INSTITUTE OF AERONAUTICAL ENGINEERING - · PDF file · 2015-12-06INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad ... 2 Write the working rule to find orthogonal trajectory

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INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad - 500 043

FRESHMAN ENGINEERING

QUESTION BANK

Course Name : ENGINEERING MATHEMATICS-I

Course Code : A10002

Class : I - B. Tech

Branch : Common for all Branches

Year : 2015 - 2016

Course Faculty : Dr.M.Anita(HOD), Mr.Ch.Kumara Swamy, Mr.J.Suresh Goud, Mrs.L.Indira,

Mrs.V.Subbalaxmi, Mrs P.Srilatha, Mr Ch .Somashekar.

OBJECTIVES

To meet the challenge of ensuring excellence in engineering education, the issue of quality needs to be addressed, debated and taken forward in a systematic manner. Accreditation is the principal means of quality assurance in higher education. The major emphasis of accreditation process is to measure the outcomes of the program that is being accredited. In line with this, Faculty of Institute of Aeronautical Engineering, Hyderabad has taken a lead in incorporating philosophy of outcome based education in the process of problem solving and career development. So, all students of the institute should understand the depth and approach of course to be taught through this question bank, which will enhance learner’s learning process. 1. Group - A (Short Answer Questions)

S. No Question Blooms Taxonomy

Level

Course

Outcome

UNIT-I THEORY OF MATRICES

1 Find the eigen values of the matrix

731

314

i

i

Apply 2

2 If A is Hermitian matrix Prove that it is skew- Hermitian matrix Analyze 1

3 State Cayley- Hamilton Theorem Analyze 1

4 Prove that 2

1

ii

ii

11

11 is a unitary matrix. Analyze 1

5 Find the value of k such that rank of

1 2 3

2 7 2

3 6 10

k is

. Apply 1

6 Find the eigen values of the matrix

243

432

i

i Apply 2

7 Apply 1

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Level

Course

Outcome

Find A if A =

ii

i

242

31

8 Define modal matrix. Remember 3

9 Find the Skew- symmetric part of the matrix

1 1 2

1 1 1

3 1 2

Apply 1

10 If 2, 3, 4 are the eigen values of A then find the eigen values of adj A

Apply 2

UNIT-II DIFFERENTIAL CALCULUS METHODS

1 Define Rolle’s Mean value theorem.

Remember 5

2 Verify Lagrange’s Mean Value theorem for f(x) = log x in [1, e] Analyze 5

3 Verify Lagrange’s Mean Value theorem for function f(x) = cos x in [0, π/2]. Analyze 5

4 Verify Cauchy’s Mean Value theorem for f(x) =x2, g(x) =x3 in[1, 2].

Analyze 5

5

Find first and second order partial derivatives of ax2+2hxy+by2 and

yx

f

yx

fverify

22

. Apply 5

6 When two functions u,v of independent variables x,y are functional dependent

Remember 5

7 If x = u(1-v), y = uv prove that JJ”=1 Analyze 5

8 Find the maximum and minimum values of

axyyx 333 Apply 6

9 If yevyeu xx cos,sin then find .),(

),(

yx

vu

Apply 5

10 Verify Rolle’s Mean value theorem for f(x)=(x+2)3(x-3)

4 in [-2,3] Analyze 5

UNIT-III IMPROPER INTEGRALS, MULTIPLE INTEGRALS AND ITS APPLICATIONS

1 Prove that ),(),( mnnm . Analyze 7

2 Prove that

anmnm nmnmbadxxxa

0

111 0,0),,()()(( Analyze 7

3 Compute )2/7(),2/1(),2/11( . Apply 7

4 Write the value of )1( . Remember 7

5 Evaluate 2

0 0

x

ydydx Evaluate 8

6 Evaluate

0

sin

0

a

rdrd . Evaluate 8

7 Evaluate 3

0

1

0)( dxdyyxxy .

Evaluate 8

8 Evaluate e y ex

zdxdydz1

log

1 2log .

Evaluate 9

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Level

Course

Outcome

9 find the value of

1

1

2

2

3

3dxdydz . Apply 9

10 Write the spherical polar coordinates Remember 9

UNIT-1V DIFFERENTIAL EQUATIONS AND APPLICATIONS

1 Solve (x+1)dy/dx –y=e3x

(x+1)2

Apply 10

2 Write the working rule to find orthogonal trajectory in Cartesian form. Remember 11

3 Form the D.E.by eliminate c in y=1+c√1-x

2

Apply 10

4 Solve (x+y+1) dy/dx =1 Apply 10

5 Prove that the system of parabolas y

2 = 4a (x+a) is self orthogonal.

Analyze 11

6 Find the O.T. of the family of curves Apply 11

7 State Newtons law of cooling Remember 10

8 A bacterial culture, growing exponentially, increases from 200 to 500 grams in the period from 6 a.m to 9 a.m. . How many grams will be present at noon.

Analyze 10

9 Solve -3 + 2y =0

Apply 10

10 Define S.H.M. and give its D.E Remember 10

UNIT-V LAPLACE TRANSFORMS AND ITS APPLICATIONS

1 Find Laplace transform of

Apply 12

2 State and prove first shifting theorem. Remember 12

3 Define change of scale property of Remember 12

4 Find Apply 12

5 If is periodic function with period T then find Apply 13

6 Find Apply 12

7 Find Apply 12

8 Find ) Apply 12

9 Define inverse L.T. of f(s) Remember 12

10 Use L.T to solve D.E. when t=0 Apply 14

1. Group - B (Long Answer Questions)


Level Course

Outcome


1

Show that only real number for which the system x+2y+3z= x ,

3x+y+2z= y , 2x+3y+z= z has non-zero solution is 6 and solve them

when =6

Analyze 1

2 Express the matrix as the sum of Hermitian

matrix and skew- Hermitian matrix.

Analyze 1

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Level Course

Outcome

3 Given that show that is unitary

matrix. Analyze 1

4 Find Inverse by elementary row operations of

Apply 1

5 Find whether the following equations are consistent if so solve them

Apply 1

6 Solve the equations using partial pivoting Guass Elimination method of

Apply 1

7 Diagonalize the matrix A=

344

120

111

and find A4 Apply 3

8 Prove The Eigen Values of Real symmetric matrix are Real.

Analyze 2

9 Reduce the Quadratic form -4xy-4yz to the Canonical

form. Analyze 4

10 Reduce the quadratic form to the

canonical form by orthogonal reduction. Analyze 4

11 Find rank by reducing to Normal form of matrix

Apply 1

12

Solve the following System of equations 4x+2y+z+3w=0,

6x+3y+4z+7w=0, 2x+y+w=0 Apply 1

13

Find and such that 12 zyx , zyx 2 ,

12 zyx has (i) no solution (ii) Unique solution (iii)

Many solutions

Apply 1

14 Find the eigen values and eigen vectors of

000

011

011

Apply 2

15 State Cayley-Hamilton theorem and verify the matrix A =

Remember 1

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Level Course

Outcome

101

210

111

16 Diagonalize the matrix

122

121

223

A Apply 3

17

Diagonalize the matrix

121

211

112

A by an orthogonal

transformation.

Apply 3

18

Reduce to sum of squares, the quadratic form

3121

2

3

2

2

2

1 8472 xxxxxxx and find the rank, index and

signature. Analyze 4

19

Prove that the eigen values of a Skew-Hermitian matrix are purely

imaginary or zeros. Analyze 2

20

If A is any square matrix then prove that i) A + A

ii) AA

, A

A

are Hermitian iii) A - A

are skew – Hermitian

Analyze 1


1 if ( )f x = x , ( )g x =

1

x prove that ‘c’ of the CMVT is the geometric

mean of a and b for any a>0,b>0

Analyze 5

2 find the minimum value of 2 2 2x y z

given 3x y z a Apply 6

3 prove using mean value theorem sin sinu v u v 5

4 if

( , , ), ,

( , , )

yz xz xy u v wu v w find

x y z x y z

Apply 5

5

using mean value theorem prove that the function 2( ) 4 7f x x x is increases when 2x decreasing when 2x

Analyze 5

6 Calculate approximately by using LMVT Apply 5

7 find the region in which ( )f x =1-4 x -

2x is increasing and the region in

which it is decreasing using mean value theorem Apply 5

8 find three positive numbers whose sum is 100 and whose product is

maximum Apply 5

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Level Course

Outcome

9

Find the volume of the greatest rectangular parallelepiped that can be

inscribed in the ellipsoid

2 2 2

2 2 21

x y z

a b c

Apply 5

10

Using mean value theorem for 0 a b prove that

1 log 1a b b

b a a and hence Show that

1 6 1log

6 5 5

Analyze 5

11 find the maxima and minima of 2 2 4 4( ) 2( )f x x y x y Apply 6

12 Using mean value theorem prove that tanx>x in 02

x

Analyze 5

13 Prove that 11 3 1

cos3 5 3 85 3

using LMVT Analyze 5

14 If Also Show that

=1

Analyze 5

15 Prove that 2 2 2, ,u xy yz zx v x y z w x y z are

functionally dependent and find the relation between them Analyze 5

16 If throughout an interval [a,b] prove using mean value theorem f(x) is a constant in that interval.

Analyze 5

17 Divide 24 into three points such that the continued product of the first,

square of the second and cube of the third is maximum Analyze 5

18 using rolle’s theorem show that 3 2( ) 8 6 2 1g x x x x has a

zero between 0 and 1 Analyze 5

19 If , ,x y z u y z uv z uvw

show that 2( , , )

( , , )

x y zu v

u v w

Analyze 5

20 If

(1 ),x u v y uv prove that

1 1JJ Analyze 5

UNIT-III IMPROPER INTEGRATION, MULTIPLE INTEGRATION AND APPLICATIONS

1

By transforming into polar coordinates Evaluate

2 2

2 2

x ydxdy

x y over

the annular region between the circles 2 2 2x y a and

2 2 2x y b with b a

Evaluate 8

2

Evaluate ( )R

x y z dzdydx where R is the region bounded by the

planes 0, 1, 0, 1, 0, 1x x y y z z

Evaluate 9

3

Evaluate

11 1

0 0 0

y zz

xyzdxdydz

Evaluate 9

4

Find the volume of the tetrahedron bounded by the plane

1x y z

a b c and the coordinate planes by triple integration

Apply 9

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Level Course

Outcome

5

Evaluate

(1 cos )

2

0 0

cos

a

r drd

Evaluate 8

6

Find the value of xydxdy taken over the positive quadrant of the

ellipse

2 2

2 21

x y

a b

Apply 8

7

Evaluate

2 2

sin2 2

0 0 0

a r

a

rdzdrd

Evaluate 9

8 ProveThat where p>0, q>0

Analyze 7

9 Show that )2/1( Analyze 7

10 Evaluate Evaluate 7

11 Evaluate the double integral

2 2

2 2

0 0

( )

a ya

x y dydx

Evaluate 8

12 Show that Analyze 9

13 Show that Analyze 8

14 Evaluate

22

2 2 2

0 0( )

rdrd

r a

Evaluate 9

15 By changing the order of integration, evaluate

1 2

0 0

x

xydxdy

Evaluate 9

16

By changing the order of integration, evaluate

2 2

2 2 2

0 0

a a x

a x y dydx

Evaluate 7

17

Express

0

cossin dqp

in terms of beta function.

Evaluate 7

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Level Course

Outcome

18

Evaluate 2x dxdy over the region bounded by hyperbola

4, 0, 1, 4xy y x x Evaluate 9

19 Evaluate

log2 log

0 0 0

x yx

x y ze dxdydz

Evaluate 9

20

Evaluate by changing the order of integration

2

4 2 20

a a

ax

ydydx

y a x Evaluate 8

UNIT-IV DIFFERENTIAL EQUATIONS AND APPLICATIONS

1 A bacterial culture, growing exponentially, in increases from 200 to 500 grams in the period from 6 am to 9 am. How many grams will be present at noon?

Analyze 10

2

Solve

2

2 2

( )a xdy ydxxdx ydy

x y

Apply 10

3 Solve 2 22 ( 1) 0xydy x y dx

Apply 11

4 Find the orthogonal trajectories of the family of curves 2 2 2x y a Apply 10

5 Solve the D.E

3 2( 1) ( 1)xdyx y e x

dx

Apply 10

6 Obtain the orthogonal trajectories of the family of curves

(1 cos ) 2r a Apply 11

7

A particle is executing S.H.M, with amplitude 5 meters time 4 sec.find the time required by the Particle in passing between points which are at distances 4 &2 meters from the centre of force and are on the same side of it.

Analyze 10

8 Solve 2 2( 3 2) 2cos(2 3) 2 xD D y x e x

Apply 10

9 Solve 2 2 2( 4) 96 sin 2D D y x x k

Apply 10

10 By using method of variation of parameters solve 2( 1) cosD y ecx Apply 10

11 Solve

3 6dyx y x y

dx

Apply 10

12

If the air is maintained at 25 cand the temperature of the body cools

from 100 cto 80 c

in 10 minutes, find the temperature of the body

after 20 minutes and when the temperature will be 40 c

Analyze 10

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Level Course

Outcome

13 Solve 2 2( 1) sin sin 2 xD y x x e x Apply 10

14 Solve

(1 ) (1 ) 0

x x

y y xe dx e dy

y

Apply 10

15

A copper ball is heated to a temperature of 80 cand time t=0, then it is

placed in water which is maintained at 30 c. If at t=3minutes,the

temperature of the ball is reduced to 50 cfind the time at which the

temperature of the ball is 40 c

Analyze 10

16 Solve 3 2 2 3( 6 11 6) x xD D D y e e Apply 10

17 Solve

3 2 2sec 3 tan cosdy

x y x y xdx

Apply 10

18

A body weighing 10kgs is hung from a spring pull of 20kgs will stretch

the spring to 10 cms. The body is pulled down to 20 cms below the

static equilibrium position and then released. Find the displacement of

the body from its equilibrium position at time‘t’ seconds, the maximum

Velocity and the period of oscillation

Analyze 10

19 Solve the D.E 4 2 2 2( 2 1) cosD D y x x

Apply 10

20

The number N of bacteria in a culture grew at a rate proportional to N.

The value of N was initially 100 and increased to 332 in one hour. What

was the value of N after 1

12

hours? Analyze 10

UNIT-V LAPLACE TRANSFORMS AND ITS APPLICATIONS

1 Using L.T Evaluate

2

0

t te edt

t

Evaluate 12

2 Find 1

2 2 2( 1)( 9)( 25)

sL

s s s

Apply 12

3 Find L f(t) where t, if0<t<b

f(t)=2b-t, if b<t<2b

of period 2b Apply 13

4 Find cos 4 sin 2

Lt t

t

Apply 12

5 Find ( )L g t where

2 2cos(t- ), if t>

3 3g(t)=

20, if t<

3

Apply 13

6 Find the .L.T of

2

3 2

2 6 5

6 11 6

S S

S S S

Apply 12

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Level Course

Outcome

7 Solve the D.E using L.T,'' ' 33 2 4 ty y y t e ,

'( ) (0) 1y o y Apply 14

8 Solve the D.E using L.T,

2

22 2 5sin ,

d y dyy t

dt dt

'(0) (0) 0y y Apply 14

9 Find the .L.T of 2

2

4log( )

9

s

s

Apply 12

10 Use Laplace transforms, solve

2 '( 1) cos2 , (0) (0) 0D x t t givenx x Apply 14

11 Find the Inverse L.T of

2

3 2

2 6 5

6 11 6

S S

S S S

Apply 12

12 Find

21

2 4 5

seL

s s

Apply 12

13 Using L.T solve 2 '( 2 3) sin , (0) (0) 0D D y x y y Apply 14

14 Find the Laplace transform of 2 sin3tte t Apply 12

15

Find the L.T of periodic function ( )f t with period T where

4ET T-E, if 0 t

T 2f(t)=

4ET T3E- , if t

T 2T

Apply 13

16 Find -1

3 2

2 3L

6 11 6

S

S S S

Apply 12

17

Solve the D.E using L.T

22

22 td x dx

x edt dt

with

(0) 2, 1dx

xdt

at t=0

Apply 14

18 Find sin3 cos2L t t t Apply 12

19

Using L.T solve 3 2 2( 4 4) 68 sin 2 , 1, 19, 37xD D D y e x y Dy D y

at 0x

Apply 14

20 Find 3 sinh3tL e t using change of scale property Apply 12

3. Group - III (Analytical Questions)

S. No Questions Blooms Taxonomy

Level Program Outcome


1 If A is n rowed matrix ijaA where .,

j

iaij denotes greatest

integer then find the value of det A

Apply 1

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2 If a=diagonal(1 -1 2)and b=diagonal (2 3 -1)then find 3a+4b Apply 1

3 If a=

i

i

0

0then find A4n

Apply 1

4 Find the value of k for which matrix A=

10

310

01

k

k

Is invertible. Apply

1

5 What is rank of 4x5 matrix Remember

1

6 If 1,2,3 are eigen value of A then find eigen value of Adj A.

Apply

2

7 If

0

1

0

&

1

0

1

are two orthogonal vectors of 3x3 matrix then find third

vector

Apply

2

8 If A is nxn matrix , rank is k and normal form is

00

0kIthen find order

of null matrix below side of kI

Apply 1

9

If A=

163

045

003

then express 3A in terms of A.

Apply 1

10 Find the rank of quadratic form whose eigen values are 0 ,0 ,6 Apply 1


1 When the Jacobian Transformation is used? Remember 5

2 Find the functional relationship between u=x + y +z, v=x y +y z + z x, w = x2+y2+z2 Apply 5

3 What are critical points? Remember 5

4

Write the relationship between yxvxy

yxu 11 tantan,

1

.

Remember 5

5 Find the stationary values of x3 y2 (1-x-y). Apply 6 6 What are saddle points? Remember 6

7 What is condition for f( x , y)to have maximum and minimum values at (a, b)?

Remember 6

8 What is the demerit of Lagrange’s method of undetermined multipliers? Remember 6

9 If f(x, y) = x y+(x-y) then find stationary points. Apply 6

10 If u=x y then x

u

. Apply 5

UNIT-III IMPROPER INTEGRALS, MULTIPLE INTEGRALS AND ITS APPLICATIONS

1 Write the relationship between beta and gamma functions. Remember 7

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2

What is the value of 2

0

24 coss in

d using -function Remember 7

3 What is the value of (p+1,q) + (p,q+1). Remember 7

4

Find (m, m) Apply 7

5

An equivalent iterated integral with order of integration reversed for

1

0 1

xe

dydx is Analyze 8

6 How to find the area of bounded region. Remember 8

7 How to find the volume of closed surface. Remember 9

8 What is difference between proper and improper integrals Remember 8

9 Convert

a xa

dydxyx0 1

2222

)( to polar co-ordinates. Analyze 8

10 What is the area of drdr

3over the region included between the

circles sin4,sin rr . Analyze 8

UNIT-IV DIFFERENTIAL EQUATIONS

1 Find the order and degree of

41

6

2

2

dx

dyy

dy

yd

Apply 10

2 A spherical rain drop evaporates at a rate proportional to its surface area at any instant t. The differential equation giving the rate of change of the radius (r) of the rain drop is.

Analyze 10

3 If 1)1(,22 yyxdx

dyx then y(2) is Apply 10

4 When the differential equation is said to be homogeneous? Remember 10

5 Mention two applications of higher order differential equations. Remember 10

6 What is general solution of higher order differential equations? Remember 10

7 what is orthogonal form of the function ),,(),,,(

d

drrf

dy

dxyxf

Remember 11

8 what is general solution of linear differential equation Remember 10

9

when the Bernoull’s differential equation becomes linear differential equation

Remember 10

10 Give the complementary function for (D2+6D+9)y=0 Analyze 10

UNIT-V LAPLACE TRANSFORMS

1 Give example where the Laplace Transforms technique is used Remember 12

2

What are the conditions that the functions has to satisfy to Apply laplace transform

Remember 12

3 Where the convolution theorem is in laplace transforms Remember 12

4 What are periodic functions Remember 13

5

Find

asL

11 Apply 12

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6

What is Laplace Transform of unit impulse function Remember 12

7 If f(0)=0 find )(' tfL Apply 12

8 Find the value of tL 2 Apply 12

9

When

nsL

11 is possible Remember 12

10 If y’’+ y=sin 3t with y=y’=0 then find )(tfL Apply 14

Prepared by : Dr M. Anitha, Mr Ch. Kumara Swamy, Ms L. Indira, Ms V. Subba Lakshmi, Mr J. Suresh Goud,

Ms P. Sri Latha and Mr Ch. Somashekar. Date: 01 August, 2015

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INSTITUTE OF AERONAUTICAL ENGINEERING - · PDF file · 2015-12-06INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad ... 2 Write the working rule to find orthogonal trajectory