Engineering Mathematics Material - Hariganesh.com

Engineering Mathematics

2017

NAME OF THE SUBJECT

: Mathematics – I

SUBJECT CODE

: MA6151

MATERIAL NAME

: University Questions

REGULATION

: R 2013

WEBSITE

: www.hariganesh.com

UPDATED ON

: June 2017

TEXT BOOK FOR REFERENCE

: Sri Hariganesh Publications (Author: C. Ganesan)

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www.hariganesh.com/textbook

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Unit – I (Matrices)  Cayley – Hamilton Theorem 1.

 1 1 1  Show that the matrix  0 1 0  satisfies the characteristics equation and hence find    2 0 3  its inverse.

(Jan 2011),(Jan 2013)

Text Book Page No.: 1.43

2.

1 3 7   Using Cayley-Hamilton theorem, find the inverse of A   4 2 3  . 1 2 1  

(N/D 2011)


3.

1 2 3    Verify Cayley-Hamilton theorem for A   2 1 4  . Hence using it find A1 .  3 1 1    (M/J 2016)

Sri Hariganesh Publications (Ph: 9841168917, 8939331876)

Page 1


4.

2017

 1 2 2    Verify Cayley Hamilton Theorem for the A  2 5 4 . Hence find A1 .    3 7 5    (Jan 2016)

5.

 1 2 3    Verify Cayley Hamilton Theorem for the matrix A   2 4 2  . (A/M 2011)  1 1 2   Text Book Page No.: 1.47

6.

 2 1 1    Find the characteristic equation of the matrix A given A   1 2 1  . Hence find  1 1 2   

A1 and A4 .

(Jan 2009)(Jan 2010)(M/J 2010) (M/J 2013)(N/D 2014)


7.

 1 0 3  Using Cayley – Hamilton theorem, find the inverse of the matrix A   8 1 7  .Text    3 0 8  Book Page No.: 1.54

(N/D 2010)

8.

2 2   1   Using Cayley-Hamilton theorem find A and A , if A   1 3 0  . (Jan 2014)  0 2 1   

9.

Use Cayley – Hamilton theorem to find the value of the matrix given by

1

4

 2 1 1   A  5 A  7 A  3 A  A  5 A  8 A  2 A  I , if the matrix A   0 1 0  .  1 1 2   8

7

6

5

4

3

2


(M/J 2009)

1 0 3    10. Verify Cayley Hamilton theorem for the matrix A   2 1 1  , hence find its A1 .  1 1 1    (M/J 2014)


Page 2


2017

 3 1 1    11. If A  1 5 1 , verify Cayley-Hamilton theorem and hence find A1 .    1 1 3    (M/J 2015)

 2 0 1    12. Verify Cayley Hamilton Theorem for the matrix  0 2 0  and hence find A1 and  1 0 2   

A4 .

(M/J 2012)

Text Book Page No.: 1.54 13.

1 4 3  . Hence find A . 2 3  

Find An using Cayley Hamilton theorem, taking A   Text Book Page No.: 1.52

(Jan 2012)

 Eigenvalues and Eigenvectors of a given matrix 1.

1 1 3 Find the eigenvalues and the eigenvectors of the matrix  1 5 1  .    3 1 1 

2.

 1 1 4    Find all the eigenvalues and eigenvectors of the matrix 3 2 1 .    2 1 1

(M/J 2016)

(Jan 2011)


3.

2 1 1   Find the eigenvalues and eigenvectors of 6 1 0 .    1 2 1

(Jan 2013)


4.

 2 2 1   Find the eigenvalues and eigenvectors of the matrix A   1 3 1  .  1 2 2   Text Book Page No.: 1.21

(M/J 2010),(N/D 2010),(Jan 2012),(Jan 2014)


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5.

2017

 6 2 2    Find the eigenvalues and eigenvectors of 2 3 1 .    2 1 3   

(M/J 2015)


6.

 2 1 1   Find the eigenvalues and eigenvectors of A   1 2 1  .  0 0 1  

7.

 11 4 7    Find the eigenvalues and eigenvectors of the matrix A   7 2 5  .  10 4 6    Text Book Page No.: 1.40

8.

(N/D 2011),(N/D 2016)

  2 2 3    Find the eigenvalues and eigenvectors for the matrix A   2 1 6  .  1 2 0    Text Book Page No.: 1.40

9.

(Jan 2009)

(M/J 2009),(Jan 1010),(M/J 2014)

 2 0 1   Find the eigenvalues and eigenvectors of the matrix A   0 2 0  .  1 0 2   Text Book Page No.: 1.39

 7 2 0    10. Find the eigenvalues and eigenvectors of  2 6 2  .  0 2 5   

(M/J 2013),(Jan 2016)

(N/D 2014)


 Diagonalisation of a Matrix  10 2 5   3  to diagonal form. 1. Reduce the matrix  2 2 5   5 3


(A/M 2017)

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2017

2. The eigenvectors of a 3X3 real symmetric matrix A corresponding to the eigenvalues

2, 3, 6 are  1, 0, 1 , 1,1,1 and  1, 2, 1 respectively. Find the matrix A . T

T

T


(A/M 2011)

 8 6 2    3. If the eigenvalues of A   6 7 4  are 0, 3, 15, find the eigenvectors of A and  2 4 3    diagonalize the matrix A . (Jan 2013) Text Book Page No.: 1.81

 Quadratic form to Canonical form 1.

Reduce the quadratic form 2 x 2  5 y 2  3z 2  4 xy to the Canonical form by orthogonal reduction and state its nature.

(M/J 2010),(Jan 2012)


Reduce the quadratic form 2 x1 x2  2 x1 x3  2 x2 x3 to a canonical form by an orthogonal reduction. Also find its nature.

(A/M 2011)


Reduce the given quadratic form Q to its canonical form using orthogonal transformation. Q  x 2  3 y 2  3z 2  2 yz .

(Jan 2009)


Reduce the quadratic form 2 x1 x2  2 x2 x3  2 x3 x1 into canonical form.(Jan 2013) Text Book Page No.: 1.114

5.

Reduce the quadratic form x12  5 x22  x32  2 x1 x2  2 x2 x3  6 x3 x1 to the canonical form through orthogonal transformation and find its nature.

6.

Reduce the quadratic form x 2  5 y 2  z 2  2 xy  2 yz  6zx into canonical form and hence find its rank.

7.

(M/J 2014)

(M/J 2015)

Reduce the quadratic form 2 x12  x22  x32  2 x1 x2  2 x1 x3  4 x2 x3 to canonical form by an orthogonal transformation. Also find the rank, index, signature and nature of the quadratic form. (N/D 2010)


Page 5


2017


Find a change of variables that reduces the quadratic form 3 x12  5 x22  3 x32  2 x1 x2

2 x1 x3  2 x2 x3 to a sum of squares and express the quadratic form in terms of new variables.

(Jan 2011)


Reduce the quadratic form 3 x 2  5 y 2  3z 2  2 xy  2 yz  2zx into canonical form through orthogonal transformation.

10.

(M/J 2013),(N/D 2014)

Reduce the quadratic form 8 x12  7 x22  3 x32  12 x1 x2  8 x2 x3  4 x3 x1 into canonical form by means of an orthogonal transformation.

(N/D 2011)


Reduce the quadratic form 6 x 2  3 y 2  3z 2  4 xy  2 yz  4zx into a canonical form by an orthogonal reduction. Hence find its rank and nature. (Jan 2014),(Jan 2016),(M/J 2016),(N/D 2016)

12.

Reduce the quadratic form 10 x12  2 x22  5 x32  6 x2 x3  10 x3 x1  4 x1 x2 to a Canonical form through an orthogonal transformation and hence find rank, index, signature, nature and also give n0n – zero set of values for x1 , x2 , x3 (if they exist), that will make the quadratic form zero.

(Jan 2010)


Reduce the quadratic form x12  2 x22  x32  2 x1 x2  2 x2 x3 to the Canonical form through an orthogonal transformation and hence show that is positive semi definite.

Also given a non – zero set of values  x1 , x2 , x3  which makes this quadratic form zero. Text Book Page No.: 1.113 14.

(M/J 2009)

Reduce the quadratic form x 2  y 2  z 2  2 xy  2 yz  2zx to canonical form through an orthogonal transformation. Write down the transformation.

(M/J 2012)


 General Problems 1. Prove that the eigenvalues of a real symmetric matrix are real.


(M/J 2014)

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2017

2. If  is an eigenvalue of a matrix, then prove that

1



is the eigenvalue of A1 . (N/D 2014)

Unit – II (Sequences and Series)  Comparison Test 1 2 3 4 (A/M 2017)     ... . 1.3 3.5 5.7 7.9 1 4 9 2. Test the convergence of the series    ... . (M/J 2014) 4.7.10 7.10.13 10.13.16 1. Test the convergence of the sum

Text Book Page No.: 2.23 3. Show by direct summation of n terms that the series

1 1 1    ... is 1.2 2.3 3.4

convergent. 4. Using comparison test, examine the convergence or divergence of

1 3 5    ... . 1.2.3 2.3.4 3.4.5

(N/D 2014)

(M/J 2015),(Jan 2016),(M/J 2016)


 Integral Test 

1. Find the nature of the series

n n 2

1

 log n 

p

by Cauchy’s integral test.

(M/J 2014)

.

(Jan 2014)

Text Book Page No.: 2.42 

2. Test the convergence of the series

 ne

 n2

n 0

Text Book Page No.: 2.45 


1

1

 n sin  n  .

(M/J 2016)

n 1

 D’Alembert’s Ratio Test 1. Using D’Alembert’s ratio test, examine the convergence or divergence of

x  2 x 2  3 x 3  ... . Text Book Page No.: 2.81


(M/J 2015)

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2017

2. Examine the convergence and the divergence of the following series

1

2 6 14 3 2 n  2 n 1 x  x2  x  ...  n  x   .... 5 9 17 2 1

 x  0 .

(Jan 2014)

Text Book Page No.: 2.72 3. Test the convergence of the series 1 

3 3.6 2 3.6.9 3 x x  x  ... by D’Alembert’s 7 7.10 7.10.13

ratio test.

(N/D 2014)

4. Test the convergence of the series 1 

p

p

p

2 3 4    ... by D’Alembert’s ratio test. 2! 3! 4!


(M/J 2014) 2


3

x x x    ... . 2 1  x 1  x 1  x3


(Jan 2016),(M/J 2016) 


n n 1

 

7. Examine convergence of the series

n xn , x  0 . 1

(N/D 2014)

2

3



n3  1  n .

n 1

(Jan 2016)


 

8. Test the series



n2  1  n .

n 1

(A/M 2017)

 Alternating Series for Leibnitz Test x x2 x3 x4     ....,  0  x  1 . 1. Test the convergence of the series 1  x 1  x2 1  x3 1  x4 Text Book Page No.: 2.95 2. Test the convergence and absolute convergence of the series

1 2 1



1 3 1



1 4 1



1 5 1

(Jan 2014)

 .... .

3. Find the interval of the convergence of the series: x 

(A/M 2017)

x2 2



x3 3



x4 4

 ... .(M/J 2016)

4. Discuss the convergence and the divergence of the following series

1 1 1 1  3  1  2   3 1  2  3   3 1  2  3  4   .... . 3 2 3 4 5

(Jan 2014)

Text Book Page No.: 2.87 5. Test for convergence or divergence of

1 1 1 1     ... . 1.2 3.4 5.6 7.8

(M/J 2015)



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2017

 Absolute and Conditional Convergence cos n for absolute and 2 1 n 1 

1. Determine convergence of an alternating series and test

n

conditional convergence.

(N/D 2014)

cos n . 2 1 n 1 

2. Test for convergence of the series

n

3. Test for absolute convergence of 1 

(A/M 2017)

x x2 x3    ... . 1! 2! 3!

(M/J 2015)


 General Problems 1. Prove that the harmonic series is divergent.

(M/J 2014)

Unit – III (Applications of Differential Calculus)  Radius of Curvature and Circle of curvature 1.

Find the radius of curvature of the curve

a a x  y  a at  ,  . 4 4

(Jan 2009)


a a ,  on x  y  a . 4 4

Find the equation of the circle of curvature at 

(M/J 2010),(N/D 2010),(A/M 2011),(N/D 2011),(Jan 2012),(M/J 2012),(Jan 2014), (N/D 2014),(Jan 2016),(M/J 2016) Text Book Page No.: 3.31 3.

Find the equation of circle of curvature of the parabola y 2  12 x at the point  3, 6  . Text Book Page No.: 3.34

4.

(Jan 2009),(N/D 2016)

Find the equation of circle of curvature of the rectangular hyperbola xy  12 at the point  3, 4  .

(Jan 2010),(A/M 2017)


x2 y2   2 at  2, 3  . (M/J 2014) Find the equation of the circle of curvature of 4 9


Page 9

Engineering Mathematics 6.

7.

2017

Find the center of curvature of the curve y  x 3  6 x 2  3 x  1 at the point  1, 1 . Text Book Page No.: 3.29

(M/J 2013)

Find the center of curvature of x 3  y 3  6 xy at (3, 3) .

(M/J 2015)


 x . c

Find the radius of curvature at the point  0, c  on the curve y  c cosh  Text Book Page No.: 3.15

9.

(M/J 2009)

 x . c

Find the radius of curvature at any point of the catenary y  c cosh  Text Book Page No.: 3.15

10.

(Jan 2016)

 3a 3a  ,  on the curve x 3  y 3  3axy . 2 2  

Find the radius of curvature at the point  Text Book Page No.: 3.27

11.

(N/D 2011)

Find the radius of curvature of the curve x 3  xy 2  6 y 2  0 at  3, 3  .

(M/J 2013)






Find the radius of curvature at the point a cos3  , a sin3  on the curve

x 2/3  y 2/3  a 2/3 .

(M/J 2009),(M/J 2015)


Find the radius of curvature at  a , 0  on y 2 

a3  x3 . x

(Jan 2010),(N/D 2014)


Prove that the radius of curvature of the curve xy 2  a 3  x 3 at the point (a , 0) is Text Book Page No.: 3.26

15.

3a . 2

(N/D 2010),(N/D 2016)

Find the radius of curvature at any point of the cycloid x  a   sin   ,

y  a 1  cos  .

(M/J 2010),(M/J 2012),(Jan 2013),(Jan 2014),(A/M 2017)



Page 10


2017

Find the radius of curvature of the curve x  a  cos t  t sint  ; y  a  sin t  t cost  at ' t ' .

(M/J 2013)


Find the radius of curvature of the curve x  3a cos   a cos 3 ,

y  3a sin  a sin 3 .

(A/M 2011)


Find the radius of curvature at any point on x  e t cos t , y  e t sin t . Text Book Page No.: 3.46

ax 19. If y  , prove that a x

(M/J 2014),(M/J 2016)

 2   a   

2/ 3

2

 x  y       , where  is the radius of  y  x 2

curvature.

(Jan 2012)


 Evolute 1. Show that the evolute of the parabola y 2  4ax is the curve 27ay 2  4( x  2a )3 . Text Book Page No.: 3.48

(Jan 2010),(M/J 2010)

2. Find the equation of the evolute of the parabola y 2  4ax . Text Book Page No.: 3.48 (Jan 2011),(Jan 2012),(M/J 2012),(Jan 2014),(Jan 2016),(M/J 2016) 3. Find the evolute of the parabola x 2  4ay .

(M/J 2013)


x2 y2 4. Find the evolute of the hyperbola 2  2  1 . a b

(N/D 2010),(N/D 2011)

Text Book Page No.: 3.56 5. Find the equation of the evolute of the curve x  a  cos t  t sin t  ,

y  a  sin t  t cos t  .

(M/J 2009),(N/D 2016)



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2017

6. Show that the evolute of the cycloid x  a   sin   , y  a  1  cos  is another cycloid.

(A/M 2011)

Text Book Page No.: 3.61 7. Find the evolute of the cycloid x  a   sin   , y  a  1  cos  .

(N/D 2014)

Text Book Page No.: 3.61 8. Obtain the evoluteof x  a   sin   , y  a  1  cos  .

(M/J 2015)

Text Book Page No.: 3.76 9. Find the evoluteof

x y a.

(M/J 2014)

 Envelope 1. Find the envelope of y  mx  a 2 m 2  b2 , where m is the parameter. Text Book Page No.: 3.79

(Jan 2016)

2. Find the envelope of the family of straight lines y  mx  2am  am 3 , where m is the parameter.

(Jan 2014),(M/J 2016)

Text Book Page No.: 3.85 3. Find the envelope of the family of straight lines x cos   y sin   c sin  cos  ,  being the parameter.

(A/M 2011)


x y   1 , where a and b are parameters that a b are connected by the relation a  b  c . (Jan 2009),(M/J 2009)

4. Find the envelope of the straight line

Text Book Page No.: 3.87 5. Find the envelope of

x y   1 , where a and b are connected by the relation a b

a 2  b2  c 2 , c being constant.

(N/D 2010),(Jan 2013),(M/J 2015),(A/M 2017)



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2017

6. Find the envelope of the family of straight lines

x y   1 , where a and b are a b

connected by a 2  b2  64 .

(N/D 2014)

Text Book Page No.: 3.89 7. Find the envelope of the straight line

x y   1 where the parameters a and b are a b

connected by the relation a n  bn  c n , c being a constant.

(N/D 2011),(M/J 2014)

Text Book Page No.: 3.91 8. Find the envelope of the straight line

x y   1 , where a and b are connected by the a b

relation ab  c 2 , c is a constant.

(Jan 2010),(M/J 2010)

Text Book Page No.: 3.100 9. Find the envelope of the ellipse

x2 y2   1 where a and b are connected by the a 2 b2

relation a 2  b2  c 2 , c being a constant.

(Jan 2014),(N/D 2016)

Text Book Page No.: 3.93 10. Find the envelope of the system of ellipses

x2 y2   1 , where a and b are connected a 2 b2

by the relation ab  4 .

(M/J 2012)


 Evolute as the envelope of normals 1.

Find the evolute of the hyperbola

x2 y2   1 considering it as the envelope of its a 2 b2

normals.

(Jan 2009)


Find the evolute of the ellipse

x2 y2   1 , considering it as the envelope of its a 2 b2

normal.


(A/M 2017)

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2017

Unit – IV (Differential Calculus of Several Variables)  Partial Derivatives 1.

If u  x y , show that uxxy  uxyx .

2.

If u  log x 2  y 2  tan1  y / x  prove that uxx  uyy  0 . (Jan 2009),(N/D 2010)



(Jan 2009)




If u  log  tan x  tan y  tan z  , find

u

 sin 2 x x .

(M/J 2015)

 Euler’s theorem and Total derivatives  x y  u u 1  cot u .  , then prove that x  y x y 2  x  y 

1.

If u  cos 1 

2.

If u  xy  yz  zx where x  , y  e t and z  e  t find

3.

If w  f  y  z , z  x , x  y  , then show that

w w w    0. x y z


(Jan 2014),(Jan 2016),(M/J 2016)

4.

1 t

dy . dt

(A/M 2017)

(M/J 2013)

If z  f ( x, y ) , where x  u2  v 2 , y  2uv , prove that 2 2z 2z 2z  2 2   z   4 u  v   2  2  . u2 v 2 y   x

(Jan 2010),(Jan 2012)


If x  u cos   v sin  , y  u sin   v cos  and V  f ( x, y ) , show that

 2V  2V  2V  2V    . x 2 y 2 u2 v 2

(Jan 2011)


6.

2 2  2 u  2 u 1  u   u        . If u  e , show that x 2 y 2 u  x   y   xy


(Jan 2013)

Page 14


2017


If F is a function of x and y and if x  e u sin v , y  e u cos v , prove that 2  2F  2F  2F  2 u   F   e   2 . x 2 y 2 v 2   u

(Jan 2013)


If u  f ( x, y ) where x  r cos  , y  r sin , prove that 2

1  u   u   u   u   x    y    r   r 2    .         2

2

2

(M/J 2010)


2 2  2u  y 2  u 2  u If u  ( x  y ) f   , then find x .  2 xy y x 2 xy y 2  x

(M/J 2014)

 Taylor’s expansion 1. Find the Taylor’s series expansion of x 2 y 2  2 x 2 y  3 xy 2 in powers of ( x  2) and

( y  1) upto 3rd degree terms.

(Jan 2010),(M/J 2010),(Jan 2012)

Text Book Page No.: 4.54 2. Use Taylor’s formula to expand the function defined by f ( x, y )  x 3  y 3  xy 2 in powers of ( x  1) and ( y  2) .

(A/M 2011),(M/J 2015),(A/M 2017)

Text Book Page No.: 4.68 3. Expand x 2 y  3 y  2 in powers of ( x  1) and ( y  2) upto 3rd degree terms. Text Book Page No.: 4.68

(M/J 2012)

4. Find the Taylor series expansion of e x sin y at the point  1,  / 4  up to 3rddegree terms.

(Jan 2009),(M/J 2009)

Text Book Page No.: 4.58 5. Expand e x sin y in powers of x and y as far as the terms of the 3rd degree using Taylor’s expansion.

(M/J 2013),(Jan 2016),(N/D 2016)



Page 15


2017     4

6. Find the Taylor’s series expansion of e x cos y in the neighborhood of the point  1, upto third degree terms.

(N/D 2010)


 

7. Expand e x cos y at  0,

 2 

upto the third term using Taylor’s series.

(M/J 2014)

Text Book Page No.: 4.68 8. Expand e x log(1  y ) in power of x and y upto terms of third degree using Taylor’s theorem.

(N/D 2011),(Jan 2014),(M/J 2016)


   upto second degree terms using Taylor’s series. (N/D 2014)  2

9. Expand sin xy at  1,


 Maxima and Minima 1. Find the extreme values of the function f ( x, y )  x 3  y 3  3 x  12 y  20 . Text Book Page No.: 4.70

(Jan 2010),(A/M 2011),(Jan 2012),(N/D 2014)

2. Test for maxima and minima of the function f ( x, y )  x 3  y 3  12 x  3 y  20 . Text Book Page No.: 4.111

(M/J 2013)

3. Examine f ( x, y )  x 3  3 xy 2  15 x 2  15 y 2  72 x for extreme values. Text Book Page No.: 4.73

(Jan 2016)

4. Find the maximum and minimum values of x 2  xy  y 2  2 x  y .

(M/J 2012)

Text Book Page No.: 4.72 5. Discuss the maxima and minima of the function f ( x, y )  x 4  y 4  2 x 2  4 xy  2 y 2 . Text Book Page No.: 4.76 6. Test for an extrema of the function f ( x, y )  x 4  y 4  x 2  y 2  1 .

(N/D 2010) (Jan 2011)



Page 16

Engineering Mathematics 7. Examine the extrema of f  x , y   x 2  xy  y 2 

2017 1 1  . x y

(N/D 2016)

8. Examine the function f  x, y   x 3 y 2  12  x  y  for extreme values. (M/J 2009) Text Book Page No.: 4.80 9. Test for the maxima and minima of the function f  x, y   x 3 y 2  6  x  y  .(Jan 2013) Text Book Page No.: 4.112 10. Discuss the maxima and minima of f  x, y   x 3 y 2  1  x  y  .

(Jan 2014)

Text Book Page No.: 4.83 11. Find the maximum value of x m y n z p subject to the condition x  y  z  a . Text Book Page No.: 4.103

(Jan 2009)

12. Find the minimum values of x 2 yz 3 subject to the condition 2 x  y  3z  a . (A/M 2017) 13. Find the extreme value of x 2  y 2  z 2 subject to the condition x  y  z  3a . Text Book Page No.: 4.112

(M/J 2014)

14. A rectangular box open at the top, is to have a volume of 32 cc. Find the dimensions of the box, that requires the least material for its construction. Text Book Page No.: 4.94 (M/J 2010),(N/D 2011),(M/J 2012),(M/J 2016),(A/M 2017) 15. A rectangular box open at the top, is to have a capacity of 108 cu. ms. Find the dimensions of the box requiring the least material for its construction. (Jan 2014) Text Book Page No.: 4.112 16. Find the dimensions of the rectangular box, open at the top, of maximum capacity whose surface area is 432 square meter. (M/J 2013) Text Book Page No.: 4.97 17. Find the volume of the greatest rectangular parallelepiped inscribed in the ellipsoid

x2 y2 z2    1. a 2 b2 c 2

(M/J 2009),(M/J 2015)



Page 17


2017  

18. Find the length of the shortest line from the point  0, 0,

25  to the surface z  xy . 9  (N/D 2014)

19. Find the shortest and longest distances from the point  1, 2, 1 to the sphere

x 2  y 2  z 2  24 .

(N/D 2016)

 Jacobians 1.

Find the Jacobian

 ( x, y, z ) of the transformation x  r sin cos  , y  r sin sin   ( r , ,  )

and z  r cos .

(Jan 2009),(A/M 2011),(Jan 2016),(M/J 2016)


If x  y  z  u, y  z  uv , z  uvw prove that

 ( x, y, z )  u2 v .  ( u, v , w )


(Jan 2010),(Jan 2012)

Find the Jacobian of u  x  y  z , v  xy  yz  zx , w  x 2  y 2  z 2 .(M/J 2015) Text Book Page No.: 4.49

4.

Find the Jacobian of y1 , y2 , y3 with respect to x1 , x2 , x3 if y1 

y3 

x1 x2 . x3

x2 x 3 x x , y2  3 1 , x1 x2 (N/D 2010)


If u 

xy yz  ( u, v , w ) zx ,v  , w , find . z x  ( x, y, z ) y

(Jan 2014),(M/J 2014)



Page 18


2017

Unit – V (Multiple Integrals)  Double integration a

1. Evaluate

a2  x2

  0

a 2  x 2  y 2 dxdy .

(N/D 2016)

0

 Change of order of integration 

1.

e y Evaluate   dxdy by changing the order of integration. y 0 x

(N/D 2010),(A/M 2011)

Text Book Page No.: 5.37  y

2.

Change the order of integration

  ye

 y2 / x

dxdy and hence evaluate it.(N/D 2014)

0 0


3.

Change the order of integration in

2

4 y 2

0

0

 

xy dxdy and evaluate it.

(N/D 2016)

4 a 2 ax

4.

Change the order of integration and hence evaluate it

  0

5.


a

a2  y2

0

a y

 

xy dydx . (A/M 2017)

2

x 4a

y dxdy and then evaluate it. (M/J 2009)

Text Book Page No.: 5.51 1 2 x

6.




xy dxdy and hence evaluate.

0 x2


(Jan 2010),(M/J 2012),(Jan 2014),(Jan 2016),(M/J 2016) a 2a  x

7.

Change the order of integration in the interval

 

xy dydx and hence evaluate it.

2

0 x /a


(M/J 2010),(Jan 2013),(M/J 2014)


Page 19


2017 1 2 y

8.

Change the order of integration and hence find the value of

 0

xy dxdy .(N/D 2011)

y

Text Book Page No.: 5.54 a a

9.


 x

2

0 y

x dxdy and hence evaluate it. (M/J 2013)  y2

Text Book Page No.: 5.35 1 1

10.

By changing the order of integration, evaluate

 x 0 y

2

x dxdy .  y2

(M/J 2015)

Text Book Page No.: 5.35 2 2 a a a  y

11.


 

xy dxdy and hence evaluate it.

0 a a  y 2

(Jan 2011)

2

Text Book Page No.: 5.44 b a

12.




a2  x2

  0



a

x 2 dydx and then evaluate it.(Jan 2012)

0


 Change into polar coordinates a a

1.

Express

 0 y

x 2 dxdy

x

2

 y2 

3/ 2

in polar coordinates and then evaluate it.

(M/J 2009)

Text Book Page No.: 5.100 

2.

Evaluate

e



 x2  y2

 dxdy by converting to polar coordinates. Hence deduce the value

0 0



of

e

 x2

dx .

(Jan 2010),(N/D 2010),(Jan 2014),(Jan 2016),(M/J 2016),(N/D 2016)

0



Page 20

Engineering Mathematics 2

3.

Transform the integral

2017 2 x  x2

  x 0

2

 y 2  dydx into polar coordinates and hence

0

evaluate it.

(A/M 2011),(N/D 2014)


By Transforming into polar coordinates, evaluate

 x2 y2     x 2  y 2  dxdy over annular

region between the circles x 2  y 2  16 and x 2  y 2  4 .

(M/J 2010)


By Transforming into polar coordinates, evaluate



x2 y2 dxdy over annular region x2  y2

between the circles x 2  y 2  a 2 and x 2  y 2  b2 , (b  a ) .

(Jan 2013)

Text Book Page No.: 5.113 a

6.

Transform the double integral

a2  x2

  0

ax  x

dxdy a  x2  y2 2

2

into polar co-ordinates and then

evaluate it.

(Jan 2012)


Transform the integral into polar coordinates and hence evaluate a

a2  x2

  0

x 2  y 2 dydx .

(Jan 2012)

0


 Area as a double integral 1. Find the area bounded by the parabolas y 2  4  x and y 2  x by double integration. Text Book Page No.: 5.68

(N/D 2010)

2. Find, by double integration, the area enclosed by the curves y 2  4ax and x 2  4ay . Text Book Page No.: 5.66

(Jan 2010),(A/M 2011),(M/J 2013)

3. Find, by double integration, the area between the two parabolas 3 y 2  25 x and

5 x2  9 y .


(M/J 2012)

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2017

Text Book Page No.: 5.94 4. Find the area common to y 2  4 x and x 2  4 y using double integration.(N/D 2011) Text Book Page No.: 5.94 5. Using double integral find the area of the ellipse

x2 y2   1 . (M/J 2013),(N/D 2016) a 2 b2

Text Book Page No.: 5.63 6. Evaluate

  xy dxdy over the positive quadrant of the circle x

Text Book Page No.: 5.19 7. Evaluate

2

 y2  a2 .

(Jan 2014), (Jan 2016),(M/J 2016)

  ( x  y) dxdy over the region between the line y  x and the parabola

y  x2 .

(Jan 2011),(A/M 2017)

Text Book Page No.: 5.17 8. Find the smaller of the areas bounded by the ellipse 4 x 2  9 y 2  36 and the straight line 2 x  3 y  6 .

(Jan 2012)

Text Book Page No.: 5.94 9. Find the surface area of the section of the cylinder x 2  y 2  a 2 made by the plane

x y z  a.

(M/J 2014)

10. Find the area inside the circle r  a sin but lying outside the cardioids

r  a  1  cos   .

(Jan 2009)

Text Book Page No.: 5.90 11. Find the area which is inside the circle r  3a cos  and outside the cardioids

r  a  1  cos   .

(Jan 2013)

Text Book Page No.: 5.88 12. Find the area of the cardioid r  a  1  cos   . (M/J 2014),(N/D 2014),(M/J 2015) Text Book Page No.: 5.80


Page 22


2017

 Triple integral a b c

1.

Evaluate

x

2

 y 2  z 2  dxdydz .

(A/M 2017)

0 0 0

log 2 x x  y

2.

Evaluate

 0

0

e x  y  z dxdydz .

(M/J 2009)

0

Text Book Page No.: 5.140 log 2 x x  log y

3.

 

Evaluate

0

0

e x  y  z dzdydx

(M/J 2013)

0

Text Book Page No.: 5.141 a

4.

Evaluate

a2  x2

  0

a2  x2  y2

1



0

a  x  y2  z2 2

0

2

dzdydx .

(N/D 2011)(AUT)


5.

Evaluate

1

1 x 2

1 x 2  y 2

0

0

0

 



dzdydx 1  x2  y2  z2

.

(Jan 2012),(Jan 2013),(M/J 2015)


Using triple integration, find the volume of the sphere x 2  y 2  z 2  a 2 . Book Page No.: 5.146

7.

Text

(N/D 2010)

Find the volume of x 2  y 2  z 2  r 2 using triple integral.

(M/J 2015)


x2 y2 z2 Find the volume of the ellipsoid 2  2  2  1 . a b c

(Jan 2010),(A/M 2011)


Find the volume of the tetrahedron bounded by the plane

x y z    1 and the a b c

coordinate plane x  0, y  0, z  0 .

(M/J 2010),(N/D 2014)



Page 23


Find the value of

2017

 xyz dxdydz through the positive spherical octant for which

x2  y2  z2  a2 . 11.

Evaluate

 x

2

(A/M 2017)

yz dxdydz taken over the tetrahedron bounded by the planes

x  0, y  0, z  0 and

x y z    1. a b c

(Jan 2011)


Evaluate

dzdydx

  x  y  z  1

3

where V is the region bounded by x  0, y  0,

z  0, x  y  z  1 .

(N/D 2011),(Jan 2014),(Jan 2016),(M/J 2016)


Find the volume of the region bounded by the paraboloid z  x 2  y 2 and the plane

z  4.

(M/J 2014)


Textbook for Reference: “ENGINEERING MATHEMATICS - I” Publication: Sri Hariganesh Publications

Author: C. Ganesan

Mobile: 9841168917, 8939331876 To buy the book visit

www.hariganesh.com/textbook

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