Welcome To Basics In Maths

Engineering Mathematics Sem – I Concept

Polytechnic Engineering Maths Feature Image for Sem 1 Concept

Mathematics Notes for Polytechnic SEM – 1 is Designed by the ” Basics in Maths” team. Here we can learn Concepts in Basic Engineering mathematics Polytechnic Sem – I.

 This Material is very Useful for Basic Engineering Mathematics Polytechnic Sem – I Students.

By learning These Notes, Basic Engineering Mathematics Polytechnic Sem – I Students can Write their Exam successfully and fearlessly.


LOGARITHMS

Logarithm: For ant two positive real numbers a, b, and a ≠ 1. If the real number x such then ax = b, then x is called logarithm of b to the base a. it is denoted by Polytechnic SEM - I image 1

Polytechnic SEM - I image 2Standard formulae of logarithms:

Polytechnic SEM - I image 3

Logarithmic Function:

Let a be a positive real number and a ≠ 1. The function f: (o, ∞) → R Defined by f(x) = Polytechnic SEM - I image 4

Polytechnic SEM - I image 5

Polytechnic SEM - I image 6


PARTIAL FRACTIONS

Fractions:

If f(x) and g(x) are two polynomials, g(x) ≠ 0, then Polytechnic SEM - I Partial Fractions1  is called rational fraction.

Ex:

Polytechnic SEM - I Partial Fractions 2   etc.  are rational fractions.

Proper Fraction:

A rational fractionPolytechnic SEM - I Partial Fractions1 is said to be a Proper fraction if the degree of g(x) is greater than the degree of f(x).

Ex:

Polytechnic SEM - I Partial Fractions 3  etc. are the proper fractions.

Improper Fraction:

A rational fractionPolytechnic SEM - I Partial Fractions1 is said to be an Improper fraction if the degree of g(x) is less than the degree of f(x).

Ex:

Polytechnic SEM - I Partial Fractions 4 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) =Polytechnic SEM - I Partial Fractions1  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:

Polytechnic SEM - I Partial Fractions 5

Case(ii): – For every factor of g(x) of the form (ax2 + bx + c) n, there will be a sum of n partial fractions of the form:

Polytechnic SEM - I Partial Fractions 6

∎ If R(x) =Polytechnic SEM - I Partial Fractions1 is improper fraction, then

Case (i): – If degree f(x) = degree of g(x),Polytechnic SEM - I Partial Fractions 6   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) = Polytechnic SEM - I Partial Fractions 8


 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:

Polytechnic SEM - I Matrices 1

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.                 

Polytechnic SEM - I Matrices 2

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

Polytechnic SEM - I Matrices 3

Principal diagonal (diagonal) Matrix: If A = [a ij] is a square matrix of order ‘n’ the elements a11, a22, a33, ………. An n is said to constitute its principal diagonal.

Polytechnic SEM - I Matrices 4

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).

Polytechnic SEM - I Matrices 5

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

Polytechnic SEM - I Matrices 6

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.

Polytechnic SEM - I Matrices 7

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

Polytechnic SEM - I Matrices 29

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

Polytechnic SEM - I Matrices 8

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.

Polytechnic SEM - I Matrices 9

Triangular matrices:

A square matrix A = [aij] is said to be upper triangular if aij = 0   ∀ i > j

A square matrix A = [aij] is said to be lower triangular matrix aij = 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.

Polytechnic SEM - I Matrices 11

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.

Polytechnic SEM - I Matrices 12

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.

Polytechnic SEM - I Matrices 13

Product of Matrices:

Let A = [aik]mxn and B = [bkj]nxp be two matrices, then the matrix C = [cij]mxp   where

Polytechnic SEM - I Matrices 14

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  . Bp x q = AB mx q; n = p

Transpose of Matrix: If A = [aij] 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 AI or AT.

Note: (i) (AI)I = A     (ii) (k AI) = k . AI    (iii)  (A + B )T = AT + BT  (iv)  (AB)T = BTAT

Symmetric Matrix: A square matrix A is said to be symmetric if AT =A

If A is a symmetric matrix, then A + AT is symmetric.

Skew-Symmetric Matrix: A square matrix A is said to be skew-symmetric if AT = -A

If A is a skew-symmetric matrix, then A – AT is skew-symmetric.

 Minor of an element: Consider a square matrix Polytechnic SEM - I Matrices 16

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 a3 is Polytechnic SEM - I Matrices 17    = b1c2 – b2c1

Minor of b2 is   = a1c3 – a3c1

Cofactor of an element: The cofactor of an element in i th row and j th column of A3×3 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 = Polytechnic SEM - I Matrices 19   ⇒ 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 =   Polytechnic SEM - I Matrices 20  ⇒ 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: –  Polytechnic SEM - I Matrices 21

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 (AI).

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

For any positive integer n Det (An) = (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 =  Polytechnic SEM - I Matrices 22    det(A) = 4 – 4 = 0

∴ A is singular matrix

B = Polytechnic SEM - I Matrices 23

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 = Polytechnic SEM - I Matrices 24 and     cofactor matrix of A = Polytechnic SEM - I Matrices 25

Then adj (A) =Polytechnic SEM - I Matrices 26

 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.

  1. (A– I)– I = A
  2. (AI)– I = (A-I)I
  3. (AB)-I = B-I A-I
  4. A-I =Polytechnic SEM - I Matrices 27

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) =Polytechnic SEM - I Compound Angles 1

tan (A − B) =Polytechnic SEM - I Compound Angles 2

cot (A + B) =Polytechnic SEM - I Compound Angles 3

cot (A − B) = Polytechnic SEM - I Compound Angles 4

tan (Polytechnic SEM - I Compound Angles 5 + A) = Polytechnic SEM - I Compound Angles 6

tan ( Polytechnic SEM - I Compound Angles 5− A) =Polytechnic SEM - I Compound Angles 7

cot (Polytechnic SEM - I Compound Angles 5+ A) =Polytechnic SEM - I Compound Angles 8

cot (Polytechnic SEM - I Compound Angles 5− 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) =Polytechnic SEM - I Compound Angles 10

⋇ cot (A + B + C) =Polytechnic SEM - I Compound Angles 11

⋇ sin (A + B) sin (A – B) = sin2 A – sin2 B = cos2 B – cos2 A

⋇ cos (A + B) cos (A – B) = cos2 A – sin2 B = cos2 B – sin2 A

Polytechnic SEM - I Compound Angles 12

 

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.

Polytechnic SEM - I Multiple and Submultiple Angles 1

∎ If Polytechnic SEM - I Multiple and Submultiple Angles 2  is not an add multiple of Polytechnic SEM - I Multiple and Submultiple Angles 3

Polytechnic SEM - I Multiple and Submultiple Angles 4 Polytechnic SEM - I Multiple and Submultiple Angles 5

Polytechnic SEM - I Multiple and Submultiple Angles 8

Polytechnic SEM - I Multiple and Submultiple Angles 7

 

PROPERTIES OF TRIANGLES

In ∆ABC,

Polytechnic SEM - I Properties of triangles 1

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,

Polytechnic SEM - I Properties of triangles 2

⟹ 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,

Polytechnic SEM - I Properties of triangles 3

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):

Polytechnic SEM - I Properties of triangles 4

 Area of the triangle:

Polytechnic SEM - I Properties of triangles 7

In ∆ABC, a, b, and c are sides

S = Polytechnic SEM - I Properties of triangles 5and area of the triangle Polytechnic SEM - I Properties of triangles 11

 

HYPERBOLIC FUNCTIONS

Polytechnic SEM - I Hyperbolic Functions 1

The function f: R→R defined by f(x) =Polytechnic SEM - I Hyperbolic Functions 2 ∀ x ∈ R is called the ‘hyperbolic sin’ function. It is denoted by Sinh x.

∴Sinh x =Polytechnic SEM - I Hyperbolic Functions 2

Similarly,

Polytechnic SEM - I Hyperbolic Functions 3

Identities:

cosh2x – sinh2 x = 1

cosh2x = 1 + sinh2 x

sinh2 x = cosh2 x – 1

sech2 x = 1 – tanh2 x

tanh2 x = 1 – sesh2 x

cosech2 x = coth2 x – 1

coth2 x = 1 + coth2 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) = Polytechnic SEM - I Hyperbolic Functions 4

tanh (x − y) = Polytechnic SEM - I Hyperbolic Functions 5

coth (x + y) = Polytechnic SEM - I Hyperbolic Functions 6

sinh 2x = 2 sinh x cosh 2x =Polytechnic SEM - I Hyperbolic Functions 7

cosh 2x = cosh2x + sinh2 x = 2 cosh2x – 1 = 1 + 2 sinh2x =Polytechnic SEM - I Hyperbolic Functions 8

tanh 2x =Polytechnic SEM - I Hyperbolic Functions 9

 

Inverse hyperbolic functions:

Sinh−1x = Polytechnic SEM - I Hyperbolic Functions 10 ∀ x ∈ R

Cosh−1x = Polytechnic SEM - I Hyperbolic Functions 11  ∀ x ∈ (1, ∞)

Tanh−1x = Polytechnic SEM - I Hyperbolic Functions 15   ∀ < 1

Polytechnic SEM - I Hyperbolic Functions 13

 

COMPLEX NUMBERS

The equation x2 + 1 = 0 has no roots in real number system.

∴ scientists imagined a number ‘i’ such that i2 = − 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 z1 = x1 + i y1, z2 = x2 + i y2, then
  • z1 + z2 = (x1 + x2, y1 + y2) = (x1 + x2) + i (y1 + y2)
  • z1 − z2 = (x1 − x2, y1 − y2) = (x1 − x2) + i (y1 − y2)
  • z1∙   z2 = (x1 x2 −y1 y2, x1y2 + x2y1) = (x1x2 −y1 y2) + i (x1y2 +x2 y1)
  • z1/ z2 = (x1x2 + y1 y2/x22 +y22, x2 y1 – x1y2/ x22 +y22)

= (x1x2 + y1 y2/x22 +y22) + i (x2 y1 – x1y2/ x22 +y22)

  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/ x2 + y2

Conjugate complex numbers:

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

Conjugate of z is denoted byPolytechnic SEM - I Complex Numbers 1

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

If z1, z2 are two complex numbers then

Polytechnic SEM - I Complex Numbers 2

 Modulus and amplitude of complex numbers:

Modulus: – If z = x + i y, then the non-negative real numberPolytechnic SEM - I Complex Numbers 3 is called modulus of z and it is denoted byPolytechnic SEM - I Complex Numbers 4 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

x2 + y2 = r2 cos2θ + r2 sin2θ = r2 (cos2θ + sin2θ) = r2(1)

⇒ x2 + y2 = r2

⇒ r =Polytechnic SEM - I Complex Numbers 3  and Polytechnic SEM - I Complex Numbers 4 = r.

• Arg (z) = tan−1(y/x)

• Arg (z1.z2) = Arg (z1) + Arg (z2) + nπ for some n ∈ { −1, 0, 1}

• Arg(z1/z2) = Arg (z1) − Arg (z2) + nπ for some n ∈ { −1, 0, 1}

Note:

∎ e = 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) = cos2θ – i2 sin2θ = cos2θ + sin2θ = 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

nth root of a complex number: let n be a positive integer and z0 ≠ 0 be a given complex number. Any complex number z satisfying z n = z0 is called an nth root of z0. It is denoted by z01/n orPolytechnic SEM - I Complex Numbers 5

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

letPolytechnic SEM - I Complex Numbers 6 Then a0, a1, a2, …, an-1 are all n distinct nth roots of z and any nth root of z is coincided with one of them.

nth root of unity:  Let n be a positive integer greater than 1 andPolytechnic SEM - I Complex Numbers 7 

Note:

  • The sum of the nth roots of unity is zero.
  • The product of nth roots of unity is (– 1) n – 1.
  • The nth roots of unity 1, ω, ω2, …, ωn-1 are in geometric progression with common ratio ω.

Cube root of unity:      

x3 – 1 = 0 ⇒ x3 = 1

x =11/3

cube roots of unity are: 1 Polytechnic SEM - I Complex Numbers 8    Polytechnic SEM - I Complex Numbers 9

ω = Polytechnic SEM - I Complex Numbers 8 , ω2 =Polytechnic SEM - I Complex Numbers 9

ω2 +ω + 1 = 0 and ω3 = 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 = 2sinPolytechnic SEM - I Transformations 1 cosPolytechnic SEM - I Transformations 2

 

⋇ sin C −sin D= 2cosPolytechnic SEM - I Transformations 1  sinPolytechnic SEM - I Transformations 2

⋇ cos C + cos D = 2 cosPolytechnic SEM - I Transformations 1   cosPolytechnic SEM - I Transformations 2

⋇ cos C − cos D = − 2sinPolytechnic SEM - I Transformations 1   sin Polytechnic SEM - I Transformations 2

If A + B + C = π or 1800, 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 = 900 or Polytechnic SEM - I Transformations 3then

⋇ sinPolytechnic SEM - I Transformations 4   = cosPolytechnic SEM - I Transformations 9;   sinPolytechnic SEM - I Transformations 5   = cosPolytechnic SEM - I Transformations 8 ;  sinPolytechnic SEM - I Transformations 6   = cosPolytechnic SEM - I Transformations 7

⋇ cos Polytechnic SEM - I Transformations 4  = sinPolytechnic SEM - I Transformations 9; cosPolytechnic SEM - I Transformations 5   = sinPolytechnic SEM - I Transformations 8; cosPolytechnic SEM - I Transformations 6   = sinPolytechnic SEM - I Transformations 7

IfPolytechnic SEM - I Transformations 1    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.

Polytechnic SEM - I Inverse Trigonometric Ratios 1

Polytechnic SEM - I Inverse Trigonometric Ratios 2

Properties of Inverse Trigonometric functions:

Polytechnic SEM - I Inverse Trigonometric Ratios 3 Polytechnic SEM - I Inverse Trigonometric Ratios 4 Polytechnic SEM - I Inverse Trigonometric Ratios 5 Polytechnic SEM - I Inverse Trigonometric Ratios 6 Polytechnic SEM - I Inverse Trigonometric Ratios 7

Polytechnic SEM - I Inverse Trigonometric Ratios 8

Solutions of Simultaneous Equations

Matrix Inversion Method:

Let a system of simultaneous equations be

a1 x + b1 y + c1z = d1

a2 x + b2 y + c2z = d2

a3 x + b3 y + c3z = d3

The matrix form of the above equations is

Polytechnic SEM - I Equations In Matrices 1

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

a1 x + b1 y + c1z = d1

a2 x + b2 y + c2z = d2

a3 x + b3 y + c3z = d3

Polytechnic SEM - I Equations In Matrices 2

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

Polytechnic SEM - I Equations In Matrices 3

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

Polytechnic SEM - I Equations In Matrices 4

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

Now Polytechnic SEM - I Equations In Matrices 5

This method is called Cramer’s Method

Gauss-Jordan Method:

Let a system of simultaneous equations be

a1 x + b1 y + c1z = d1

a2 x + b2 y + c2z = d2

a3 x + b3 y + c3z = d3

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

[AD] = Polytechnic SEM - I Equations In Matrices 6

This Matrix is reduced to the standard form ofPolytechnic SEM - I Equations In Matrices 7by using row operations

  1. Interchanging any two rows
  2. Multiplying the elements of any two elements by a constant.
  3. 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:
  1. Take the coefficient of x as the unity as a first equation.
  2. If 1 is there in the first-row first column, then make the remaining two elements in the first column zero.
  3. After that, if one element in R2 or R3 is 1, then make the remaining two elements in that column C2 or C3 as zeroes.
  4. 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.

 

 

PDF FILE: Mathematics Notes 4 Polytechnic SEM – I


Visit my YouTube Channel: Click on Below Logo

My Youtube channel Logo

Leave a Comment

Your email address will not be published.