Avccs

2021-09-03 19:10:49 Channel photo updated Open / Comment

2021-09-03 11:26:55 Mathematics formulas

1. (α+в)²= α²+2αв+в²
2. (α+в)²= (α-в)²+4αв
3. (α-в)²= α²-2αв+в²
4. (α-в)²= (α+в)²-4αв
5. α² + в²= (α+в)² - 2αв.
6. α² + в²= (α-в)² + 2αв.
7. α²-в² =(α + в)(α - в)
8. 2(α² + в²) = (α+ в)² + (α - в)²
9. 4αв = (α + в)² -(α-в)²
10. αв =1. (α + в + ¢)² = α² + в² + ¢² + 2(αв + в¢ + ¢α)
12. (α + в)³ = α³ + 3α²в + 3αв² + в³
13. (α + в)³ = α³ + в³ + 3αв(α + в)
14. (α-в)³=α³-3α²в+3αв²-в³
15. α³ + в³ = (α + в) (α² -αв + в²)
16. α³ + в³ = (α+ в)³ -3αв(α+ в)
17. α³ -в³ = (α -в) (α² + αв + в²)
18. α³ -в³ = (α-в)³ + 3αв(α-в)
ѕιη0° =0
ѕιη30° = 1/2
ѕιη45° = 1/√2
ѕιη60° = √3/2
ѕιη90° = 1
¢σѕ ιѕ σρρσѕιтє σƒ ѕιη
тαη0° = 0
тαη30° = 1/√3
тαη45° = 1
тαη60° = √3
тαη90° = ∞
¢σт ιѕ σρρσѕιтє σƒ тαη
ѕє¢0° = 1
ѕє¢30° = 2/√3
ѕє¢45° = √2
ѕє¢60° = 2
ѕє¢90° = ∞
¢σѕє¢ ιѕ σρρσѕιтє σƒ ѕє¢
2ѕιηα¢σѕв=ѕιη(α+в)+ѕιη(α-в)
2¢σѕαѕιηв=ѕιη(α+в)-ѕιη(α-в)
2¢σѕα¢σѕв=¢σѕ(α+в)+¢σѕ(α-в)
2ѕιηαѕιηв=¢σѕ(α-в)-¢σѕ(α+в)
ѕιη(α+в)=ѕιηα ¢σѕв+ ¢σѕα ѕιηв.
» ¢σѕ(α+в)=¢σѕα ¢σѕв - ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв+ѕιηαѕιηв.
» тαη(α+в)= (тαηα + тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1+ тαηαтαηв)
» ¢σт(α+в)= (¢σтα¢σтв −1) / (¢σтα + ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв + 1) / (¢σтв− ¢σтα)
» ѕιη(α+в)=ѕιηα ¢σѕв+ ¢σѕα ѕιηв.
» ¢σѕ(α+в)=¢σѕα ¢σѕв +ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв+ѕιηαѕιηв.
» тαη(α+в)= (тαηα + тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1+ тαηαтαηв)
» ¢σт(α+в)= (¢σтα¢σтв −1) / (¢σтα + ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв + 1) / (¢σтв− ¢σтα)
α/ѕιηα = в/ѕιηв = ¢/ѕιη¢ = 2я
» α = в ¢σѕ¢ + ¢ ¢σѕв
» в = α ¢σѕ¢ + ¢ ¢σѕα
» ¢ = α ¢σѕв + в ¢σѕα
» ¢σѕα = (в² + ¢²− α²) / 2в¢
» ¢σѕв = (¢² + α²− в²) / 2¢α
» ¢σѕ¢ = (α² + в²− ¢²) / 2¢α
» Δ = αв¢/4я
» ѕιηΘ = 0 тнєη,Θ = ηΠ
» ѕιηΘ = 1 тнєη,Θ = (4η + 1)Π/2
» ѕιηΘ =−1 тнєη,Θ = (4η− 1)Π/2
» ѕιηΘ = ѕιηα тнєη,Θ = ηΠ (−1)^ηα

1. ѕιη2α = 2ѕιηα¢σѕα
2. ¢σѕ2α = ¢σѕ²α − ѕιη²α
3. ¢σѕ2α = 2¢σѕ²α − 1
4. ¢σѕ2α = 1 − ѕιη²α
5. 2ѕιη²α = 1 − ¢σѕ2α
6. 1 + ѕιη2α = (ѕιηα + ¢σѕα)²
7. 1 − ѕιη2α = (ѕιηα − ¢σѕα)²
8. тαη2α = 2тαηα / (1 − тαη²α)
9. ѕιη2α = 2тαηα / (1 + тαη²α)
10. ¢σѕ2α = (1 − тαη²α) / (1 + тαη²α)
11. 4ѕιη³α = 3ѕιηα − ѕιη3α
12. 4¢σѕ³α = 3¢σѕα + ¢σѕ3α

» ѕιη²Θ+¢σѕ²Θ=1
» ѕє¢²Θ-тαη²Θ=1
» ¢σѕє¢²Θ-¢σт²Θ=1
» ѕιηΘ=1/¢σѕє¢Θ
» ¢σѕє¢Θ=1/ѕιηΘ
» ¢σѕΘ=1/ѕє¢Θ
» ѕє¢Θ=1/¢σѕΘ
» тαηΘ=1/¢σтΘ
» ¢σтΘ=1/тαηΘ
» тαηΘ=ѕιηΘ/¢σѕΘ Open / Comment

2021-09-01 08:12:53 You are preparing for? Open / Comment

2021-09-01 03:54:55 Some Basic Concepts of Chemistry

Matter:
Anything that exhibits inertia is called matter. The quantity of matter is its mass.

Classification of Matter:-
Based on chemical composition of various substances.

Elements:
It is the simplest form of the matter.
Smallest unit of an element is known as atom.
Total number of the known elements is 118 out of which 98 elements occur naturally and 20 are formed by artificial transmutation.
Examples: Na, K, Mg. Al, Si, P, C, F, Br etc.


Compound:
It is a non-elemental pure compound.
Formed by chemical combination of two or more atoms of different elements in a fixed ratio.
Examples: H2O, CO2, C6H12O6 etc.

Mixture:
Formed by physical combination of two or more pure substances in any ratio.
Chemical identity of the pure components remains maintained in mixtures.
Homogeneous mixtures are those whose composition for each part remains constant.
Example, Aqueous and gaseous solution.
Heterogeneous mixtures are those whose composition may vary for each and every part.
Example, Soil and concrete mixtures.


Dalton’s Atomic Theory:
Every matter consists of indivisible atoms.
Atoms can neither be created nor destroyed.
Atoms of a given element are identical in properties
Atoms of different elements differ in properties.
Atoms of different elements combine in a fixed ratio to form molecule of a compound.

Precision and Accuracy:
Precision: Closeness of outcomes of different measurements taken for the same quantity.
Accuracy: Agreement of experimental value to the true value

Laws of Chemical Combination:
Law of conservation of mass:
“For any chemical change total mass of active reactants are always equal to the mass of the product formed”

Law of constant proportions:
“A chemical compound always contains same elements in definite proportion by mass and it does not depend on the source of compound”.

Law of multiple proportions:
“When two elements combine to form two or more than two different compounds then the different masses of one element B which combine with fixed mass of the other element bear a simple ratio to one another”

Law of reciprocal proportion:
“ If two elements B and C react with the same mass of a third element (A), the ratio in which they do so will be the same or simple multiple if B and C reacts with each other”.

Gay Lussac’s law of combining volumes:
“At given temperature and pressure the volumes of all gaseous reactants and products bear a simple whole number ratio to each other”. Open / Comment

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2021-08-31 16:13:43 Notes on Quadratic Equations


In order to solve a quadratic equation of the form ax2 + bx + c, we first need to calculate the discriminant with the help of the formula D = b2 – 4ac.

The solution of the quadratic equation ax2 + bx + c= 0 is given by x = [-b ± √ b2 – 4ac] / 2a

If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then we have the following results for the sum and product of roots:

α + β = -b/a

α.β = c/a

α – β = √D/a

It is not possible for a quadratic equation to have three different roots and if in any case it happens, then the equation becomes an identity.

Nature of Roots:

Consider an equation ax2 + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then we have the following cases:

D > 0 iff the roots are real and distinct i.e. the roots are unequal

D = 0 iff the roots are real and coincident i.e. equal

D < 0 iffthe roots are imaginary

The imaginary roots always occur in pairs i.e. if a+ib is one root of a quadratic equation, then the other root must be the conjugate i.e. a-ib, where a, b ∈ R and i = √-1.

Consider an equation ax2 + bx + c = 0, where a, b and c ∈Q and a ≠ 0, then

If D > 0 and is also a perfect square then the roots are rational and unequal.

If α = p + √q is a root of the equation, where ‘p’ is rational and √q is a surd, then the other root must be the conjugate of it i.e. β = p - √q and vice versa.

If the roots of the quadratic equation are known, then the quadratic equation may be constructed with the help of the formula
x2 – (Sum of roots)x + (Product of roots) = 0.

So if α and β are the roots of equation then the quadratic equation is

x2 – (α + β)x + α β = 0

For the quadratic expressiony = ax2 + bx + c, where a, b, c ∈ R and a ≠ 0, then the graph between x and y is always a parabola.

If a > 0, then the shape of the parabola is concave upwards

If a < 0, then the shape of the parabola is concave upwards

Inequalities of the form P(x)/ Q(x) > 0 can be easily solved by the method of intervals of number line rule.

The maximum and minimum values of the expression y = ax2 + bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0.

y ∈[(4ac-b2) / 4a, ∞] if a > 0

If a < 0, then y ∈ [-∞, (4ac-b2) / 4a]

The quadratic function of the form f(x, y) = ax2+by2 + 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh –af2 – bg2 – ch2 = 0

In general, if α1,α2, α3, …… ,αn are the roots of the equation

f(x) = a0xn +a1xn-1 + a2xn-2 + ……. + an-1x + an, then

1.Σα1 = - a1/a0

2.Σ α1α2 = a2/a0

3.Σ α1α2α3 = - a3/a0

……… ……….

Σ α1α2α3 ……αn= (-1)n an/a0

Every equation of nth degree has exactly n roots (n ≥1) and if it has more than n roots then the equation becomes an identity.

If there are two real numbers ‘a’ and ‘b’ such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between ‘a’ and ‘b’.

Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term. Open / Comment

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2021-08-29 07:32:50 Electromagnetic Induction

Electromagnetic induction refers to the production of a voltage or a potential difference across a conductor when it is exposed to a changing magnetic field. The name of Faraday is generally acknowledged with the discovery of induction.

We first discuss the terms which will be used frequently in this topic and later move on to the Faraday Law of electromagnetic Induction.

Flux: Flux is defined as the rate of flow of a property per unit area. For example, the magnitude of a river's current which gives the quantity of water flowing through a cross-section of the river each second is a kind of flux.

Emf: Emf is an abbreviation of electromagnetic force which is the voltage developed by any source of electrical energy such as a battery. It is denoted by E and is measured in volts. The EMF is also given by the rate of change of the magnetic flux:

E = - dφB/ dt,

where E is the electromagnetic force Emf in volts and ΦB is the magnetic flux.

Faraday's laws of electromagnetic induction is a basic law of electromagnetism which describes the interaction of a magnetic field with an electric circuit to produce an emf I. It is the prime operating principle of various kinds of motors and generators.

Christian Oersted’s discovery of magnetic field around a current carrying conductor was quite accidental. If a flow of electric current can produce a magnetic field then why can't a Magnetic field produce an electric current? While searching for an answer to this Michel Faraday ended up inventing generators.

Relationship between Induced Emf and Flux
In this experiment Faraday took a magnet and a coil and connected a galvanometer across the coil. In the beginning the magnet is at rest so there is no deflection in the galvanometer and hence the needle of galvanometer is at center or zero position. When the magnet is moved toward the coil, the needle of galvanometer deflects in one direction. When the magnet is held stationary at that position, the needle of galvanometer returns back to zero position. Now when the magnet is moved away from the coil , there is some deflection in the needle but in opposite direction and again when the magnet becomes stationary at that point with respect to coil , the needle of galvanometer return back to zero position.

Also if the magnet is held stationary and the coil is moved away and towards the magnet, the galvanometer shows deflection in a similar manner. It is also observed that faster the change in the magnetic field, the greater will be the induced emf or voltage in the coil.

The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. This version of the Faraday’s law is valid only when the closed circuit is a loop of indefinitely thin wire. Open / Comment