# Mathematics -1

OBJECTIVES

• To acquaint the students with the basic terminology of Algebra.

• To be able to understand the complex numbers (J-operator) which are being used in electrical engineering

• To be able to understand the binomial expansion.

• To be able to use the knowledge of trigonometry in solving problems of engineering importance.

SHORT DESCRIPTION

Algebra: Set, Indices, Logarithms, AP & GP, Polynomials & polynomial equations, Complex number, Permutation & Combination, Binomial theorem for positive integral Index and negative & fractional index.

Trigonometry: Ratio of associated angles, Compound angles, Transformation formulae, multiple angles and Sub-multiple angles.

DETAIL DESCRIPTION

Algebra:

1 Apply the concept of set in solving problem.

1.1 Define set, sub-set and universal set.

1.2 Define the different types of number set.

1.3 Define union of set, intersection of set, complement of set, power set, disjoint set.

1.4 Prove (using Venn diagram) the relation of following types where A, B and C are any set.

i) AU (BUC) = (AUB) (AUC)

ii) (AUB)c = AcB c

iii) (AB) c = Ac UB c

1.5 Find the number of elements in the union of two sets.

1.6 Solve the problems using above.

2 Apply the laws of indices in solving mathematical problem.

2.1 State the laws of indices.

2.2 Apply the laws of indices to solve the problem.

2.3 Perform algebraic operation on surd.

2.4 Use the scientific calculator in solving the problems of indices.

LOGARITHIMS

3 Apply the concept of logarithms.

3.1 Define logarithm.

3.2 Prove the following laws of logarithm.

a) (m x n) = m + n

b) = m n

c) = n m

d) a X b = 1

e) 1 = O

3.3 Solve problems using 3.2.

3.4 State the difference between Naperion and common logarithms.

4 Understand the concept of AP & GP.

4.1 Define AP and common difference.

4.2 Find last term and sum of n terms, given first term and common difference.

4.3 Define GP and common ratio.

4.4 Find the sum of n terms given first and common ratio.

5 Apply the concept of polynomial in solving the problems.

5.1 Define polynomials and polynomial equation.

5.2 Explain the roots and co-efficient of polynomial equations.

5.3 Find the relation between roots and co-efficient of the polynomial equations.

5.4 Determine the roots and their nature of quadratic polynomial equations.

5.5 Form the equation when the roots of the quadratic polynomial equations are given.

5.6 Find the condition of the common roots of quadratic polynomial equations.

5.7 Solve the problems related to the above.

6 Understand the concept of complex numbers.

6.1 Define complex numbers.

6.2 Perform algebraic operation (addition, subtraction, multiplication, division, square root) with complex number of the form a+jb.

6.3 Find the cube roots of unity.

6.4 Apply the properties of cube root of unity in solving problems.

7 Apply the concept of permutation & Combination.

7.1 Explain permutation.

7.2 Find the number of permutation of n things taken r at a time when,

i) things are all different.

ii) things are not all different.

7.3 Solve problems of the related to permutation :

i) be arranged so that the vowels may never be separated.

From 10 man and 6 women a committee of 7 is to be formed. In how many ways can this be done so as to include at least two women in the committee.

7.4 Explain combination.

7.5 Find the number of combination of n different things taken r at a time.

7.6 Explain , ,

7.7 Find the number of combination of n things taken r at a time in which p particular things

i) Always occur ii) never occur.

7.8 Establish i) =

ii) + =

7.9 Solve problems related to combination.

8 Apply the concept of binomial theorem.

8.1 State binomial expression.

8.2 Find the general term, middle term, equidistant term and term independent of x.

8.3 Use binomial theorem to find the value of

i) (0.9998) 2 , correct to six places of decimal .

ii) (1 + 2 ) 5 (1 2 ) 5

8.4 Express the binomial theorem for negative and fractional index.

8.5 Solve problems of the following types:

Expand (i) (1-nx) -1n (ii)

9 Apply the concept of associated angles.

9.1 Define associated angles.

9.2 Find the sign of trigonometrical function in different quadrants.

9.3 Calculate trigonometrical ratios of associated angle.

9.4 Solve the problems using above.

10 Apply the principle of trigonometrical ratios of compound angles.

10.1 Define compound angles.

10.2 Establish the following relation geometrically for acute angles.

i) sin (A B) = sin A cos B cos A sin B.

ii) cos (A B) = cosA cosB sinAsinB.

10.3 Deduce formula for tan (A B), Cot (A B).

10.4 Apply the identities to work out the problems:

i) find the value of sin 75 0, tan 75 0.

ii) show that

iii) if + = , tan + tan = b, cot + cot = a,

show that (a b) = ab cot .

11 Apply sum and product formula of trigonometrical ratios.

11.1 Express sum or difference of two sines and cosines as a product and vice-versa

11.2 Solve problems of the followings types:

i) show that, sin 55 o + cos 55O = 2 cos 100

ii) prove that, cos 800 cos 600 cos 400 cos 200 =116

12 Apply the concept of ratios of multiple angles.

12.1 State the identities for sin 2A, cos 2A and tan 2A.

12.2 Deduce formula for sin 3A, cos 3A and tan 3A.

12.3 Solve the problems of the followings types.

i) express cos 5 in terms of cos .

ii) if tan = 2 tan , show that, tan ( + ) =

13 Apply the concept of ratios of sub-multiple angles.

13.1 Find mathematically the identities for sin , cos and tan in terms of

13.2 Solve the problems of the type :

find the value of cos 30 , cos 60 , cos 90 , cos 18 0, cos 360, etc.