Basic Math Formulas for 11th and 12th continue..............

● Surds:

(i) The surd conjugate of √a + √b (or a + √b) is √a - √b (or a - √b) and conversely.

(ii) If a is rational, √b is a surd and a + √b (or, a - √b) = 0 then a = 0 and b = 0.

(iii) If a and x are rational, √b and √y are surds and a + √b = x + √y then a = x and b = y.

● Complex Numbers:

(i) The symbol z = (x, y) = x + iy where x, y are real and i = √-1, is called a complex (or, imaginary) quantity;x is called the real part and y, the imaginary part of the complex number z = x + iy.
(ii) If z = x + iy then z = x - iy and conversely; here, z is the complex conjugate of z.

(iii) If z = x+ iy then

(a) mod. z (or, | z | or, | x + iy | ) = + √(x² + y²) and

(b) amp. z (or, arg. z) = Ф = tan^-1 y/x (-π < Ф ≤ π).

(iv) The modulus - amplitude form of a complex quantity z is

z = r (cosф + i sinф); here, r = | z | and ф = arg. z (-π < Ф <= π).

(v) | z | = | -z | = z ∙ z = √ (x² + y²).

(vi) If x + iy= 0 then x = 0 and y = 0(x,y are real).

(vii) If x + iy = p + iq then x = p and y = q(x, y, p and q all are real).

(viii) i = √-1, i² = -1, i³ = -i, and i⁴ = 1.

(ix) | z₁ + z₂| ≤ | z₁ | + | z₂ |.

(x) | z₁ z₂ | = | z₁ | ∙ | z₂ |.

(xi) | z₁/z₂| = | z₁ |/| z₂ |.

(xii) (a) arg. (z₁ z₂) = arg. z₁ + arg. z₂ + m

(b) arg. (z₁/z₂) = arg. z₁ - arg. z₂ + m where m = 0 or, 2π or, (- 2π).

(xiii) If ω be the imaginary cube root of unity then ω = ½ (- 1 + √3i) or, ω = ½ (-1 - √3i)

(xiv) ω³ = 1 and 1 + ω + ω² = 0

● Variation:

(i) If x varies directly as y, we write x ∝ y or, x = ky where k is a constant of variation.

(ii) If x varies inversely as y, we write x ∝ 1/y or, x = m ∙ (1/y) where m is a constant of variation.

(iii) If x ∝ y when z is constant and x ∝ z when y is constant then x ∝ yz when both y and z vary. 

● Theory of Quadratic Equation :

ax² + bx + c = 0 ... (1)

(i) Roots of the equation (1) are x = {-b ± √(b² – 4ac)}/2a.

(ii) If α and β be the roots of the equation (1) then,

sum of its roots = α + β = - b/a = - (coefficient of x)/(coefficient of x² );

and product of its roots = αβ = c/a = (Constant term /(Coefficient of x²).

(iii) The quadratic equation whose roots are α and β is

x² - (α + β)x + αβ = 0

i.e. , x² - (sum of the roots) x + product of the roots = 0.

(iv) The expression (b² - 4ac) is called the discriminant of equation (1).

(v) If a, b, c are real and rational then the roots of equation (1) are

(a) real and distinct when b² - 4ac > 0;

(b) real and equal when b² - 4ac = 0;

(c) imaginary when b² - 4ac < 0;

(d) rational when b²- 4ac is a perfect square and

(e) irrational when b² - 4ac is not a perfect square.

(vi) If α + iβ be one root of equation (1) then its other root will be conjugate complex quantity α - iβ and conversely (a, b, c are real).

(vii) If α + √β be one root of equation (1) then its other root will be conjugate irrational quantity α - √β (a, b, c are rational).

● Permutation:

(i) ⌊n (or, n!) = n (n – 1) (n – 2) ∙∙∙∙∙∙∙∙∙ 3∙2∙1.

(ii) 0! = 1.

(iii) Number of permutations of n different things taken r ( ≤ n) at a time ⁿP_r = n!/(n - 1)! = n (n – 1)(n - 2) ∙∙∙∙∙∙∙∙ (n - r + 1).

(iv) Number of permutations of n different things taken all at a time = ⁿP_n = n!.

(v) Number of permutations of n things taken all at a time in which p things are alike of a first kind, q things are alike of a second kind, r things are alike of a third kind and the rest are all different, is ^n!/_(p!q!r!)

(vi) Number of permutations of n different things taken r at a time when each thing may be repeated upto r times in any permutation, is n^r . 

 

● Combination:

(i) Number of combinations of n different things taken r at a time = ⁿC_r = ^n!/_{(r!(n – r)!)}.

(ii) ⁿP_r = r!∙ ⁿC_r.

(iii) ⁿC₀ = ⁿC_n = 1.

(iv) ⁿC_r = ⁿC_{n - r}.

(v) ⁿC_r + ⁿC_{n - 1} = ^{n + 1}C_r

(vi) If p ≠ q and ⁿC_p = ⁿC_p then p + q = n.

(vii) ⁿC_r/ⁿC_{r - 1}= (n - r + 1)/r.

(viii) The total number of combinations of n different things taken any number at a time = ⁿC₁ + ⁿC₂ + ⁿC₃ + …………. + ⁿC_n = 2ⁿ – 1.

(ix) The total number of combinations of (p + q + r + . . . .) things of which p things are alike of a first kind, q things are alike of a second kind r things are alike of a third kind and so on, taken any number at a time is [(p + 1) (q + 1) (r + 1) . . . . ] - 1.

● Binomial Theorem:

(i) Statement of Binomial Theorem : If n is a positive integer then
(a + x)ⁿ = aⁿ + ⁿC₁ a^{n - 1} x + ⁿC₂ a^{n - 2} x² + …………….. + ⁿC_r a^{n - r} x^r + ………….. + xⁿ …….. (1)

(ii) If n is not a positive integer then

(1 + x)ⁿ = 1 + nx + [n(n - 1)/2!] x² + [n(n - 1)(n - 2)/3!] x³ + ………… + [{n(n-1)(n-2)………..(n-r+1)}/r!] x^r+ ……………. ∞ (-1 < x < 1) ………….(2)

(iii) The general term of the expansion (1) is (r+ 1)th term

= t_{r + 1} = ⁿC_r a^{n - r} x^r

(iv) The general term of the expansion (2) is (r + 1) th term

= t_{r + 1} = [{n(n - 1)(n - 2)....(n - r + l)}/r!] ∙ x^r.

(v) There is one middle term is the expansion ( 1 ) when n is even and it is (n/2 + 1)th term ; the expansion ( I ) will have two middle terms when n is odd and they are the {(n - 1)/2 + 1} th and {(n - 1)/2 + 1} th terms.

(vi) (1 - x)^-1 = 1 + x + x² + x³ + ………………….∞.

(vii) (1 + x)^-1 = I - x + x² - x³ + ……………∞.

(viii) (1 - x)^-2 = 1 + 2x + 3x² + 4x³ + . . . . ∞ .

(ix) (1 + x)^-2 = 1 - 2x + 3x² - 4x³ + . . . . ∞ .

● Logarithm:

(i) If a^x = M then log_a M = x and conversely.

(ii) log_a 1 = 0.

(iii) log_a a = 1.

(iv) a ^log_am = M.

(v) log_a MN = log_a M + log_a N.

(vi) log_a (M/N) = log_a M - log_a N.

(vii) log_a Mⁿ = n log_a M.

(viii) log_a M = log_b M x log_a b.

(ix) log_b a x 1og_a b = 1.

(x) log_b a = 1/log_b a.

(xi) log_b M = log_b M/log_a b.

● Exponential Series:

(i) For all x, e^x = 1 + x/1! + x²/2! + x³/3! + …………… + x^r/r! + ………….. ∞.

(ii) e = 1 + 1/1! + 1/2! + 1/3! + ………………….. ∞.

(iii) 2 < e < 3; e = 2.718282 (correct to six decimal places).

(iv) a^x = 1 + (log_e a) x + [(log_e a)²/2!] ∙ x² + [(log_e a)³/3!] ∙ x³ + …………….. ∞.

● Logarithmic Series:

(i) log_e (1 + x) = x - x²/2 + x³/3 - ……………… ∞ (-1 < x ≤ 1).

(ii) log_e (1 - x) = - x - x²/ 2 - x³/3 - ………….. ∞ (- 1 ≤ x < 1).

(iii) ½ log_e [(1 + x)/(1 - x)] = x + x³/3 + x⁵/5 + ……………… ∞ (-1 < x < 1).

(iv) log_e 2 = 1 - 1/2 + 1/3 - 1/4 + ………………… ∞.

(v) log₁₀ m = µ log_e m where µ = 1/log_e 10 = 0.4342945 and m is a positive num

Basic Math Formulas for 11th and 12th standard students

The list of basic math formulas which is very useful for mainly 11 grade, 12 grade and college grade students. Math formulas are very important and necessary to know the correct formula while solving the questions on different topics. If we remember math formulas we can solve any type of math questions.

● Laws of Indices:

(i) a^m ∙ aⁿ = a^{m + n}

(ii) a^m/aⁿ

(iii) (a^m)ⁿ = a^mn

(iv) a⁰ = 1 (a ≠ 0).

(v) a^{- n} = 1/aⁿ

(vi) ⁿ√a^m = a^m/n

(vii) (ab)^m = a^m ∙ bⁿ.

(viii) (a/b) ^m = a^m/bⁿ

(ix) If a^m = b^m (m ≠ 0), then a = b.

(x) If a^m = aⁿ then m = n. 

● Arithmetical Progression (A.P.):

(i) The general form of an A. P. is a, a + d, a + 2d, a+3d,.....

where a is the first term and d, the common difference of the A.P.

(ii) The nth term of the above A.P. is t_n = a + (n - 1)d.

(iii) The sum of first n terns of the above A.P. is s = n/2 (a + l) = (No. of terms/2)[1st term + last term] or, S = ⁿ/₂ [2a + (n - 1) d]

(iv) The arithmetic mean between two given numbers a and b is (a + b)/2.

(v) 1 + 2 + 3 + ...... + n = [n(n + 1)]/2.

(vi) 1² + 2² + 3² +……………. + n² = [n(n+ 1)(2n+ 1)]/6.

(vii) 1³ + 2³ + 3³ + . . . . + n³ = [{n(n + 1)}/2 ]².

● Geometrical Progression (G.P.) : 

(i) The general form of a G.P. is a, ar, ar², ar³, . . . . . where a is the first term and r, the common ratio of the G.P.

(ii) The n th term of the above G.P. is t_n = a.r^{n - 1} .

(iii) The sum of first n terms of the above G.P. is S = a ∙ [(1 - rⁿ)/(1 – r)] when -1 < r < 1

or, S = a ∙ [(rⁿ – 1)/(r – 1) ]when r > 1 or r < -1.

(iv) The geometric mean of two positive numbers a and b is √(ab) or, -√(ab).

(v) a + ar + ar² + ……………. ∞ = a/(1 – r) where (-1 < r < 1).

● Arithmetical Progression (A.P.):

(i) The general form of an A. P. is a, a + d, a + 2d, a+3d,.....

where a is the first term and d, the common difference of the A.P.

(ii) The nth term of the above A.P. is t_n = a + (n - 1)d.

(iii) The sum of first n terns of the above A.P. is s = n/2 (a + l) = (No. of terms/2)[1st term + last term] or, S = ⁿ/₂ [2a + (n - 1) d]

(iv) The arithmetic mean between two given numbers a and b is (a + b)/2.

(v) 1 + 2 + 3 + ...... + n = [n(n + 1)]/2.

(vi) 1² + 2² + 3² +……………. + n² = [n(n+ 1)(2n+ 1)]/6.

(vii) 1³ + 2³ + 3³ + . . . . + n³ = [{n(n + 1)}/2 ]².

to be continue...............

8 Grade Math Formula Chart

Grade 8^th Math formula chart provides complete solution towards solving the problem using formulas.

By simply substituting the data into the formula the unknown values can be found. Formulas are given based on the syllabus of the 8^th class board.

Grade 8 Math formula:
A list of formulas for 8^th grade class are given in pdf form for easy memorization and better understanding of the subject .

Geometry Figure	Area	Perimeter
Triangle	12 bh	a+b+c	b= breadth  h = height  a,b,c sides of triangle
Rectangle	bh	2l + 2w	w = width
Square	s2		S = side of square
Parallelogram	bh
Circle	πr2		r = radius
Trapezoid	12 (b1+b2)h
Three-Dimensional Figure	surface area	Lateral surface area
Rectangular prism	2ab+2bc+2ca
Cylinder	2πrh
Cube	6a2
Sphere	4πr2
Prism		ph	P = perimeter
Pyramid		12 pl+b
Cylinder		2πrh

Algebraic  Expansions:

(a + b)² = a² + b² + 2ab

(a - b)² = a² + b² - 2ab

( a² - b²) = ( a + b)  ( a - b)

( a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

( a + b - c)² = a² + b² + c² + 2ab - 2bc -2ca

Simple Interest = PTR100P = principal,T = time ,R = rate of interest

Customary Conversions :

1 mile = 5,280 feet (ft)

1 yard (yd) = 3 feet

1 foot = 12 inches (in.)

1 ton (T) = 2,000 pounds (lb)

1 pound = 16 ounces (oz.)

1 gallon (gal) = 4 quarts (qt)

1 quart = 2 pints (pt)

1 pint = 2 cups (c)

1 cup = 8 fluid ounces

1 day = 24 hours (hr)

1 hour = 60 minutes (min)

1 minute = 60 seconds (sec)

Metric Conversions :

1 kilometer = 1000 meters

1 hectometer = 100 meters

1 decimeter = 0.1 meter

1 centimeter = 0.01 meter

1 millimeter = 0.001 meter

1 meter = 100 cm = 10 dm = 0.01 m

Pythagorean Theorem:
a² + b²  = c²

Laws of Exponents
Quotient Law: aman=am−n, if m > n
              = 1an−m, if n > m

Power law: (a^m)ⁿ = a^mn

Product law: a^m*aⁿ = a^m+n

The Compound Interest Equation

P = C (1 + r/n) ^nt

where
    P = future value
    C = initial deposit
    r = interest rate (expressed as a fraction: eg. 0.06)
    n = # of times per year interest is compounded
    t = number of years invested

Simplified Compound Interest Equation

When interest is only compounded once per year (n=1), the equation simplifies to:

P = C (1 + r) ^t

Continuous Compound Interest

When interest is compounded continually (i.e. n --> ), the compound interest equation takes the form:

P = C e ^rt

Demonstration of Various Compounding

The following table shows the final principal (P), after t = 1 year, of an account initially with C = $10000, at 6% interest rate, with the given compounding (n). As is shown, the method of compounding has little effect.

n	P
1 (yearly)	$ 10600.00
2 (semiannually)	$ 10609.00
4 (quarterly)	$ 10613.64
12 (monthly)	$ 10616.78
52 (weekly)	$ 10618.00
365 (daily)	$ 10618.31
continuous	$ 10618.37

Loan Balance

Situation: A person initially borrows an amount A and in return agrees to make n repayments per year, each of an amount P. While the person is repaying the loan, interest is accumulating at an annual percentage rate of r, and this interest is compounded n times a year (along with each payment). Therefore, the person must continue paying these installments of amount P until the original amount and any accumulated interest is repaid. This equation gives the amount B that the person still needs to repay after t years.

B = A (1 + r/n)NT - P

(1 + r/n)NT - 1 (1 + r/n) - 1

where B = balance after t years
A = amount borrowed
n = number of payments per year
P = amount paid per payment

Hierarchy of Decimal Numbers

Number	Name	How many
0	zero
1	one
2	two
3	three
4	four
5	five
6	six
7	seven
8	eight
9	nine
10	ten
20	twenty	two tens
30	thirty	three tens
40	forty	four tens
50	fifty	five tens
60	sixty	six tens
70	seventy	seven tens
80	eighty	eight tens
90	ninety	nine tens

Number	Name	How Many
100	one hundred	ten tens
1,000	one thousand	ten hundreds
10,000	ten thousand	ten thousands
100,000	one hundred thousand	one hundred thousands
1,000,000	one million	one thousand thousands

Some people use a comma to mark every 3 digits. It just keeps track of the digits and makes the numbers easier to read.
Beyond a million, the names of the numbers differ depending where you live. The places are grouped by thousands in America and France, by the millions in Great Britain and Germany.

Name	American-French	English-German
million	1,000,000	1,000,000
billion	1,000,000,000 (a thousand millions)	1,000,000,000,000 (a million millions)
trillion	1 with 12 zeros	1 with 18 zeros
quadrillion	1 with 15 zeros	1 with 24 zeros
quintillion	1 with 18 zeros	1 with 30 zeros
sextillion	1 with 21 zeros	1 with 36 zeros
septillion	1 with 24 zeros	1 with 42 zeros
octillion	1 with 27 zeros	1 with 48 zeros
googol	1 with 100 zeros
googolplex	1 with a googol of zeros

Fractions
Digits to the right of the decimal point represent the fractional part of the decimal number. Each place value has a value that is one tenth the value to the immediate left of it.

Number	Name	Fraction
.1	tenth	1/10
.01	hundredth	1/100
.001	thousandth	1/1000
.0001	ten thousandth	1/10000
.00001	hundred thousandth	1/100000

Examples:
0.234 = 234/1000 (said - point 2 3 4, or 234 thousandths, or two hundred thirty four thousandths)
4.83 = 4 83/100 (said - 4 point 8 3, or 4 and 83 hundredths)
SI Prefixes

Number Prefix Symbol 10 1 deka- da 10 2 hecto- h 10 3 kilo- k 10 6 mega- M 10 9 giga- G 10 12 tera- T 10 15 peta- P 10 18 exa- E 10 21 zeta- Z 10 24 yotta- Y

Number Prefix Symbol 10 -1 deci- d 10 -2 centi- c 10 -3 milli- m 10 -6 micro- u (greek mu) 10 -9 nano- n 10 -12 pico- p 10 -15 femto- f 10 -18 atto- a 10 -21 zepto- z 10 -24 yocto- y

Roman Numerals

I=1		(I with a bar is not used)
V=5		_ V=5,000
X=10		_ X=10,000
L=50		_ L=50,000
C=100		_ C = 100 000
D=500		_ D=500,000
M=1,000		_ M=1,000,000

Roman Numeral Calculator

Examples:

1 = I 2 = II 3 = III 4 = IV 5 = V 6 = VI 7 = VII 8 = VIII 9 = IX 10 = X

11 = XI 12 = XII 13 = XIII 14 = XIV 15 = XV 16 = XVI 17 = XVII 18 = XVIII 19 = XIX 20 = XX 21 = XXI

25 = XXV 30 = XXX 40 = XL 49 = XLIX 50 = L 51 = LI 60 = LX 70 = LXX 80 = LXXX 90 = XC 99 = XCIX

There is no zero in the roman numeral system.
The numbers are built starting from the largest number on the left, and adding smaller numbers to the right. All the numerals are then added together.
The exception is the subtracted numerals, if a numeral is before a larger numeral, you subtract the first numeral from the second. That is, IX is 10 - 1= 9.
This only works for one small numeral before one larger numeral - for example, IIX is not 8, it is not a recognized roman numeral.
There is no place value in this system - the number III is 3, not 111.
Number Base Systems

Decimal(10)	Binary(2)	Ternary(3)	Octal(8)	Hexadecimal(16)
0	0	0	0	0
1	1	1	1	1
2	10	2	2	2
3	11	10	3	3
4	100	11	4	4
5	101	12	5	5
6	110	20	6	6
7	111	21	7	7
8	1000	22	10	8
9	1001	100	11	9
10	1010	101	12	A
11	1011	102	13	B
12	1100	110	14	C
13	1101	111	15	D
14	1110	112	16	E
15	1111	120	17	F
16	10000	121	20	10
17	10001	122	21	11
18	10010	200	22	12
19	10011	201	23	13
20	10100	202	24	14

Each digit can only count up to the value of one less than the base. In hexadecimal, the letters A - F are used to represent the digits 10 - 15, so they would only use one character.