18
1441
APPLIED MATHEMATICS  I
T P*
C
2
3
3
AIMS


To enable to solve the simultaneous equations with the help of matrix.
To make understand the basic concept and techniques of composition and resolution
of vectors and computing the resultant of vectors.
 To enable to apply knowledge of differential calculus in solving minimum cost of
production, maximum bending moment, rate of flow of liquids, efficiency when
motor does
maximum work.
 To enable to apply the process of integration in solving practical problems like calculation
of area of regular figures in two dimensions and volumes of regular solids of different
shapes.
 To enable to use the knowledge of differential equations to solve problems of
SHM, fluid mechanics, strength of materials, work, energy, power, impedance and
reactance.
SHORT DESCRIPTION
Algebra :
Matrix.
Trigonometry: De Moiver’s theorem.
Geometry:
Three dimensional rectangular co-ordinate system and plane.
Vector :
Addition and subtraction, dot product and cross product.
Calculus :
Expansion of function by Maclaurin’s and Taylor’s Theorem, tangent and normal,
maximum and minimum, application of integration for finding area and volume.
Differential Equations: First order and first degree differential equations,
Fourier’s series
.
ALGEBRA
1
Solve simultaneous equations with the help of matrix.
1.1 Define matrix, null matrix, unit matrix, square matrix, column matrix, row matrix,
inverse matrix, transpose matrix, adjoin matrix, rank of a matrix, singular matrix.
1.2 Explain equality, addition and multiplication of matrix.
1.3 Find the rank of a matrix.
1.4 Solve the problems of the following types :
i)
Solve the given set of linear equations with the help of matrix.
ii)
Find transpose and adjoin matrix of a given matrix.
TRIGONOMETRY
2
Apply De Moiver’s theorem to solve the problems linking with the complex roots.
2.1
State and prove De Moiver’s theorem for positive, negative and fractional index.
2.2
Find the complex roots applying De Moiver’s theorem of the type:
19
ii) 1 i  3
1
i)
x7 + 1 = 0,
GEOMETRY
3
Understand three dimensional co-ordinate system.
3.1
Discuss the co-ordinate axes (three dimensional).
3.2
Find the distance between two points whose co-ordinates are given.
3.3
Find the co-ordinates of a point dividing the join of two points in a given ratio.
3.4
Explain direction cosines and direction ratios .
3.5
Define orthogonal projection .
3.6
Find the projection of two points on a line.
3.7
Find the angle between two lines.
3.8
Find the condition of perpendicularity and condition of parallelism.
3.9
Find the area of a triangle.
3.10 Solve the problems involving 3.2 to 3.8
4
Understand the plane.
4.1
Define the plane.
4.2
Show that the general equation of first degree in x, y, z represents a plane.
4.3
Reduce the general equation of the plane in the intercept and normal form.
4.4
Find the equation of the plane through three given points.
4.5
Find the angles between two planes whose equations are given.
4.6
Find the length of the perpendicular from a point to a given plane.
4.7
Solve the problems related to plane and it’s equations.
VECTOR
5
Apply the theorems of vector algebra.
5.1
Define scalar and vector.
5.2
Explain null vector, free vector, like vector, equal vector, collinear vector, unit
vector, position vector, addition and subtraction of vectors, linear combination,
direction cosines and direction ratios, dependent and independent vectors, scalar
field and vector field.
5.3
Prove the laws of vector algebra.
5.4
Resolve a vector in space along three mutually perpendicular directions .
5.5
Solve problems involving addition and subtraction of vectors.
6
Apply the concept of dot product and cross product of vectors.
6.1
Define dot product and cross product of vectors.
6.2
Interpret dot product and cross product of vector geometrically.
6.3
Deduce the condition of parallelism and perpendicularity of two vectors.
6.4
Prove the distributive law of dot product and cross product of vector.
6.5
Explain the scalar triple product and vector triple product.
6.6
Solve problems involving dot product and cross product.
CALCULUS
7
Apply Maclaurin’s Theorem and Tailor’s Theorem to expand function.
7.1
State Maclaurin’s and Tailor’s theorem.
7.2
Expand the function of the following types by Maclaurin’s Theorem and Tailor’s
theorem :
ln(1+x), sinx, cosx, etc.
20
8
9
10
11.
12.
Apply the concept of tangent and normal to a curve in solving problems.
8.1
Deduce the equation of tangent and normal in the form:
dy
dy
i) Y - y = dx (X - x)
ii) (X - x) + (Y - y) dx =0
8.2
Find the formula for angle between two curves.
8.3
Define sub-tangent and sub-normal.
8.4
Find the length of sub-tangent and sub-normal .
8.5
Solve the problems involving tangent, normal, sub-tangent and sub-normal.
Apply the knowledge of derivatives in finding the maximum and minimum values
of a function.
9.1 Define increasing and decreasing function .
9.2 Understand the maximum and minimum values of a function.
9.3 Discuss the working rule to find the maximum and minimum values of a function.
9.4 Solve the problems involving maximum and minimum values.
Apply the concept of integration to find out area and volume.
10.1 Find the area between two lines.
10.2 Find the area
i)
of a circle
ii)
Bounded by parabola y 2 = 4ax and any double ordinate.
x2
y2
iii)
of quadrant of the ellipse 2 + 2 =1 between major and minor axes.
a
b
10.3 Find the volume of a sphere of certain radius.
10.4 Solve the problems of areas and volumes by integration.
Understand differential equation of first order and first degree.
11.1 Define order and degree of differential equation.
11.2 Solve the following forms of differential equations.
i)
Variable separable.
ii)
Homogeneous equation.
iii)
Exact equations.
iv)
Linear equations.
Understand Fourier's theorem.
12.1 State Fourier's theorem in the form
y=f(x)=Ao+A1sinx+A2sin2x+.........+Ansinnx
+B1cosx+B2cos2x+........+Bncosnx.
12.2 Determine the co-efficient of Fourier's series.
12.3 Find the Fourier's series for square wave form and saw-tooth wave form.
12.4 Solve simple problems using Fourier's series.
P* = Practical continuous assessment .