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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,542

hi Temporary username

Welcome to the forum.

I think the poster didn't intend the LaTex to show here; but rather on the Dr Who site given.

Years ago MIF-Forum used a different server but had to change. Some LaTex that worked on the orignal now fails because the commands aren't implemented now. Cannot do a lot about it I'm afraid

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

A quadratic equation solution is

or

`x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}`

`x = (-b \pm \sqrt(b^2 - 4ac))/(2a).`

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Square, cube, and nth power; square root, cube root and nth root**

`a^2`

gives

`a^3`

gives

`a^n`

gives

`\sqrt{n}`

gives

`\sqrt[3]{n}`

gives

`\sqrt[a]{n}`

gives

For example,

`\sqrt[6]{64}`

gives

which is 2.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Some Algebraic Expansions**

`(a + b)^2 = a^2 + 2ab + b^2`

gives

`(a - b)^2 = a^2 - 2ab + b^2`

gives

`(a + b)^3 = a^3 + 3a^2b + 3ab^3 + b^3`

gives

`(a - b)^3 = a^3 - 3a^3 + 3ab^2 - b^3`

gives

`a^3 + b^3 = (a + b)(a^2 - ab + b^2)`

gives

`a^3 - b^3 = (a - b)(a^2 + ab + b^2`

gives

.It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Factorial, Permutations, and Combinations**

`n! = n \times (n - 1) \times (n - 2) \times .... 3 \times 2 \times 1 = n!`

gives

`nP_r = \dfrac{n!}{(n - r)!}`

gives

`nC_r = \dfrac{n!}{(n - r)!r!}`

gives

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**Summation**

`\Sigma \ n = \dfrac{n(n+ 1)}{2}`

gives

`\Sigma \ n^2 = \dfrac{n(n + 1)(2n + 1)}{6}`

gives

`\Sigma \ n^3 = \left[\dfrac{n(n +1)}{2}\right]^2`

gives

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Arithmetic Progression**

nth term of a Arithmetic Progression is

`a_n = a + (n - 1)d`

given by

where a is the first term, n is the number of terms, d is the common difference, and

is the nth term.Sum of n terms of an Arithmetic Progression :

`S_n = n/2[2a + (n - 1)d]`

is

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Geometric Progression**

nth term is

`a_n = ar^{n - 1}`

gives

**Sum of the terms**

`\Sigma = a\left(\dfrac{1 - r^n}{1 - r}\right)`

gives

or

`\Sigma = a\left(\dfrac{r^n - 1}{r - 1}\right)`

gives

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Circle, Hemisphere, and Sphere**

Area of a Circle:

`\pi{r^2}`

written as

is the Area of a circle.`2\pi{r}`

written as

is the Circumference of a circle.`\dfrac{2}{3}`

written as

is the Volume of a Hemisphere.`\dfrac{4}{3}\pi{r^3}`

written as

is the Volume of a Sphere.`{3}\pi{r^2}`

written as

is the Surface area of a Hemisphere.`4\pi{r^2}`

written as

is the Surface area of a sphere.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Two Dimensions**

Rectangle :

`lb`

is

Square:

`a^2`

is

Triangle:

`\dfrac{1}{2}bh`

is

where b is base and h is height.

Hero's formula for Area of a Triangle:

`\sqrt{s(s - a)(s - b)(s - c)}`

is

where a, b, and c are side lengths and s is semi-perimeter (half of perimeter).

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**Parallelogram, Trapezium, Kite, and Quadrilateral**

**Area of a Parallelogram:**

`Area = Base \times Height`

is

where b is the base and h is the height.

Perimeter:

`P = 2(a + b)`

is

where a and b are the two sides of a Parallelogram.

Area of Rhombus:

`Area = \dfrac{d_1 \times d_2}{2}`

is

where d1 and d2 are length of diagonals.

Perimeter:

`4a`

is

.Kite:

`Area = \dfrac{pq}{2}`

is

where p and q are the diagonals.

Perimeter :

`Perimeter: 2 x (sum of lengths of the sides)`

is

Trapezium:

`Area = \dfrac{a + b}{2}h`

is

where a, b are sides and h is the height.

Quadrilateral:

`Area: 1/2 x diagonal x (sum of perpendicular heights)`

is

`Perimeter: a + b + c + d`

is

.`Perimeter: sum of lengths sides of the quadrilateral.`

is

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**Cylinder and Cone**

**Right Circular Cylinder**

Volume:

`\pi{r^2}h`

gives

.Surface Area:

`2\pr{r}h`

gives

where r is radius, h is height.

**Right Circular Cone**

Volume:

`\dfrac{1}{3}\pi{r^2}h`

gives

.Area:

`\pi{r}(r + l)`

gives

where l is slant height.

Slant height:

`\sqrt(r^2 + h^2)`

gives

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**Hollow Sphere, Hollow Right Cylinder, Hollow Right Circular Cone**

**Volume**

`Sphere = \dfrac{4}{3}\pi(R^3 - r^3)`

gives

where R and r are external and internal radii.

`Right Circular Cylinder = \pi({R^2 - r^2})h`

gives

.`Right Circular Cone = \dfrac{1}{3}\pi(R^2 - r^2)h`

gives

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Exponents**

`a^m \times a^2 = a^{m + n}`

gives

`\dfrac{a^m}{a^n} = a^{m - n}`

gives

`(a^m)^{n}) = a^{mn}`

gives

`a^0 = 1, a \neq 0`

gives

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Rational Numbers**

`\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}`

gives

`\dfrac{a}{b} - \dfrac{c}{d} = \dfrac{ad - bc}{bd}`

gibes

`\frac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}`

gives

`\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{ad}{bc}`

gives

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Quadratic Equation**

`Standard form : ax^2 + bx + c = 0`

gives

.**Forming a Quadratic Equation:**

We will learn the formation of the quadratic equation whose roots are given.

To form a quadratic equation, let

`\alpha and \beta`

is

and be the two roots.Let us assume that the required equation be

`ax^2 + bx + c = 0, a \neq 0`

is

According to the problem, roots of this equation are

`\alpha and \beta`

gives

and .Therefore,

`\alpha + \beta = -\dfrac{b}{a} and \alpha\beta = \dfrac{c}{a}`

gives

andNow,

`ax^2 + bx + c = 0`

gives

`x^2 + \dfrac{b}{a}x + \dfrac{c}{a}a = 0 (Since, a \neq 0)`

gives

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Surface Area**

**Sphere**:

**Curved surface area (CSA) :**

The Curved surface area of hollow sphere is the area of the paper that can completely cover the surface of the hollow sphere. It is equal to the CSA of inner sphere subtracted from the CSA of outer sphere.

CSA of hollow sphere, = CSA of outer sphere - CSA of inner sphere

`= 4\pi(R^2) - 4\pi(r^2)`

is

`= 4 \pi(R^2-r^2)`

is

.**Total surface area of hollow sphere :**

The total surface area of a hollow sphere is equal to the CSA of hollow sphere as a hollow sphere has only one surface that constitutes it.

Thus CSA=TSA for a hollow sphere

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Surface Area**

**Cylinder**:

**Total**:

`Area = 2\pi{r^2} + 2\pi{r}h`

gives

where r is radius, h is the height.

**Curved Surface Area:**

`Area = 2\pi{r}h`

gives

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**Cone**

**Surface Area**

The surface area of a cone is equal to the curved surface area plus the area of the base:

`\pi{r^2} + \pi{l}r`

gives

, where r denotes the radius of the base of the cone, and L denotes the slant height of the cone. The curved surface area is also called the lateral area.`l = \sqrt{r^2 + h^2}`

gives

where r is radius and h height.

**Curved Surface Area:**

`Area = \pi{r}l`

gives

lNothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Set Formulas**

If A, B, and C are three sets, then the number of elements

`n(A \cup B) = n(A) + n(B) - n(A \cap B)`

gives

.If

`A \cap B = \phi, then n(A \cup B) = n(A) + n(B)`

gives

`n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)`

gives

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Simple Interest**

`Simple Interest = Prt where P is Principal, r is rate of Interest, and t time (months, quarters, years etc.)`

gives

,**Compound Interest**

`Compound Interest`

gives

`A = P\left(1 + \dfrac{r}{n}\right)^{nt}`

gives

where:

* A is the final amount

* P is the original principal sum

* r is the nominal annual interest rate

* n is the compounding frequency

* t is the overall length of time the interest is applied (expressed using the same time units as r, usually years).

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Profit and Loss**

`Profit = Selling Price - Cost Price`

gives

`Loss = Cost Price - Selling Price`

gives

`Profit Percentage = \dfrac{Profit}{Cost \ Price} \ times \ 100\%`

gives

`Loss Percentage = \dfrac{Loss}{Cost \ Price} \times 100\%`

gives

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Pythagoras Theorem**

*Pythagoras Theorem: In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two sides.*

`A right angled Triangle ABC, right angled at B. `

gives

`{AC}^2 = {AB}^2 + {BC}^2`

gives

`3^2 square units+ 4^2 square units = 5^2 square units`

gives

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,741

**Trigonometric Ratios Formulas**

Trigonometric ratios can be calculated by taking the ratio of any two sides of the right-angled triangle. We can evaluate the third side using the Pythagoras theorem, given the measure of the other two sides. We can use the abbreviated form of trigonometric ratios to compare the length of any two sides with the angle in the base. The angle θ is an acute angle (θ < 90º) and in general is measured with reference to the positive x-axis, in the anticlockwise direction. The basic trigonometric ratios formulas are given below,

`sin\theta = Perpendicular / Hypotenuse`

gives

`cos\theta = Base / Hypotenuse`

gives

`tan\theta = Perpendicular / Base`

gives

`sec\theta = Hypotenuse / Base`

gives

`cosec\theta = Hypotenuse / Perpendicular`

gives

`cot\theta = Base / Perpendicular`

gives

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**Trigonometric Ratios of Complementary Angles Identities**

The complementary angles are a pair of two angles such that their sum is equal to 90°. The complement of an angle θ is (90° - θ). The trigonometric ratios of complementary angles are:

`sin (90°- \theta) = cos \theta`

gives

`cos (90°- \theta) = sin \theta`

gives

`cosec (90°- \theta) = sec \theta`

gives

`sec (90°- \theta) = cosec \theta`

gives

`tan (90°- \theta) = cot \theta`

gives

`cot (90°- \theta) = tan \theta`

gives

.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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