Engineering Mathematics. 2017. Sri Hariganesh Publications (Ph: 9841168917, 8939331876). Page 7. 2. If β is an eigenval
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)
Text Book Page No.: 1.45
3.
1 2 3 Verify Cayley-Hamilton theorem for A 2 1 4 . Hence using it find A1 . 3 1 1 (M/J 2016)
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2017
1 2 2 Verify Cayley Hamilton Theorem for the A 2 5 4 . Hence find A1 . 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
A1 and A4 .
(Jan 2009)(Jan 2010)(M/J 2010) (M/J 2013)(N/D 2014)
Text Book Page No.: 1.54
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
Text Book Page No.: 1.50
(M/J 2009)
1 0 3 10. Verify Cayley Hamilton theorem for the matrix A 2 1 1 , hence find its A1 . 1 1 1 (M/J 2014)
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3 1 1 11. If A 1 5 1 , verify Cayley-Hamilton theorem and hence find A1 . 1 1 3 (M/J 2015)
2 0 1 12. Verify Cayley Hamilton Theorem for the matrix 0 2 0 and hence find A1 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)
Text Book Page No.: 1.14
3.
2 1 1 Find the eigenvalues and eigenvectors of 6 1 0 . 1 2 1
(Jan 2013)
Text Book Page No.: 1.17
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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6 2 2 Find the eigenvalues and eigenvectors of 2 3 1 . 2 1 3
(M/J 2015)
Text Book Page No.: 1.35
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)
Text Book Page No.: 1.40
Diagonalisation of a Matrix 10 2 5 3 to diagonal form. 1. Reduce the matrix 2 2 5 5 3
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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
Text Book Page No.: 1.85
(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)
Text Book Page No.: 1.95 2.
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)
Text Book Page No.: 1.100 3.
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)
Text Book Page No.: 1.113 4.
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)
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Text Book Page No.: 1.113 8.
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)
Text Book Page No.: 1.113 9.
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)
Text Book Page No.: 1.113 11.
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)
Text Book Page No.: 1.106 13.
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)
Text Book Page No.: 1.114
General Problems 1. Prove that the eigenvalues of a real symmetric matrix are real.
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2. If is an eigenvalue of a matrix, then prove that
1
is the eigenvalue of A1 . (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)
Text Book Page No.: 2.20
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
3. Test the convergence of the series
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
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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!
Text Book Page No.: 2.59
(M/J 2014) 2
5. Test the convergence of the series
3
x x x ... . 2 1 x 1 x 1 x3
Text Book Page No.: 2.62
(Jan 2016),(M/J 2016)
6. Test the convergence of the series
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)
Text Book Page No.: 2.36
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)
Text Book Page No.: 2.97
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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)
Text Book Page No.: 2.108
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)
Text Book Page No.: 3.31 2.
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)
Text Book Page No.: 3.36 5.
x2 y2 2 at 2, 3 . (M/J 2014) Find the equation of the circle of curvature of 4 9
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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)
Text Book Page No.: 3.46 8.
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)
Text Book Page No.: 3.13 12.
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)
Text Book Page No.: 3.17 13.
Find the radius of curvature at a , 0 on y 2
a3 x3 . x
(Jan 2010),(N/D 2014)
Text Book Page No.: 3.26 14.
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)
Text Book Page No.: 3.21
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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)
Text Book Page No.: 3.23 17.
Find the radius of curvature of the curve x 3a cos a cos 3 ,
y 3a sin a sin 3 .
(A/M 2011)
Text Book Page No.: 3.19 18.
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)
Text Book Page No.: 3.24
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)
Text Book Page No.: 3.50
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)
Text Book Page No.: 3.69
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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)
Text Book Page No.: 3.84
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)
Text Book Page No.: 3.89
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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)
Text Book Page No.: 3.95
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)
Text Book Page No.: 3.107 2.
Find the evolute of the ellipse
x2 y2 1 , considering it as the envelope of its a 2 b2
normal.
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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 tan1 y / x prove that uxx uyy 0 . (Jan 2009),(N/D 2010)
(Jan 2009)
Text Book Page No.: 4.6 3.
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
Text Book Page No.: 4.22
(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)
Text Book Page No.: 4.26 5.
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)
Text Book Page No.: 4.29
6.
2 2 2 u 2 u 1 u u . If u e , show that x 2 y 2 u x y xy
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Text Book Page No.: 4.32 7.
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)
Text Book Page No.: 4.33 8.
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)
Text Book Page No.: 4.24 9.
2 2 2u y 2 u 2 u If u ( x y ) f , then find x . 2 xy y x 2 xy 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)
Text Book Page No.: 4.68
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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)
Text Book Page No.: 4.68
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)
Text Book Page No.: 4.61
upto second degree terms using Taylor’s series. (N/D 2014) 2
9. Expand sin xy at 1,
Text Book Page No.: 4.63
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)
Text Book Page No.: 4.111
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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)
Text Book Page No.: 4.100
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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)
Text Book Page No.: 4.44 2.
If x y z u, y z uv , z uvw prove that
( x, y, z ) u2 v . ( u, v , w )
Text Book Page No.: 4.46 3.
(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)
Text Book Page No.: 4.47 5.
If u
xy yz ( u, v , w ) zx ,v , w , find . z x ( x, y, z ) y
(Jan 2014),(M/J 2014)
Text Book Page No.: 4.47
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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
Text Book Page No.: 5.61
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.
Change the order of integration in
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.
Change the order of integration
xy dxdy and hence evaluate.
0 x2
Text Book Page No.: 5.60
(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
Text Book Page No.: 5.47
(M/J 2010),(Jan 2013),(M/J 2014)
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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.
Change the order of integration in
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.
Change the order of integration
xy dxdy and hence evaluate it.
0 a a y 2
(Jan 2011)
2
Text Book Page No.: 5.44 b a
12.
Change the order of integration in
a2 x2
0
a
x 2 dydx and then evaluate it.(Jan 2012)
0
Text Book Page No.: 5.41
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
Text Book Page No.: 5.110
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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)
Text Book Page No.: 5.102 4.
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)
Text Book Page No.: 5.118 5.
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)
Text Book Page No.: 5.106 7.
Transform the integral into polar coordinates and hence evaluate a
a2 x2
0
x 2 y 2 dydx .
(Jan 2012)
0
Text Book Page No.: 5.104
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 .
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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
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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)
Text Book Page No.: 5.133
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)
Text Book Page No.: 5.133 6.
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)
Text Book Page No.: 5.146 8.
x2 y2 z2 Find the volume of the ellipsoid 2 2 2 1 . a b c
(Jan 2010),(A/M 2011)
Text Book Page No.: 5.148 9.
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)
Text Book Page No.: 5.150
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Engineering Mathematics 10.
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)
Text Book Page No.: 5.161 12.
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)
Text Book Page No.: 5.160 13.
Find the volume of the region bounded by the paraboloid z x 2 y 2 and the plane
z 4.
(M/J 2014)
Text Book Page No.: 5.155
Textbook for Reference: “ENGINEERING MATHEMATICS - I” Publication: Sri Hariganesh Publications
Author: C. Ganesan
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