing, decreasing, maxima and minima Class 12

Increasing and decreasing
f′(x)>0⇒increasing,f′(x)<0⇒decreasingf'(x) > 0 \Rightarrow \text{increasing},\qquad f'(x) < 0 \Rightarrow \text{decreasing}
Stationary points
f′(x)=0f'(x) = 0
First derivative test
f′ changes +→−⇒max⁡,−→+⇒min⁡f' \text{ changes } + \to - \Rightarrow \max,\qquad - \to + \Rightarrow \min
Second derivative test
f′(c)=0, f′′(c)<0⇒max⁡;f′′(c)>0⇒min⁡f'(c) = 0,\ f''(c) < 0 \Rightarrow \max;\quad f''(c) > 0 \Rightarrow \min
Concavity and inflection
f′′>0: concave up,f′′<0: concave downf'' > 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)f(a),\ f(b),\ f(c)

Optimisation Class 12

Fixed perimeter P
A(x)=x(P2−x),x=P4A(x) = x\left(\dfrac{P}{2} - x\right),\qquad x = \dfrac{P}{4}
Pen against a wall, fencing L
A(x)=x(L−2x),x=L4A(x) = x(L - 2x),\qquad x = \dfrac{L}{4}
Open box from a square sheet s
V(x)=x(s−2x)2,x=s6V(x) = x(s - 2x)^2,\qquad x = \dfrac{s}{6}
Can of volume V, least metal
S(r)=2πr2+2Vr,r=V2π3, h=2rS(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)=2xR2−x2,x=R2A(x) = 2x\sqrt{R^2 - x^2},\qquad x = \dfrac{R}{\sqrt{2}}
The four steps
write f(x); f′(x)=0; check f′′(x); state the answer\text{write } f(x);\ f'(x) = 0;\ \text{check } f''(x);\ \text{state the answer}

Integrals Class 12

Riemann sum
∑i=1nf(xi) Δx,Δx=b−an\sum_{i=1}^{n} f(x_i)\,\Delta x,\qquad \Delta x = \dfrac{b - a}{n}
Definite integral
∫abf(x) dx=lim⁡n→∞∑i=1nf(xi) Δx\int_a^b f(x)\,dx = \lim_{n \to \infty}\sum_{i=1}^{n} f(x_i)\,\Delta x
Fundamental theorem
∫abf(x) dx=F(b)−F(a),F′(x)=f(x)\int_a^b f(x)\,dx = F(b) - F(a),\qquad F'(x) = f(x)
Standard integrals
∫xndx=xn+1n+1,∫exdx=ex,∫dxx=ln⁡x\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
∫sin⁡x dx=−cos⁡x,∫cos⁡x dx=sin⁡x\int \sin x\,dx = -\cos x,\qquad \int \cos x\,dx = \sin x
Properties
∫abf=−∫baf,∫abf=∫acf+∫cbf\int_a^b f = -\int_b^a f,\qquad \int_a^b f = \int_a^c f + \int_c^b f
Trapezoid rule
Δx2 [f(x0)+2f(x1)+⋯+f(xn)]\dfrac{\Delta x}{2}\,\bigl[f(x_0) + 2f(x_1) + \dots + f(x_n)\bigr]

Differential equations Class 12

Order and degree
order=highest derivative,degree=its power\text{order} = \text{highest derivative},\qquad \text{degree} = \text{its power}
Variable separable
dydx=f(x)g(y)⇒∫dyg(y)=∫f(x) dx\dfrac{dy}{dx} = f(x)g(y) \Rightarrow \int \dfrac{dy}{g(y)} = \int f(x)\,dx
Growth and decay
dydx=ky⇒y=Cekx\dfrac{dy}{dx} = ky \Rightarrow y = Ce^{kx}
Linear first order
dydx+Py=Q,integrating factor e∫P dx\dfrac{dy}{dx} + Py = Q,\qquad \text{integrating factor } e^{\int P\,dx}
Euler's method
yn+1=yn+h f(xn,yn)y_{n+1} = y_n + h\,f(x_n, y_n)
Particular solution
use y(x0)=y0 to find the constant C\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×nm \times n
Identity matrix
AI=IA=AAI = IA = A
Symmetric
AT=AA^T = A
Skew-symmetric
AT=−AA^T = -A

Matrix arithmetic Class 12

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

Transpose Class 12

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

Determinant Class 12

2 × 2
∣abcd∣=ad−bc\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc
Expansion along row i
det⁡A=∑jaijCij\det A = \sum_j a_{ij}C_{ij}
Cofactor
Cij=(−1)i+jMijC_{ij} = (-1)^{i+j} M_{ij}
Product rule
det⁡(AB)=det⁡A⋅det⁡B\det(AB) = \det A \cdot \det B
Scalar multiple (n × n)
det⁡(kA)=kndet⁡A\det(kA) = k^n \det A
Transpose
det⁡(AT)=det⁡A\det(A^T) = \det A

Row operations and the determinant Class 12

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

Adjoint and inverse Class 12

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

Solving linear systems Class 12

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

Matrices as transformations (beyond the core syllabus) Class 12

Image of a point
(x′y′)=(abcd)(xy)\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}
Rotation by θ
(cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}
Area scale factor
∣det⁡M∣|\det M|
Eigenvalues
λ2−(tr⁡M)λ+det⁡M=0\lambda^2 - (\operatorname{tr} M)\lambda + \det M = 0
Composition (B after A)
M=BAM = BA

Maths Explainers Open the lab →

Number systems Class 9

A rational number
pq,p,q integers, q≠0\dfrac{p}{q},\qquad p, q \text{ integers},\ q \ne 0
Terminating decimal
q=2m⋅5n ⇒ pq terminatesq = 2^m \cdot 5^n \ \Rightarrow\ \dfrac{p}{q} \text{ terminates}
Repeating decimal to a fraction
0.ab‾=ab99,0.a‾=a90.\overline{ab} = \dfrac{ab}{99},\qquad 0.\overline{a} = \dfrac{a}{9}
Rationalising
1a+bc=a−bca2−b2c\dfrac{1}{a + b\sqrt{c}} = \dfrac{a - b\sqrt{c}}{a^2 - b^2 c}
Laws of surds
ab=ab,(a+b)(a−b)=a−b\sqrt{a}\sqrt{b} = \sqrt{ab},\qquad (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b

Arithmetic progressions Class 10

nth term
an=a+(n−1)da_n = a + (n - 1)d
Sum of n terms
Sn=n2 [2a+(n−1)d]S_n = \dfrac{n}{2}\,[2a + (n - 1)d]
Sum from the last term l
Sn=n2 (a+l)S_n = \dfrac{n}{2}\,(a + l)
A term from the sums
an=Sn−Sn−1a_n = S_n - S_{n-1}
Three numbers in AP
2b=a+c2b = a + c

Complex numbers Class 11

The imaginary unit
i2=−1,i4=1i^2 = -1,\qquad i^4 = 1
Modulus and conjugate
∣z∣=a2+b2,zzˉ=∣z∣2|z| = \sqrt{a^2 + b^2},\qquad z\bar{z} = |z|^2
Polar form
z=r(cos⁡θ+isin⁡θ),r=∣z∣z = r(\cos\theta + i\sin\theta),\qquad r = |z|
Product and quotient
∣z1z2∣=∣z1∣∣z2∣,arg⁡(z1z2)=arg⁡z1+arg⁡z2|z_1z_2| = |z_1||z_2|,\qquad \arg(z_1z_2) = \arg z_1 + \arg z_2
Quotient by the conjugate
z1z2=z1z2ˉ∣z2∣2\dfrac{z_1}{z_2} = \dfrac{z_1\bar{z_2}}{|z_2|^2}
Triangle inequality
∣z1+z2∣≤∣z1∣+∣z2∣|z_1 + z_2| \le |z_1| + |z_2|

Roots of complex numbers Class 11

De Moivre's theorem
[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)[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
r1/n,θ+2πknr^{1/n},\quad \dfrac{\theta + 2\pi k}{n}
Roots of unity
ω=e2πi/n,1+ω+⋯+ωn−1=0\omega = e^{2\pi i/n},\qquad 1 + \omega + \dots + \omega^{n-1} = 0
Cube roots of unity
ω3=1,1+ω+ω2=0\omega^3 = 1,\qquad 1 + \omega + \omega^2 = 0

Counting Class 11

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

Binomial theorem Class 11

Expansion
(a+b)n=∑r=0nnCr an−rbr(a + b)^n = \sum_{r=0}^{n} {}^nC_r\, a^{n-r}b^r
General term
Tr+1=nCr an−rbrT_{r+1} = {}^nC_r\, a^{n-r}b^r
Number of terms
n+1n + 1
Sum of the coefficients
nC0+nC1+⋯+nCn=2n{}^nC_0 + {}^nC_1 + \dots + {}^nC_n = 2^n
Alternating sum
nC0−nC1+nC2−⋯=0{}^nC_0 - {}^nC_1 + {}^nC_2 - \dots = 0

Geometric progressions and means Class 11

nth term
an=arn−1a_n = ar^{n-1}
Sum of n terms
Sn=a(1−rn)1−r,r≠1S_n = \dfrac{a(1 - r^n)}{1 - r},\qquad r \ne 1
Sum to infinity
S∞=a1−r,∣r∣<1S_\infty = \dfrac{a}{1 - r},\qquad |r| \lt 1
The three means
AM=a+b2,GM=ab,HM=2aba+bAM = \dfrac{a + b}{2},\quad GM = \sqrt{ab},\quad HM = \dfrac{2ab}{a + b}
Order and link
AM≥GM≥HM,GM2=AM⋅HMAM \ge GM \ge HM,\qquad GM^2 = AM \cdot HM

Standard sums Class 11

Natural numbers
1+2+⋯+n=n(n+1)21 + 2 + \dots + n = \dfrac{n(n + 1)}{2}
Squares
12+22+⋯+n2=n(n+1)(2n+1)61^2 + 2^2 + \dots + n^2 = \dfrac{n(n + 1)(2n + 1)}{6}
Cubes
13+23+⋯+n3=[n(n+1)2]21^3 + 2^3 + \dots + n^3 = \left[\dfrac{n(n + 1)}{2}\right]^2
Fibonacci numbers
Fn=Fn−1+Fn−2,F1+⋯+Fn=Fn+2−1F_n = F_{n-1} + F_{n-2},\qquad F_1 + \dots + F_n = F_{n+2} - 1

Sets Class 11

Two sets
∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B|
Three sets
∣A∪B∪C∣=∑∣A∣−∑∣A∩B∣+∣A∩B∩C∣|A \cup B \cup C| = \sum |A| - \sum |A \cap B| + |A \cap B \cap C|
De Morgan's laws
(A∪B)′=A′∩B′,(A∩B)′=A′∪B′(A \cup B)' = A' \cap B',\qquad (A \cap B)' = A' \cup B'
Difference and complement
A−B=A∩B′,A′=U−AA - B = A \cap B',\qquad A' = U - A
Number of subsets
∣P(A)∣=2n|P(A)| = 2^n

Linear programming Class 12

Objective function
Z=ax+byZ = ax + by
Constraints
a1x+b1y≤c1,a2x+b2y≥c2,x,y≥0a_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
a corner point\text{a corner point}
Two optimal corners: the optimum is
the whole edge\text{the whole edge}
Unbounded region: check that
Z>M meets itZ \gt M \text{ meets it}

Mensuration & 3D Shapes Open the lab →

Area of a triangle Class 9

Base and height
A=12 b hA = \tfrac{1}{2}\,b\,h
Semi-perimeter
s=a+b+c2s = \dfrac{a + b + c}{2}
Heron's formula
A=s(s−a)(s−b)(s−c)A = \sqrt{s(s - a)(s - b)(s - c)}
Equilateral triangle of side a
A=34 a2A = \dfrac{\sqrt{3}}{4}\,a^2
Isosceles (equal sides a, base b)
A=b44a2−b2A = \dfrac{b}{4}\sqrt{4a^2 - b^2}

Quadrilaterals Class 9

Rectangle and square
A=lb,A=a2A = lb,\qquad A = a^2
Parallelogram
A=b×hA = b \times h
Trapezium
A=12(a+b) hA = \tfrac{1}{2}(a + b)\,h
Rhombus
A=12 d1d2,side=(d12)2+(d22)2A = \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=12×perimeter×apothemA = \tfrac{1}{2} \times \text{perimeter} \times \text{apothem}

Cuboid and cube Class 9

Volume of a cuboid
V=l×b×hV = l \times b \times h
Lateral surface area
2h(l+b)2h(l + b)
Total surface area
2(lb+bh+hl)2(lb + bh + hl)
Space diagonal
l2+b2+h2\sqrt{l^2 + b^2 + h^2}
Cube of edge a
V=a3,TSA=6a2,d=a3V = a^3,\quad \text{TSA} = 6a^2,\quad d = a\sqrt{3}

Cylinder Class 9

Volume
V=πr2hV = \pi r^2 h
Curved surface area
CSA=2πrh\text{CSA} = 2\pi r h
Total surface area
TSA=2πr(r+h)\text{TSA} = 2\pi r(r + h)
Hollow cylinder
V=πh (R2−r2)V = \pi h\,(R^2 - r^2)

Cone Class 9

Slant height
l=r2+h2l = \sqrt{r^2 + h^2}
Volume
V=13πr2hV = \tfrac{1}{3}\pi r^2 h
Curved surface area
CSA=πrl\text{CSA} = \pi r l
Total surface area
TSA=πr(r+l)\text{TSA} = \pi r(r + l)
The unrolled sector
θ=360∘×rl\theta = \dfrac{360^\circ \times r}{l}

Sphere and hemisphere Class 9

Sphere
V=43πr3,S=4πr2V = \tfrac{4}{3}\pi r^3,\qquad S = 4\pi r^2
Hemisphere volume
V=23πr3V = \tfrac{2}{3}\pi r^3
Hemisphere surface areas
CSA=2πr2,TSA=3πr2\text{CSA} = 2\pi r^2,\qquad \text{TSA} = 3\pi r^2
Spherical shell
V=43π(R3−r3)V = \tfrac{4}{3}\pi(R^3 - r^3)
Cone : sphere : cylinder (r, 2r)
1:2:31 : 2 : 3

Combinations of solids Class 10

Volume
V=V1+V2+…V = V_1 + V_2 + \dots
Surface area
visible surfaces only\text{visible surfaces only}
Ice-cream cone (cone + hemisphere)
V=13πr2h+23πr3,S=πrl+2πr2V = \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=πr2h+43πr3,S=2πrh+4πr2V = \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
Vold=VnewV_{\text{old}} = V_{\text{new}}
Number of small solids
n=VbigVsmalln = \dfrac{V_{\text{big}}}{V_{\text{small}}}
Sphere into spheres
n=(Rr)3n = \left(\dfrac{R}{r}\right)^3
Surface area
changes: n small solids have more\text{changes: } n \text{ small solids have more}

Frustum of a cone Class 10

Slant height
l=h2+(R−r)2l = \sqrt{h^2 + (R - r)^2}
Volume
V=13πh (R2+Rr+r2)V = \tfrac{1}{3}\pi h\,(R^2 + Rr + r^2)
Curved surface area
CSA=π(R+r) l\text{CSA} = \pi(R + r)\,l
Total surface area
TSA=π(R+r) l+πR2+πr2\text{TSA} = \pi(R + r)\,l + \pi R^2 + \pi r^2
Big cone minus small cone
V=13πR2H−13πr2(H−h)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=Δx2+Δy2+Δz2AB = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}
Distance from the origin
OP=x2+y2+z2OP = \sqrt{x^2 + y^2 + z^2}
Midpoint (the x-coordinate; y and z are the same)
x=x1+x22x = \dfrac{x_1 + x_2}{2}
Internal section m : n (the x-coordinate)
x=mx2+nx1m+nx = \dfrac{mx_2 + nx_1}{m + n}
External section m : n
m b⃗−n a⃗m−n\dfrac{m\,\vec{b} - n\,\vec{a}}{m - n}

Vectors Class 12

Magnitude
∣a⃗∣=x2+y2+z2|\vec{a}| = \sqrt{x^2 + y^2 + z^2}
Unit vector
a^=a⃗∣a⃗∣\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}
Direction cosines
l=x∣a⃗∣,m=y∣a⃗∣,n=z∣a⃗∣l = \dfrac{x}{|\vec{a}|},\quad m = \dfrac{y}{|\vec{a}|},\quad n = \dfrac{z}{|\vec{a}|}
The identity
l2+m2+n2=1l^2 + m^2 + n^2 = 1
Vector from A to B
AB→=b⃗−a⃗\overrightarrow{AB} = \vec{b} - \vec{a}
Triangle inequality
∣a⃗+b⃗∣≤∣a⃗∣+∣b⃗∣|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|

Dot product Class 12

Definition
a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta
By components
a⃗⋅b⃗=x1x2+y1y2+z1z2\vec{a}\cdot\vec{b} = x_1x_2 + y_1y_2 + z_1z_2
Angle between vectors
cos⁡θ=a⃗⋅b⃗∣a⃗∣∣b⃗∣\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}
Projection of a on b
a⃗⋅b⃗∣b⃗∣\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}
Perpendicular
a⃗⋅b⃗=0\vec{a}\cdot\vec{b} = 0

Cross product Class 12

Magnitude
∣a⃗×b⃗∣=∣a⃗∣∣b⃗∣sin⁡θ|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta
By components
∣i^j^k^a1a2a3b1b2b3∣\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
∣a⃗×b⃗∣|\vec{a}\times\vec{b}|
Area of a triangle
12 ∣a⃗×b⃗∣\tfrac{1}{2}\,|\vec{a}\times\vec{b}|
Order matters
a⃗×b⃗=− b⃗×a⃗\vec{a}\times\vec{b} = -\,\vec{b}\times\vec{a}
Parallel vectors
a⃗×b⃗=0⃗\vec{a}\times\vec{b} = \vec{0}

Scalar triple product Class 12

Definition
[a⃗ b⃗ c⃗]=a⃗⋅(b⃗×c⃗)[\vec{a}\ \vec{b}\ \vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c})
Cyclic
[a⃗ b⃗ c⃗]=[b⃗ c⃗ a⃗][\vec{a}\ \vec{b}\ \vec{c}] = [\vec{b}\ \vec{c}\ \vec{a}]
Volume of a parallelepiped
V=∣[a⃗ b⃗ c⃗]∣V = |[\vec{a}\ \vec{b}\ \vec{c}]|
Volume of a tetrahedron
V=16 ∣[a⃗ b⃗ c⃗]∣V = \tfrac{1}{6}\,|[\vec{a}\ \vec{b}\ \vec{c}]|
Coplanar vectors
[a⃗ b⃗ c⃗]=0[\vec{a}\ \vec{b}\ \vec{c}] = 0

Lines in space Class 12

Vector form
r⃗=a⃗+λb⃗\vec{r} = \vec{a} + \lambda\vec{b}
Cartesian form
x−x1a=y−y1b=z−z1c\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}
Through two points
r⃗=a⃗+λ(b⃗−a⃗)\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})
Angle between two lines
cos⁡θ=∣b⃗1⋅b⃗2∣∣b⃗1∣∣b⃗2∣\cos\theta = \dfrac{|\vec{b}_1\cdot\vec{b}_2|}{|\vec{b}_1||\vec{b}_2|}
Perpendicular and parallel
b⃗1⋅b⃗2=0,b⃗1×b⃗2=0⃗\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=∣(a⃗2−a⃗1)⋅(b⃗1×b⃗2)∣∣b⃗1×b⃗2∣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=∣(a⃗2−a⃗1)×b⃗∣∣b⃗∣d = \dfrac{|(\vec{a}_2 - \vec{a}_1)\times\vec{b}|}{|\vec{b}|}
Lines that meet
(a⃗2−a⃗1)⋅(b⃗1×b⃗2)=0(\vec{a}_2 - \vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2) = 0
Parallel directions
b⃗1×b⃗2=0⃗\vec{b}_1\times\vec{b}_2 = \vec{0}

Planes Class 12

Vector form
r⃗⋅n⃗=d\vec{r}\cdot\vec{n} = d
Cartesian form
ax+by+cz=dax + by + cz = d
Through three points
n⃗=(b⃗−a⃗)×(c⃗−a⃗)\vec{n} = (\vec{b} - \vec{a})\times(\vec{c} - \vec{a})
Intercept form
xp+yq+zr=1\dfrac{x}{p} + \dfrac{y}{q} + \dfrac{z}{r} = 1
Distance of a point from a plane
∣n⃗⋅Q⃗−d∣∣n⃗∣\dfrac{|\vec{n}\cdot\vec{Q} - d|}{|\vec{n}|}

Angles with planes Class 12

Line and plane
sin⁡θ=∣n⃗⋅b⃗∣∣n⃗∣∣b⃗∣\sin\theta = \dfrac{|\vec{n}\cdot\vec{b}|}{|\vec{n}||\vec{b}|}
Two planes
cos⁡θ=∣n⃗1⋅n⃗2∣∣n⃗1∣∣n⃗2∣\cos\theta = \dfrac{|\vec{n}_1\cdot\vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}
Line of intersection
direction n⃗1×n⃗2\text{direction } \vec{n}_1\times\vec{n}_2
Perpendicular planes
n⃗1⋅n⃗2=0\vec{n}_1\cdot\vec{n}_2 = 0

Vedic Math Shortcuts Open the lab →

Multiplication shortcuts

Multiply by 11 (2-digit)
a b‾×11=a (a+b) b‾\overline{a\,b} \times 11 = \overline{a\ (a+b)\ b}
Same leading digit, units add to 10
(10t+u)(10t+(10−u))=100 t(t+1)+u(10−u)(10t+u)\bigl(10t+(10-u)\bigr) = 100\,t(t+1) + u(10-u)
Nikhilam: two numbers near a base B
(B+d1)(B+d2)=B(B+d1+d2)+d1d2(B+d_1)(B+d_2) = B(B+d_1+d_2) + d_1d_2
Multiply by 9, 99, 999, …
n×(10k−1)=n×10k−nn \times (10^k - 1) = n\times 10^k - n
Multiply by 5, 25, 50, 125
n×5=n2×10,n×25=n4×100,n×125=n8×1000n\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(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(10a+5)^2 = 100\,a(a+1) + 25
Square any number (base method)
n2=(n+d)(n−d)+d2n^2 = (n+d)(n-d) + d^2

Divisibility checks

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

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