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# Mathematics – 3

AIMS
• To make understand the basic concept and techniques of composition and resolution of vectors and computing the resultant of vectors.
• To enable to use the knowledge of gradient of a straight line in finding speed, acceleration etc.
• To enable to use the knowledge of conic in finding the girder of a railway bridge, cable of a suspension bridge and maximum height of an arch.
• To provide ability to apply the knowledge of differential calculus in solving problem like slope, gradient of a curve, velocity, acceleration, rate of flow of liquid etc.
• To enable to apply the process of integration in solving practical problems like calculation of area of a regular figure in two dimensions and volume of regular solids of different shapes.

SHORT DESCRIPTION
Vector : Addition and subtraction, dot and cross product.
Co-ordinate Geometry : Co-ordinates of a point, locus and its equation, straight lines, circles and conic.
Differential Calculus : Function and limit of a function, differentiation with the help of limit, differentiation of functions, geometrical interpretation of dydx , successive differentiation and Leibnitz theorem, partial differentiation.
Integral Calculus : Fundamental integrals, integration by substitutions, integration by parts, integration by partial fraction, definite integrals.
DETAIL DESCRIPTION
Vector
1 Apply the theorems of vector algebra.
1.1 Define scalar and vector.
1.2 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 fields and vector field.
1.3 Prove the laws of vector algebra.
1.4 Resolve a vector in space along three mutually perpendicular directions
1.5 solve problems involving addition and subtraction of vectors.

2 Apply the concept of dot product and cross product of vectors.
2.1 Define dot product and cross product of vectors.
2.2 Interpret dot product and cross product of vector geometrically.
2.3 Deduce the condition of parallelism and perpendicularity of two vectors.
2.4 Prove the distributive law of dot product and cross product of vector.
2.5 Explain the scalar triple product and vector triple product.
2.6 Solve problems involving dot product and cross product.

CO-ORDINATE GEOMETRY

3 Apply the concept of co-ordinates to find lengths and areas.
3.1 Explain the co-ordinates of a point.
3.2 State different types of co-ordinates of a point.
3.3 Find the distance between two points (x1, y1) and (x2, y2 ).
3.4 Find the co-ordinates of a point which divides the straight line joining two points in certain ratio.
3.5 Find the area of a triangle whose vertices are given.
3.6 Solve problems related to co-ordinates of points and distance formula.

4 Apply the concept of locus.
4.1 Define locus of a point.
4.2 Find the locus of a point.
4.3 Solve problems for finding locus of a point under certain conditions.

5 Apply the equation of straight lines in calculating various parameter.
5.1 Describe the equation x=a and y=b and slope of a straight line.
5.2 Find the slope of a straight line passing through two point (x1, y1,) and (x2, y2 ).
5.3 Find the equation of straight lines:
i) Point slope form.
ii) Slope intercept form.
iii) Two points form.
iv) Intercept form.
v) Perpendicular form.
5.4 Find the point of intersection of two given straight lines.
5.5 Find the angle between two given straight lines.
5.6 Find the condition of parallelism and perpendicularity of two given straight lines.
5.7 Find the distances of a point from a line.

6 Apply the equations of circle, tangent and normal in solving problems.
6.1 Define circle, center and radius .
6.2 Find the equation of a circle in the form:
i) x2 + y2 =a 2
ii) (x  h) 2 + (y  k) 2 =a 2
iii) x 2 + y 2 + 2gx + 2fy + c=0
6.3 Find the equation of a circle described on the line joining (x1, y1) and (x2, y2).
6.4 Define tangent and normal.
6.5 Find the condition that a straight line may touch a circle.
6.6 Find the equations of tangent and normal to a circle at any point.
6.7 Solve the problems related to equations of circle, tangent and normal.
7. Understand conic or conic sections.
7.1 Define conic, focus, directrix and eccentricity.
7.2 Find the equations of parabola, ellipse and hyperbola.
7.3 Solve problems related to parabola, ellipse and hyperbola.

DIFFERENTIAL CALCULUS
FUNCTION AND LIMIT
8. Understand the concept of functions and limits.
8.1 Define constant, variable, function, domain, range and continuity of a function.
8.2 Define limit of a function
8.3 Distinguish between f(x) and f(a).

8.4 Establish i) lim sinxx =1
x

. ii) lim tanxx =1.
x

9. Understand differential co-efficient and differentiation.
9.1 Define differential co-efficient in the form of
dydx = lim f(x+h)-f(x)h
h
9.2 Find the differential co-efficient of algebraic and trigonometrical functions from first principle.

10. Apply the concept of differentiation.
10.1 State the formulae for differentiation:
i) sum or difference
ii) product
iii) quotient
iv) function of function
v) logarithmic function
Find the differential co-efficient using the sum or difference
formula, product formula and quotient formula.
10.2 Find the differential co-efficient function of function and logarithmic function.

11. Apply the concept of geometrical meaning of dydx
11.1 Interpret dydx geometrically.
11.2 Explain dydx under different conditions
11.3 Solve the problems of the type:
A circular plate of metal expands by heat so that its radius increases at the rate of 0.01 cm per second. At what rate is the area increasing when the radius is 700 cm ?
12 Use Leibnitz’s theorem to solve the problems of successive differentiation.
12.1 Find 2nd, 3rd and 4th derivatives of a function and hence find n-th derivatives.
12.2 Express Leibnitz’s theorem
12.3 Solve the problems of successive differentiation and Leibnitz’s theorem.

13 Understand partial differentiation.
13.1 Define partial derivatives.
13.2 State formula for total differential.
13.3 State formulae for partial differentiation of implicit function and homogenous function.
13.4 State Euler’s theorem on homogeneous function.
13.5 Solve the problems of partial derivatives.

INTEGRAL CALCULUS
14 Apply fundamental indefinite integrals in solving problems.
14.1 Explain the concept of integration and constant of integration.
14.2 State fundamental and standard integrals.
14.3 Write down formulae for:
i) Integration of algebraic sum.
ii) Integration of the product of a constant and a function.
14.4 Integrate by method of substitution, integrate by parts and by partial fractions.
14.5 Solve problems of indefinite integration.

15 Apply the concept of definite integrals.
15.1 Explain definite integration.
15.2 Interpret geometrically the meaning of
15.3 Solve problems of the following types:
i) ii)

P* =Practical continuous assessment

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