JAHANGIRABAD INSTITUTE

OF

TECHNOLOGY

ENGINEERING MATHEMATICS – I

(QUESTION BANK)

PREPRAED BY

MOHAMMAD IMRAN

(ASISTANT PROFESSOR, JIT)

E-mail: mohammad.imran@jit.edu.in

Website: www.jit.edu.in

DIFFERENTIAL CALCULUS

Differential Calculus – I

Successive Differentiation, Leibnitz’s

theorem, Limit , Continuity and Differentiability of functions of several

variables, Partial derivatives, Euler’s theorem for homogeneous

functions, Total derivatives, Change of variables.

Differential Calculus - II

Taylor’s and Maclaurin’s Theorem,

Expansion of function of several variables, Jacobian, Approximation of

errors, Extrema of functions of several variables, Lagrange’s method of

multipliers (Simple applications).

1. If 1 3sin ( ), 3 , 4u x y x t y t−= − = = show that

2

3

1

du

dt t=

−. (UPTU 2005)

2. If log( )u x xy= where 3 3 3 1x y xy+ + = find

du

dx. (special exam 2001)

3. If ( , , ),u f y z z x x y= − − − prove that 0du du du

dx dy dz+ + = . (UPTU 2003 2005)

4. If ( , ) 0, ( , ) 0,f x y y zφ= = show that . . .df d dz df df

dy dz dx dx dy

φ= (UPTU ,AG 2005)

5. IF ( , ) 0,xy zφ = , show that 1x y z

y z x

z x y

∂ ∂ ∂ = −

∂ ∂ ∂ (UPTU 2004)

6. If ,y x z x

u uyx xz

− −=

,show that

2 2 2 0u u u

x y zx y z

∂ ∂ ∂+ + =

∂ ∂ ∂. (UPTU 2005)

7. Expand 1

tany

x

− in the neighborhood of (1,1) upto and inclusive of second degree terms. Hence compute

(1.1,0.1)f approximately. (UPTU 2003)

8. Expand 2 3 2x y y+ − in powers of ( 1)x − and ( 2)y + using Taylor’s Theorem. (UPTU 2001)

9. Expand cosxe y at 1,4

π

. (UPTU ,AG, 2005)

10. Expand yx in the powers of ( 1)x − and ( 1)y − upto the third terms and hence evaluate ( )

1.021.1

(UPTU 2004)

11. If u, v are the function of r, s where r,s are functions of x, y then ( , ) ( , ) ( , )

( , ) ( , ) ( , )

u v u v r s

x y r s x y

∂ ∂ ∂= ×

∂ ∂ ∂

12. If 1J is the Jacobian of u, v with respect to x, y and 2J is the Jacobian of x, y with respect to u,v then

1 2

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

( , ) ( , )

u v x yJ J i e

x y u v

∂ ∂= =

∂ ∂ ( Q. bank UPTU 2001)

13. If sin cos , sin sin , cos ,x r y r z rθ φ θ φ θ= = = show that 2( , ) ( , , )

. sin( , ) ( , , )

u v x y zr

x y rθ

θ φ

∂ ∂=

∂ ∂ and find

( , , )

( , , )

r

x y z

θ φ∂

∂.

(UPTU 2001)

14. If 2 3 1 3 1 21 2

1 2 3

, ,x x x x x x

y y yx x x

= = = , then show that 1 2 3

1 2 3

( , , )4

( , , )

y y y

x x x

∂=

∂.

(UPTU special exam 2001, UPTU 2005)

15. If 2 2 2, ,u xyz v x y z w x y z= = + + = + +

(i) Find the Jacobian ( , , )

( , , )

x y z

u v w

∂

∂ (UPTU 2003, UPTU (CO), 2003)

(ii) , ,x y z u y x uv z uvw+ + = + = = then show that 2( , , )

( , , )

x y zu v

u v w

∂=

∂ (UPTU 2001)

16. (i) if u,v,w are the roots of the cubic ( ) ( ) ( )3 3 3

0x y zλ λ λ− + − + − = in λ , find ( , , )J u v w .

(ii) if u,v,w are the roots of the equation ( ) ( ) ( )3 3 3

0x a x b x c− + − + − = then find ( , , )

( , , )

u v w

a b c

∂

∂.

(UPTU 2002)

17. If 1 1 2 2sin sin , 1 1u x y v x y y x

− −= + = − + − find ( , )

( , )

u v

x y

∂

∂. Is u, v functionally related? If so find the

relationship. ( Q. bank UPTU 2001)

18. Find the percentage error in the area of an ellipse when an error of +1 percent is made in measuring the major and

minor axes. ( Q. bank UPTU 2001)

19. The period of the simple pendulum is 2l

Tg

π= . Find the maximum error in T due to the possible errors upto 1%

in l and 2.5% in g. (UPTU 2004)

20. If the base radius and high of a cone are measured as 4 and 8 inches with a possible error of 0.04 and 0.08 inches

respectively. Calculate the percentage error in calculating volume of cone. (UPTU CO 2003)

21. A balloon is in the form of right circular cylinder of radius 1.5m and length 4m and is surmounted by hemispherical

ends. If the radius is increased by 0.01m and length by 0.05m, find the percentage change in the balloon.

(UPTU special exam 2001)

22. In estimating the number of bricks in a pile which is measured to be (5m x 10m x 5m), the count of the bricks is taken

as 100 bricks per 3

m . Find the error in the cost when the tape is stretched 2% beyond its standard. The cost of the

bricks is Rs. 2000 per thousand bricks. (UPTU 2001)

23. In estimating the cost of a pile of bricks measured as 6m x 50m x 4m, the tape is stretched 1% beyond the standard

length. If the count is 12 bricks in 1 3m and bricks cost Rs. 100 per 1,000, find the approximate error in the cost.

(UPTU 2005)

24. Find the extreme values of the function3 3 3x y axy+ − . (UPTU 2005)

25. A rectangular box, is to have a given capacity. Find the dimensions of the box requiring least material for its

construction. ( Q. bank UPTU 2001)

26. Find the semi- vertical angle of thecone of maximum volume and of a given slant height. (UPTU 2001)

27. Show that the rectangular solid of maximum volume that can be inscribed in the given sphere ia a cube.

(UPTU 2004)

28. find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid

2 2 2

2 2 21

x y z

a b c+ + = .

(UPTU 2001,2003)

29. find the maximum and minimum distances of the point (3,4,12) from the sphere 2 2 2 1x y z+ + =

(UPTU special exam 2001)

30. find the maximum and minimum distances from the origin to the curve 2 24 6 140x xy y+ + =

(UPTU CO 2003)

31. find the minimum value of 2 2 2x y z+ + , given that ax by cz p+ + = (UPTU,AG, 2005)

32. A tent of given volume has a square base of side 2a and has its four sides of height b vertical and surmounted by a

pyramid of height h. find the values of a and b in the terms of h so that the canvas required fir its construction be

minimum. ( Q. bank UPTU 2001)

33. If ( ) ( )

2

21 2

xy

x x=

− +, find th

n derivative of y. (UPTU 2000)

34. find thn derivative of 1

2

2tan

1

x

x

−

− . (UPTU Special exam 2001)

35. if ( )

1log

1

xy x

x

−=

+, show that ( )

( ) ( )

2!

1 11 ( 2)

n

n nn

x n x n

xy n

x

− − +

− − +

−

−

=

. (UPTU 2003)

36. if 1sin( sin )y m x−= , then prove that 2 2

2 1(1 ) 0x y xy m y− − + = and 2 2 2

2 1(1 ) (2 1) ( )n n n

x y n xy n m y+ +− = + + − .

(UPTU Special exam 2001, UPTU 2005)

37. if cos(log ) sin(log )y a x b x= + ,show that 2

2 1 0x y xy y+ + = and 2 2

2 1(2 1) ( 1) 0n n n

x y n xy n y+ ++ + + + = .

(UPTU 2004)

38. if ( )2 1n

y x= − , prove that 2 2

2 1( 1) 2 ( 1) 0n n n

x y xy n n y+ +− + − + = hence if 2( 1)

nn

n n

dP x

dx= − show that

2 2(1 ) ( 1)nn n

dPdP x n n P

dx dx

= − + +

. (UPTU 2001, 2003)

39. if 1tan x

y e−

= , prove that [ ]2

2 1(1 ) (2 2) 1 ( 1) 0n n nx y n x y n n y+ +− + + − + + = . (UPTU CO, 2003)

40. if 1 2sin( sin )y m x−= , find (0)ny . (UPTU 2005)

41. if 1tany x−= ,show that [ ]2

2 1(1 ) 2( 1) 1 ( 1) 0n n nx y n x y n n y+ ++ + + − + + = and hence find 3 4,y y and

5y at

0x = . (UPTU CO, 2003)

42. if 3 3 3log( 3 )u x y z xyz= + + − ,show that ( )

2

2

9u

x y z x y z

∂ ∂ ∂+ + = −

∂ ∂ ∂ + + . (UPTU 2004)

43. if ( )u f r= ,where 2 2 2r x y= + ,prove that

2 2'' '

2 2

1( ) ( )

u uf r f r

x y r

∂ ∂+ = +

∂ ∂ . (UPTU 2001)

44. if 2 2 2

2 2 21

x y z

a u b u c u+ + =

+ + +,prove that

22 2

2u u u u u u

x y zx y z x y z

∂ ∂ ∂ ∂ ∂ ∂ + + = + +

∂ ∂ ∂ ∂ ∂ ∂ (UPTU 2001)

45. if 2 1 2 1tan tany x

u x yx y

− −= − , find the value of

2u

x y

∂

∂ ∂. (UPTU Special exam 2001)

MATRIX ALGEBRA

Types of Matrices, Inverse of a matrix

by elementary transformations, Rank of a matrix (Echelon & Normal

form), Linear dependence, Consistency of linear system of equations and

their solution, Characteristic equation, Eigen values and Eigen vectors,

Cayley-Hamilton Theorem, Diagonalization, Complex and Unitary

Matrices and its properties

1. Transform

1 3 3

2 4 10

2 8 4

into a unit matrix by using elementary transformations. (UPTU 2011)

2. Define elementary transformations. Using elementary transformations, find the inverse of the matrix

1 2 3

2 4 5

3 5 6

(UPTU 2007)

3. find the inverse of the matrix

1 1 1

3 5 7

1 1 1

5 7 11

1 1 1

7 11 13

(UPTU 2008)

4. use elementary transformations to reduce the following matrix a to triangular form and hence find the

rank of A

2 3 1 1

1 1 2 4

3 1 3 2

6 3 0 7

− − − − − −

−

. (UPTU 2006)

5. find the rank of the matrix

2 3 2 4

3 2 1 2

3 2 3 4

2 4 0 5

− − −

. (UPTU 2007)

6. find the rank of the matrices by reducing it to normal form

(i)

0 1 2 2

4 0 2 6

2 1 3 1

−

(UPTU 2007)

(ii)

1 2 1 0

2 4 3 0

1 0 2 8

− −

(UPTU 2015)

(iii)

1 3 3 1

1 1 1 0

2 5 2 3

− − − − − −

(UPTU 2014)

7. Find all the values of µ for which rank of the matrix

1 0 0

0 1 0

0 0 1

6 11 6 1

A

µ

µ

µ

− − = − − −

is equal to 3. (UPTU CO 2009)

8. Use elementary transformations to reduce the matrix A to triangular form and hence find the rank of A

2 3 1 1

1 1 2 4

3 1 3 2

6 3 0 7

A

− − − − − = −

−

(UPTU 2006)

9. Reduce A to echelon form and then to its row canonical form where

1 3 1 2

0 11 5 3

2 5 3 1

4 1 1 5

A

− − = −

hence find the rank of A. (UPTU 2015)

10. Find the non-singular matrices P and Q such that PAQ is in normal of the matrix and hence the rank of matrix

1 2 3 2

2 2 1 3

3 0 4 1

A

− = −

(UPTU 2015)

11. Find the rank of the following matrices by reducing it to normal form (canonical form)

(i)

0 1 2 2

4 0 2 6

2 1 3 1

A

− =

(UPTU 2007)

(ii)

1 2 1 0

2 4 3 0

1 0 2 8

A

= −

(UPTU 2007)

(iii)

1 3 3 1

1 1 1 0

2 5 2 3

A

− − − = − − −

(UPTU 2014)

12. Test the consistency of the following system of linear equations and hence find the solution, if exists:

(i) 1 2 1 2 3 2 34 12, 5 2 0, 2 4 8x x x x x x x− = − + − = − + = −

(UPTU 2006)

(ii) 1 2 3 1 2 3 1 2 37 2 3 16, 2 11 5 25, 3 4 13x x x x x x x x x+ + = + + = + + = (UPTU 2008)

13. Apply the matrix method to solve the system of the equations (UPTU ,CO 2003, 2003,2008)

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

14. Investigate, for what values of λ and µ do the system of the euations

6, 2 3 0, 2x y z x y z x y zλ µ+ + = + + = + + = (UPTU 2014)

15. Solve the system of linear equations using the matrix method

2 3 5

7 11 13 17

19 23 29 31

x y z

x y z

x y z

+ + =

+ + =

+ + =

(UPTU 2009)

16. Test the consistency and solve the following system of equations

2 3 8, 2 4,3 4 0x y z x y z x y z− + = − + − = + − = (UPTU 2014)

Find the value of k so that the equations 3 0, 4 3 0, 2 2 0x y z x y kz x y z+ + = + + = + + = have a non- trivial

solution. (UPTU 2011)

17. Determine “b” such that the system of homogenous equations (UPTU 2009)

2 2 0, 3 0, 4 3 0x y z x y z x y bz+ + = + + = + + = has

(i) Trivial solution (ii) non-trivial solution. Find the non-trivial solution by using the matrix method.

18. Using the matrix method ,show that the equations (UPTU 2011)

3 3 2 1, 2 4,10 3 2, 2 3 5x y z x y y z x y z+ + = + = + = − − − = are consistent and hence obtain the solutions for x,y

and z.

19. Investigation, for what values of λ and µ do the system of the equations (UPTU 2002)

6, 2 3 10, 2x y z x y z x y zλ µ+ + = + + = + + = have

(i) Unique solution

(ii) No solution

(iii) Many infinite solution

20. Find the value of λ such that the following equations have unique solution : (UPTU 2004)

2 2 1 0, 4 2 2 0,6 6 3 0x y z x y z x y zλ λ λ+ − − = + − − = + + − = and use the matrix method to solve these

equations when 2λ =

21. Show that the vectors 1 2 3(1, 2,4), (2, 1,3), (0,1,2)x x x= = − = and 4 ( 3,7, 2)x = − are linearly dependent and

find the relation between them. (UPTU 2004)

22. Find the Eigen value and corresponding Eigen vectors of the matrix5 2

2 2A

− = −

. (UPTU 2004)

23. Define Eigen vectors. (UPTU ,CO,2003)

24. Show that the characteristic roots of a unitary matrix are of unit modulus. (UPTU special exam 2001)

25. Show that the latent roots of a harmitian matrix are all real. (UPTU co, 2003)

26. The sum of the eigen values of a square matrix is equal to the sum of the elements of its principal diagonal.

(UPTU 2003)

27. Find the Eigen value and Eigen vectors of the following matrices

(i)

8 6 2

6 7 4

2 4 3

A

− = − − −

(UPTU 2005)

(ii)

3 1 4

0 2 6

0 0 5

A

=

(UPTU special 2001, 2005)

28. Show that every square matrix satisfies its own characteristic equation. (UPTU special 2001)

29. Verify Cayley Hamilton theorem for the matrix

2 1 1

1 2 1

1 1 2

A

− = − − −

. (UPTU 2005)

30. Find the characteristic equation of the matrix

2 1 1

0 1 0

1 1 2

A

=

and hence compute 1

A−

. Also find the matrix

represented by 8 7 6 5 25 7 3 8 2A A A A A A I− + − + − + . (UPTU 2005)

31. Find the characteristic equation of the matrix

2 1 1

1 2 1

1 2 2

A

− = − − −

. Show that the equation is satisfied by A and

hence obtain the inverse of the given matrix. (UPTU 2005)

32. Verify the Cayley Hemilton for the matrix

3 4 1

2 1 6

1 4 7

A

= −

. (UPTU 2005)

33. Using Cayley Hemilton theorem , find te inverse of

4 3 1

2 1 2

1 2 1

A

= −

. (UPTU 2002)

34. Show that the matrix

3 10 5

2 3 4

3 5 7

− − −

has less then there linearly independent eigen vectors. Also find them.

( UPTU 2002)

Vector Calculus

Point function, Gradient, Divergence and Curl of a vector and their

physical interpretations, Vector identities, Tangent and Normal,

Directional derivatives. Line, Surface and Volume integrals, Applications

of Green’s, Stoke’s and Gauss divergence theorems (without proof).

1. If 2 2 2, ,u x y z v x y z w yz zx xy= + + = + + = + + , prove that grad u, grad v and grad w are coplanar.

(UPTU 2002)

2. Find the directional derivative of ( )1

2 2 2 2x y zφ = + + at the point (3,1, 2)P in the direction of the vector.

(UPTU 2001) special

3. If the directional derivative of 2 2 2ax y by z cz xφ = + + at the point (1,1,1) has maximum magnitude 15 in the direction

parallel to the line 1 3

2 2 1

x y z− −= =

−,find the values of a, b and c. (UPTU 2002)

4. Find the directional derivative of 2 2 25

5 52

x y by z z xφ = − + at the point (1,1,1)P in the direction of the line

1 3

2 2 1

x y z− −= =

−. (UPTU 2004)

5. Find the directional derivative of 1

r in the direction of r

�, where . (UPTU 2003)

6. Define curl of a vector point Function. (UPTU special exam 2001, CO 2003)

7. Define physical interpretation of curl. Or prove that the angular velocity at any point is equal to half the curl of the linear

velocity at that point of the body. (UPTU 2001)

8. If a�

is a vector function and u is a scalar function then

( ) ( ).div ua u diva grad u a= +� � �

(UPTU 2004)

9. If a�

is a vector function and u is a scalar function then ( ) . .div a b b curl a a curl b× = −� � � � � �

(UPTU CO 2003

10. If a�

is a vector function and u is a scalar function then ( ) . ( )curl ua b u curl a grad u a× = + ×� � � �

(UPTU CO 2003

11. Prove that 2( ) ( )curl curl v grad div v v= − ∇

� � � (UPTU special exam 2001, UPTU 2003)

12. A fluid motion is given by � �( ) ( ) ( )V y z i z x j x y k= + + + + +��

� (UPTU 2004)

(i) Is this motion irrotational ? if so, find the velocity potential.

(ii) Is the motion possible for an incompressible fluid ?

13. If a�

is a constant vector, evaluate ( )div r a� �

and ( )curl r a� �

where � �r xi y j zk= + +�� (UPTU 2002)

14. Prove that 2 2( ) ( ) ( 1)n n ndiv grad r r n n r −= ∇ = + where � �r xi y j zk= + +

�� . Hence show that

2 10

r

∇ =

. Hence or

otherwise evaluate2

r

r

∇×

���

. (UPTU , 2005; UPTU CO, 2003)

15. Show that

16. � �1 1. 0curl k grad grad k grad

r r

× + =

� where r is the distance of a point (x, y, z) from the origin and �k is a unit

vector in the dirction of OZ. (UPTU 2001)

17. If f and g are two scalar point functions prove that 2( ) .div f g f g f g∇ = ∇ + ∇ ∇

(UPTU special exam 2001)

18. Show that the vector field 3

rF

r=

���

is irrotational as well as solenoidal. Find the scalar potential.

19. If � �

2 2 2

xi y j zkV

x y z

+ +=

+ +

��� show that

2 2 2

2.V

x y z∇ =

+ +

��� and 0V∇ × =

��� � (UPTU 2001)

20. Evaluate � �( ).s

yzi zx j xyk d S+ +∫∫��

� where s is the surface of the sphere 2 2 2 2x y z a+ + = in the first octant.

(UPTU 2005)

21. Use divergence theorem to show that 2. 6

s

r d S V∇ =∫��

� , where S is any closed surface enclosing a volume V.

(UPTU 2003)

22. Find �.s

F n dS∫∫��

, where � �2(2 3 ) ( ) ( 2 )F x z i xz y j y z k= + − + + +

��� and S is the surface of the sphere having centre at (3,-

1,2) and radius 3. (UPTU 2001)

23. The vector field � �2F x i z j yzk= + +��

� is defined over the volume of the cuboid given by 0 ,0 ,0x a y b z c≤ ≤ ≤ ≤ ≤ ≤

enclosing the surface S, evaluate the surface integral �.s

F n dS∫∫��

. (UPTU 2002)

24. Use the divergence theorem to evaluate the surface integral ( )s

xdydz ydzdx z dx dy+ +∫∫ where S is the plane

2 3 6x y z+ + = which lies in the first octant. (UPTU 2004)

25. Verify the divergence theorm for � �2 2 2( ) ( ) ( )F x yz i y zx j z xy k= − + − + −��

� taken over the rectangular parallelopiped

0 ,0 ,0x a y b z c≤ ≤ ≤ ≤ ≤ ≤.

(UPTU special exam 2001)

26. Verify stoke’s theorem for �2 2( ) 2F x y i xy j= + −

��� taken round the rectangle bounded by the lines , 0,x a y y b= ± = = .

(UPTU 2003)

27. Verify Stoke’s theorem for the vector field � �2 2(2 )F x y i yz j y zk= − − −��

� over the upper half surface 2 2 2 1x y z+ + =

bounded by its projection on the xy- plane. (UPTU 2002)

28. Evaluate .c

F dr∫�� ���

� by the stock’s theorem, where � �2 2( )F y i x j x z k= + − +

��� and C is the boundary of the triangle with the

vertices at ( 0,0,0), (1,0,0), and (1,1,0). (UPTU 2001)

29. Verify the stock’s theorem for the vector � �F zi x j yk= + +��

� taken over the half of the sphere 2 2 2 2x y z a+ + =

Lying above xy-plane. (UPTU CO, 2003)

30. Verify the stock’s theorem for the function � �F zi x j yk= + +��

� where C is a unit circle in xy plane bounded by the

hemisphere2 21z x y= − − . (UPTU special exam 2001)

31. Apply Green’s theorem to evaluate ( ) ( )2 2 2 22

c

x y dx x y dy − + + ∫ where C is the boundary of the area enclosed by

the x-axis and the upper half of the circle 2 2 2x y a+ =

(UPTU 2005)

32. Using the green’s theorem, evaluate ( )2 2

c

x ydx x dy+∫ where is the boundarydescribed counter clockwise of the triangle

with the vertices (0,0), (1,0), (1,1).

(UPTU 2004)