1. ARITHMETIC

Number System

Natural numbers: N={1,2,3,… }N=\{1,2,3,\dots\}
Whole numbers: W={0,1,2,3,… }W=\{0,1,2,3,\dots\}
Integers: Z={…,−2,−1,0,1,2,… }Z=\{\dots,-2,-1,0,1,2,\dots\}
Rational number:

pq,q≠0\frac pq,\quad q\neq0

HCF and LCM

For two positive integers a,ba,b:

HCF×LCM=a×b\boxed{HCF\times LCM=a\times b}

Euclid's division lemma:

a=bq+r,0≤r<b\boxed{a=bq+r,\quad 0\le r<b}

Fractions

ab+cd=ad+bcbd\frac ab+\frac cd=\frac{ad+bc}{bd}ab−cd=ad−bcbd\frac ab-\frac cd=\frac{ad-bc}{bd}ab×cd=acbd\frac ab\times\frac cd=\frac{ac}{bd}ab÷cd=ab×dc=adbc\frac ab\div\frac cd=\frac ab\times\frac dc=\frac{ad}{bc}

Reciprocal:

ab→ba\frac ab\rightarrow\frac ba

2. PERCENTAGE

Percentage=PartWhole×100\boxed{\text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times100}x% of y=x100×yx\%\text{ of }y=\frac{x}{100}\times y

Percentage increase:

% Increase=IncreaseOriginal×100\boxed{\%\text{ Increase}= \frac{\text{Increase}}{\text{Original}}\times100}

Percentage decrease:

% Decrease=DecreaseOriginal×100\boxed{\%\text{ Decrease}= \frac{\text{Decrease}}{\text{Original}}\times100}

New value after x%x\% increase:

New=Original(1+x100)\boxed{\text{New}=\text{Original}\left(1+\frac{x}{100}\right)}

After x%x\% decrease:

New=Original(1−x100)\boxed{\text{New}=\text{Original}\left(1-\frac{x}{100}\right)}

3. PROFIT AND LOSS

Profit=SP−CP\boxed{\text{Profit}=SP-CP}Loss=CP−SP\boxed{\text{Loss}=CP-SP}Profit%=ProfitCP×100\boxed{\text{Profit\%}=\frac{\text{Profit}}{CP}\times100}Loss%=LossCP×100\boxed{\text{Loss\%}=\frac{\text{Loss}}{CP}\times100}

Selling price with profit:

SP=CP(100+Profit%100)\boxed{SP=CP\left(\frac{100+\text{Profit\%}}{100}\right)}

Selling price with loss:

SP=CP(100−Loss%100)\boxed{SP=CP\left(\frac{100-\text{Loss\%}}{100}\right)}

Cost price:

CP=SP×100100+Profit%CP=\frac{SP\times100}{100+\text{Profit\%}}

or

CP=SP×100100−Loss%CP=\frac{SP\times100}{100-\text{Loss\%}}

4. DISCOUNT

Discount=MP−SP\boxed{\text{Discount}=MP-SP}Discount%=DiscountMP×100\boxed{\text{Discount\%}= \frac{\text{Discount}}{MP}\times100}SP=MP(1−d100)\boxed{SP=MP\left(1-\frac{d}{100}\right)}

5. SIMPLE INTEREST

SI=PRT100\boxed{SI=\frac{PRT}{100}}

where:

PP = Principal
RR = Rate %
TT = Time

Amount:

A=P+SI\boxed{A=P+SI}

6. COMPOUND INTEREST

Annual compounding:

A=P(1+R100)n\boxed{A=P\left(1+\frac{R}{100}\right)^n}CI=A−P\boxed{CI=A-P}

Half-yearly:

A=P(1+R/2100)2nA=P\left(1+\frac{R/2}{100}\right)^{2n}

Quarterly:

A=P(1+R/4100)4nA=P\left(1+\frac{R/4}{100}\right)^{4n}

Depreciation:

A=P(1−R100)n\boxed{A=P\left(1-\frac{R}{100}\right)^n}

7. RATIO AND PROPORTION

a:b=aba:b=\frac ab

If

ab=cd\frac ab=\frac cd

then

ad=bc\boxed{ad=bc}

Continued proportion:

a:b=b:ca:b=b:c

then

b2=ac\boxed{b^2=ac}

Direct proportion:

y∝xy\propto xy=kxy=kx

Inverse proportion:

y∝1xy\propto\frac1xxy=kxy=k

8. AVERAGE

Average=Sum of observationsNumber of observations\boxed{\text{Average}= \frac{\text{Sum of observations}} {\text{Number of observations}}}

Sum:

Sum=Average×Number\boxed{\text{Sum}=\text{Average}\times\text{Number}}

Weighted average:

xˉ=∑wx∑w\boxed{ \bar{x}= \frac{\sum wx}{\sum w} }

9. TIME, SPEED AND DISTANCE

Speed=DistanceTime\boxed{\text{Speed}=\frac{\text{Distance}}{\text{Time}}}Distance=Speed×Time\boxed{\text{Distance}=\text{Speed}\times\text{Time}}Time=DistanceSpeed\boxed{\text{Time}=\frac{\text{Distance}}{\text{Speed}}}

Average speed:

Average Speed=Total DistanceTotal Time\boxed{\text{Average Speed}= \frac{\text{Total Distance}}{\text{Total Time}}}

Conversion:

1 m/s=185 km/h1\text{ m/s}=\frac{18}{5}\text{ km/h}1 km/h=518 m/s1\text{ km/h}=\frac{5}{18}\text{ m/s}

10. BOATS AND STREAMS

Let:

bb = speed of boat in still water
ss = speed of stream

Downstream:

D=b+s\boxed{D=b+s}

Upstream:

U=b−s\boxed{U=b-s}

Boat speed:

b=D+U2\boxed{b=\frac{D+U}{2}}

Stream speed:

s=D−U2\boxed{s=\frac{D-U}{2}}

11. TIME AND WORK

If A completes work in xx days:

A’s one-day work=1x\boxed{\text{A's one-day work}=\frac1x}

If B takes yy days:

1y\frac1y

Together:

1T=1x+1y\boxed{\frac1T=\frac1x+\frac1y}

So:

T=xyx+y\boxed{T=\frac{xy}{x+y}}

Work:

Work=Efficiency×Time\boxed{\text{Work}=\text{Efficiency}\times\text{Time}}

12. ALGEBRA — BASIC IDENTITIES

(a+b)2=a2+2ab+b2\boxed{(a+b)^2=a^2+2ab+b^2}(a−b)2=a2−2ab+b2\boxed{(a-b)^2=a^2-2ab+b^2}(a+b)(a−b)=a2−b2\boxed{(a+b)(a-b)=a^2-b^2}(x+a)(x+b)=x2+(a+b)x+ab\boxed{(x+a)(x+b)=x^2+(a+b)x+ab}(x−a)(x−b)=x2−(a+b)x+ab\boxed{(x-a)(x-b)=x^2-(a+b)x+ab}(a+b+c)2=a2+b2+c2+2ab+2bc+2ca\boxed{(a+b+c)^2 =a^2+b^2+c^2+2ab+2bc+2ca}

13. CUBE IDENTITIES

(a+b)3=a3+3a2b+3ab2+b3\boxed{(a+b)^3=a^3+3a^2b+3ab^2+b^3}(a−b)3=a3−3a2b+3ab2−b3\boxed{(a-b)^3=a^3-3a^2b+3ab^2-b^3}a3+b3=(a+b)(a2−ab+b2)\boxed{a^3+b^3=(a+b)(a^2-ab+b^2)}a3−b3=(a−b)(a2+ab+b2)\boxed{a^3-b^3=(a-b)(a^2+ab+b^2)}a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−bc−ca)\boxed{ a^3+b^3+c^3-3abc =(a+b+c)(a^2+b^2+c^2-ab-bc-ca) }

If

a+b+c=0a+b+c=0

then

a3+b3+c3=3abc\boxed{a^3+b^3+c^3=3abc}

14. EXPONENTS / INDICES

am×an=am+n\boxed{a^m\times a^n=a^{m+n}}aman=am−n\boxed{\frac{a^m}{a^n}=a^{m-n}}(am)n=amn\boxed{(a^m)^n=a^{mn}}(ab)n=anbn\boxed{(ab)^n=a^nb^n}(ab)n=anbn\boxed{\left(\frac ab\right)^n=\frac{a^n}{b^n}}a0=1\boxed{a^0=1}a−n=1an\boxed{a^{-n}=\frac1{a^n}}a1/n=an\boxed{a^{1/n}=\sqrt[n]{a}}

15. LINEAR EQUATION

ax+b=0ax+b=0x=−ba\boxed{x=-\frac ba}

General equation in two variables:

ax+by+c=0\boxed{ax+by+c=0}

16. QUADRATIC EQUATION

Standard form:

ax2+bx+c=0\boxed{ax^2+bx+c=0}

Quadratic formula:

x=−b±b2−4ac2a\boxed{ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} }

Discriminant:

D=b2−4ac\boxed{D=b^2-4ac}

If:

D>0D>0: two real roots
D=0D=0: equal roots
D<0D<0: no real roots

Sum of roots:

α+β=−ba\boxed{\alpha+\beta=-\frac ba}

Product:

αβ=ca\boxed{\alpha\beta=\frac ca}

Equation from roots:

x2−(α+β)x+αβ=0\boxed{x^2-(\alpha+\beta)x+\alpha\beta=0}

17. SIMULTANEOUS EQUATIONS

a1x+b1y=c1a_1x+b_1y=c_1a2x+b2y=c2a_2x+b_2y=c_2

Cross multiplication:

xb1c2−b2c1=yc1a2−c2a1=1a1b2−a2b1\boxed{ \frac{x}{b_1c_2-b_2c_1} = \frac{y}{c_1a_2-c_2a_1} = \frac1{a_1b_2-a_2b_1} }

18. POLYNOMIALS

Remainder theorem:

If p(x)p(x) is divided by (x−a)(x-a),

Remainder=p(a)\boxed{\text{Remainder}=p(a)}

Factor theorem:

If

p(a)=0p(a)=0

then

(x−a) is a factor\boxed{(x-a)\text{ is a factor}}

19. BINOMIAL THEOREM

(a+b)n=∑r=0n(nr)an−rbr\boxed{ (a+b)^n= \sum_{r=0}^{n} {n\choose r}a^{n-r}b^r }

where:

(nr)=n!r!(n−r)!\boxed{ {n\choose r} = \frac{n!}{r!(n-r)!} }

General term:

Tr+1=(nr)an−rbr\boxed{ T_{r+1}={n\choose r}a^{n-r}b^r }

20. GEOMETRY — ANGLE RULES

Right angle:

90∘\boxed{90^\circ}

Straight angle:

180∘\boxed{180^\circ}

Complete angle:

360∘\boxed{360^\circ}

Complementary angles:

A+B=90∘\boxed{A+B=90^\circ}

Supplementary angles:

A+B=180∘\boxed{A+B=180^\circ}

Angles around a point:

A+B+C+⋯=360∘\boxed{A+B+C+\cdots=360^\circ}

Linear pair:

A+B=180∘\boxed{A+B=180^\circ}

Vertically opposite angles:

∠1=∠3\boxed{\angle1=\angle3}∠2=∠4\boxed{\angle2=\angle4}

21. PARALLEL LINES CUT BY A TRANSVERSAL

Corresponding angles

Corresponding angles are equal\boxed{\text{Corresponding angles are equal}}

Alternate interior angles

Alternate interior angles are equal\boxed{\text{Alternate interior angles are equal}}

Alternate exterior angles

Alternate exterior angles are equal\boxed{\text{Alternate exterior angles are equal}}

Same-side / co-interior angles

A+B=180∘\boxed{A+B=180^\circ}

These four rules solve most parallel-line angle questions.

22. TRIANGLE ANGLE RULES

Interior angles:

A+B+C=180∘\boxed{A+B+C=180^\circ}

Exterior angle:

Exterior angle=sum of two opposite interior angles\boxed{ \text{Exterior angle} = \text{sum of two opposite interior angles} }

For example:

x=A+B\boxed{x=A+B}

23. ISOSCELES TRIANGLE

If two sides are equal:

AB=ACAB=AC

then opposite angles are equal:

∠B=∠C\boxed{\angle B=\angle C}

Converse:

If

∠B=∠C\angle B=\angle C

then

AB=AC\boxed{AB=AC}

24. EQUILATERAL TRIANGLE

All sides equal:

AB=BC=CAAB=BC=CA

Every angle:

60∘\boxed{60^\circ}

25. PYTHAGORAS THEOREM

For a right triangle:

c2=a2+b2\boxed{c^2=a^2+b^2}

Hypotenuse:

c=a2+b2\boxed{c=\sqrt{a^2+b^2}}

Other side:

a=c2−b2\boxed{a=\sqrt{c^2-b^2}}

26. TRIANGLE CONGRUENCE RULES

Two triangles are congruent by:

SSS — Side-Side-Side
SAS — Side-Angle-Side
ASA — Angle-Side-Angle
AAS — Angle-Angle-Side
RHS — Right angle-Hypotenuse-Side

For congruent triangles:

Corresponding sides and angles are equal\boxed{\text{Corresponding sides and angles are equal}}

27. SIMILAR TRIANGLES

Criteria:

AA
SAS
SSS

If:

△ABC∼△DEF\triangle ABC\sim\triangle DEF

then

ABDE=BCEF=ACDF\boxed{ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} }

Corresponding angles:

∠A=∠D\angle A=\angle D∠B=∠E\angle B=\angle E∠C=∠F\angle C=\angle F

Area ratio:

Area of △ABCArea of △DEF=(ABDE)2\boxed{ \frac{\text{Area of }\triangle ABC} {\text{Area of }\triangle DEF} = \left(\frac{AB}{DE}\right)^2 }

28. POLYGONS

Sum of interior angles:

(n−2)×180∘\boxed{(n-2)\times180^\circ}

Each interior angle of a regular polygon:

(n−2)180∘n\boxed{ \frac{(n-2)180^\circ}{n} }

Sum of exterior angles:

360∘\boxed{360^\circ}

Each exterior angle of regular polygon:

360∘n\boxed{\frac{360^\circ}{n}}

Number of diagonals:

n(n−3)2\boxed{\frac{n(n-3)}2}

29. QUADRILATERALS

All interior angles:

360∘\boxed{360^\circ}

Parallelogram

Opposite sides equal
Opposite sides parallel
Opposite angles equal
Adjacent angles supplementary

A+C=180∘A+C=180^\circ

for adjacent angles appropriately:

A+B=180∘\boxed{A+B=180^\circ}

Diagonals bisect each other.

Rectangle

All angles:

90∘\boxed{90^\circ}

Diagonals are equal and bisect each other.

Square

All sides equal.

All angles:

90∘90^\circ

Diagonals are equal, perpendicular and bisect each other.

Rhombus

All sides equal.

Diagonals are perpendicular bisectors.

30. CIRCLE THEOREMS

Central angle:

Angle at center=2×angle at circumference\boxed{ \text{Angle at center} = 2\times \text{angle at circumference} }

Angle in a semicircle:

90∘\boxed{90^\circ}

Angles in the same segment:

Equal\boxed{\text{Equal}}

Opposite angles of cyclic quadrilateral:

A+C=180∘\boxed{A+C=180^\circ}B+D=180∘\boxed{B+D=180^\circ}

Radius and tangent:

Radius⊥Tangent\boxed{\text{Radius}\perp\text{Tangent}}

Tangents from the same external point:

PA=PB\boxed{PA=PB}

31. COORDINATE GEOMETRY

Distance between:

(x1,y1),(x2,y2)(x_1,y_1),(x_2,y_2)d=(x2−x1)2+(y2−y1)2\boxed{ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }

Midpoint:

(x1+x22,y1+y22)\boxed{ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) }

Slope:

m=y2−y1x2−x1\boxed{ m=\frac{y_2-y_1}{x_2-x_1} }

Equation of straight line:

y=mx+c\boxed{y=mx+c}

Point-slope form:

y−y1=m(x−x1)\boxed{y-y_1=m(x-x_1)}

Section formula for ratio m:nm:n:

(mx2+nx1m+n,my2+ny1m+n)\boxed{ \left( \frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n} \right) }

32. MENSURATION — TRIANGLE

Perimeter:

P=a+b+c\boxed{P=a+b+c}

Area:

A=12bh\boxed{A=\frac12 bh}

Heron's formula:

A=s(s−a)(s−b)(s−c)\boxed{ A=\sqrt{s(s-a)(s-b)(s-c)} }

where:

s=a+b+c2s=\frac{a+b+c}{2}

Equilateral triangle area:

A=34a2\boxed{ A=\frac{\sqrt3}{4}a^2 }

33. SQUARE

P=4a\boxed{P=4a}A=a2\boxed{A=a^2}

Diagonal:

d=a2\boxed{d=a\sqrt2}

34. RECTANGLE

P=2(l+b)\boxed{P=2(l+b)}A=lb\boxed{A=lb}

Diagonal:

d=l2+b2\boxed{d=\sqrt{l^2+b^2}}

35. PARALLELOGRAM

P=2(a+b)\boxed{P=2(a+b)}A=bh\boxed{A=bh}

36. RHOMBUS

P=4a\boxed{P=4a}A=12d1d2\boxed{A=\frac12d_1d_2}

37. TRAPEZIUM / TRAPEZOID

P=a+b+c+d\boxed{P=a+b+c+d}A=12(a+b)h\boxed{A=\frac12(a+b)h}

where a,ba,b are parallel sides.

38. CIRCLE

Diameter:

d=2r\boxed{d=2r}

Circumference:

C=2πr=πd\boxed{C=2\pi r=\pi d}

Area:

A=πr2\boxed{A=\pi r^2}

39. SEMICIRCLE

Area:

A=12πr2\boxed{A=\frac12\pi r^2}

Perimeter:

P=πr+2r\boxed{P=\pi r+2r}

40. ARC AND SECTOR

Arc length:

L=θ360∘×2πr\boxed{ L=\frac{\theta}{360^\circ}\times2\pi r }

Sector area:

A=θ360∘×πr2\boxed{ A=\frac{\theta}{360^\circ}\times\pi r^2 }

41. CUBE

LSA:

4a2\boxed{4a^2}

TSA:

6a2\boxed{6a^2}

Volume:

a3\boxed{a^3}

Diagonal:

a3\boxed{a\sqrt3}

42. CUBOID

LSA:

2h(l+b)\boxed{2h(l+b)}

TSA:

2(lb+bh+hl)\boxed{2(lb+bh+hl)}

Volume:

lbh\boxed{lbh}

Diagonal:

l2+b2+h2\boxed{\sqrt{l^2+b^2+h^2}}

43. CYLINDER

CSA:

2πrh\boxed{2\pi rh}

TSA:

2πr(r+h)\boxed{2\pi r(r+h)}

Volume:

πr2h\boxed{\pi r^2h}

44. CONE

Slant height:

l=r2+h2\boxed{l=\sqrt{r^2+h^2}}

CSA:

πrl\boxed{\pi rl}

TSA:

πr(l+r)\boxed{\pi r(l+r)}

Volume:

13πr2h\boxed{\frac13\pi r^2h}

45. SPHERE

Surface area:

4πr2\boxed{4\pi r^2}

Volume:

43πr3\boxed{\frac43\pi r^3}

46. HEMISPHERE

CSA:

2πr2\boxed{2\pi r^2}

TSA:

3πr2\boxed{3\pi r^2}

Volume:

23πr3\boxed{\frac23\pi r^3}

47. TRIGONOMETRY BASICS

For a right triangle:

sin⁡θ=OppositeHypotenuse\boxed{\sin\theta= \frac{\text{Opposite}}{\text{Hypotenuse}}}cos⁡θ=AdjacentHypotenuse\boxed{\cos\theta= \frac{\text{Adjacent}}{\text{Hypotenuse}}}tan⁡θ=OppositeAdjacent\boxed{\tan\theta= \frac{\text{Opposite}}{\text{Adjacent}}}

Reciprocals:

csc⁡θ=1sin⁡θ\boxed{\csc\theta=\frac1{\sin\theta}}sec⁡θ=1cos⁡θ\boxed{\sec\theta=\frac1{\cos\theta}}cot⁡θ=1tan⁡θ\boxed{\cot\theta=\frac1{\tan\theta}}

Important identities:

sin⁡2θ+cos⁡2θ=1\boxed{\sin^2\theta+\cos^2\theta=1}1+tan⁡2θ=sec⁡2θ\boxed{1+\tan^2\theta=\sec^2\theta}1+cot⁡2θ=csc⁡2θ\boxed{1+\cot^2\theta=\csc^2\theta}

Common values

Angle0∘0^\circ30∘30^\circ45∘45^\circ60∘60^\circ90∘90^\circsin⁡θ\sin\theta01/21/21/21/\sqrt23/2\sqrt3/21cos⁡θ\cos\theta13/2\sqrt3/21/21/\sqrt21/21/20tan⁡θ\tan\theta01/31/\sqrt313\sqrt3Undefined

Very Important Geometry Problem-Solving Rules

When solving an unknown angle, check these first:

Straight line → 180∘180^\circ
Around a point → 360∘360^\circ
Right angle → 90∘90^\circ
Triangle → 180∘180^\circ
Vertically opposite angles → equal
Corresponding angles → equal when lines are parallel
Alternate angles → equal when lines are parallel
Same-side interior angles → 180∘180^\circ
Isosceles triangle → base angles equal
Exterior triangle angle → sum of two opposite interior angles
Quadrilateral → 360∘360^\circ
Angle in semicircle → 90∘90^\circ
Cyclic quadrilateral opposite angles → 180∘180^\circ
Pythagoras → c2=a2+b2c^2=a^2+b^2 for a right triangle
Radius to tangent → 90∘90^\circ

https://goodone.club/prashnaya/papers/PAP-NP-00000018

MASTER MATHEMATICS FORMULA SHEET

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