Maths Formula for 10th class
Math formula Basic , Math formula all
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Basic formula (a ± b)² =
a² ± 2ab + b² (a ± b)³ =
a³ ± 3a²b ± 3ab² ± b (a ± b)⁴ =
a⁴ ± 4a³b ± 6a²b² ± 4ab³ ± b³ (a + b +
c)² =
a² + b² + c² + 2ab + 2bc + 2ca (a + b -
c)² =
a² + b² + c² + 2ab - 2bc - 2ca (a - b -
c)² = a² + b² + c² - 2ab - 2ac - 2bc (a + b +
c)³ =
a³ + b³ + c³ + 6abc + 3(a²b + ab² + bc² + b²c + c²a +
ca²) (a² -
b²) = (a + b)(a - b) (a³ +
b³) = (a + b)(a² - ab + b²) (a³ -
b³) = (a - b)(a² + ab + b²) (a⁴ + b⁴) =
(a² + b²)² - 2a²b² = (a² + √2ab + b²) (a² - √2ab + b²) (a⁴ -
b⁴) = (a² - b²)(a² + b²) = (a +
b)(a - b)(a²+ b²) (a⁵ +
b⁵) = (a + b)(a⁴ - a³b + a²b² -
ab³ + b⁴) (a⁵ + b⁵) = (a - b)(a⁴ + a³b + a²b² + ab³ + b⁴) |
Math formula of algebra
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Exponential Aᵐ = a . a…..a upto m Aᵐ / aⁿ = aᵐ⁻ⁿ (a/b)ᵐ = (aᵐ / bᵐ) ( b ≠ 0) Aᵐ * aⁿ = aᵐ⁺ⁿ A⁰ = 1 (ab)ᵐ = aᵐ bᵐ A⁻ᵐ = 1/aᵐ Polynomial (a + b + c)x
= ax + bx + cx (a + b) (m +
n) = a(m + n) + (m + n) = am + an +
bm + bn (a + b+ c) / x = a/x + b/x + c/x Fraction a/b ± c/d =
(ad ± cb)/bd a/b × c/d =
ac/bd a/b : c/d =
ad/bc a(b/c) = (ac
+ b) / c 1 × ¾ = (1 × 4 × 3)/4 = 7/4 Rate formula a/b = c/d = ad=bc
a = bc/d , b = ad/c a/c = b/d ;. d/b = c/a ; d/c = b/a ; (a ± b) / b ; (c±d)/c a + b/a - b = c + d/c - d
; a ± b/a ; c ± d /c a/a ± b = c
/c ± d ; b/a ± b d/c ± d i² = -1 i³ =
i² . I = -1 i⁴ = i³ , i= -I
=1 , ….. i⁴n =1 i⁴ⁿ⁺¹ = I , i⁴ⁿ⁺² = -1 i⁴ⁿ⁺³ , = -1 (a + bi) +
(c + di) = (a + c) + (b + di) (a + bi) – (c
+ di) = (a - c) + (b - di) (a + bi)(a - bi)
= a² + b² (a + bi)(c + di)
= (ac - bd) + (ad +bc)i |
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Percentage By a certain per cent we mean that many hundredths. We denote x per cent by x%. x% = x hundredths = 100 SOLVED EXAMPLES EXAMPLE 1. Express each of the following as a fraction: (1) 36% (lt) 120% (ttt) 0.8% We have (1) 36% = 36/100 =9/25 (2) 120%= 120/100 = 6/5 What per cent of 120 is 90? (2) What per cent of 1 kg is 5 g? (ill) What per cent of 3.5 L is 150 mL? (1) Required percentage = Profit and loss Cost price (CP) The amount for which an article is bought is called its cost price. Selling price (SP) The amount for which an article is sold is called its selling price. Profit or gain When (SP) > (CP) then there is a gain. Gain = (SP) – (CP). When (SP) < (CP) then there is a loss. Loss =
(CP) - (SP).
 Various Types of Algebraic Expressions (1) Monomials: An algebraic expression which contains only one term, is called a monomial. Thus 5x, 2xy. -3a b.-7. etc., are all monomials. (i) Binomials: An algebraic expression containing two terms is called a binomial. Thus, (2a +3b). (8-3x), (x² - 4xy), etc., are all binomials. (iii) Trinomials: An algebraic expression containing three terms is called a trinomial. Thus (a + 2b +5c), (x+2y-3z), (x²-y-2°), etc., are all trinomials. (iv) Quadrinomials: An algebraic expression containing four terms is called a quadrinomial. Thus (x+y+z-51. (x+y+z²+3xyz), etc., are all quadrinomials. (v) Polynomials: An expression containing two or more terms is called a polynomial. ADDITION OF ALGEBRAIC EXPRESSIONS While adding algebraic expressions, we collect the like terms and add them. The sum of several like terms is another like term whose coefficient is the sum of the coefficients of those like terms. EXAMPLE 1. Add: 5x² - 7x +3,-8x²+2x-5 and 7x²-x-2. = (5x² - 7x+3)+(-8x²+2x-5)+(7x²-x-2) = 5x² - 8x² +7x²-7x+2x-x+3-5-2 =(5-8+7) x² +(-7+2-1)x+(3-5-2) = 4x² - 6x-4. The difference of two like terms is a like term whose coefficient is the difference of the numerical coefficients of the two like terms. EXAMPLE (4x² - 6x²) = (4-6) x² = -2x². Rule for Subtraction of Algebraic Expressions Change the sign of each term of the expression to be subtracted and then add. Subtract (2x² -5x+7) from (3x² + 4x-6). We have: (3x² + 4x-6)-(2x² - 5x+7) = 3x² + 4x-6-2x² +5x-7 = (3-2) x² +(4+5)x+(-6-7) = x² +9x-13. |
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Arithmetic progression A₁ , a₂ ,
…….. , aₙ ,…… A₂ = a₁ +d , a₃ = a₁+ 3d , …….., aₙ = a +(n-1)d Sₙ = a₁ + a₂ +…….+ aₙ ={(a + aₙ)n}/2 = [2a + (n -1)d]n/2 Geometric progression A₁ , a₂ , a₃
, ……. , aₙ⁻₁ , aₙ ,….. A₂ = a₁q , a₃
= a₁q² ,……. , an =a₁qⁿ⁻¹ Sₙ = a₁ + a₂
+……. Aₙ = a₁ (q-1/q-1) Sₙ = na₁ (q=1) 1 + 2 + 3 + …..+ (n - 1)
= {n(n - 1)}\2 P + (q + 1) +……+
(q - 1) + q ={(q + p)(q - p + 1)}/2 1 + 3 + 5 + …..(2n - 3)
+ (2n - 1) =n² 2 + 4 + 6 +….+ (2n - 2)
+ 2n = n(n + 1) 1² + 2² + 3²
+ …….. + (n - 1)² + n² = {n(n + 1)(2n + 1)}/6 1² + 2² + 3³
+ ……. + (n - 1)³ = {n²(n + 1)}/4 1² + 3² + 5² +…….
+ (2n - 3)² + (2n - 1)² = {n(4n² - 1)}/3 1³ + 3³ + 5³ +………
+ (2n - 3)³ + (2n - 1)³ = n²(2n² - 1) 1⁴ + 2⁴ + 3⁴ + ……+ (n - 1)⁴ = [n(n + 1)(2n + 1)(3n² + 3n - 1)]/30 Logarithm Logₐ= x bˣ = N (N > 0 ,
b > 0 b ≠ 1 Logₐ (N₁ N₂) = logₐ |N₁| + logₐ |N₂| (N₁ N₂ >0) Logₐ (N₁/N₂) = logₐ |N₁| - logₐ |N₂| (N₁ N₂ > 0) Logₐ N = a logₐ
N (N > 0) Logₐ ᵃ√N = 1/a
logₐ N (N > 0) Logₐ N = logₐ
a logₐ N (a > 0,a ≠ 0 , N > 0) Logₐ N = logₐ N/logₐ b Logₐ 1 =0 Logₐ a =1 Logₑ N = lgN
(b = 10) IgN = x 10ˣ =
N Logₐ N = lgN lnN = x = eˣ = N |
0⁰ | ||||||
30⁰ | 1/2 | |||||
45⁰ | ||||||
60⁰ | ||||||
90⁰ | 1 | |||||
120⁰ | √3/2 | |||||
180⁰ | ||||||
360⁰ | 0 |
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Sin a = a/c opposite/hypotenuse Cos a = b/c adjacent/hypotenuse Tan a = a/b opposite/adjacent Cot a = b/a adjacent/opposite Sec a c/b. Cosec a c/a Sine = seno = sen = sim Cosine = cosseno = cos Tangent = tan = tg Cotangent = cottan = cot = ctg Secant = sec Cosecant = cosec = csc |
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90⁰- α 90⁰+ α 180⁰- α 180⁰+ α 270⁰- α 270⁰+ α 360⁰k-α 360⁰k+α |
-sin α +cosα +cosα +sinα -sin α -cosα -cosα -sin α +sin α |
+sin α -sin α +sin α +cosα +cosα |
-tan α +cotα -cot α -tan α +tanα +cot α -cot α +tanα +tanα |
-cotα +tanα -tanα -cot α +cotα +tanα -tan α -cot α +cotα |
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Sin²a +cos² = 1 tan a.cot.tana = 1 tan a =sin
a/cos a =1/cos tan a Cot tan a =
cos a/ sin a =1/tan a 1 + tan²a =
1/cos²a = sec²a 1+ cot tan²a = 1/sin²a = cos sec²a Sin 2a = 2 sin
a cos a Cos 2a =
2Cos²a -1 = 1 – 2sin²a = Cos²a - sin²a tan 2a = 2tan
a/1 - tan²a Cot tan 2a =
cot tan²a - 1/2 cot tan a = (cot tan a – tan a)/2 Sin 3a 3sin a
– 4sin³a Cos 3 a =
4cos³a – 3cos a tan 3a (3tan
a – tan³a)/1 - 3tan²a Cot tan 3a (cot tan³a – 3cot tan a)/3cot tan²a - 1 Sin x = 2t/1
+ t² Cos x (1 -
t²)/(1 + t²) Tan x = 2t /1 - t² Cotan x (1 - t²)/2t Additional formula Sin(a ± b) =
sin a cos b ± cos a sin b Cos(a ± b) =
cos a cos b ± sin a sin b Tan (a ± b) =
(tan a ± tan b)/1 ± tan a tan b Cot(a ± b) (cot a cot b ± 1)/cot b ± cot a Product of trigonometric function Cos a cos b =
½[ cos (a - b) + cos (a + b)] Sin a sin b =
½[cos(a - b) + cos(a + b)] Sina cos b = ½[sin(a + b) + sin (a - b)] Sin a + sin b
+ sin c = 4cos a/2cosb/2cosc/2 Cos a + cos b
+ cos c = 4sin a/2sin b/2 sin c/2 Sin a + sinb
- sin c = 4sin a/2sin b/2 cos c/2 Cos a + cos b
– cos c = 4cos a/2 cos b/2 sin y/2 - 1 Sin²a + sin²b + sin²c
=2cos a cos b cos c + 2 Sin²a + sin²b - sin²c
= 2sin.a sin b cis c Sin 2a + sin
2b + sin 2c = 4sin a sin b sinc Sin 2a + sin
2b -b sin 2c 4 cos a cos b sin c Tan a + tanb
+ tan c = tan a tan b tanc |
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Math formula integration  [ f(x) dx = F(x) + F'(x)=f(x). [dx= x + c [kf(x)dx = k[ f(x) dx INTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS sin xdx = -cos x + c cos xdx = sin x + C tan xdx = ln secx + C = - In cos x + C cotxdx = In sin x + C sin² xdx = x/2 - 1/4 sin 2x + C cos² xdx = x/2 + 1/4 sin 2x + c tan ²xdx = tan x - x + C cot ² xdx =-cotx - x + C sin³ xdx == cos³x - cos x + C [ cos³xdx = sinx - 1/3 sin³ x + C |
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1 km = 1000 m
1m = 100 cm 1 cm = 0.01m
= 10⁻² m 1 mm = 10⁻³ m 1 μm= 10⁻⁶ m 1 mμ =1nm = 10⁻⁹ 1 nm = 10 m 1 angstrom (Å) = 10⁻¹⁰ m 1 inch (in) = 2.54 cm 1 foot (ft) = 30.48 cm 1 cm = 0.3937 in 1 m = 39.37
in |
10m | dm | decimetre | 10m | dam | decametre |
10m | cm | centimetre | 10m | hm | hectometre |
10m | mm | millimetre | 10m | km | kilometre |
10m | um | micrometre | 10m | Mm | megametre |
10m | nm | nanometre | 10m | Gm | gigametre |
10m | pm | picometre | 10m | Tm | terametre |
10m | fm | femtometre | 10m | Pm | petametre |
10m | am | attometre | 10m | Em | exametre |
10m | zm | zeptometre | 10m | Zm | zettametre |
10m | ym | yoctometre | 10m | Ym | yottametre |
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1 nautical mile (NM)= 1.852 km AREA 1 m² = 10.76 ft² 1 ft² = 929 cm² 1 mi² = 640 acres 1 ac (acre) 4046.85 m² Ia (are) = 100 m² = 10m x 10 m 1 ha (hectare) = 0,01 km² = 10.000 m² 1 ha = 100 a = 2.471 ac 1 km² = 100 ha = 10⁶ m² 1 acre = 66 feet x 660 feet = 43,560 ft² 1 acre = 1.62 acres (6,600 m²) (in Ireland) VOLUME 1 litter (l) 1000 cm³= 1.057 quart (qt) = 61.02 in³ = 0.03532 ft³ 1 m³= 1000 l = 35.32 ft³ 1 ft³ = 7.481 U.S. gal = 0.02832 m³ = 28.32 L 1 U.S. gallon (gal)=231 in³ ⁴ 3.785L British gallon = 1.201 U.S. gallon = 277.4 in³ MASS 1 ton = 1000 kg 1 kilogram (kg) 2.2 pounds (lb) = 0.0685 slug 1 lb 453.6 gm = 0.031 slug 1 slug = 32.174 lb = 14.59 kg 1 lb = 16 ounce (oz) 1 oz = 28.35 g 1 troy ounce = 31.1034768 gram TIME 1 hour = 60 minutes = 3600 seconds 1 day = 24 hours. 1 month 30 days 1 year 365 days = 52 weeks = 12 months SPEED 1 km/h = 0.2778 m/sec = 0.6214 mi/h = 0,9113 ft/sec 1 mi/h = 1.609 km/h = 1.467 ft/sec = 0.4470 m/sec I knot = 1 nautical mile/hour = 1.852 km/h. DENSITY I gm/cm³ = 10³ kg/m³ = 62.43 lb/ft³ = 1.940 slug/ft³¹ 1 lb/ft=0.01602 gm/cm³ 1 slug/ft³ = 0.5154 gm/cm³ FORCE 1 newton (nt) 10$ dynes = 0.1020 kgwt = 0.2248 lbwt 1 pound
weight (lbwt) = 4.448 nt = 0.4536 kgwt = 32.17 = poundals = 1 kilogram weight
(kgwt)=2.205 lbwt I U.S. short ton = 2000 lbwt; 1 long ton 2240 lbwt; 1 metric ton = 2205 lbwt ENERGY 1 joule 1 nt m = 107 ergs = 0.7376 ft lbwt = 0.2389 cal = 9.481 x 10 Btu 1 ft lbwt = 1.356 joules = 0.3239 cal = 1.285 x 10-³ Btu 1 calorie (cal) = 4.186 joules 3.087 ft lbwt = 3.968 × 10³ Btu 1 Btu (British thermal unit) = 778 ft lbwt = 1055 joules = 0.293 watt hr 1 kilowatt hour (kw hr) - 3.60 x 10 joules = 860.0 kcal = 3413 Btu I electron volt (ev) = 1.602x 10-19 joule POWER 1 watt 1 joule/sec = 107 ergs/sec = 0.2389 cal/sec 1 horsepower (hp) = 550 ft lbwt/sec = 33,000 ft lbwt/min = 745.7 watts. 1 kilowatt (kw) 1.341 hp-737.6 ft lbwt/sec = 0.9483 Btu/sec PRESSURE 1 nt/m² 10 dynes/cm² = 9.869 × 10 atmosphere = 2.089 x 10-² lbwt/ft² 1 lbwt/in²=6895 nt/m² = 5.171 cm mercury = 27.68 in water I atm 1.013 x 105 nt/m² = 1.013 106 dynes/cm² = 14.70 lbwt/in2 = 76 cm
mercury =406.8 in water |
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