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**LOGARITHMS**

**Logarithm:** For ant two positive real numbers a, b, and a ≠ 1. If the real number x such then a^{x} = b, then x is called logarithm of b to the base a. it is denoted by

**Standard formulae of logarithms:**

**Logarithmic Function:**

Let a be a positive real number and a ≠ 1. The function f: (o, ∞) → R Defined by f(x) =

**PARTIAL FRACTIONS**

**Fractions:**

If f(x) and g(x) are two polynomials, g(x) ≠ 0, then is called rational fraction.

Ex:

_{ }etc. are rational fractions.

**Proper Fraction:**

A rational fraction is said to be a Proper fraction if the degree of g(x) is greater than the degree of f(x).

Ex:

_{ }etc. are the proper fractions.

**Improper Fraction:**

A rational fraction is said to be an Improper fraction if the degree of g(x) is less than the degree of f(x).

Ex:

_{ }etc. are the Improper fractions.

**Partial Fractions:**

Expressing rational fractions as the sum of two or more simpler fractions is called resolving a given fraction into a partial fraction.

∎ If R(x) = is proper fraction, then

**Case(i)**: – For every factor of g(x) of the form (ax + b)^{ n}, there will be a sum of n partial fractions of the form:

**Case(ii): – **For every factor of g(x) of the form (ax^{2} + bx + c)^{ n}, there will be a sum of n partial fractions of the form:

∎ If R(x) = is improper fraction, then

**Case (i): – **If degree f(x) = degree of g(x), where k is the quotient of the highest degree term of f(x) and g(x).

**Case (ii): – **If f(x) > g(x)

R(x) =

## **MATRICES AND DETERMINANTS**

**Matrix**: A set of numbers arranged in the form of a rectangular array having rows and columns is called Matrix.

•Matrices are generally enclosed by brackets like

•Matrices are denoted by capital letters A, B, C, and so on

•Elements in a matrix are real or complex numbers; real or complex real-valued functions.

**Oder of Matrix:** A matrix having ‘m’ rows and ‘n’ columns is said to be of order m x n read as m by n.

Ex:

**Types Of Matrices**

**Rectangular Matrix: **A matrix in which the no. of rows is not equal to the no. of columns is called a rectangular matrix._{ }

** Square Matrix:** A matrix in which the no. of rows is equal to no. of columns is called a square matrix.

**Principal diagonal (diagonal) Matrix:** If A = [a _{ij}] is a square matrix of order ‘n’ the elements a_{11}, a_{22}, a_{33}, ………. A_{n n} is said to constitute its principal diagonal.

**Trace Matrix:** The sum of the elements of the principal diagonal of a square matrix A is called the trace of the matrix. It is denoted by Tr (A).

**Diagonal Matrix:** If each non-diagonal element of a square matrix is ‘zero’ then the matrix is called a diagonal matrix.

**Scalar Matrix:** If each non-diagonal element of a square matrix is ‘zero’ and all diagonal elements are equal to each other, then it is called a scalar matrix.

**Identity Matrix or Unit Matrix:** If each of the non-diagonal elements of a square matrix is ‘zero’ and all diagonal elements are equal to ‘1’, then that matrix is called unit matrix

**Null Matrix or Zero Matrix:** If each element of a matrix is zero, then it is called a null matrix.

**Row matrix & column Matrix:** A matrix with only one row s called a row matrix and a matrix with only one column is called a column matrix.

**Triangular matrices: **

A square matrix A = [a_{ij}] is said to be upper triangular if a_{ij} = 0 ∀ i > j

A square matrix A = [a_{ij}] is said to be lower triangular matrix a_{ij} = 0 ∀ i < j

**Equality of matrices:**

matrices A and B are said to be equal if A and B are of the same order and the corresponding elements of A and B are equal.

**Addition of matrices:**

If A and B are two matrices of the same order, then the matrix obtained by adding the corresponding elements of A and B is called the sum of A and B. It is denoted by A + B.

**Subtraction of matrices:**

If A and B are two matrices of the same order, then the matrix obtained by subtracting the corresponding elements of A and B is called the difference from A to B.

**Product of Matrices: **

Let A = [a_{ik}]_{mxn} and B = [b_{kj}]_{nxp} be two matrices, then the matrix C = [c_{ij}]_{mxp} where

**Note:** Matrix multiplication of two matrices is possible when no. of columns of the first matrix is equal to no. of rows of the second matrix.

A _{m x n } . B_{p x q} = AB _{mx q}; n = p

**Transpose of Matrix:** If A = [a_{ij}] is an m x n matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of A. It is denoted by A^{I} or A^{T}.

**Note:** (i) (A^{I})^{I} = A (ii) (k A^{I}) = k . A^{I} (iii) (A + B )^{T} = A^{T} + B^{T} (iv) (AB)^{T} = B^{T}A^{T}

**Symmetric Matrix:** A square matrix A is said to be symmetric if A^{T }=A

If A is a symmetric matrix, then A + A^{T} is symmetric.

**Skew-Symmetric Matrix:** A square matrix A is said to be skew-symmetric if A^{T }= -A

If A is a skew-symmetric matrix, then A – A^{T} is skew-symmetric.

** Minor of an element:** Consider a square matrix

the minor element in this matrix is defined as the determinant of the 2×2 matrix obtained after deleting the rows and the columns in which the element is present.

Ex: – minor of a_{3} is = b_{1}c_{2} – b_{2}c_{1}

Minor of b_{2 }is = a_{1}c_{3} – a_{3}c_{1}

**Cofactor of an element**: The cofactor of an element in i ^{th} row and j ^{th} column of A_{3×3 }matrix is defined as its minor multiplied by (- 1) ^{i+j}.

**Properties of determinants: **

If each element of a row (column) f a square matrix is zero, then the determinant of that matrix is zero.

Ex: – A = ⇒ det(A) = 0

If A is a square matrix of order 3 and k is scalar then.

If two rows (columns) of a square matrix are identical (same), then Det. Of that matrix is zero.

Ex: – A = ⇒ det(A) = 0

If each element in a row (column) of a square matrix is the sum of two numbers then its determinant can be expressed as the sum of the determinants.

Ex: –

If each element of a square matrix are polynomials in x and its determinant is zero when x = a, then (x-a) is a factor of that matrix.

For any square matrix A Det(A) = Det (A^{I}).

Det (AB) = Det(A). Det(B).

For any positive integer n Det (A^{n}) = (DetA)^{n}.

**Singular and non-singular matrices:**

A Square matrix is said to be singular if its determinant is is zero, otherwise it is said to be non-singular matrix.

Ex: – A = det(A) = 4 – 4 = 0

∴ A is singular matrix

B =

Det(B) = 4 + 4 = 8≠ 0

∴ B is non-singular

** ****Adjoint of a matrix:** The transpose of the matrix formed by replacing the elements of a square matrix A with the corresponding cofactors is called the adjoint of A.

Let A = and cofactor matrix of A =

Then adj (A) =

** ****Invertible matrix:** Let A be a square matrix, we say that A is invertible if there exists a matrix B such that AB =BA = I, where I is a unit matrix of the same order as A and B.

- (A
^{– I})^{– I} = A
- (A
^{I})^{– I} = (A^{-I})^{I}
- (AB)
^{-I} = B^{-I} A^{-I}
- A
^{-I} =

**Compound Angles**

The algebraic sum of two or more angles is called a ‘compound angle’. Thus, angles A + B, A – B, A + B + C etc., are Compound Angles

For any two real numbers A and B

**⋇ **sin (A + B) = sin A cos B + cos A Cos B

**⋇ **sin (A − B) = sin A cos B − cos A Cos B

**⋇ **cos (A + B) = cos A cos B − sin A sin B

**⋇ **cos (A − B) = cos A cos B + sin A sin B

**⋇ **tan (A + B) =

**⋇ **tan (A − B) =

**⋇ **cot (A + B) =

**⋇** cot (A − B) =

**⋇ **tan ( + A) =

**⋇ **tan ( − A) =

**⋇ **cot (+ A) =

**⋇** cot (− A) =

**⋇ **sin (A + B + C) = ∑sin A cos B cos C − sin A sin B sin C ** **

**⋇ **cos (A + B + C) = cos A cos B cos C− ∑cos A sin B sin C** **

**⋇ **tan (A + B + C) =

⋇ cot (A + B + C) =

⋇ sin (A + B) sin (A – B) = sin^{2 }A – sin^{2} B = cos^{2 }B – cos^{2} A

⋇ cos (A + B) cos (A – B) = cos^{2 }A – sin^{2} B = cos^{2 }B – sin^{2} A

**Multiple and Sub Multiple Angles**

If A is an angle, then its integral multiples 2A, 3A, 4A, … are called ‘multiple angles ‘of A and the multiple of A by fraction like are called ‘submultiple angles.

∎ If is not an add multiple of

**PROPERTIES OF TRIANGLES**

In ∆ABC,

Lengths AB = c; BC = a; AC =b

The area of the triangle is denoted by ∆.

Perimeter of the triangle = 2s = a + b + c

A = ∠CAB; B = ∠ABC; C = ∠BCA.

R is circumradius.

**Sine rule:**

In ∆ABC,

⟹ a = 2R sin A; b = 2R sin B; c = 2R sin C

Where R is the circumradius and a, b, c, are lengths of the sides of ∆ABC.

**Cosine rule:**

In ∆ABC,

**Projection rule:**

In ∆ABC,

a = b cos C + c cos B

b = a cos C + c cos A

c = a cos B + b cos A

**Tangent rule (Napier’s analogy):**

** Area of the triangle:**

In ∆ABC, a, b, and c are sides

S = and area of the triangle

**HYPERBOLIC FUNCTIONS**

**⨂ **

**⨂ **The function f: R→R defined by f(x) = ∀ x ∈ R is called the ‘hyperbolic sin’ function. It is denoted by Sinh x.

∴Sinh x =

Similarly,

**Identities:**

**⨂ **cosh^{2}x – sinh^{2} x = 1

cosh^{2}x = 1 + sinh^{2} x

sinh^{2} x = cosh^{2} x – 1

**⨂ **sech^{2} x = 1 – tanh^{2} x

tanh^{2} x = 1 – sesh^{2} x

**⨂ **cosech^{2} x = coth^{2} x – 1

coth^{2} x = 1 + coth^{2} x

**Addition formulas of hyperbolic functions:**

**⨂ **Sinh (x + y) = Sinh x Cosh y + Cosh x Sinh y

**⨂ **Sinh (x − y) = Sinh x Cosh y − Cosh x Sinh y

**⨂ **Cosh (x + y) = Cosh x Cosh y + Sinh x Sinh y** **

**⨂ **Cosh (x − y) = Cosh x Cosh y − Sinh x Sinh y** **

**⨂ **tanh (x + y) =

**⨂ **tanh (x − y) =

**⨂ **coth (x + y) =

**⨂ **sinh 2x = 2 sinh x cosh 2x** = **

**⨂ **cosh 2x = cosh^{2}x + sinh^{2} x = 2 cosh^{2}x – 1 = 1 + 2 sinh^{2}x =

**⨂ **tanh 2x =

**Inverse hyperbolic functions:**

**⨂ **Sinh^{−1}x = ∀ x ∈ R

**⨂ **Cosh^{−1}x = ∀ x ∈ (1, ∞)

**⨂** Tanh^{−1}x = ∀ < 1

**COMPLEX NUMBERS**

The equation x^{2} + 1 = 0 has no roots in real number system.

∴ scientists imagined a number ‘i’ such that i^{2} = − 1.

**Complex number:**

if x, y are any two real numbers then the general form of the complex number is

z = x + i y; where x real part and y is the imaginary part.

3 + 4i, 2 – 5i, – 3 + 2i are the examples for Complex numbers.

- z = x +i y can be written as (x, y).
- If z
_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}, then
- z
_{1 + }z_{2} = (x_{1} + x_{2}, y_{1} + y_{2}) = (x_{1} + x_{2}) + i (y_{1} + y_{2})
- z
_{1 − }z_{2} = (x_{1} − x_{2}, y_{1} − y_{2}) = (x_{1} − x_{2}) + i (y_{1} − y_{2})
- z
_{1∙ }z_{2} = (x_{1} x_{2} −y_{1} y_{2}, x_{1}y_{2 }+ x_{2}y_{1}) = (x_{1}x_{2} −y_{1} y_{2}) + i (x_{1}y_{2} +x_{2} y_{1})
- z
_{1/ }z_{2 = (}x_{1}x_{2} + y_{1} y_{2}/x_{2}^{2} +y_{2}^{2}, x_{2} y_{1 }– x_{1}y_{2}/ x_{2}^{2} +y_{2}^{2})

= (x_{1}x_{2} + y_{1} y_{2}/x_{2}^{2} +y_{2}^{2}) + i (x_{2} y_{1 }– x_{1}y_{2}/ x_{2}^{2} +y_{2}^{2})

** Multiplicative inverse of complex number:**

** The multiplicative** inverse of the complex number z is 1/z.

z = x + i y then 1/z = x – i y/ x^{2} + y^{2}

**Conjugate complex numbers:**

The complex numbers x + i y, x – i y are called conjugate complex numbers.

Conjugate of z is denoted by

The sum and product of two conjugate complex numbers are real.

If z_{1}, z_{2} are two complex numbers then

** Modulus and amplitude of complex numbers:**

**Modulus: –** If z = x + i y, then the non-negative real number is called modulus of z and it is denoted by or ‘r’.

**Amplitude: – **The complex number z = x + i y represented by the point P (x, y) on the XOY plane. ∠XOP = θ is called amplitude of z or argument of z.

x = r cos θ, y = r sin θ

x^{2} + y^{2} = r^{2} cos^{2}θ + r^{2} sin^{2}θ = r^{2} (cos^{2}θ + sin^{2}θ) = r^{2}(1)

⇒ x^{2} + y^{2} = r^{2}

⇒ r = and = r.

• Arg (z) = tan^{−1}(y/x)

• Arg (z_{1}.z_{2}) = Arg (z_{1}) + Arg (z_{2}) + nπ for some n ∈ { −1, 0, 1}

• Arg(z_{1}/z_{2}) = Arg (z_{1}) − Arg (z_{2}) + nπ for some n ∈ { −1, 0, 1}

Note:

∎ e^{iθ} = cos θ + i sin θ

∎ e^{−iθ} = cos θ − i sin θ

**De- Moiver’s theorem**

For any integer n and real number θ, (cos θ + i sin θ)^{ n} = cos n θ + i sin n θ.

**→** cos α + i sin α can be written as cis α

**→** cis α.cis β= cis (α + β)

**→ **1/cisα = cis(-α)

**→ **cisα/cisβ = cis (α – β)

**⟹** (cos θ + i sin θ)^{ -n} = cos n θ – i sin n θ

**⟹** (cos θ + i sin θ) (cos θ – i sin θ) = cos^{2}θ – i^{2} sin^{2}θ = cos^{2}θ + sin^{2}θ = 1.

**→ **cos θ + i sin θ = 1/ cos θ – i sin θ and cos θ – i sin θ = 1/ cos θ + i sin θ

**⟹** (cos θ – i sin θ)^{ n} = (1/ (cos θ –+i sin θ))^{ n} = (cos θ + i sin θ)^{-n} = cos n θ – i sin n θ

**n**^{th} root of a complex number: let n be a positive integer and z_{0} ≠ 0 be a given complex number. Any complex number z satisfying z ^{n} = z_{0} is called an n^{th} root of z_{0}. It is denoted by z_{0}^{1/n} or

**⟹** let z = r (cos θ + i sin θ) ≠ 0 and n be a positive integer. For k∈ {0, 1, 2, 3…, (n – 1)}

let Then a_{0}, a_{1}, a_{2, …, }a_{n-1} are all n distinct n^{th} roots of z and any n^{th} root of z is coincided with one of them.

**n**^{th} root of unity: Let n be a positive integer greater than 1 and* *

**Note:**

- The sum of the n
^{th} roots of unity is zero.
- The product of n
^{th} roots of unity is (– 1)^{ n – 1}.
- The n
^{th} roots of unity 1, ω, ω^{2}, …, ω^{n-1} are in geometric progression with common ratio ω.

**Cube root of unity: **** **

x^{3} – 1 = 0 ⇒ x^{3} = 1

x =1^{1/3}

cube roots of unity are: 1

ω = , ω^{2} =

ω^{2} +ω + 1 = 0 and ω^{3} = 1

**TRANSFORMATIONS**

For A, B∈ R

⋇ sin (A + B) + sin (A – B) = 2sin A cos B

⋇ sin (A + B) −sin (A – B) = 2cos A sin B

⋇ cos (A + B) + cos (A – B) = 2 cos A cos B

⋇ cos (A + B) − cos (A – B) = − 2sin A sin B

For any two real numbers C and D

⋇ sin C + sin D = 2sin cos

⋇ sin C −sin D= 2cos sin

⋇ cos C + cos D = 2 cos cos

⋇ cos C − cos D = − 2sin sin

If A + B + C = π or 180^{0}, then

⋇ sin (A + B) = sin C; sin (B + C) = sin A; sin (A + C) = sin B

⋇ cos (A + B) = − cos C; cos (B + C) = −cos A; cos (A + C) = − cos B

If A + B + C = 90^{0} or then

⋇ sin = cos; sin = cos ; sin = cos

⋇ cos = sin; cos = sin; cos = sin

If then

⋇ sin (A + B) = cos C; sin (B + C) = cos A; sin (A + C) = cos B

⋇ cos (A + B) = sin C; cos (B + C) = sin A; cos (A + C) = sin B

**INVERSE TRIGONOMETRIC RATIOS**

If A, B are two sets and f: A→ B is a bijection, then f^{-1 }is existing and f^{-1}: B → A is an inverse function.

**Properties of Inverse Trigonometric functions:**

**Solutions of Simultaneous Equations**

**Matrix Inversion Method:**

Let a system of simultaneous equations be

a_{1 }x + b_{1} y + c_{1}z = d_{1}

a_{2 }x + b_{2} y + c_{2}z = d_{2}

a_{3 }x + b_{3} y + c_{3}z = d_{3}

The matrix form of the above equations is

Therefore, the matrix equation is AX = B

If Det A ≠ 0, A^{-1 }is exists

X = A^{-1 }B

By using above Condition, we get the values of x, y and z

This Method is called as Matrix Inversion Method

**Cramer’s Method:**

Let system of simultaneous equations be

a_{1 }x + b_{1} y + c_{1}z = d_{1}

a_{2 }x + b_{2} y + c_{2}z = d_{2}

a_{3 }x + b_{3} y + c_{3}z = d_{3}

∆_{1 }is obtained by replacing the coefficients of x (1^{st} column elements of ∆) by constant values

∆_{2 }is obtained by replacing the coefficients of y (2^{nd} column elements of ∆) by constant values

∆_{3 }is obtained by replacing the coefficients of z (3^{rd} column elements of ∆) by constant values

Now

This method is called Cramer’s Method

**Gauss-Jordan Method:**

Let a system of simultaneous equations be

a_{1 }x + b_{1} y + c_{1}z = d_{1}

a_{2 }x + b_{2} y + c_{2}z = d_{2}

a_{3 }x + b_{3} y + c_{3}z = d_{3}

**Augmented matrix:** The coefficient matrix (A) augmented with the constant column matrix (B) is called the augmented matrix. It is denoted by [AD].

[AD] =

This Matrix is reduced to the standard form ofby using row operations

- Interchanging any two rows
- Multiplying the elements of any two elements by a constant.
- Adding to the elements of one row with the corresponding elements of another row multiplied by a constant.

∴ The solution of a given system of simultaneous equations is x = α, y = β, and z = γ.

**Procedure to get the standard form:**

- Take the coefficient of x as the unity as a first equation.
- If 1 is there in the first-row first column, then make the remaining two elements in the first column zero.
- After that, if one element in R
_{2} or R_{3} is 1, then make the remaining two elements in that column C_{2} or C_{3} as zeroes.
- If any row contains two elements as zeros and only non-zero divide that row elements with the non-zero element to get unity and make the remaining two elements in that column as zeros.

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