DRONACHARYA COLLEGE OF ENGINEERING, GURGAONDepartment of Applied Sciences And Humanities

Subject: MATHEMATICS – I (MATH-101-F)ASSIGNMENTS

Lecture-1

Introduction to syllabus.

Assignment-1

Nil

Lecture-2

Convergence, divergence of an infinite series.

Assignment-2

Test the nature of the following series:

(i) 12 + 32 + 52 + ………… ∞(ii) 1 / 1.3 + 1/ 3.5 + 1/5.7 + ……….∞

Lecture-3

Convergence, divergence.

Assignment-3

Examine the convergence of the series

1 + 1/42/3 + 1/92/3 + 1/162/3 + ---------------------------

Lecture-4

Convergence of Harmonic series.

Assignment-4

Test the convergence and divergence of the following series

(i) ∑ cot-1 n2

(ii) ∑ (2n3 + 5)/(4n5 + 1)

Lecture-5

D’Alembert’s ratio test.

Assignment-5

Test the convergence and divergence of the following series

(i) ∑ n 1/2 /(n2 + 1)

(ii) ∑ (n2 -1) 1/2 / (n3 +1)

Lecture-6

Rabbes’s test, Logarithmic, Gauss’s tests.

Assignment-6

Discuss the convergence of the following series :

(i) 1+ 2x /2! + (32x2) /3! + (43x3)/4! +…………(ii) x /1.2 + x2 / 3.4 + x3 / 5.6 + …………. (x >0)

Lecture-7

Integral test,Cauchy root tests.

Assignment-7

Discuss the convergence of the following series :

(i) ∑ (1 + nx)n / (n)n

(ii) ∑ ((n + 1) / 3n)n

Lecture-8

Convergence of Alternating series by Leibnitz’s test.

Assignment-8

Test for the convergence of the following series :

(i) 2 – 3/2 + 4/3 – 5/4 + -------------------------------

(ii) ∑(-1)n -1 . n / (2n -1)

Lecture-9

Absolute and conditional convergence.

Assignment-9

Test for the conditional convergence and absolute convergence of the following series :

(i) 1 – ½ + ¼ -1/8 +………………….

(ii) ∑(-)n -1 . n / (n2 +1)

Lecture-10

Elementary Transformations and Inverse using Gauss-Jordan Method

Assignment-10

Use Gauss - Jordan Method to find inverse of matrices:

(i) A = [3 −3 42 −3 40 −1 1]

Lecture-11

Minor of Matrix , Rank of Matrix ,Normal form of Matrix

Assignment-11

Reduce the Matrix into Normal form and also find the Rank of the matrix using Minor

A = [ 8 3 60 2 2

−8 −3 4]Lecture-12

Consistency of linear system of non-homogeneous equation.

Assignment-12

(i)Find the values of a and b for which the equations x + ay +z=3 , x+2y + 2z=b, x+5y + 3z=9 are consistent. When will these equations have unique solution?

(ii)Test for consistency and solve the equation 2x-3y+7z=5,3x+y-3z=13 , 2x+19y-47z=32.

Lecture-13

Consistency of linear system of homogeneous equation.

Assignment-13

Solve the equations x+3y+2z=0, 2x-y+3z=0, 3x-5y+4z=0, x+17y+4z=0.

Lecture-14

Linear dependence and independence of vectors.

Assignment-14

Are the vectors Linearly dependent ? If so find a relation between them.

(i) x1= (1,1,1, 3) x2 = (1, 2, 3, 4 ) x3 = (2,3,4, 9 )

Lecture-15

Linear and Orthogonal transformation

Assignment-15

If =xcos - ysin, = xsin+ ycos ,

write the matrix A of transformation and Prove that A-1 = A’.Hence write the inverse transformation.

Lecture-16

Characteristic Equation, Eigen values and Eigen vectors.

Assignment-16

Find the characteristic eq. , eigen values and eigen vectors of the matrix

A = [ 6 −2 2−2 3 −12 −1 3 ]

Lecture-17

Properties of Eigen values.

Assignment-17

Define properties of eigen values and its proof.

Lecture-18

Caley-Hamilton theorem and its application.

Assignment-18

Verify Cayley -Hamilton theorem and also find its inverse

A = [ 2 −1 1−1 2 −11 −1 2 ]

Lecture-19

Diagonalization of Matries

Assignment-19

by diagonalising the matrix

A= [−1 3−2 4] and find A4

Lecture-20

Quadratic forms

Assignment-20

Reduce the Quadratic corresponding to the matrix

A = [ 2 −1 1−1 2 −11 −1 2 ]

Lecture-21

Successive differentiation.

Assignment-21

(i) If y = aemx + be-mx , show that y2 = m2y(ii) Find the nth derivative of sinx sin3x

Lecture-22

Leibnitz’s theorem and its applications.

Assignment-22

(i) Find the nth derivative of x2ex cosx (ii) If y = cos(logx), prove that x2yn+2 + (2n + 1)xyn+1 +( n2 + 1)yn = 0

Lecture-23

Taylor’s series.

Assignment-23

Using Taylor’s series , prove that

log(x+h) = log x + h / x – h2 / 2x2 + h3 / 3 x3 - …………..

Lecture-24

Maclaurin’s series.

Assignment-24

Expand the sinx using maclaurine’s series.

Prove that ecosx = 1 + x + x2 /2 + x3 /3 +……..

Lecture-25

Curvature and Radius of Curvature for Cartesian curves.

Assignment-25

Find the radius of curvature of the curve y = ex at the point

where it crosses the y – axis.

Show that the radius of curvature r at any point (x,y) of the curve

x2/3 +y2/3 = a2/3 satisfies r3 = 27axy.

Lecture-26

Radius of curvature for polar equations, Radius of curvature at the origin.

Assignment-26

Find the radius of curvature at the origin for the curves :

(i) x3 = a( y2 – x2)

(ii) r = a sin n θ

Lecture-27

Centre of Curvature,Evolute and Involute and chord of curvature.

Assignment-27

Show that the Evolute of the cycloid

x = a(cosθ + θ sinθ) , y = a(sin θ−¿ θ cosθ) is x2 + y2 = a2.

Find the length of chord of curvature through the pole for the curves rn = an cosnθ

Lecture-28

Asymptote parallel to co-ordinate axes and oblique asymptotes.

Assignment-28

Find all the asymptote of curves:

(i) x2 (x – y)2 – a2 (x2 + y2) = 0

(ii) x3 + ax2 = y3

Lecture-29

Asymptote of polar curves.

Assignment-29

Find the asymptotes of the curves :

(i) r = a tan θ(ii) r = a(sec θ + tanθ).

Lecture-30

Function of two or more variables,Partial derivatives .

Assignment-30

If u = xy , show that

∂3u / (∂x2∂ y) = ∂3u / (∂x∂y∂z)

Lecture-31

Euler’s theorem on homogeneous functions.

Assignment-31

Verify Euler’s theorem for the function

U = (x1/4 + y1/4 ) / ( x1/5 y1/5 )

If u = sin-1 {( x2 + y2 ) / (x+y)} prove that

x ∂u / ∂x + y ∂u / ∂y = tanu

Lecture-32

Total differential,Derivatives of composite and implicit function.

Assignment-32

If u = x3 + y3 , where x = a cos t , y = b sin t , find du / dt and

verify the result.

Lecture-33

Jacobians and its properties.

Assignment-33

If u = yz / x , v = zx /y , w = xy / z

Show that ∂ (u,v,w) / ∂(x,y,z)

Lecture-34

Taylor’s theorem for a function of two variables.

Assignment-34

Expand eax sin by in power of x and y as far as the terms of third degree.

Lecture-35

Maxima and minima of function of two variables.

Assignment-35

Show that of all triangles in a circle , the one with maximum area is equilateral.

Lecture-36

Lagrange’s method of undermined multipliers.

Assignment-36

Given x + y + z = a , find the maximum value of xm yn zn.

Lecture-37

Differentiation under integral sign.

Assignment-37

Prove that

∫a

∞

e−x¿)dx/x= log( 1 + a) , (a > -1 )

Lecture-38

Application of Single integration and Surface area of solids of revolution.

Assignment-38

Find the volume generated by revolving about y = 2 the area in

the first quadrant bounded by the parabola 8y = x2 , the y –axis and the line y = 2.

Lecture-39

Double integral double integration in polar coordinates.

Assignment-39

Evaluate the integral

∫0

π

∫0

a sinθ

r dr dθ

Lecture-40

Double integration in polar coordinates.

Assignment-40

Evaluate the integral

∫0

π2

∫0

a cosθ

rsin θdr dθ

Lecture-41

Change of order of integration.

Assignment-41

Evaluate the integral by changing the order of Integration:

∫0

∞

∫x

∞e− y

y dy dx

Lecture-42

Triple integral.

Assignment-42

Evaluate 0∫a 0∫x 0∫x+y e(x + y + z) dzdydx

Lecture-43

Change of Variables ( in double integral or triple integral )

Assignment-43

Transform the integral to Cartesian form and hence evaluate

0∫Л 0∫a r3 sinθ cosθ dr dθ

Lecture-44

Area and Volume as a Double integral.

Assignment-44

Find the area bounded by the circles r = 2 sinθ and r = 4 sinθ

Find the volume bounded by the cylinder x2 + y2 = 4 and the hyperboloid x2 +

y2 - z2 =1

Lecture-45

Volume as a triple integral.

Assignment-45

Find , by triple integration , the volume in the positive octant bounded by the co-ordinate

planes and the plane x +2y + 3z = 4

Lecture-46

Gamma function , Beta function.

Assignment-46

Prove that

0 ∫1 x m (log x ) n dx = (-1)n n ! )/ (m+1) n + 1

Where n is a positive integer and m > 1

Lecture-47

Relationship between Gamma function , Beta function.

Assignment-47

Prove that B (m+1,n) / B (m,n) = m / (m+n)