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ENGINEERING MATHEMATICS – I

C.Ganesan, M.Sc., M.Phil.,

Assistant Professor of Mathematics

Dhanalakshmi College of Engineering

Mobile: 9841168917

Website: www.hariganesh.com

6

2. MA2111 MATHEMATICS – I 3 1 0 4

UNIT I MATRICES 12 Characteristic equation – Eigen values and eigen vectors of a real matrix – Properties – Cayley-Hamilton theorem (excluding proof) – Orthogonal transformation of a symmetric matrix to diagonal form – Quadratic form – Reduction of quadratic form to canonical form by orthogonal transformation. UNIT II THREE DIMENSIONAL ANALYTICAL GEOMETRY 12 Equation of a sphere – Plane section of a sphere – Tangent Plane – Equation of a cone – Right circular cone – Equation of a cylinder – Right circular cylinder. UNIT III DIFFERENTIAL CALCULUS 12 Curvature in Cartesian co-ordinates – Centre and radius of curvature – Circle of curvature – Evolutes – Envelopes – Evolute as envelope of normals. UNIT IV FUNCTIONS OF SEVERAL VARIABLES 12 Partial derivatives – Euler’s theorem for homogenous functions – Total derivatives – Differentiation of implicit functions – Jacobians – Taylor’s expansion – Maxima and Minima – Method of Lagrangian multipliers. UNIT V MULTIPLE INTEGRALS 12 Double integration – Cartesian and polar coordinates – Change of order of integration – Change of variables between Cartesian and polar coordinates – Triple integration in Cartesian co-ordinates – Area as double integral – Volume as triple integral

TOTAL : 60 PERIODS

TEXT BOOK:

1. Bali N. P and Manish Goyal, “Text book of Engineering Mathematics”, Third

edition, Laxmi Publications(p) Ltd.,(2008).

2. Grewal. B.S, “Higher Engineering Mathematics”, 40th

Edition, Khanna Publications, Delhi, (2007).

REFERENCES:

1. Ramana B.V, “Higher Engineering Mathematics”, Tata McGraw Hill

Publishing Company, New Delhi, (2007).

2. Glyn James, “Advanced Engineering Mathematics”, 7th

Edition, Pearson Education, (2007).

3. Jain R.K and Iyengar S.R.K,” Advanced Engineering Mathematics”,

3rd

Edition, Narosa Publishing House Pvt. Ltd., (2007).

Engineering Mathematics Material 2012

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 1

SUBJECT NAME : Engineering Mathematics - I

SUBJECT CODE : MA 2111

MATERIAL NAME : University Questions

MATERIAL CODE : JM08AM1004

Name of the Student: Branch: Unit – I (Matrices)

• Cayley – Hamilton Theorem

1. Find the characteristic equation of the matrix A given

2 1 1

1 2 1

1 1 2

A

−−−− = − −= − −= − −= − − −−−−

. Hence find

1A−−−−ande

4A . (Jan 2009)

2. Show that the matrix

1 1 1

0 1 0

2 0 3

−−−−

satisfies the characteristics equation and hence find

its inverse. (Jan 2011)(AUT)

3. Using Cayley-Hamilton theorem, find the inverse of

1 3 7

4 2 3

1 2 1

A

====

. (N/D 2011)(AUT)

4. Using Cayley – Hamilton theorem, find the inverse of the matrix

1 0 3

8 1 7

3 0 8

A

−−−− ==== −−−−

.

(N/D 2010)

5. Using Cayley – Hamilton theorem, find 1A−−−−

when

2 1 2

1 2 1

1 1 2

A

−−−− = − −= − −= − −= − − −−−−

. (M/J 2010)



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

8 7 6 5 4 3 25 7 3 5 8 2A A A A A A A A I− + − + − + − +− + − + − + − +− + − + − + − +− + − + − + − + , if the matrix

2 1 1

0 1 0

1 1 2

A

====

.

(M/J 2009)

7. Verify Cayley Hamilton Theorem and hence find 1A−−−−

for

2 1 1

1 2 1

1 1 2

A

−−−− = − −= − −= − −= − − −−−−

.

(Jan 2010)

8. Verify Cayley Hamilton Theorem for the matrix

1 2 3

2 4 2

1 1 2

A

−−−− = −= −= −= − −−−−

. (A/M 2011)

9. Verify Cayley Hamilton Theorem for the matrix

2 0 1

0 2 0

1 0 2

−−−− −−−−

and hence find 1A−−−−

and

4A . (M/J 2012)

10. Find nA using Cayley Hamilton theorem, taking A

====

1 4

2 3. Hence find A3

.

(Jan 2012)

• Eigen Values and Eigen Vectors of a given matrix

1. Find the eigen values and eigen vectors of

2 1 1

1 2 1

0 0 1

A

====

. (Jan 2009)

2. Find all the eigenvalues and eigenvectors of the matrix

1 1 4

3 2 1

2 1 1

−−−− −−−− −−−−

. (Jan 2011)(AUT)

3. Find the eigen values and eigen vectors of the matrix

11 4 7

7 2 5

10 4 6

A

− −− −− −− − = − −= − −= − −= − − − −− −− −− −

.

(N/D 2011)(AUT)



4. Find the eigen values and eigen vectors of the matrix

2 2 1

1 3 1

1 2 2

A

====

.

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

5. Find the eigen values and eigen vectors for the matrix

2 2 3

2 1 6

1 2 0

A

− −− −− −− − = −= −= −= − − −− −− −− −

.

(M/J 2009),(Jan 1010)

• Diagonalisation of a Matrix

1. The eigen vectors of a 3X3 real symmetric matrix A corresponding to the eigen-values

2,3,6 are (((( )))) (((( ))))1,0, 1 , 1,1,1T T−−−− and (((( ))))1,2, 1

T−−−− respectively. Find the matrix A .

(A/M 2011)

• Quadratic form to Canonical form

1. Reduce the given quadratic form Q to its canonical form using orthogonal

transformation. 2 2 23 3 2Q x y z yz= + + −= + + −= + + −= + + − . (Jan 2009)

2. Reduce the quadratic form 2 2 22 5 3 4x y z xy+ + ++ + ++ + ++ + + to the Canonical form by orthogonal

reduction and state its nature. (M/J 2010),(Jan 2012)

3. Reduce the quadratic form 1 2 1 3 2 32 2 2x x x x x x+ −+ −+ −+ − to a canonical form by an

orthogonal reduction. Also find its nature. (A/M 2011)

4. Reduce the quadratic form 2 2 21 2 3 1 2 1 3 2 32 2 2 4x x x x x x x x x+ + + − −+ + + − −+ + + − −+ + + − − to canonical form

by an orthogonal transformation. Also find the rank, index, signature and nature of the

quadratic form. (N/D 2010)

5. Find a change of variables that reduces the quadratic form 2 2 21 2 3 1 23 5 3 2x x x x x+ + −+ + −+ + −+ + −

1 3 2 32 2x x x x+ −+ −+ −+ − to a sum of squares and express the quadratic form in terms of new

variables. (Jan 2011)(AUT)

6. Reduce the quadratic form 2 2 21 2 3 1 2 2 3 3 18 7 3 12 8 4x x x x x x x x x+ + − − ++ + − − ++ + − − ++ + − − + into canonical

form by means of an orthogonal transformation. (N/D 2011)(AUT)



7. Reduce the quadratic form 2 2 21 2 3 1 2 2 32 2 2x x x x x x x+ + − ++ + − ++ + − ++ + − + to the Canonical form

through an orthogonal transformation and hence show that is positive semi definite.

Also given a non – zero set of values (((( ))))1 2 3, ,x x x which makes this quadratic form zero.

(M/J 2009)

8. Reduce the quadratic form 2 2 21 2 3 2 3 3 1 1 210 2 5 6 10 4x x x x x x x x x+ + + − −+ + + − −+ + + − −+ + + − − to a

Canonical form through an orthogonal transformation and hence find rank, index,

signature, nature and also give n0n – zero set of values for 1 2 3, ,x x x (if they exist), that

will make the quadratic form zero. (Jan 2010)

9. Reduce the quadratic form 2 2 2 2 2 2x y z xy yz zx+ + − − −+ + − − −+ + − − −+ + − − − to canonical form through

an orthogonal transformation. Write down the transformation. (M/J 2012)

Unit – II (Three Dimensional Analytical Geometry)

• Sphere

1. Find the equation of the sphere passing through the points

(((( )))) (((( ))))0,0,0 , 0,1, 1 ,−−−− (((( ))))1,2,0−−−− and (((( ))))1,2,3 . (N/D 2011)(AUT)

2. Obtain the equation of the sphere having the circle 2 2 2 9x y z+ + =+ + =+ + =+ + = , 3x y z+ + =+ + =+ + =+ + = as

a great circle. (Jan 2009)

3. Obtain the equation of the sphere having the circle x y z y z+ + + − − =+ + + − − =+ + + − − =+ + + − − =2 2 2 10 4 8 0,

x y z+ + =+ + =+ + =+ + = 3 as the greatest circle. (Jan 2012),(M/J 2012)

4. Find the equation to the sphere passing through the circle 2 2 2 9,x y z+ + =+ + =+ + =+ + =

1x y z+ + =+ + =+ + =+ + = and cuts orthogonally the sphere

2 2 2 2 4 16 17 0x y z x y z+ + + − − + =+ + + − − + =+ + + − − + =+ + + − − + = . (M/J 2010)

5. Find the equation of the sphere passing through the circle

2 2 2 3 2 1 0,x y z x y z+ + + − + − =+ + + − + − =+ + + − + − =+ + + − + − = 2 5 7 0x y z+ − + =+ − + =+ − + =+ − + = and cuts orthogonally the

sphere 2 2 2 3 5 7 6 0x y z x y z+ + − + − − =+ + − + − − =+ + − + − − =+ + − + − − = . (N/D 2010)

6. Find the equation of the sphere having its centre on the plane 4 5 3x y z− − =− − =− − =− − = and

passing through the circle 2 2 2 2 3 4 8 0 ;x y z x y z+ + − − + + =+ + − − + + =+ + − − + + =+ + − − + + = 2 8x y z− + =− + =− + =− + = .

(N/D 2010)



7. Find the equation of the sphere described on the line joining the points (((( ))))2, 1,4−−−− and

(((( ))))2,2, 2− −− −− −− − as diameter. Find the area of the circle in which this sphere is cut by the

plane 2 3x y z+ − =+ − =+ − =+ − = . (Jan 2009)

8. Find the equation of the sphere of radius 3 and whose centre lies on the

line1 1

1 2 2x y z− −− −− −− −= == == == = at a distance 2 from (((( ))))1,1,0 . (A/M 2011)

9. Find the equations of the spheres which pass through the circle 2 2 2 5x y z+ + =+ + =+ + =+ + = and

2 3 3x y z+ + =+ + =+ + =+ + = and touch the plane 4 3 15x y+ =+ =+ =+ = . (M/J 2009)

10. Find the equation of the sphere which passes through the circle

2 2 2 2 4 0x y z x y+ + − − =+ + − − =+ + − − =+ + − − = , 2 3 0x y z+ + =+ + =+ + =+ + = and touch the plane 4 3 25x y+ =+ =+ =+ = .

(Jan 2011)(AUT)

11. Show that the plane 2 2 12 0x y z− + + =− + + =− + + =− + + = touches the sphere

2 2 2 2 4 2 3x y z x y z+ + − − + =+ + − − + =+ + − − + =+ + − − + = and find also the point of contact.

(M/J 2009), (N/D 2011)(AUT)

12. Find the two tangent planes to the sphere 2 2 2 4 2 6 5 0x y z x y z+ + − + − + =+ + − + − + =+ + − + − + =+ + − + − + = , which

are parallel to the plane 4 8 0x y z+ + =+ + =+ + =+ + = . Find their point of contact.

(Jan 2010),(M/J 2012)

13. Obtain the equation of the tangent planes to the sphere

x y z x y z+ + + − + − =+ + + − + − =+ + + − + − =+ + + − + − =2 2 2 2 4 6 7 0, which intersect in the line

x y z− − = = +− − = = +− − = = +− − = = +6 3 23 0 3 2. (Jan 2012)

14. Find the equation of the tangent lines to the circle

2 2 23 3 3 2 3 4 22 0x y z x y z+ + − − − − =+ + − − − − =+ + − − − − =+ + − − − − = , 3 4 5 26 0x y z+ + − =+ + − =+ + − =+ + − = at the point

(1,2,3). (Jan 2011)(AUT)

15. Find the centre, radius and area of the circle given by

2 2 2 2 2 4 19 0x y z x y z+ + + − − − =+ + + − − − =+ + + − − − =+ + + − − − = , 2 2 7 0x y z+ + + =+ + + =+ + + =+ + + = . (Jan 2010),(M/J 2010)

• Cone

1. Find the equation of the right circular cone whose vertex is at the origin and axis is the

line 1 2 3x y z= == == == = and which has semi vertical angle of 30°. (Jan 2009),(N/D 2010)



2. Find the equation of the right circular cone generated by revolving the line

0, 0x y z= − == − == − == − = about the axis 0, 2x z= == == == = . (M/J 2009)

3. Find the equation of the right circular cone generated when the straight line which is

the intersection of the planes 2 3 6y z+ =+ =+ =+ = and 0x ==== revolves about the z axis−−−− with

constant angle. (Jan 2011)(AUT),(M/J 2012)

4. Find the equation of the cone whose vertex is (((( ))))1,2,3 and whose guiding curve is the

circle2 2 2 4, 1x y z x y z+ + = + + =+ + = + + =+ + = + + =+ + = + + = . (M/J 2009), (N/D 2011)(AUT)

5. Find the equation of the cone with vertex at (((( ))))1,1,1 and passing through curve of

intersection of 2 2 2 1x y z+ + =+ + =+ + =+ + = and 1x y z+ + =+ + =+ + =+ + = . (A/M 2011)

6. Find the equation of the cone formed by rotating the line 2 3 5,x y+ =+ =+ =+ = 0z ==== about the

y – axis. (Jan 2010)

7. Find the equation of the cone whose vertex is the point (((( ))))1,1,0 and whose base is the

curve2 20, 4y x z= + == + == + == + = . (M/J 2010)

8. Find the equation of the cone formed by rotating the line x y+ =+ =+ =+ =2 3 6, 0z ==== about the

y – axis. (Jan 2012)

• Cylinder

1. Find the equation of the right circular cylinder whose axis is the line 2x y z= = −= = −= = −= = − and

radius 4. (Jan 2009)

2. Find the equation of the right circular cylinder of radius 3 and axis

1 3 52 2 1

x y z− − −− − −− − −− − −= == == == =−−−−

. (Jan 2010),(M/J 2010),(A/M 2011),(M/J 2012)

3. Find the equation of the right circular cylinder whose axis is 1 2 3

2 1 2x y z− − −− − −− − −− − −= == == == = and

radius 2. (N/D 2010),(Jan 2012)

4. Find the equation of the right circular cylinder of radius 5 whose axis is the line

1 2 32 1 2

x y z− − −− − −− − −− − −= == == == = . (N/D 2011)(AUT)

5. Find the equation of the right circular cylinder which passes through the circle

2 2 2 9, 3x y z x y z+ + = − + =+ + = − + =+ + = − + =+ + = − + = . (Jan 2011)(AUT)



Unit – III (Differential Calculus)

• Radius of Curvature and Circle of curvature

1. Find the radius of curvature of the curve x y a+ =+ =+ =+ = at ,4 4a a

. (Jan 2009)

2. Find the circle of curvature at ,4 4a a

on x y a+ =+ =+ =+ = .

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

3. Find the equation of circle of curvature of the parabola 2 12y x==== at the point (((( ))))3,6 .

(Jan 2009)

4. Find the equation of circle of curvature of the rectangular hyperbola 12xy ==== at the

point (((( ))))3,4 . (Jan 2010)

5. Find the radius of curvature at the point (((( ))))0,c on the curve coshx

y cc

====

.

(M/J 2009)

6. Find the radius of curvature at the point3 3

,2 2a a

on the curve 3 3 3x y axy+ =+ =+ =+ = .

(N/D 2011)(AUT)

7. Find the radius of curvature at the point (((( ))))3 3cos , sina aθ θθ θθ θθ θ on the curve

2/3 2/3 2/3x y a+ =+ =+ =+ = . (M/J 2009)

8. Find the radius of curvature at (((( )))),0a on

3 32 a x

yx−−−−==== . (Jan 2010)

9. Prove that the radius of curvature of the curve 2 3 3xy a x= −= −= −= − at the point ( ,0)a is

32a

.

(N/D 2010)

10. Find the radius of curvature at any point of the cycloid (((( ))))sinx a θ θθ θθ θθ θ= += += += + ,

(((( ))))1 cosy a θθθθ= −= −= −= − . (M/J 2010),(M/J 2012)



11. Find the radius of curvature of the curve 3 cos cos 3 ,x a aθ θθ θθ θθ θ= −= −= −= −

3 sin sin 3y a aθ θθ θθ θθ θ= −= −= −= − . (A/M 2011)

12. If ax

ya x

====++++

, prove that x y

a y xρρρρ = += += += +

22/3 22

, where ρρρρ is the radius of

curvature. (Jan 2012)

• Evolute

1. Show that the evolute of the parabola 2 4y ax==== is the curve

2 327 4( 2 )ay x a= −= −= −= − .

(Jan 2010),(M/J 2010)

2. Find the equation of the evolute of the parabola2 4y ax==== .

(Jan 2011)(AUT),(Jan 2012),(M/J 2012)

3. Find the evolute of the hyperbola

2 2

2 2 1x ya b

− =− =− =− = . (N/D 2010),(N/D 2011)(AUT)

4. Obtain the equation of the evolute of the curve (((( ))))cos sinx a θ θ θθ θ θθ θ θθ θ θ= += += += + ,

(((( ))))sin cosy a θ θ θθ θ θθ θ θθ θ θ= −= −= −= − . (M/J 2009)

5. Show that the evolute of the cycloid (((( ))))sinx a θ θθ θθ θθ θ= −= −= −= − , (((( ))))1 cosy a θθθθ= −= −= −= − is another

cycloid. (A/M 2011)

• Envelope

1. Find the envelope of the family of straight lines cos sin sin cosx y cα α α αα α α αα α α αα α α α+ =+ =+ =+ = ,

αααα being the parameter. (A/M 2011)

2. Find the envelope of the straight line 1x ya b

+ =+ =+ =+ = , where a and b are parameters that

are connected by the relation a b c+ =+ =+ =+ = . (Jan 2009),(M/J 2009)

3. Find the envelope of 1x ya b

+ =+ =+ =+ = , where a and b are connected by the relation

2 2 2a b c+ =+ =+ =+ = , c being constant. (N/D 2010)


+ =+ =+ =+ = where the parameters a and b are

connected by the relationn n na b c+ =+ =+ =+ = , c being a constant. (N/D 2011)(AUT)



5. Find the envelope of x yl m

+ =+ =+ =+ = 1, where the parameters l and m are connected by the

relation l ma b

+ =+ =+ =+ = 1( a and b are constants). (Jan 2012)


+ =+ =+ =+ = , where a and b are connected by the

relation 2ab c==== , c is a constant. (Jan 2010),(M/J 2010)

7. Find the envelope of the system of ellipses

2 2

2 2 1x ya b

+ =+ =+ =+ = , where a and b are connected

by the relation 4ab ==== . (M/J 2012)

• Evolute as the envelope of normals

1. Find the evolute of the hyperbola

2 2

2 2 1x ya b

− =− =− =− = considering it as the envelope of its

normals. (Jan 2009)

Unit – IV (Functions of several variables)

• Euler’s Theorem

1. If yu x==== , show that xxy xyxu u==== . (Jan 2009)

2. If (((( )))) (((( ))))2 2 1log tan /u x y y x−−−−= + += + += + += + + prove that 0xx yyu u+ =+ =+ =+ = .

(Jan 2009),(N/D 2010)

3. If 1cos

x yu

x y−−−− ++++====

++++, prove that

1cot

2u u

x y ux y

∂ ∂ −∂ ∂ −∂ ∂ −∂ ∂ −+ =+ =+ =+ =∂ ∂∂ ∂∂ ∂∂ ∂

. (N/D 2011)(AUT)

4. If 1sin

x yu

x y−−−− ++++====

++++, prove that

2 2 22 2

2 2 3

sin cos 22

4cosu u u u u

x xy yx x y y u

∂ ∂ ∂ −∂ ∂ ∂ −∂ ∂ ∂ −∂ ∂ ∂ −+ + =+ + =+ + =+ + =∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂

.

(A/M 2011)

5. If

2 21sin

x yu

x y−−−− ++++==== ++++

, prove that (1) tanu u

x y ux y

∂ ∂∂ ∂∂ ∂∂ ∂+ =+ =+ =+ =∂ ∂∂ ∂∂ ∂∂ ∂

and (2)

2 2 22 2 2

2 22 tanu u u

x xy y ux x y y

∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂+ + =+ + =+ + =+ + =∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂

. (Jan 2011)(AUT)



• Total derivatives

1. If , ,x y z

u fy z x

====

, prove that 0

u u ux y z

x y z∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂+ + =+ + =+ + =+ + =∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂

. (M/J 2009)

2. If ( , )z f x y==== , where 2 2, 2x u v y uv= − == − == − == − = , prove that

(((( ))))2 2 2 2

2 22 2 2 24z z z z

u vu v x y

∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂+ = + ++ = + ++ = + ++ = + + ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂ . (Jan 2010),(Jan 2012)

3. If cos sin , sin cosx u v y u vα α α αα α α αα α α αα α α α= − = += − = += − = += − = + and ( , )V f x y==== , show that

2 2 2 2

2 2 2 2

V V V Vx y u v

∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂+ = ++ = ++ = ++ = +∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂

. (Jan 2011)(AUT)

4. If ( , )u f x y==== where cos , sinx r y rθ θθ θθ θθ θ= == == == = , prove that

22 2 2

2

1u u u ux y r r θθθθ

∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂ + = ++ = ++ = ++ = + ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂ . (M/J 2010)

5. If2 2 2u x y z= + += + += + += + + and

2 2 2, cos 3 , sin 3t t tx e y e t z e t= = == = == = == = = , Find dudt

.

(N/D 2011)(AUT)

• Taylor’s expansion

1. Find the Taylor series expansion of sinxe y at the point (((( ))))1, / 4ππππ−−−− up to 3rd

degree

terms. (Jan 2009),(M/J 2009)

2. Find the Taylor’s series expansion of cosxe y in the neighborhood of the point 1,4ππππ

upto third degree terms. (N/D 2010)

3. Expand log(1 )xe y++++ in power of x and y upto terms of third degree using Taylor’s

theorem. (N/D 2011)(AUT)

4. Find the Taylor’s series expansion of 2 2 2 22 3x y x y xy+ ++ ++ ++ + in powers of ( 2)x ++++ and

( 1)y −−−− upto 3rd

degree terms. (Jan 2010),(M/J 2010),(Jan 2012)

5. Use Taylor’s formula to expand the function defined by 3 3 2( , )f x y x y xy= + += + += + += + + in

powers of ( 1)x −−−− and ( 2)y −−−− . (A/M 2011)



6. Expand 2 3 2x y y+ −+ −+ −+ − in powers of ( 1)x −−−− and ( 2)y ++++ upto 3

rd degree terms.

(M/J 2012)

• Maxima and Minima

1. Find the extreme values of the function3 3( , ) 3 12 20f x y x y x y= + − − += + − − += + − − += + − − + .

(Jan 2010),(A/M 2011),(Jan 2012)

2. Find the maximum and minimum values of 2 2 2x xy y x y− + − +− + − +− + − +− + − + . (M/J 2012)

3. Discuss the maxima and minima of the function 4 4 2 2( , ) 2 4 2f x y x y x xy y= + − + −= + − + −= + − + −= + − + − .

(N/D 2010)

4. Test for an extrema of the function 4 4 2 2( , ) 1f x y x y x y= + − − −= + − − −= + − − −= + − − − . (Jan 2011)(AUT)

5. Examine the function (((( )))) (((( ))))3 2, 12f x y x y x y= − −= − −= − −= − − for extreme values. (M/J 2009)

6. Find the maximum value of m n px y z subject to the condition x y z a+ + =+ + =+ + =+ + = .

(Jan 2009)

7. 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.

(M/J 2010), (N/D 2011)(AUT),(M/J 2012)

8. Find the volume of the greatest rectangular parallelepiped inscribed in the ellipsoid

whose equation is

2 2 2

2 2 2 1x y za b c

+ + =+ + =+ + =+ + = . (M/J 2009)

• Jacobians

1. Find the Jacobian ( , , )( , , )x y zr θ φθ φθ φθ φ

∂∂∂∂∂∂∂∂

of the transformation sin cos ,x r θ φθ φθ φθ φ==== sin siny r θ φθ φθ φθ φ====

and cosz r θθθθ==== . (Jan 2009),(A/M 2011)

2. If , , x y z u y z uv z uvw+ + = + = =+ + = + = =+ + = + = =+ + = + = = prove that 2( , , )

( , , )x y z

u vu v w

∂∂∂∂ ====∂∂∂∂

.

(Jan 2010),(Jan 2012)

3. Find the Jacobian of 1 2 3, ,y y y with respect to 1 2 3, ,x x x if 2 3 3 11 2

1 2

, ,x x x x

y yx x

= == == == =

1 23

3

x xy

x==== . (N/D 2010)



Unit – V (Multiple Integrals)

• Simple problems on double integral

No recent problem from this topic

• Change of order of integration

1. Evaluate

0

y

x

edxdy

y

∞ ∞∞ ∞∞ ∞∞ ∞ −−−−

∫ ∫∫ ∫∫ ∫∫ ∫ by changing the order of integration. (N/D 2010),(A/M 2011)

2. Change the order of integration in

2 2

0

a ya

a y

y dxdy−−−−

−−−−∫ ∫∫ ∫∫ ∫∫ ∫ and then evaluate it. (M/J 2009)

3. Change the order of integration 2

1 2

0

x

x

xy dxdy−−−−

∫ ∫∫ ∫∫ ∫∫ ∫ and hence evaluate.

(Jan 2010),(M/J 2012)

4. Change the order of integration in the interval 2

2

0 /

a a x

x a

xy dydx−−−−

∫ ∫∫ ∫∫ ∫∫ ∫ and hence evaluate it.

(M/J 2010)

5. Change the order of integration and hence find the value of

21

0

y

y

xy dxdy−−−−

∫ ∫∫ ∫∫ ∫∫ ∫ .

(N/D 2011)(AUT)

6. Change the order of integration and hence evaluate

3 6/2

1 0

x

y

x dydx====

∫ ∫∫ ∫∫ ∫∫ ∫ . (Jan 2009)

7. Change the order of integration

2 2

2 20

a a ya

a a y

xy dxdy+ −+ −+ −+ −

− −− −− −− −∫ ∫∫ ∫∫ ∫∫ ∫ and hence evaluate it.

(Jan 2011)(AUT)

8. Change the order of integration in

(((( ))))b a x

a a

x dydx

−−−−

∫ ∫∫ ∫∫ ∫∫ ∫

2 2

2

0 0

and then evaluate it.

(Jan 2012)



• Change into polar coordinates

1. Express

(((( ))))2

3/22 20

a a

y

x dxdy

x y++++∫ ∫∫ ∫∫ ∫∫ ∫ in polar coordinates and then evaluate it. (M/J 2009)

2. Evaluate (((( ))))2 2

0 0

x y

e dxdy∞ ∞∞ ∞∞ ∞∞ ∞

− +− +− +− +

∫ ∫∫ ∫∫ ∫∫ ∫ by converting to polar coordinates. Hence deduce the value

of 2

0

xe dx∞∞∞∞

−−−−∫∫∫∫ . (Jan 2010),(N/D 2010)

3. Transform the integral (((( ))))22 2

2 2

0 0

x x

x y dydx−−−−

++++∫ ∫∫ ∫∫ ∫∫ ∫ into polar coordinates and hence

evaluate it. (A/M 2011)

4. By Transforming into polar coordinates, evaluate

2 2

2 2 x y

dxdyx y

++++

∫ ∫∫ ∫∫ ∫∫ ∫ over annular

region between the circles 2 2 16x y+ =+ =+ =+ = and

2 2 4x y+ =+ =+ =+ = . (M/J 2010)

5. Transform the double integral

a a x

ax x

dxdy

a x y

−−−−

−−−− − −− −− −− −∫ ∫∫ ∫∫ ∫∫ ∫2 2

22 2 2

0

into polar co-ordinates and then

evaluate it. (Jan 2012)

6. Transform the integral into polar coordinates and hence evaluate

2 2

2 2

0 0

a a x

x y dydx−−−−

++++∫ ∫∫ ∫∫ ∫∫ ∫ . (Jan 2012)

• Area as a double integral

1. Find the area bounded by the parabolas 2 4y x= −= −= −= − and

2y x==== by double integration.

(N/D 2010)

2. Find, by double integration, the area enclosed by the curves 2 4y ax==== and

2 4x ay==== .

(Jan 2010),(A/M 2011)

3. Find, by double integration, the area between the two parabolas 23 25y x==== and

25 9x y==== . (M/J 2012)

4. Find the area common to2 4y x==== and

2 4x y==== using double integration.



(N/D 2011)(AUT)

5. Evaluate ( ) x y dxdy−−−−∫ ∫∫ ∫∫ ∫∫ ∫ over the region between the line y x==== and the

parabola2y x==== . (Jan 2011)(AUT)

6. Find the smaller of the areas bounded by the ellipse x y+ =+ =+ =+ =2 24 9 36and the straight line

x y+ =+ =+ =+ =2 3 6. (Jan 2012)

7. Find the area inside the circle sinr a θθθθ==== but lying outside the cardioids

(((( ))))1 cosr a θθθθ= −= −= −= − . (Jan 2009)

8. Evaluate (((( )))) (((( ))))2 3 3 23 3C

xy y dx x xy dy + + ++ + ++ + ++ + + ∫∫∫∫ where C is the parabola 2 4y ax==== from

(((( ))))0,0 to (((( )))), 2a a . (M/J 2009)

• Triple integral

1. Evaluate (((( ))))2 2 2

0 0 0

a b c

x y z dxdydz+ ++ ++ ++ +∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫ . (Jan 2009)

2. Evaluate

log2

0 0 0

x yx

x y ze dxdydz++++

+ ++ ++ ++ +∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫ . (M/J 2009)

3. Evaluate

2 2 22 2

2 2 2 20 0 0

1a x ya a x

dzdydxa x y z

− −− −− −− −−−−−

− − −− − −− − −− − −∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫ . (N/D 2011)(AUT)

4. Evaluate

x yx dxdydz

x y z

− −− −− −− −−−−−

− − −− − −− − −− − −∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫2 22 11 1

2 2 20 0 0 1

. (Jan 2012)

5. Using triple integration, find the volume of the sphere 2 2 2 2x y z a+ + =+ + =+ + =+ + = .

(N/D 2010)

6. Find the volume of the ellipsoid

2 2 2

2 2 2 1x y za b c

+ + =+ + =+ + =+ + = . (Jan 2010),(A/M 2011)

7. Find the volume of the tetrahedran bounded by the plane 1x y za b c

+ + =+ + =+ + =+ + = and the

coordinate plane 0, 0, 0x y z= = == = == = == = = . (M/J 2010)



8. Evaluate 2 x yz dxdydz∫∫∫∫∫∫∫∫∫∫∫∫ taken over the tetrahedron bounded by the planes

0, 0, 0x y z= = == = == = == = = and 1x y za b c

+ + =+ + =+ + =+ + = . (Jan 2011)(AUT)

9. Change to spherical polar co – ordinates and hence evaluate 2 2 2

1

V

dxdydzx y z+ ++ ++ ++ +∫∫∫∫∫∫∫∫∫∫∫∫ ,

where V is the volume of the sphere 2 2 2 2x y z a+ + =+ + =+ + =+ + = . (Jan 2009)

10. Find the value of xyz dxdydz∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫ through the positive spherical octant for which

2 2 2 2x y z a+ + ≤+ + ≤+ + ≤+ + ≤ . (M/J 2010)

11. Evaluate(((( ))))3

1

dzdydx

x y z+ + ++ + ++ + ++ + +∫∫∫∫∫∫∫∫∫∫∫∫ whereV is the region bounded by 0, 0,x y= == == == =

0, 1z x y z= + + == + + == + + == + + = . (N/D 2011)(AUT)

---- All the BestAll the BestAll the BestAll the Best ----


Prepared by Mr.C.Ganesan, M.Sc.,M.Phil., (Ph: 9841168971) Page 1

NAME OF THE SUBJECT : Engineering Mathematics – I

SUBJECT CODE : 181101/ MA 2111

NAME OF THE METERIAL : Formula Material


Unit – I (Matrices)

1. The Characteristic equation of matrix A is

a) 2

1 20S S if A is 2 X 2 matrix

1

2

Where Sum of the main diagonal elements.S

S A

b) 3 2

1 2 30S S S if A is 3 X 3 matrix

1

2

3

Where Sum of the main diagonal elements.

Sum of the minors of the main diagonal elements.

S

S

S A

2. To find the eigen vectors solve 0A I X .

3. Property of eigen values:

Let A be any matrix then

a) Sum of the eigen values = Sum of the main diagonal.

b) Product of the eigen values = A

c) If the matrix A is triangular then diagonal elements are eigen values.

d) If is an eigen value of a matrix A, the 1

is the eigen value of 1

A .

e) If 1 2, , ...

n are the eigen values of a matrix A, then

1 2, , ...

n

m m m are

eigen values of mA .( m being a positive integer)

f) The eigen values of A & TA are same.

4. Cayley-Hamilton Theorem:

Every square matrix satisfies its own characteristic equation. (ie) 0A I .



5. Matrix of Q.F

1

2

3

2

1 2 1 3

2

2 1 2 3

2

3 1 3 2

1 1( ) ( ) ( )

2 2

1 1( ) ( ) ( )

2 2

1 1( ) ( ) ( )

2 2

coeff x coeff x x coeff x x

coeff x x coeff x coeff x x

coeff x x coeff x x coeff x

6. Index = p = Number of positive eigen values

Rank = r = Number of non-zero rows

Signature = s = 2p-r

7. Diagonalisation of a matrix by orthogonal transformation (or) orthogonal

reduction:

Working Rules:

Let A be any square matrix of order n.

Step:1 Find the characteristic equation.

Step:2 Solve the characteristic equation.

Step:3 Find the eigen vectors.

Step:4 Form a normalized model matrix N, such that the eigen vectors are orthogonal.

Step:5 Find TN .

Step:6 Calculate TD=N AN .

Note:

We can apply orthogonal transformation for symmetric matrix only. If any two eigen values are equal then we must use a, b, c method for third eigen vector.


1. Equation of the sphere, general form 2 2 22 2 2 0x y z ux vy wz d ,

centre , ,u v w , radius 2 2 2r u v w d .

2. Equation of the sphere with centre , ,a b c , radius r is

2 2 2 2

x a y b z c r .

3. Equation of the sphere with centre origin and radius r is 2 2 2 2x y z r .

4. Equation of circle:



The curve of intersection of a sphere by a plane is a circle. So, a circle can be represented by two equations, one being the equation of a sphere and the

other that of a plane. Thus, the equation 2 2 22 2 2 0,x y z ux vy wz d

x my nz p taken together represent a circle.

5. Tangent plane:

Equation of tangent plane of sphere at the point 1 1 1, ,x y z is

1 1 11 1 10u x x v y y w z z dxx yy zz .

6. Condition for the plane x my nz p to be a tangent plane to the

sphere 2 2 2 2 2 2 2

u mv nw p m n u v w d .

7. Condition for the spheres to cut orthogonally 1 2 1 2 1 2 1 22 2 2u u v v w w d d .

8. Equation of Right Circular Cone is

2 2 2 22 2 2 2

cosx m y n z m n x y z

9. Equation of Right Circular Cylinder is

2 2 2

2 2 2 2n y m z z n x m x y r m n

If radius is not given

2

2 2 22

2 2 2

x m y n zr x y z

m n

.


1. Curvature of a circle = Reciprocal of it’s radius

2. Radius of curvature with Cartesian form

3

2 21

2

1 y

y

3. Radius of curvature if 1y ,

3

2 21

2

1 x

x

, where 1

dxx

dy

4. Radius of curvature in implicit form

3

2 22

2 22

x y

xx y xy x y yy y

f f

f f f f f f f



5. Radius of curvature with paramatic form

3

2 2 2x y

x y x y

6. Centre of curvature is ,x y .

7. Circle of curvature is 2 2 2

x x y y .

where 2

1 1

2

1y yx x

y

,

2

1

2

1 yy y

y

8. Evolute: The locus of centre of curvature of the given curve is called evolute of

the curve. 2

1 1

2

1y yx x

y

,

2

1

2

1 yy y

y

9. Envelope: The envelope is a curve which meets each members of a family of

curve.

If the given equation can be rewrite as quadratic equation in parameter, (ie)

20A B C where , , A B C are functions of x and y then the envelope is

24 0B AC .

10. Evolute as the envelope of normals.

Equations Normal equations

24y ax 3

2y xt at at

24x ay 3

2x yt at at

2 2

2 21

x y

a b

2 2

cos sin

ax bya b

2 2

2 21

x y

a b

2 2

sec tan

ax bya b

2 2 2

3 3 3x y a cos sin cos2x y a

2xy c 2 3c

y xt ctt




1. Euler’s Theorem:

If f is a homogeneous function of x and y in degree n , then

(i) f f

x y nfx y

(first order)

(ii) 2 2 2

2 2

2 22 1

f f fx xy y n n f

x x y y

(second order)

2. If ( , , )u f x y z , 1 2 3( ), ( ), ( )x g t y g t z g t then

du u dx u dy u dz

dt x dt y dt z dt

3. If 1 2( , ), ( , ), ( , )u f x y x g r y g r then

(i) u u x u y

r x r y r

(ii)

u u x u y

x y

4. Maxima and Minima :

Working Rules:

Step:1 Find xf and y

f . Put 0x

f and 0y

f . Find the value of x and y.

Step:2 Calculate , ,xx xy yy

r f s f t f . Now 2rt s

Step:3 i. If 0 , then the function have either maximum or minimum.

1. If 0r Maximum

2. If 0r Minimum

ii. If 0, then the function is neither Maximum nor Minimum, it is

called Saddle Point.

iii. If 0, then the test is inconclusive.

5. Maxima and Minima of a function using Lagrange’s Multipliers:

Let ( , , )f x y z be given function and ( , , )g x y z be the subject to the condition.

Form ( , , ) ( , , ) ( , , )F x y z f x y z g x y z , Putting 0x y z

F F F F and

then find the value of x,y,z. Next we can discuss about the Max. and Min.

6. Jacobian:



Jacobian of two dimensions: , ( , )

, ( , )

u v u vJ

x y x y

u u

x y

v v

x y

7. The functions u and v are called functionally dependent if( , )

0( , )

u v

x y

.

8. ( , ) ( , )

1( , ) ( , )

u v x y

x y u v

9. Taylor’s Expansion:

2 2

3 2 2 3

1 1( , ) ( , ) ( , ) ( , ) ( , ) 2 ( , ) ( , )

1! 2!

1 ( , ) 3 ( , ) 3 ( , ) ( , ) ...

3!

x y xx xy yy

xxx xxy xyy yyy

f x y f a b hf a b kf a b h f a b hkf a b k f a b

h f a b h kf a b hk f a b k f a b

where h x a and k y b

Unit – V (Multiple Integrals)

1. 0

( , )b x

af x y dxdy x : a to b and y : o to x (Here the first integral is w.r.t. y)

2. 0

( , )b y

af x y dxdy x : 0 to y and y : a to b (Here the first integral is w.r.t. x)

3. Area R

dxdy (or) R

dydx

To change the polar coordinate

cos

sin

x r

y r

dxdy rdrd

4. Volume V

dxdydz (or) V

dzdydx

GENERAL:

1. 1

2 2sin

dx x

aa x

(or) 1

2sin

1

dxx

x

2. 2 2

2 2log

dxx a x

a x

(or) 2

2log 1

1

dxx x

x



3. 1

2 2

1tan

dx x

a x a a

(or) 1

2tan

1

dxx

x

4. 2

2 2 2 2 1 sin

2 2

x a xa x dx a x

a

5. /2 /2

0 0

1 3 2sin cos . ... .1 if is odd and 3

2 3

n n n nx dx x dx n n

n n

6. /2 /2

0 0

1 3 1sin cos . ... . if is even

2 2 2

n n n nx dx x dx n

n n

Engineering Mathematics 2012


SUBJECT NAME : Engineering Mathematics – I

SUBJECT CODE : 181101/MA 2111

MATERIAL NAME : Part – A questions


Name of the Student: Branch:

Unit – I (Matrices)

1. Given :

1 0 0

2 3 0

1 4 2

A

. Find the eigen values of 2A .

2. If 3 and 6 are two eigen values of

1 1 3

1 5 1

3 1 1

A

, write down all the eigen values

of 1A

.

3. Write down the quadratic form corresponding to the matrix

0 5 1

5 1 6

1 6 2

A

.

4. The product of two eigenvalues of the matrix A

6 2 2

2 3 1

2 1 3

is 16. Find the third

eigenvalue of A .

5. For a given matrix A of order 3, 32A and two of its eigen values are 8 and 2.

6. Check whether the matrix B is orthogonal? Justify.

cos sin 0

sin cos 0

0 0 1

B

.

7. Can1 0

0 1A

be diagonalized? Why?



8. If the sum of two eigen values and trace of a 3 X 3 matrix A are equal, Find the value

of A .

9. Use Cayley – Hamilton theorem to find

4 3 24 5 2A A A A I when

1 2

4 3A

.

10. If 1 and 2 are the eienvalues of a 2 X 2 matrix A, what are the eigenvalues of A2 and A-1?

11. State Cayley – Hamilton theorem.

12. Find the nature of the Quadratic Form 2 2 2

1 2 3 1 2 2 32 2 2x x x x x x x .


1. Write the equation of the tangent plane at 1,5,7 to the sphere

2 2 2

2 3 4 14x y z .

2. Find the equation of the tangent plane at 1,4,2 on the sphere x y z 2 2 2

x y z 2 4 2 3 0 .

3. Find the equation of the tangent plane to the sphere 2 2 2x y z 2 4x y

6 6 0z at 1,2,3 .

4. Find the equation of the sphere concentric with x y z x y z 2 2 24 6 8 4 0

and passing through the point 1,2,3 .

5. Find the equation of the sphere having the points 2, 3,4 and 1,5,7 as the ends

of a diameter.

6. Check whether the two spheres 2 2 26 2 8 0x y z y z and

2 2 26 8 4 20 0x y z x y z are orthogonal.

7. Find the centre and radius of the sphere 2 2 22 6 6 8 9 0x y z x y z .

8. Find the equation of the right circular cone whose vertex is at the origin and axis is the

line 1 2 3

x y z having semi vertical angel of 45°.



9. Find the equation of the cone whose vertex is the origin and guiding curve is 2 2 2

1, 14 9 1

x y zx y z .

10. Find the equation of the right circular cone whose vertex is the origin, axis is the y –

axis, and semi – vertical angle is 30⁰.

11. Write down the equation of the right circular cone whose vertex is at the origin, semi

vertical angel is and axis is along z-axis.


1. For the catenary coshx

y cc

, find the curvature.

2. Find the radius of curvature for xy e at the point where it cuts the y – axis.

3. Define the circle of curvature at a point 1 1,p x y on the curve ( )y f x .

4. Find the curvature of the curve 2 22 2 5 2 1 0x y x y .

5. Write down the formula for Radius of curvature in terms of Parametric Coordinates

System.

6. Find the envelope of the lines 2 2 2y mx a m b where m is the parameter.

7. Find the envelope of family of straight linesa

y mxm

, m being the parameter.

8. Find the envelope of the family of straight lines1

y mxm

, where m is a parameter.

9. Write the properties of Evolutes.

10. Find the envelope of the family of straight lines cos sinx y where is the

parameter.

11. Find the envelope of the family of circles 2 2 2

x y r , being the parameter.


1. Using Euler’s theorem, given ( , )u x y is a homogeneous function of degree n , prove

that 2 22 ( 1)

xx xy yyx u xyu y u n n u .



2. Using the definition of total derivative, find the value of du

dtgiven 2

4u y ax ;

2, 2x at y at .

3. If 3 2 2 3u x y x y where 2

x at and 2y at then finddu

dt?

4. Find du

dt if sin( / )u x y , where 2

,t

x e y t .

5. If 2 2 2

, ,2 2

y x yu v

x x

find

( , )

( , )

u v

x y

.

6. If x u v 2 2 and y uv 2 , find the Jacobian of x and y with respect to u and v .

7. If 2 22 , , cos , sinu xy v x y x r y r then compute

( , )

( , )

u v

r

?

8. Write the sufficient condition for ( , )f x y to have a maximum value at (a,b).

9. If cos , sinx r y r find ( , )

( , )

x y

r

.

10. If yu x , show that

u u

x y y x

2 2

.

11. Given2 1

( , ) tany

u x y xx

, find the value of 2 2

2xx xy yy

x u xyu y u .

12. If x y z

uy z x

, findu u u

x y zx y z

.

Unit – V (Multiple Integral)

1. Write down the double integral, to find the area between the circles 2sinr and

4sinr

2. Evaluate

sin

0 0

r drd

.



3. Evaluate 1

0

x

x

xy x y dxdy .

4. Evaluate 21

2 2

0 0

x

x y dydx .

5. Evaluate0 0

( )

a b

x y dxdy .

6. Evaluate 2 2

C

x dy y dx where C is the path y x from 0,0 to 1,1 .

7. * EvaluateR

dxdy , where R is the shaded region in the figure.

8. Change the order of integration in2

1 2

0

( , )

x

x

I f x y dxdy

.

9. Change the order of integration for the double integral

1

0 0

( , )

x

f x y dxdy .

10. Change the order of integration in0

( , )

a a

x

f x y dydx .

11. Change the order of integration

1 1

0 y

dxdy .

12. Express f x y dxdy

0 0

( , ) in polar co-ordinates.

13. Evaluate

y x y

dxdydz

1

0 0 0

.

---- All the Best ----



SUBJECT NAME : Engineering Mathematics – I

SUBJECT CODE : 181101/ MA 2111

MATERIAL NAME : Problem Material


Name of the Student: Branch: Unit – I (Matrices)

• Cayley – Hamilton Theorem

1) Verify that the matrix

−−−− = − −= − −= − −= − − −−−−

2 1 2

1 2 1

1 1 2

A satisfies its characteristic equation and

hence find4A .

2) Verify Cayley – Hamilton theorem and find the inverse of the matrix

1 1 1

0 1 0

2 0 3

A

−−−− ====

.

3) Using Cayley – Hamilton theorem, find 4A of the matrix

2 1 1

0 1 2

1 0 1

−−−−

.

4) If 1 2

3 4A

====

find 1A−−−−

and 3A using Cayley – Hamilton theorem.

5) Find nA using Cayley Hamilton theorem, taking A

====

1 4

2 3. Hence find A3

.

6) Use Cayley – Hamilton theorem to find the value of 8 7 6 55 7 3A A A A− + −− + −− + −− + −

4 3 25 8 2A A A A I+ − + − ++ − + − ++ − + − ++ − + − + where

2 1 1

0 1 0

1 1 2

A

====

. Ans.:

8 5 5

0 3 0

5 5 8

• Eigen Values and Eigen Vectors of a given matrix

1) Find the Eigen values and Eigen vectors of the matrix

2 2 1

1 3 1

1 2 2

.



2) Find the eigen values and eigen vectors of

2 1 1

1 2 1

0 0 1

A

====

.

3) Find the eigen values and eigen vectors of

2 2 3

2 1 6

1 2 0

A

− −− −− −− − = −= −= −= − − −− −− −− −

.

4) Find all the eigenvalues and eigenvectors of the matrix

1 1 4

3 2 1

2 1 1

−−−− −−−− −−−−

.

5) Find the eigen values and eigen vectors of the matrix

11 4 7

7 2 5

10 4 6

A

− −− −− −− − = − −= − −= − −= − − − −− −− −− −

.

• Diagonalisation of a Matrix

1) Diagonalise the matrix

2 1 1

1 2 1

1 1 2

−−−− − −− −− −− − −−−−

by orthogonal transformation.

Ans.:

4 0 0

0 1 0

0 0 1

2) Diagonalise the matrix

8 6 2

6 7 4

2 4 3

A

−−−− = − −= − −= − −= − − −−−−

by orthogonal transformation.

Ans.:

0 0 0

0 3 0

0 0 15

• Quadratic form to Canonical form

1) Reduce the quadratic form 2 2 26 3 3 4 2 4x y z xy yz xz+ + − − ++ + − − ++ + − − ++ + − − + to canonical form

by orthogonal reduction.

2) Reduce the quadratic form 2 2 21 2 3 1 2 2 32 2 2x x x x x x x+ + − ++ + − ++ + − ++ + − + to the canonical form

through an orthogonal transformation. Also find the rank, index, signature and

nature of the quadratic form.

3) Reduce the quadratic form 2 2 2xy yz zx+ ++ ++ ++ + into canonical form by means of

orthogonal transformation. Find its nature.



4) Reduce the quadratic form x y z xy+ + ++ + ++ + ++ + +2 2 22 5 3 4 to canonical form by orthogonal

reduction and state its nature.

5) The eigen vectors of a 3 X 3 real symmetric matrix A corresponding to the eigen

values 2, 3, 6 are [[[[ ]]]] [[[[ ]]]] [[[[ ]]]]1, 0, 1 , 1, 1, 1 , 1, 2, 1T T T− − −− − −− − −− − − respectively, find the

matrix A . Ans.:

3 1 1

1 5 1

1 1 3

−−−− − −− −− −− − −−−−

Unit – II (Three Dimensional Analytical Geometry) • Sphere

1) Find the equation of the sphere which passes through the points (((( ))))0,0,0 ,

(((( ))))0,1, 1−−−− , (((( ))))1,2,0−−−− and (((( ))))1,2,3 .

2) Find the equation of the sphere passing through the points (((( ))))1,1, 1−−−− , (((( ))))5,4,2−−−− ,

(((( ))))0,2,3 and having its centre on the plane 3 4 2 6x y z+ + =+ + =+ + =+ + = .

3) Find the centre and radius of the circle given by 2 2 2 2 2 4 19 0x y z x y z+ + + − − − =+ + + − − − =+ + + − − − =+ + + − − − = and 2 2 7 0x y z+ + + =+ + + =+ + + =+ + + = .

4) Find the equation of the sphere through the circle 2 2 2 2 3 6 0, 2 4 9 0x y z x y x y z+ + + + + = − + − =+ + + + + = − + − =+ + + + + = − + − =+ + + + + = − + − = and the centre of the sphere

2 2 2 2 4 6 5 0x y z x y z+ + − + − + =+ + − + − + =+ + − + − + =+ + − + − + = .

5) Find the equations of the spheres which passes through the circle 2 2 2 5x y z+ + =+ + =+ + =+ + =

and 2 3 3x y z+ + =+ + =+ + =+ + = and touch the plane 4 3 15x y+ =+ =+ =+ = .

6) Find the equation of the sphere having the circle x y z y z+ + + − − =+ + + − − =+ + + − − =+ + + − − =2 2 2 10 4 8 0, x y z+ + =+ + =+ + =+ + = 3 as a great circle.

7) Find the equation of the sphere that passes through the circle 2 2 2 3 2 1 0, 2 5 7 0x y z x y z x y z+ + + − + − = + − + =+ + + − + − = + − + =+ + + − + − = + − + =+ + + − + − = + − + = and cuts orthogonally the

sphere2 2 2 3 5 7 6 0x y z x y z+ + − + − − =+ + − + − − =+ + − + − − =+ + − + − − = .

Equation of tangent plane to a sphere:

8) Show that the plane 2 2 12 0x y z− + + =− + + =− + + =− + + = touches the sphere x y z+ ++ ++ ++ +2 2 2 x y z− − + =− − + =− − + =− − + =2 4 2 3 also find the point of contact.

9) Show that the line 6 7 3

3 4 5x y z− − −− − −− − −− − −= == == == = touches the sphere

2 2 2 2 4 4 0x y z x y+ + − − − =+ + − − − =+ + − − − =+ + − − − = and find the coordinates of the point of contact.

10) Find the equations of the tangent planes to the sphere 2 2 2 4 2 6 5 0x y z x y z+ + − − − + =+ + − − − + =+ + − − − + =+ + − − − + = which are parallel to the plane

4 8 0x y z+ + =+ + =+ + =+ + = . Find also their points of contact.



11) Find the equations of the two tangent planes to the sphere 2 2 2 4 2 6 11 0x y z x y z+ + − + − − =+ + − + − − =+ + − + − − =+ + − + − − = which are parallel to the coordinate plane

0x ==== .

• Cone 1) Find the equation of the cone whose vertex is (((( ))))3,1,2 and base curve

+ = =+ = =+ = =+ = =2 22 3 1, 1x y z .

2) Find the equation of the cone with vertex at (((( ))))1,2,3 and the guiding curve is the

circle2 2 2 4, 1x y z x y z+ + = + + =+ + = + + =+ + = + + =+ + = + + = .

3) Find the equation of the right circular cone whose vertex is the origin, whose axis is

the line = == == == =1 2 3x y z

and which has semi vertical angle of 30°.

4) Find the equation of the right circular cone whose vertex is (((( ))))3,2,1 , semi – vertical

angle 30° and the axis the line3 2 1

4 1 3x y z− − −− − −− − −− − −= == == == = .

Ans.: 2 2 27 37 21 16 12 48 38 88 126 32 0x y z xy yz zx x y z+ + − − − + − + − =+ + − − − + − + − =+ + − − − + − + − =+ + − − − + − + − =

5) Find the semi vertical angle and the equation of the right circular cone having its

vertex at the origin and passing through the circle2 2 25, 4y z x+ = =+ = =+ = =+ = = .

Ans.: 2 2 225 16 16 0x y z− − =− − =− − =− − =

6) Find the equation of the right circular cone generated by the straight lines drawn

from the origin to cut the circle through the three points (((( )))) (((( ))))1,2,2 , 2,1, 2−−−− and

(((( ))))2, 2,1−−−− . Ans.: 2 2 28 4 4 5 5 0x y z xy yz zx− − + + + =− − + + + =− − + + + =− − + + + =

• Cylinder 1) Find the equation of the right circular cylinder of radius 2 and having as axis of the

line1 2 3

2 1 2x y z− − −− − −− − −− − −= == == == = .

Ans.: 2 2 25 8 5 4 4 8 22 16 14 10 0x y z xy yz zx x y z+ + − − − + − − − =+ + − − − + − − − =+ + − − − + − − − =+ + − − − + − − − =

2) Find the equation of the right circular cylinder whose axis is 2 1 0

2 1 3x y z− − −− − −− − −− − −= == == == =

and which passes through the point (((( ))))0,0,3 .

Ans.: 2 2 210 13 5 4 6 12 36 18 30 135 0x y z xy yz zx x y z+ + − − − − − + − =+ + − − − − − + − =+ + − − − − − + − =+ + − − − − − + − =

3) Find the right circular cylinder which has the circle 2 2 2 2 4 4 1 0x y z x y z+ + − − − − =+ + − − − − =+ + − − − − =+ + − − − − = , 2 2 13 0x y z− − + =− − + =− − + =− − + = as the guiding curve.

Ans.: 2 2 25 8 5 4 4 8 34 28 20 56 0x y z xy yz zx x y z+ + + − + − − − + =+ + + − + − − − + =+ + + − + − − − + =+ + + − + − − − + =



Unit – III (Differential Calculus) • Radius of Curvature and Circle of curvature

1) Find the circle of the curvature of the curve x y a+ =+ =+ =+ = at (((( ))))/ 4, / 4a a .

2) Find the radius of curvature at the point (((( ))))3 / 2,3 / 2a a on the curve

3 3 3x y axy+ =+ =+ =+ = .

3) Find the radius of curvature of the curve

3 32 a x

yx−−−−==== at( ,0)a .

4) Find the radius of curvature at the origin for the curve 3 3 22 4 3 0x y x y x+ + − + =+ + − + =+ + − + =+ + − + = .

5) For the curve ax

ya x

====++++

if ρρρρ is the radius of curvature any point ( , )x y , show

that

22/3 22 y xa x yρρρρ = += += += +

6) Show that the radius of curvature at the point 3 3( cos , sin )a aθ θθ θθ θθ θ on the curve

2/3 2/3 2/3x y a+ =+ =+ =+ = is 3 sin cosa θ θθ θθ θθ θ .

7) Prove that the radius of curvature at any point of the cycloid ( sin )x a θ θθ θθ θθ θ= += += += + ,

(1 cos )y a θθθθ= −= −= −= − is4 cos2

a θθθθ.

8) Find the radius of curvature of the curve (cos sin )x a t t t= += += += + ,

(sin cos )x a t t t= −= −= −= − at any point t.

9) Find the circle of curvature at (3,4)on 12xy ==== .

10) Fin the equation of circle of curvature at (3,6)on2 12xy ==== .

11) Find the equation of the circle of curvature at ( , )c c on2xy c==== .

12) Find the radius of curvature and centre of curvature at any point ( , )x y on the

curve logsecxc

y c==== .



13) Find the radius of curvature of the curve coshxc

y c==== at the point(0, )c .

• Evolute

1) Find the equation of evolute of the parabola2 4y ax==== .

2) Find the equation of evolute of the parabola 2 4x ay==== .

3) Find the equation of the evolute of the ellipse

2 2

2 2 1x ya b

+ =+ =+ =+ = .

4) Find the equation of the evolute of the hyperbola

2 2

2 2 1x ya b

− =− =− =− = .

5) Show that the evolute of the cycloid ( sin )x a θ θθ θθ θθ θ= −= −= −= − , (1 cos )y a θθθθ= −= −= −= − is

another cycloid.

6) Find the evolute of the curve 2/3 2/3 2/3x y a+ =+ =+ =+ = .

7) Find the evolute of the rectangular hyperbola2xy c==== .

8) Show that the evolute of the tractrix cos log tan2t

tx a ++++

==== , sin ty a==== is

the catenary coshx

y aa

==== .

• Envelope

1) Find the envelope of the family lines a m by mx ++++= += += += + 2 2 2, where m is the

parameter.

2) Find the envelope of the family of straight lines 2 2

cos sinax by

a bθ θθ θθ θθ θ

− = −− = −− = −− = − ,θθθθ being

the parameter.

3) Find the envelope of cos sin

1x y

a bα αα αα αα α+ =+ =+ =+ = , where αααα is the parameter.

4) Find the envelope of the family of straight lines 1x ya b

+ =+ =+ =+ = , where the

parameters a and b are related by the equation a b c+ =+ =+ =+ = , c being a constant.




+ =+ =+ =+ = , where the

parameters a and b are related by the equation2 2 2a b c+ =+ =+ =+ = , c being a

constant.


+ =+ =+ =+ = , where the

parameters a and b are related by the equationn n na b c+ =+ =+ =+ = , c being a

constant.


+ =+ =+ =+ = , where2ab c==== ,

,a b are parameters.

8) Find the envelope of

2 2

2 2 1x ya b

+ =+ =+ =+ = , wherea b c+ =+ =+ =+ = , a and b are the

parameters and c is a constant.


2 2

2 2 1x ya b

+ =+ =+ =+ = , where2 2 2a b c+ =+ =+ =+ = , a and b are the



2 2

2 2 1x ya b

+ =+ =+ =+ = , wheren n na b c+ =+ =+ =+ = , a and b are the



2 2

2 2 1x ya b

+ =+ =+ =+ = , where2ab c==== , a and b are the parameters

and c is a constant.

• Evolute as the envelope of normals

1) Find the evolute of the parabola2 4axy ==== , treating it as the envelope of

normals.

2) Find the evolute of the parabola 2 4x ay==== , treating it as the envelope of

normals.



3) Find the evolute of the ellipse

2 2

2 2 1x ya b

+ =+ =+ =+ = as envelope of it’s normals.

4) Find the evolute of the hyperbola

2 2

2 2 1x ya b

− =− =− =− = as envelope of it’s normals.

5) Find the evolute of the curve 2xy c==== as envelope of normals.

Unit – IV (Functions of several variables) • Euler’s Theorem 1) If u is a homogeneous function of degree n in x and y . Show that

(((( ))))∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂+ + = −+ + = −+ + = −+ + = −∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂

2 2 22 2

2 22 1u u u

x xy y n n ux x y y

.

2) If−−−− ++++==== ++++

1cosx y

ux y

, Prove that∂ ∂∂ ∂∂ ∂∂ ∂+ + =+ + =+ + =+ + =∂ ∂∂ ∂∂ ∂∂ ∂

1cot 0

2u u

x y ux y

.

3) If

2 21sin

x yu

x y−−−− ++++==== ++++

, prove that

(1) tanu u

x y ux y

∂ ∂∂ ∂∂ ∂∂ ∂+ =+ =+ =+ =∂ ∂∂ ∂∂ ∂∂ ∂

and

(2)

2 2 22 2 2

2 22 tanu u u

x xy y ux x y y

∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂+ + =+ + =+ + =+ + =∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂

4) If−−−− ++++==== ++++

1sinx y

ux y

, Prove that

∂ ∂ ∂ −∂ ∂ ∂ −∂ ∂ ∂ −∂ ∂ ∂ −+ + =+ + =+ + =+ + =∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂

2 2 22 2

2 2 3

sin cos 22

4cosu u u u u

x xy yx x y y u

.

• Total derivatives

1) If (((( )))) (((( ))))2 2 1log tan /u x y y x−−−−= + += + += + += + + prove that 0xx yyu u+ =+ =+ =+ = .

2) If ==== ( , )z f x y where = == == == =cos , sinx r y rθ θθ θθ θθ θ , Show

that ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂ + = ++ = ++ = ++ = + ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂

22 2 2

2

1z z z zx y r r θθθθ

.

3) u is a function of x and y , = == == == =cos , sinx r y rθ θθ θθ θθ θ Show that

∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂+ = + ++ = + ++ = + ++ = + +∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

2 2 2 2

2 2 2 2 2

1 1u u u u ux y r r r r θθθθ

.



4) If ( , , )u f x y y z z x= − − −= − − −= − − −= − − − , Show that 0u u ux y z

∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂+ + =+ + =+ + =+ + =∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂

.

5) If − −− −− −− −====

,y x z x

uxy xz

, Show that∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂+ + =+ + =+ + =+ + =∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂

2 2 2 0u u u

x y zx y z

.

6) If z be a function of u & v and u & v are other two variables x & y , such that

= + = −= + = −= + = −= + = −, u x my v y mx� �� . Show that (((( )))) ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂+ = + ++ = + ++ = + ++ = + + ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂

2 2 2 22 2

2 2 2 2

z z z zm

x y u v�� .

7) Given that the transformations ==== cosxu e y , ==== sinxv e y and that φφφφ is the

function of u and v and also of x and y , Prove that

(((( )))) ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂+ = + ++ = + ++ = + ++ = + + ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂

2 2 2 22 2

2 2 2 2u vx y u vφ φ φ φφ φ φ φφ φ φ φφ φ φ φ

.

8) If z f x y==== ( , ) , where x u v= −= −= −= −2 2and y uv==== 2 . Prove that

(((( ))))z z z zu v

u v x y

∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂+ = + ++ = + ++ = + ++ = + + ∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂

2 2 2 22 2

2 2 2 24 .

• Taylor’s expansion

1) Expand by Taylor’s series the function 2 2 2 2( , ) 2 3f x y x y x y xy= + += + += + += + + in powers

of (((( ))))2x ++++ and (((( ))))1y −−−− upto the third powers.

2) Expand cosxe y about 0,2ππππ

upto the third term using Taylor’s series.

3) Obtain terms upto the third degree in the Taylor series expansion of

sinxe y around the point 1,

2ππππ

.

4) Find the Taylor series expansion of sinxe y at the point 1,4ππππ −−−−

upto 3

rd degree

terms.

5) Expand (((( ))))++++log 1xe y in powers of x and y upto the terms of third degree.

6) Expand ( , ) xyf x y e==== in Taylor series in power of 1x −−−− and 1y −−−− upto second

dagree.

7) Expand the function sin xy in powers of −−−− 1x and −−−−2

yππππ

upto second degree

terms.



• Maxima and Minima

1) Find the extreme values of the function = + − − += + − − += + − − += + − − +3 3( , ) 3 12 20f x y x y x y .

2) Examine the function (((( )))) (((( ))))3 2, 12f x y x y x y= − −= − −= − −= − − for extreme values.

3) Find the maxima and minima of 4 4 2 22 4 2x y x xy y+ − + −+ − + −+ − + −+ − + − .

4) Discuss the maxima and minima of the function + −+ −+ −+ −3 3 3x y axy .

5) In a plane triangle ABC , find the maximum value of cos cos cosA B C .

Problems of Lagrangian Multipliers:

6) A rectangular box open at the top, is to have a volume of 32 cc. Find the dimension

of the box, that requires the least material for its construction.

7) Find the dimension of the rectangular box without top of maximum capacity with

surface area 432 square meter.

8) Find the Maximum value of the largest rectangular parallelepiped that can be

inscribed in an ellipsoid + + =+ + =+ + =+ + =2 2 2

2 2 2 1x y za b c

.

9) The temperature ( , , )u x y z at any point in space is ==== 2400u xyz . Find the highest

temperature on surface of the sphere + + =+ + =+ + =+ + =2 2 2 1x y z .

10) Find the maximum and minimum values of + ++ ++ ++ +2 2 2x y z subject to the condition

+ + =+ + =+ + =+ + = 3x y z a .

11) Find the maximum value ofm n px y z , when + + =+ + =+ + =+ + =x y z a .

12) Find the shortest and the longest distance from the point (((( ))))−−−−1,2, 1 to the sphere

+ + =+ + =+ + =+ + =2 2 2 24x y z , using Lagrange’s method of constrained maxima and minima.

• Jacobians

1) If = = −= = −= = −= = −2 22 , u xy v x y while = == == == =cos , sinx r y rθ θθ θθ θθ θ . Prove that∂∂∂∂ = −= −= −= −∂∂∂∂

3( , )4

( , )u v

rr θθθθ

.

2) If , , x y z u y z uv z uvw+ + = + = =+ + = + = =+ + = + = =+ + = + = = prove that 2( , , )

( , , )x y z

u vu v w

∂∂∂∂ ====∂∂∂∂

.

3) Find the Jacobian ∂∂∂∂∂∂∂∂( , , )( , , )x y zr θ φθ φθ φθ φ

of the transformation ==== sin cos ,x r θ φθ φθ φθ φ

= == == == =sin sin , cosy r z rθ φ θθ φ θθ φ θθ φ θ .

4) Find the Jacobian of 1 2 3, ,y y y with respect to 1 2 3, ,x x x if ==== 2 31

1

,x x

yx

= == == == =3 1 1 22 3

2 3

, x x x x

y yx x

.



Unit – V (Multiple Integrals) • Simple problems on double integral

1) Evaluate

−−−−

− −− −− −− −∫ ∫∫ ∫∫ ∫∫ ∫2 2

2 2 20 0

1a a x

dydxa x y

Ans.:2aππππ

2) Evaluate

++++

+ ++ ++ ++ +∫ ∫∫ ∫∫ ∫∫ ∫21 1

2 20 0

11

x

dydxx y

Ans.: (((( ))))++++log 1 24ππππ

• Change of order of integration

1) Change the order of integration of ++++∫ ∫∫ ∫∫ ∫∫ ∫ 2 2

0

a a

y

xdxdy

x yand evaluate it. Ans.:

4a

ππππ

2) Change the order of integration in

+ −+ −+ −+ −

− −− −− −− −∫ ∫∫ ∫∫ ∫∫ ∫

2 2

2 20

a a ya

a a y

xy dxdy and evaluate it. Ans.: 42

3a


∞∞∞∞ −−−−

∫ ∫∫ ∫∫ ∫∫ ∫2

0 0

y y

xye dxdy and evaluate it. Ans.: 12


−−−−

−−−−∫ ∫∫ ∫∫ ∫∫ ∫

2 2

0

a ya

a y

y dxdy and evaluate it. Ans.:

3

6a

5) By change of order of integration evaluate ∫ ∫∫ ∫∫ ∫∫ ∫2

4 2

0

4

a ax

xa

xy dydx . Ans.: 464

3a

Note: Do the same problem by putting ==== 1a .

6) Change the order of integration and evaluate

−−−−

∫ ∫∫ ∫∫ ∫∫ ∫2

2

0

a a x

xa

xy dxdy . Ans.: 43

8a

Note: Do the same problem by putting ==== 1a .

7) Change the order of integration and evaluate

−−−−

∫ ∫∫ ∫∫ ∫∫ ∫21

0

y

y

xy dxdy . Ans.: 13

8) Evaluate by changing the order of integration

−−−−

++++∫ ∫∫ ∫∫ ∫∫ ∫21 2

2 20

x

x

xdydx

x y. Ans.:

−−−−2 22

• Change into polar coordinates

1) By changing into polar coordinates, evaluate

−−−−

++++∫ ∫∫ ∫∫ ∫∫ ∫22 2

2 20 0

x x xdxdyx y

. Ans.: 2ππππ



2) Evaluate by changing to polar coordinates++++∫ ∫∫ ∫∫ ∫∫ ∫2

2 20

a a

y

xdxdy

x y.

Ans.: (((( ))))++++3

log 2 13a

3) Evaluate by changing to polar coordinates

(((( ))))++++∫ ∫∫ ∫∫ ∫∫ ∫

2

3/22 20

a a

y

xdxdy

x y. Ans.:

2

a

4) By changing into polar coordinates, evaluate (((( ))))−−−−

++++∫ ∫∫ ∫∫ ∫∫ ∫22 2

2 2

0 0

a ax x

x y dydx . Ans.:

434aππππ

5) By changing into polar coordinates, evaluate(((( ))))∞ ∞∞ ∞∞ ∞∞ ∞

− +− +− +− +

∫ ∫∫ ∫∫ ∫∫ ∫2 2

0 0

x ye dxdy . Hence prove that

∞∞∞∞−−−− ====∫∫∫∫

2

0 2te dt

ππππ. Ans.:

4ππππ

• Area as a double integral

1) Using double integration find the area enclosed by the curves ==== 22y x and

====2 4y x . Ans.: 23

2) Using double integral, find the area bounded by ====y x and ==== 2y x . Ans.: 16

3) Find the smaller of the areas bounded by = −= −= −= −2y x and + =+ =+ =+ =2 2 4x y . Ans.: −−−− 2ππππ

4) Evaluate (((( ))))++++∫∫∫∫∫∫∫∫2 2

R

x y dxdy where R is the region enclosed by = == == == =0, 0x y and

+ =+ =+ =+ = 1x y . Ans.: 16

5) Evaluate ∫∫∫∫∫∫∫∫ R

xy dxdy where R is the domain bounded by X – axis, ordinate ==== 2x a

and the curve ====2 4x ay . Ans.:

4

3a

6) Evaluate ( ) R

xy x y dxdy++++∫∫∫∫∫∫∫∫ over the area between ==== 2y x and ====y x .



• Triple integral

1) Evaluate

2 22 11 1

2 2 20 0 0 1

x yx dxdydz

x y z

− −− −− −− −−−−−

− − −− − −− − −− − −∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫ .

2) Find the volume of the portion of the ellipsoid

2 2 2

2 2 2 1x y za b c

+ + =+ + =+ + =+ + = which lies in the

first octant using triple integral.

3) Find the volume of the sphere 2 2 2 2x y z a+ + =+ + =+ + =+ + = using triple integrals.

4) Find the volume of the tetrahedron bounded by the plane 1x y za b c

+ + =+ + =+ + =+ + = and the

coordinate planes.

5) Evaluate

log2

0 0 0

x yx

x y ze dxdydz++++

+ ++ ++ ++ +∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫ .

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