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Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

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MA131 MATHEMATICS I QUESTION BANK Question Bank MATRICES: PART –A 1. Find the eigen values of 2A+I given that 2 3 . = 1 4 A 2. Find the eigen values and eigen vectors of = 0 0 a a A . 3. Find the sum of the eigen values of 2A if = 3 4 2 4 7 6 2 6 8 A . 4. Verify Cayley- Hamilton theorem for . = 3 1 3 5 A 5. Find the matrix of the quadratic form yz xz z y x 2 4 2 2 2 2 + + + + . PART - B 6. (i) Using Cayley-Hamilton theorem, find the inverse of A where (8 Marks) = 4 4 2 3 3 1 3 1 1 A (ii) Using diagonalization, find A 6 given that . (8 Marks) = 2 1 4 5 A 7. Find an orthogonal transformation which reduces the quadratic form xz yz xy 2 2 2 + + to a canonical form. Also find its nature, index, signature and rank. (16 Marks) 7. Reduce the quadratic form xz to a canonical form by an orthogonal transformation. Also find its nature, index, signature and rank. (16 Marks) yz xy z y x 2 4 2 2 2 2 2 + + + 8. (i) Using Cayley-Hamilton theorem, find A n given that . Hence find A 3 . (8 Marks) = 3 2 4 1 A
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Page 1: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

MA131 MATHEMATICS I QUESTION BANK

Question Bank

MATRICES: PART –A

1. Find the eigen values of 2A+I given that ⎟⎟⎞

⎜⎜ 23. ⎠⎝

⎛=

14A

2. Find the eigen values and eigen vectors of ⎟⎟⎠

⎞⎜⎜⎝

⎛=

00a

aA .

3. Find the sum of the eigen values of 2A if ⎟⎟⎟

⎜⎜⎜

−−−

−=

342476

268A .

4. Verify Cayley- Hamilton theorem for ⎟⎟ . ⎠

⎞⎜⎜⎝

⎛=

3135

A

5. Find the matrix of the quadratic form yzxzzyx 242222 ++++ .

PART - B 6. (i) Using Cayley-Hamilton theorem, find the inverse of A where

(8 Marks) ⎟⎟⎟

⎜⎜⎜

−−−=

442331

311A

(ii) Using diagonalization, find A6 given that . (8 Marks) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

2145

A

7. Find an orthogonal transformation which reduces the quadratic form

xzyzxy 222 ++ to a canonical form. Also find its nature, index, signature and rank. (16 Marks)

7. Reduce the quadratic form xz to a canonical

form by an orthogonal transformation. Also find its nature, index, signature and rank. (16 Marks)

yzxyzyx 2422 222 −−+++

8. (i) Using Cayley-Hamilton theorem, find An given that ⎟⎟ . Hence

find A3. (8 Marks) ⎠

⎞⎜⎜⎝

⎛=

3241

A

Page 2: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

(ii) Diagonalize A by an orthogonal transformation where . ⎟⎟⎟

⎜⎜⎜

⎛=

204060402

A

(8 Marks) 9. Find the characteristic equation of the matrix A and hence find the matrix

respresented by IAAAAAAAA +−+−+−+− 285375 2345678

⎟⎟⎟

⎞where

. (16 Marks) ⎜⎜⎜

⎛=

211010112

A

ANALYTICAL GEOMETRY: PART-A

1. A, B, C, D are the points (-3, 2, k), (4, 1, 6), (-1, -2, -3) and (13, -4, -1). Find the value of k if AB is parallel to CD.

2. Find the angle between the planes 072 =++− zyx and 0112 =−++ zyx .

3. Find the value of k if the lines

kzyxz

kyx

−=

+=

−++

=−

=−−

31

41&

513

11

are perpendicular.

4. Find the equation of the sphere having the points (2,-3,4) and (-1,5,7) as the ends of a diameter.

5. Find the equation of the tangent plane at the point (1,-1,2) to the sphere 012 . 642222 =−++−++ zyxzyx

PART-B 6. (i) Prove that the four points A(2,5,3), B(7,9,1), C(3,-6,2), D(13,2,-2) are

coplanar. (8 Marks) (ii) Find the equation of the plane that contains the parallel lines

34

22

13&

33

22

11 +

=+

=−−

=−

=− zyxzyx

. (8 Marks)

7. (i) Find the length of the shortest distance line between the lines

32306532;43

12

2+−−==−−+=

+=

− zyxzyxzyx. (8 Marks)

(ii) Find the co-ordinates of the foot, the length and the equations of the

perpendicular from the point (-1,3,9) to the line 1

3188

513 −

=−+

=− zyx

.

(8 Marks)

2

Page 3: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

8. (i) Show that plane 01222 =++− zyx touches the sphere

32 and find also the point of contact. (8 Marks) 42222 +−−++ yxzyx =z(ii) Find the equation of the sphere having the circle

as a great circle. (8 Marks) 3;08410222 =++=−−+++ zyxzyzyx

9. (i) Find the equation of the sphere that passes through the circle 01 and cuts

orthogonally the sphere 02 . (8 Marks)

32;0232222 =−−−=−+++++ zyxzyxzyx3222 =−+−++ yxzyx

(ii) Find the centre and radius of the circle given by

. (8 Marks) 0722;019422222 =+++=−−−+++ zyxzyxzyx

(i) Find the length and equations of the shortest distance line between the lines

33206523;12

23

1−+−==−−+=

−=

+ zyxzyxzyx. (8 Marks)

(ii) Find the equation of the sphere passing through the circle given by and

and the point (1,-2,3). (8 Marks) 0343222 =−+++++ zyxzyx 0632222 =+++++ yxzyx

GEOMETRICAL APPLICATIONS OF DIFFERENTIAL CALCULUS

PART-A 1. Find the curvature of the curve given by xcy tan= at x = 0.

2. Find the radius of curvature of the curve xy = c2. 3. Find the radius of curvature at the point (p,r) of the ellipse

22

2

222111

bar

bap−+= .

4. Find the envelope of the family of lines mamxy += , where m is a parameter.

5. Show that the radius of curvature of a circle is its radius. PART-B

6. (i) Find the measure of curvature of the curve 1=+by

ax

at any point (x,y)

on it. (8 Marks)

(ii) Find the centre of curvature at 2πθ = on the curve

ttyttx 2sinsin2,2coscos2 +=+= . (8 Marks)

3

Page 4: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

7. (i) Find the equation of the circle of curvature of the parabola x at the point (3,6).

y 122 =

(ii) Find the radius of curvature of the curve )cos1( θ+= ar at the point

2πθ = . (8 Marks)

8. (i) If ϕ be the angle which the radius vector of the curve )(θfr = makes with

the tangent, then prove that ⎟⎠⎞

⎜⎝⎛ +=

θϕϕ

ρ ddr 1sin , where ρ is the radius of

curvature. Apply this result to show that ρ = a/2 for the circle θcosar = . (8 Marks) (ii) Find the radius of curvature at the origin for the cycloid

)cos1();sin( θθθ −=+= ayax . (8 Marks) 9. (i) Find the envelope of the straight lines

)2/tan(logcoscossin ttaatxty +=+ . (8 Marks)

(ii) Find the radius of curvature at the point θ on θθθθ 3sinsin3,3coscos3 aayaax −=−= . (8 Marks)

10. (i) Find the evolute of the astroid θ . (8 Marks) θ 33 sin,cos ayax ==

(ii) Show that the envelope of a circle whose centre lies on the parabola and which passes through its vertex is the cissoid

. (8 Marks)

axy 42 =

2(2 + xay 0) 3 =+ x

FUNCTIONS OF SEVERAL VARIABLES:

PART-A

1. Evaluate yz

xz

∂∂

∂∂ & if zzyx log=++ .

2. If tztyexzyxu t cos,sin,&)log( ===++= − , then find dtdu

.

3. Find dxdu

, when 9 . 4&)sin( 2222 =++= yxyxu

4. Write down the Maclaurin’s series for sin(x + y)

5. If y

xvxyu

22, == , then find

),(),(

vuyx

∂∂

.

4

Page 5: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

PART-B

6. (i) Expand 23 in powers of (x-1) and (y+2) using Taylor’s expansion. 2 −+ yyx (8 Marks) (ii) Find the maximum and minimum values of . yxyxyx +−+− 222

(8 Marks) 7. (i) Find the volume of the greatest rectangular parallelepiped that can be inscribed

in the ellipsoid 12

2

2

2

2

2=++

cz

by

ax

. (8 Marks)

(ii) Prove that )1log()1(0

adxex

e axx

+=−∫∞

−−

where a>-1 using differentiation

under the integral sign. (8 Marks)

8. (i) If z = f(u,v) where xyv , then show that

) . (8 Marks)

yxu 2,22 =−=

vv)((4 22uuyyxx zzyxzz ++=+

(ii) By using the transformations yxvyxu −=+= ,

0, change the independent

variables x and y in the equation =− yyxx zz to u and v. (8 Marks)

9. (i) If 1, show that 2222 =−++ xyzzyx

0111 222

=−

+−

+− z

dz

y

dy

x

dx. (8 Marks)

(ii) Expand in powers of x and y as far as the terms of third degree. yex cos (8 Marks) 10. (i)A rectangular box, open at the top, is having a volume of 32c.c. Find the

dimensions of the box, that requires the least material for its construction. (8 Marks)

(ii) Expand in powers of x and y as far as the terms of third degree. )1log( yex + (8 Marks) DIFFERENTIAL EQUATIONS:

PART-A

1. Solve . xeyD 22)2( =−

2. Find the Particular Integral of xeyD x cos)2( 2 −=+ .

3. Solve x . yD 2sin)4( 2 =+

5

Page 6: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

4. Solve xndt

xd 44

4= .

5. Find the Particular Integral of xxeyD 22)3( −=− .

PART-B

6. (i) Solve xx sin3 . (8 Marks) eeydxdy

dxyd x44 22

2+=++ −

(ii) Solve by the method of variation of parameters xydx

yd 2tan442

2=+ .

(8 Marks)

7. (i) Solve the simultaneous equations txdtdyty

dtdx cos;sin =+=+ given that

x = 2, y = 0 when t = 0. (8 Marks)

(ii) Solve xydx

yd sec2

2=+ by the method of variation of parameters.(8 Marks)

8. Solve teyxdtdytyx

dtdx 2223;532 =+−=−+ . (16 Marks)

9. Solve the equation xxydxdyx

dxyd 22

2sincot)cot1( =−−+ by the method of

reduction of order. (16 Marks)

10. Solve )sin(log42

22 xy

dxdyx

dxydx =++ . (16 Marks)

--End--

6

Page 7: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

MA132 MATHEMATICS – II QUESTION BANK

MULTIPLE INTEGRALS:

PART- A

1. Evaluate1 1

a b dxdyxy∫ ∫ .

2. Evaluate . ∫ ∫π θ

θ0

sin

0

addrr

3. Change the order of integration in ∫ ∫−0

0

1

0

2

),(x

dydxyxf .

4. Define Gamma function and Beta function.

5. Show that ),(),( mnnm ββ = .

PART-B

6. (i) Prove that)()()(),(

nmnmnm

+ΓΓΓ

=β .

(ii) Evaluate . ∫∞

0

7dxxe x

7. (i) Find the volume bounded by the cylinder 42 and the planes 4=+ z and 0=z .

2 =+ yxy

(ii) Evaluate over the positive quadrant of the circle ∫∫ dydxyx .122 =+ yx

8. (i) Change the order of integration dydxyx

xIa a

y∫ ∫ +

=0

22 and then evaluate.

(ii) Evaluate dydxyxIa

a

xa

∫ ∫−

−+=

22

0

22 )( .

9. (i) Evaluate . ∫=2/

0

76 cossinπ

θθθ dI

(ii) Evaluate ∫=2/

0cot

πθθ dI .

10. (i) Evaluate and prove∫∞

−=0

2dxeI x π=⎟

⎠⎞

⎜⎝⎛Γ

21

.

(ii) Prove that !)1( nn =+Γ , when n is a positive integer.

Page 8: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

VECTOR ANALYSIS: PART – A

1. Find a unit normal to the surface zy at )5,2,1( . x =+ 22

2. Find directional directive of 22 at (1,0,1) in the direction 2 zyx ++ ki 32 + .

3. Prove that 0)( =φgradcurl .

4. Evaluate )(log . 2 r∇

5. If φ . φ find,kxyjzxiyz ++=∇

PART – B

6. (i) Show that kyxzjzxizxyF )3()3()6( 223 −+−++= is irrotational. Find

φ such that φ∇=F .

(ii) Find the work done when a force jyxyixyxF )2()( 22 +−+−= moves a

particle in the xy plane from to along the curve . )0,0( )1,1( xy =2

7. (i) Prove that the area bounded by a simple closed curve C is given by

∫ −C

dxydyx ).(21

Hence find the area of the ellipse.

(ii) Find the area between and by using Green’s theorem. xy 42 = yx 42 =

8. Verify Gauss’s divergence theorem for kzjyixF 2 taken over the cube bounded by planes

22 ++=.10,1 and,0,1,0 ====== y zzyxx

9. Verify Green’s theorem for ]where C is the boundary of the

common area between 2xy = and

)[( 22 dyxdxyxyC

++∫

.xy =

10. Verify Stoke’s theorem for a vector field jxyiyxF 2 in the rectangular region of the plane 0=z bounded by the lines by

)( 22 +−=0,,0 andya .xx ====

Page 9: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

ANALYTIC FUNCTIONS: PART – A

1. Find the invariant points of the transformation 3 51

zwz−

=+

.

2. Prove that an analytic function with constant real part is constant.

3. Find the Bilinear transformation which maps 0,1, 2z = into the points respectively. 2, 1,3w = −

4. Test whether the function z

zf 1)( = is analytic or not.

5. Define: Conformal mapping.

PART-B

6. (i) Verify that 2 2

2 2

xv x yx y

= − ++

is harmonic and find u such that

is analytic. Express w as a function of z . w u iv= +(ii) Find the bi-linear transformation which maps 1, , 1i − in -plane to of the

-plane. z 0,1,∞

w7. (i) Under the transformation 2w z= , obtain the map in the w -plane of the square

with vertices )2,0( in z -plane. and)2,2),0,2(),0,0(

(ii) Under the transformation 1wz

= find the image of the circle and11 =+z

22 =− iz .

8. If )(zf is an analytic function of z , prove 2

22

)()()( zfzfy

zfx

′=⎭⎬⎫

⎩⎨⎧∂∂

+⎭⎬⎫

⎩⎨⎧∂∂

9. If ivuzf +=)( is an analytic function find )(zf and v if yx

xu2cosh2cos

2sin+

= .

10. If ivuzf +=)( is an analytic function find )(zf given that

xyh

xvu2cos2cos

2sin−

=+ .

Page 10: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

COMPLEX INTEGRATION: PART – A

1. Evaluate dzz

z

c∫ + 2

where C is z = 3.

2. Find the singular points ofzz

zfsin1)( = .

3. Expand zez in a Taylor’s series about 0f =)( =z .

4. Find the residues at the isolated singularities of the functions)2)(1( −+ zz

z.

5. Define essential singularity with an example.

PART – B 6. Evaluate the following integrals, using Cauchy’s residue theorem

(i) ,42

)1(2∫ +++

c zzdzz

where C is 2i1z =++ .

(ii) ∫ +c zdz

32 )9(, where C is 3iz =− .

7. Find the Laurent’s series of )1(

1)(zz

zf−

= valid in the region

(i) 11 <+z , (ii) 211 <+< z and (iii) 21 >+z

8. Use Cauchy’s integral formula to evaluate

(i) dzzz

zz

C∫ −−

+)3)(2(

cossin 22 ππ , where C is the circle 4=z .

(ii) dzzz

z

C∫ −−

4317

2 , where c is the ellipse 114

22=+

yx .

9. (i) Evaluate 2

0 cosdπ θ

θ2 +∫ using contour integration.

(ii) ) Evaluate ∫∞

+022 )1( x

dx using contour integration

10. Evaluate 2

4 2

210 9

x x dxx x

−∞

− ++ +∫ using contour integration.

Page 11: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

STATISTICS: PART – A

1. Prove that the first moment about mean is always zero.

2. What is the difference between t distribution and Normal distribution?

3. What is correlation coefficient?

4. How is accuracy of regression equation ascertained?

5. Give two uses for 2χ distribution.

PART-B 6. (i) Find the coefficient of correlation between X and Y using the following data

3026231916:252015105:

YX

(ii) A study of prices of rice at Chennai and Madurai gave the following data:

Chennai Madurai

Mean 19.5 17.75

S.D. 1.75 2.5

Also the coefficient of correlation between the two is 0.8. Estimate the most likely price of rice (i) at Chennai corresponding to the price of 18 at Madurai and (ii) at Madurai corresponding to the price of 17 at Chennai.

7. (i) In a large city a, 20 % of a random sample of 900 school boys had a slight physical defect. In another large city B, 18.5 % of a random sample of 1600 school boys had the same defect. Is the difference between the proportions significant?

(ii) A sample of 100 students is taken from a large population. The mean height of the students in this sample is 160cm. Can it be reasonably regarded that, in the population, the mean height is 165 cm, and the S.D. is 10 cm?

8. (i) Samples of two types of electric bulbs were tested for length of life and the following data were obtained.

Size Mean S.D.

Sample I 8 1234 hours 36 hours

Sample II 7 1036 hours 40 hours

Is the difference in the means sufficient to warrant that type I bulbs are superior to type II bulbs?

(ii) Two samples of sizes nine and eight gave the sums of squares of deviations from their respective means equal to 160 and 91 respectively. Can they be regarded as drawn from the same normal population?

Page 12: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

9. (i) Theory predicts that the proportion of beans in four groups DCBA ,,, should be 1:3:3: . In an experiment among 1600 beans, the numbers in the four groups were 882, 313, 287 and 118. Does the experiment support the theory?

9

(ii) A sample of size 13 gave an estimated population variance of 3.0, while another sample of size 15 gave an estimate of 2.5. Could both samples be from populations with the same variance?

10. (i) A number of automobile accidents per week in a certain community are as follows: 12, 8, 20, 2, 14, 10, 15, 6, 9, 4. Are these frequencies in agreement with the belief that accident conditions were the same during this 10 week period?

(ii) The mean of two random samples of size 9 and 7 are 196.42 and 198.82 respectively. The sums of the squares of the deviation from the mean are 26.94 and 18.73 respectively. Can the sample be considered to have been drawn from the same normal population?

--End--

Page 13: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

MA231 MATHEMATICS-III

QUESTION BANK

PARTIAL DIFFERENTIAL EQUATIONS

PART-A

1. Find the partial differential equation from (x−a)2 +(y−b)2 +z2 = r2 by eliminating the arbitraryconstants a and b.

2. Find the partial differential equation from z = x + y + f(xy) by eliminating the arbitrary functionf .

3. Find the complete integral of√

p +√

q = 1.

4. Find the general solution of the Lagrange’s equation px2 + qy2 = z2.

5. Solve (D + D′)(D + D′ + 1)z = 0.

PART-B

6.(i) Find the singular integral of the partial differential equation z = px + qy + p2 + pq + q2.

(ii) Solve z2(p2 + q2) = x + y.

7.(i) Solve the equation x2(y − z)p + y2(z − x)q = z2(x− y).

(ii) Form the partial differential equation by eliminating f from f(z − xy, x2 + y2) = 0.

8.(i) Solve the equation (D2 + 4DD′ + D′2)z = e2x−y + 2x.

(ii) Solve the equation (D2 −D′2)z = sin(x + 2y) + ex−y + 1.

9.(i) Solve the equation pq + p + q = 0.

(ii) Solve the Lagrange’s equation (y + z)p + (z + x)q = x + y.

(iii) Solve (D3 −D2D′ − 8DD′2 + 12D′3)z = 0.

10.(i) Solve the equation√

p +√

q = x + y.

(ii) Formulate the partial differential equation by eliminating arbitrary functions f and g from

z = f(x + ay) + g(x− ay).

(iii) Solve p tan x + q tan y = tan z.

FOURIER SERIES

PART-A

1. State the Dirichlet’s conditions for the existence of Fourier series of f(x).

2. Find the Fourier sine series of f(x) = x, 0 < x < π.

3. Define Fourier series of f(x) in (c, c + 2l).

4. Define the root mean square value of a function f(x) in (0, 2π).

5. Find the Fourier coefficient an, given that f(x) = x2 in (−π, π).

PART-B

6.(i) Find the Fourier series of f(x) = e−x in (−π, π).

(ii) Find the half range cosine series of f(x) ={

x, 0 < x < 12− x 1 < x < 2 .

1

Page 14: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

7.(i) Obtain the Fourier series of the function given by f(x) =

1 +2x

l, −l ≤ x ≤ 0

1− 2x

l, 0 ≤ x < l

.

(ii) Find the Fourier series of periodicity 2π for f(x) = x2, in −π < x < π. Hence show that114

+124

+134

+ . . . =π4

90.

8.(i) Find the Fourier sine series of f(x) = x(π − x), 0 < x < π.

(ii) Compute the fundamental and first harmonics of the Fourier series of f(x) given by the table.

x 0 π/3 2π/3 π 4π/3 5π/3 2πf(x) 1.0 1.4 1.9 1.7 1.5 1.2 1.0

9.(i) Express f(x) = (π − x)2 as a Fourier series in 0 < x < 2π.

(ii) Find the Fourier series of periodicity 2 for the function f(x) =

k, −l ≤ x ≤ 0

x, 0 ≤ x < l.

10.(i) Find the half-range sine series of f(x) = l−x in (0, l). Hence prove that112

+122

+132

+ . . . =π2

6.

(ii) Find the constant term and the first two harmonics of the Fourier cosine series of y = f(x) usingthe following table.

x 0 π/6 π/3 π/2 2π/3 5π/6f(x) 10 12 15 20 17 11

BOUNDARY VALUE PROBLEMS

PART-A

1. Classify the partial differential equation

(x + 1)zxx +√

2(x + y + 1)zxy + (y + 1)zyy + yzx − xzy + 2 sin x = 0.

2. Write down all possible solutions of one dimensional wave equation.

3. A taut string of length 50 cm fastened at both ends, is disturbed from its position of equilibriumby imparting to each of its points an initial velocity of magnitude kx for 0 < x < 50. Formulatethe problem mathematically.

4. Write down all possible solutions of the one dimensional heat flow equation.

5. If the temperature at one end of a bar, 50 cm long and with insulated sides, is kept at 0◦C andthat the other end is kept at 100◦C until steady state conditions prevail, find the steady statetemperature in the rod.

PART-B

6. A tightly stretched string with fixed end points x = 0 and x = 50 is initially at rest in its

equilibrium position. If it is set to vibrate by giving each point a velocity v = v0 sinπx

50cos

2πx

50,

find the displacement of the string at any subsequent time.

7. A tightly stretched string with end points x = 0 and x = L is initially in a position given byy(x, 0) = kx. If it is released from this position, find the displacement y(x, t) at any point of thestring.

8. A rod 30cm. long has its ends A and B kept at 20◦C and 80◦C, respectively, until steady stateconditions prevail. The temperature at each end is then suddenly reduced to 0◦C and kept so.Find the temperature function u(x, t) taking x = 0 at A.

2

Page 15: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

9. A rod of length l has its ends A and B kept at 0◦C and 100◦C respectively until steady stateconditions prevail. If the temperature of A is suddenly raised to 50◦C and that of B to 150◦C,find the temperature distribution at any point in the rod.

10. A rod of length 30cm has its ends A and B kept at 20◦C and 80◦C respectively until steady stateconditions prevail. If the temperature of A is suddenly raised to 40◦C while that the other end Bis reduced to 60◦C, find the temperature distribution at any point in the rod.

LAPLACE TRANSFORMS

PART-A

1. Find the Laplace transform of f(t) = te−t.

2. State and prove the scaling property of Laplace transform.

3. Find the inverse Laplace transform of F (s) =s

(s + a)2 + b2.

4. State the convolution theorem of Laplace transform.

5. State the initial and final value theorem of Laplace transform.

PART-B

6.(i) Find the Laplace transform of the function f(t) ={

t, 0 < t < 33, t > 3 .

(ii) Find the Laplace transform of the periodic function

f(t) ={

a, 0 ≤ t < a−a, a ≤ x ≤ 2a

and f(t + 2a) = f(t).

7.(i) Find the inverse Laplace transform of3s + 7

s2 − 2s− 3.

(ii) Solve the differential equation y′′ − 2y′ − 8y = 0, y(0) = 3 and y′(0) = 6 using Laplace transform.

8.(i) Using convolution theorem, find L−1

{16

(s− 2)(s + 4)

}.

(ii) Solve the initial value problem y′′ − 6y′ + 9y = t2e2t, y(0) = 2 and y′(0) = 6, using Laplacetransform.

9.(i) Prove that L{f ′′(t)} = s2F (s)− sf(0)− f ′(0) where F (s) = L{f(t)}.

(ii) Find the inverse Laplace transform of2

(s + 1)(s2 + 1).

10.(i) Find the Laplace transforms of f(t) = e4t cosh 5t.

(ii) Verify the initial value theorem for the function f(t) = 5 + 4 cos 2t.

(iii) Solve x′ = 2x− 3y; y′ = y − 2x, x(0) = 8, and y(0) = 3.

FOURIER TRANSFORMS

PART-A

1. State the Fourier integral theorem.

2. Find the Fourier transform of f(x), defined as f(x) ={

1, |x| < a0, |x| > a

.

3. Find the Fourier sine transform of f(x) = e−ax (a > 0).

3

Page 16: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

4. If Fs(s) is the Fourier sine transform of f(x), prove that the Fourier cosine transform of f ′(t) is

FC{f ′(t)} = sFs(s)−√

f ′(0).

5. Show that FC{f(t) cos at} =12

[Fc(s + a) + Fc(s− a)], where Fc(s) is the Fourier cosine transform

of f(x).

PART-B

6.(i) Find the Fourier integral representation of f(x) ={

0, x < 0e−x, x > 0

(ii) Find the Fourier transform of f(x) = e−a|x|, a > 0.

7. Find the Fourier transform of f(x) ={

1− x2, |x| < 10, |x| > 1 . Hence evaluate

∫ ∞

0

(x cos x− sin x

x3

)cos

x

2dx.

8.(i) Evaluate∫ ∞

0

dx

(x2 + a2)(x2 + b2)using transform methods.

(ii) Show that (1) Fs[xf(x)] = − d

dsFc(s) and (2) Fc[xf(x)] =

d

dsFs(s).

9. Find the Fourier transform of f(x) ={

a− |x|, |x| < a0, |x| > a > 0 and hence evaluate

∫ ∞

0

(sinx

x

)2

dx.

10. Find the Fourier sine and cosine transform of xn−1.

−End−

4

Page 17: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

MA034 - RANDOM PROCESSES QUESTION BANK

PROBABILITY AND RANDOM VARIABLES:

PART A

1. Suppose that 75% of all investors invest in traditional annuities and 45% of them invest in the stock market. If 85% invest in the stock market and / or traditional annuities, what percentage invests in both?

2. A factory produces its entire output with three machines. Machines I, II and III produce 50%, 30% and 20% of the output, but 4%, 2% and 4% of their outputs are defective respectively. What fraction of the total output is defective?

3. Find the moment generating function of a random variable X which is uniformly distributed over (-2.3) and hence find its mean.

4. If A and B ate events such that ( ) ( ) ( )3 1, ,4 4

P A B P A B P A∪ = ∩ =2 .3

= Find ( )|P A B

5. Suppose that for a RV X, 1,2,3....Calculate its moment generating function.

2 ,n nE X n⎡ ⎤ = =⎣ ⎦

PART B

6. (i) Let X be a continuous RV with pdf ( ) 2 ,1 2.Xf x xx2

= < < Find [ ]logE X (4)

(ii) The average IQ score on a certain campus is 110. If the variance of these scores is 15, what can be said about the percentage of students with an IQ above 140 ? (6)

(iii) The MGF of a RV X is what is the MGF of Y=3X+2. Also find the mean and variance of X. (6)

( )2,t3 7e⋅ + ⋅

7. (i) If the continuous RV X has pdf ( ) ( )1 , 1 29

0 otherwiX

xf x

2

se

x⎧ + − < <⎪= ⎨⎪⎩

, find the pdf

of Y=X2. (4)

(ii) If the probability that an applicant for a driver’s license will pass the road test on any given trial is 0.8, what is the probability that he will finally pass the test

(a) on the fourth trial (b) in fewer than four trials ? (8)

Page 18: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

(iii) The cumulative distribution function for a RV X is given by , find Var(3X+2). (4) ( ) 31 , 0xF x e x−= − ≥

1

X

8. (i) Let X be an exponential RV with parameter λ = . Use Chebyshev’s inequality, to find { }1 3 .P X− ≤ ≤ Also, find the actual probability. (6)

(ii) Let X be a continuous RV with pdf ( ) 1 2, 24Xf x x= − < < .Find { }1P X >

{ }2 1 2and P X + > 6)

(iii) Let X be a RV with the pdf

(

( ) ( )2.

1Xf x xxπ

1= −∞ < < ∞

+Find the pdf of

1tanZ X−= . (4)

9. (i) The time that it takes for a computer system to fail is exponential with mean

(ii) The Pap test makes a correct diagnosis with probability 95%. Given that the test is

(iii) Experience has shown that while walking in a certain park, the time X, in

700 hours. If a lab has 20 such computer systems what is the probability that atleast two fail before 1700 hours of use ? (6)

positive for a lady, what is the probability that she really has the disease? Assume that one in every 2000 women has the disease (on an average). (5)

minutes, between seeing two people smoking has a density function of the form ( ) , 0.x

Xf x xe xλ −= > Calculate the value of λ . Find the cumulative distribution s the probability that George who has just seen a person

smoking will see another person smoking in 2 to 5 minutes? In at least 7 minutes? (5)

function of X.What i

10. (i) Let X be a Gamma RV with parameters n and λ . Find the moment

(ii) Suppose that, on an average, a post office handles 10,000 letters a day with a

(iii) Peter and Xavier play a series of backgammon games until one of them wins five

(a) Find the probability that the series ends in seven games

(b) If the series ends in seven games, what is the probability that Peter wins.

generating function of X and use it to find E[X] and Var(X). (6)

variance of 2000. What can be said about the probability that this post office will handle between 8000 and 12000 letters tomorrow? (6)

games. Suppose that the games are independent and the probability that Peter wins a game is 0.58.

(4)

Page 19: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

TWO-DIMENSIONAL RANDOM VARIABLES:

PART A

1. The joint pdf of a bivariate RV (X,Y) is given by f(x,y)= kxy, 0<x<1, 0<y<1,where k is a constant. Find the value of k.

2. The joint pdf of a RV (X,Y) is given by ,( , ) ,0yf x y e−= <

( , ) 2 , 0 1, 0 1.f x y x y x y

x y≤ find the conditional cumulative distribution function of Y given that X=x.

3. Let X and Y be two independent RVs, show that Cov(X,XY)=E[Y] VarX

4. Let the jont pdf of (X,Y) be f(x,y)=2, 0<x<y<1. Find the marginal density function of the RV X.

5. Given that X=4Y+5 and Y=kX+4 are the lines of regression of X on Y and Y on X respectively. Show that 0<4k<1. If k=1/16 , find the means of the two variables.

PART B

=6. Two RVs X and Y have the following joint pdf − − ≤ ≤ ≤ ≤ Find (i) Marginal pdfs of X and Y (ii) Conditional density functions (iii) Var (X) and Var(Y) (iv) Correlation (v) Lines of regression (16)

7.(i) What is the probability that the average of 150 random points from the interval (0,1) is within 0.02 of the midpoint of the interval? (8)

(ii) Let X and Y be independent (strictly positive) exponential RVs each with parameter λ . Are the RVs X+Y and X | Y independent? (8)

( , ) ,0yf x y e x y−= < < < ∞

( ) , 0.xf x e x−= >

. Find r(X,Y). 8. Let the joint pdf of (X,Y) be

9. (i) Let X and Y be independent RVs with common pdf Find the joint pdf of U=X+Y and V=eX. (8)

X

(ii) Suppose that, whenever invited to a party, the probability that a person attends with his or her guest is 1/3, attends alone is 1/3, and does not attend is 1/3. A company invited all 300 of its employees and their guests to a Christmas party. What is the probability that atleast 320 will attend? (8)

10. (i) The joint pdf of (X,Y) is |2( , ) , 0,0 2 [ | 1]2

xy Xf x y e x y find E e Y−y= > < < = (8)

(ii) If X and Y are continuous RVs with joint pdf 2( , ) 2 ,0 1,0 1

23f x y xy y x y= + < < < < , find the conditional pdfs X and Y. (8)

Page 20: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

RANDOM PROCESSES:

PART A

1. Given the random process x(t) = cos at + B sin at where a is a constant and A and B are uncorrelated zero-mean RVs having different density functions but common variance 2σ . Is x(t) wide-sense stationary?

2. Show that a Binomial process is a Markov process.

3. Prove that the sum of two independent Poisson process is a Poisson process.

4. Define (i) Ergodic process (ii) Weakly stationary random process.

5. Show that the interarrival times of a Poisson process with intensity λ obeys an exponential probability distribution.

PART B

( ) cos( ),x t t= φ φ6.(i) Consider the random process + is a RV with pdf 1( ) ,

2 2f xφ

π π

( ) sin cos

φπ

= − < < check whether x(t) is first order stationary. (8)

x t A t B t=(ii) A stochastic process is described by + where A and B are independent RVs with zero means and equal variances. Find the variance and covariance of the given process. (8)

7. (i) Let N(t) be a Poisson process with parameter λ . Determine the coefficient of correlation between N(t) and N(t + 0)> ; t>0 , 0τ > . τ

ii) Let { }( ), 0x t t ≥ be a Poisson process with parameterλ . Suppose each arrival is

registered with probability p independent of other arrivals. Let { }( ), 0y t t ≥ be the process of registered arrivals. Prove that Y(t) is a Poisson process with parameter pλ . (6)

( ) 1;0 1fA =8.(i) Prove that the process x(t)=8+A with α α< < is (a) first order stationary (b) Second-order stationary (c) strictly stationary (d) not ergodic. (8)

(ii) x(t) is a random telegraph type process composed of pulses of heights +1 and -1 respectively. The number of transactions of the t-axis in a time 2 is given by

4 4 .( )!

Ke−

( ) cos( )x t t

P k transitionsK

= Classify the above process. (8)

= φ+ where φ is uniformly distributed in 9. Show that the random process ( )0,2π is (a) first order stationary (b) stationary in wide sense (c) Ergodic. (16)

Page 21: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

10.(i) Let { }( ), 0x t t ≥

(0) 0x

be a random process with stationary independent increments,

and assume that ( ) 2cov ( ), ( ) min( , )1x= . Show that t x s t sσ=2σ

( ) cos

, where

. (8) ( )(1)Var x=1

x t A t(ii) Classify the random process ω= where A and ω are RVs with joint

pdf 18Af ω α β α= < <( , ) , 0 2, 8 12.β< < (8)

CORRELATION FUNCTION:

PART A

1. Statistically independent zero-mean random processes x(t) and y(t) have autocorrelation functions ( )xx ( )R cos 2yyR e ττ −= and τ πτ respectively. Find the autocorrelation function of the sum Z(t)=x(t)+y(t).

=

2. Suppose that a random process is wide sense stationary with autocorrelation function

( ) 2xxR e

τ

τ−

= Find the second moment of the random variable x(5) –x(3)

3. Show that ( ) 20xxR x= .

4. Show that the autocorrelation function ( )xxR τ is maximum at 0τ = .

5. If x(t) is a stationary random process having mean value E[x(t)]=3 and autocorrelation function ( ) 10xx 3R e ττ −= + . Find the variance of x(t).

PART B

6. Given the random process ( ) ( )nn

x t A g t nb φ∞

=−∞

= − +∑ where t b< <

, 'iA s

and

1 0⎧( )0

g totherwise

= ⎨⎩

φ are independent RVs with density functions ( ) ( ) ( )δ α1 11 12 2Af α δ α= − + + and

1 , 0( ) .f bα < The joint probability mass function of Ai is given by bφ α = <

1iA + iA -1 +1

-1 0.2 0.3

+1 0.3 0.2

Page 22: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

Find the autocorrelation function ( )xxR τ in 0 bτ< < for the above process.

7. Given ( )( ) kk

x n X δ=∑ n k− where the joint mass function for the RVs kX is given

below:-

1kX +

kX 0 1

0 0.2

0.3

1 0.3

0.2

Find the autocorrelation function ( )xxR k and covariance function for

( )xxL k0,1,2,3.k =

8. Consider the random process ( )( ) kk

x n X nδ=−∞

= ∑ k− where the 'k

X s are characterized

by the joint mass function

1kX +

kX 0 1

0 0.5

0.1

1 0.1

0.3

Find the autocorrelation function ( )xxR k and covariance function for 0,1,2,3.k = for the above process.

9. Given the random process ( ) ( )nn

x t A g t nb φ∞

=−∞

= − +∑ where t b1 0

( )0

g totherwise< <

and

the 'nA s and

⎧= ⎨⎩

φ are independent RVs with density functions 1( ) , 0 .f bbφ α α= < < and

Page 23: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

( ) ( ) ( )δ α + Evaluate the autocorrelation function 1 11 12 2iAf α δ α= − + ( )xxR τ in

0 bτ< < for the above process. Also the joint probability mass function for the 'iA s is given by

1iA + iA -1 +1

-1 14

14

+1 14

14

10. Given the random process ( ) ( )nx t A g t nb φ∞

−∞

= − +∑ where 'nA s

and

1 0( )

0g t

otherwise⎧

= ⎨⎩

t b< <

φ are independent RVs with density functions ( ) ( ) ( )1 11 1δ α+2 2iAf α δ α= − +

and 1( ) , 0 .bα < Evaluate the autocorrelation function ( )xxRfb

α = <φ τ in 0 bτ< < .

The joint probability mass function of iA is given by

1iA + iA -1 +1

-1 13

16

+1 16

13

Page 24: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

SPECTRAL DENSITIES:

PART A

1. State and prove any one of the properties of cross spectral density functions.

2. The autocorrelation function of a random process x(t) is 24( ) 3 2xxR e ττ = +

( )

− . Find the power spectral density of x(t).

3. A widesense stationary noise process N(t) has an autocorrelation function ,R Pe ϑττ −= −∞NN τ< < ∞ with P as a constant. Find its power density spectrum.

4. The power spectral density of a stationary random process is a constant in a symmetrical interval about zero and zero outside the interval. Compute the autocorrelation function.

5. Which of the following functions could be a power spectral density?

(i) 2b f+a (ii) 2

af b−

PART B

6. (i) Consider two independent zero-mean random processes x(t) and y(t) with power spectral densities and respectively. Define new random processes z(t)=x(t)+y(t), x(t)= x(t)-y(t) and

( )S jw ( )S jwxx yy

( )tω =x(t)y(t). Find formulas for, , and . (8)

( )S jw ( )S jw( )S jw

zz uu

ωω

(ii) Given the power spectral density 21 ω+

4 24 4ω ω+ +. Use residue theory to find the average

power in the process x(t). (8)

( ) cos( )x t A t7.(i) Two random processes x(t) and y(t) are given by ω θ= +( ) sin( )y t A t

and ω θ= + where A and ω are constants and θ is a uniform RV over ( )20, π .

Find the cross-spectral density functions ( )Sxy ω and ( )yxS ω and verify

( )xyS ω = ( )yxS ω− .

(ii) Find the cross-correlation function corresponding to the cross-power spectrum

( ) ( ) ( )229 3xySj

ω 6ω ω

=+ + +

.

Page 25: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

8.(i) The autocorrelation function of a signal is 2

22keτ

− where k is a constant. Find the power

spectral density and average power. (8)

(ii) Show that in an input-output system the energy of a signal is equal to the energy of its spectrum. (8)

9.(i) Define convolution and correlation integrals for an input-output system. State and prove Wiener-Khinchine theorem. (8)

( ) cos( )x t b t (ii) A random process x(t) is given by ω θ+ were θ is a RV, b and = ω are constants. Find the autocorrelation and power spectral density functions. (8)

10. (i) For a linear system with random input x(t), the impulse response h(t) and output y(t), obtain the cross correlation function and cross power spectral density functions.

(8) (ii) The power spectrum density function of a wide sense stationary process x(t) is given

by ( )22

1( )4

xxS ωω

=+

. Find its autocorrelation and average power.

(8)

--End--

Page 26: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

MA 035 DISCRETE MATHEMATICS QUESTION BANK

LOGIC:

PART-A 1. What are the possible truth values for an atomic statement?

2. Symbolize the following statement with and without using the set of positive

integers as the universe of discourse. “Given any positive integer, there is a greater

positive integers”.

3. When a set of formulae is consistent and inconsistent?

4. What are free and bound variables in predicate logic?

5. Show that { }↑ and { }↓ are functionally complete sets.

PART-B

6. Without constructing truth table show that .R)RP()RQ())RQ(P( ⇔∧∨∧∨∧¬∧¬

7. Without constructing truth table verify whether the formula

)Q is a contradiction or tautology. P()QP(Q ¬∧¬∨¬∧∨

8. Without constructing truth table obtain PCNF of ( )∧∧→ )RQ(P ( ))RQ(P ¬∧¬→¬

and hence find its PDNF.

9. Use rule CP to show that

( )( ) ( )( ) ( )( ))x(P)x(Rx)x(Q)x(Rx,)x(Q)x(Px ¬→∀⇒¬→∀→∀ .

10. Use indirect method of proof to show that ( ) )z(Qz∃ is not valid conclusion from the

premises ))x(Q and ( )( )x(Px →∀ ( ) )y(Qy∃ .

COMBINATORICS

PART-A

1. Use mathematical induction to prove that , where n > 1.

2. State and prove Pigeon hole principle.

3. How many positive integers not exceeding 100 that is divisible by 5?

4. What is the minimum number of students required in discrete mathematics class

to be sure that at least six will receive the same grade, if there are five possible

grades?

5. Find the recurrence relation for the sequence for

Page 27: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

PART-B

6. Find the number of integers between 1 and 250 that are not divisible by any of

the integers 2, 3, 5 and 7.

7. Write the recurrence relation for Fibonacci numbers and hence solve it.

8. Solve the recurrence relation with

9. Find the generating function of Fibonacci sequence F(n) = F(n-1) + F(n-2) for

with F(0) = F(1) = 1.

10. Solve, by using generating function, the recurrence relation

with

GROUPS

PART-A

1. Is the set N = { 1, 2, ………..} under the binary operation * defined by x * y = max

{ x , y } semi group (or) monoid ? Justify your claim.

2. Show that the inverse of an identity element in a group ∗,G is itself..

3. If ∗,G is an abelian group show that for all a, b in G, ( ) nbna . nba ∗=∗

4. If every element in a group is its own inverse, verify whether G is an abelian

group or not.

5. What is meant by ring with unity?

PART-B

6. Show that the set of all permutations of three distinct elements with right

composition of permutation is a permutation group. Is it an abelian group ?

7. Show that every finite group of order n is isomorphic to a permutation group of

degree n.

8. Show that the order of a subgroup of a finite group G divides the order of the

group G.

9. Let H be a nonempty subset of a group ∗,G . Show that H is a subgroup of G if

and only if .Hb,a allforHba ∈∈∗ −1

10. Show that the Kernel of a group homomorphism is a normal subgroup of a group.

Page 28: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

LATTICES PART- A

1. Let N be the set of all natural numbers with the relation R as follows: x R y if and

only if x divides y. Show that R is a partial order relation on N.

2. Draw the Hasse diagram of the set of all positive divisors of 45.

3. If the least element and greatest element in a poset exist, then show that they are

unique.

4. If A = (1,2) is a subset of the set of all real numbers, find least upper bound and

greatest lower bound of A.

5. Show that absorption laws are valid in a Boolean algebra.

PART-B

6. Show that in a complemented distributive lattice, the De Morgan’s laws hold.

7. If L is a distributive lattice with 0 and 1 , show that each element has atmost one

complement.

8. Show that every distributive lattice is modular. Is the converse true? Justify the

claim.

9. Show that a lattice L is modular if and only if for all x,y,z L∈ , =

)zx( .

( ))zx(yx ∨∧∨

)yx( ∨∧∨

10. Which of the following lattices given by the Hasse diagrams are complemented,

distributive and modular?

1

35

9

3

15

5

0

a

c

d e

1

b

30

15

3

1

2

106

5

d

cb

e

a (a) (b) (c) (d)

Page 29: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

GRAPHS

PART-A

1. How many edges are there in a graph with 10 vertices each of degree 5?

2. If the simple graph G has n vertices and m edges, how many edges does have?

3. Define regular graph and a complete graph.

4. What is meant by isomorphism of graphs?

5. Define Euler and Hamilton paths.

PART-B 6. If G is a simple graph with n vertices with minimum degree , show

that G is connected. 7. Show that if g is a self complementary simple graph with vertices,

then . 8. Verify the following graphs are isomorphic.

9. If G is a connected simple graph with n vertices ( and if the degree of each vertex is at least n/2, then show that G is Hamiltonian.

10. For what value of n the following graphs are Eulerian

--End--

Page 30: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

MA039 PROBABILITY AND STATISTICSQUESTION BANK

PROBABILITY AND RANDOM VARIABLES

PART A

1. The probabilities of A, B and C solving a problem are 1/3, 2/7 and 3/8 respectively.If all three try to solve the problem simultaneously, find the probability that exactlyone of them will solve it.

2. A continuous random variable X has the following probability density function

f(x) =

x2

3−1 < x < 2

0 otherwise

Find the distribution function F (x) and use it to evaluate P (0 < X ≤ 1).

3. If the random variable X has the moment generating function Mx(t) =3

3− t, obtain

the standard deviation of X.

4. For a binomial distribution with mean 6 and standard deviation√

2, find the firsttwo terms of the distribution.

5. If X is a Poisson random variable such that P (X = 2) = 23P (X = 1), find P (X = 0).

PART B

6. (a) The probability function of an infinite distribution is given by

P (X = j) =1

2jfor j = 1, 2, · · · ,∞. Verify if it is a legitimate

probability mass function and also find P (Xis even), P (X ≥ 5) andP (Xis divisible by 3). (8)

(b) If a random variable X has a pdf

f(x) =

1

3, −1 < x < 2

0, otherwise

find the moment generating function of X. Hence find the mean andvariance of X. (8)

7. (a) Find the first four moments about the origin for a random variable X having

the pdf f(x) =4x(9− x2)

81, 0 ≤ x ≤ 3. (8)

(b) In a bolt factory machines A, B, C manufacture respectively 25,35 and 40 per-cent of the total. Of their output 5,4 and 2 percent are defective bolts respec-tively. A bolt is drawn at random from the product and is found to be defective.What is the probability that it was manufactured bymachine A? (8)

1

Page 31: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

8. (a) Find the moment generating function of a Poisson random variable and hencefind its mean and variance. (8)

(b) Suppose that a trainee soldier shoots a target in an independent fashion. Ifthe probability that the target is shot on any one shot is 0.7,

i. What is the probability that the target would be hit on tenth attempt?

ii. What is the probability that it takes him less than 4 shots?

iii. What is the probability that it takes him an even number of shots?

(8)

9. (a) Find the moment generating function of a Geometric random variable andhence find its mean and variance. (8)

(b) The time required to repair a machine is exponentially distributed withparameter 1/2

i. What is the probability that the repair time exceeds 2 hours?

ii. What is the conditional probability that a repair takes at least 10 hoursgiven that its duration exceeds 9 hours?

(8)

10. (a) State and prove the memoryless property of an exponential distribution. (8)

(b) If X and Y are two independent random variables having density functionsfX(x) = 2e−2x, x ≥ 0 and fY (y) = 3e−3y, y ≥ 0, find the density function ofU = X + Y . (8)

TWO DIMENSIONAL RANDOM VARIABLES

PART A

11. The joint pdf of two random variables X and Y is given by

f(x, y) =

{c(1− x)(1− y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1

0, otherwise

Find the constant c.

12. The joint pdf of (X, Y ) is given by f(x, y) = xy2 +x2

8, 0 ≤ x ≤ 2, 0 ≤ y ≤ 1. Find

P (X < Y ).

13. Let (X, Y ) be a two dimensional non-negative continuous random variable havingthe joint density function

f(x, y) =

{xy

36x, y = 1, 2, 3

0 otherwise

If U = X + Y and V = X − Y then obtain the joint pdf of U and V .

14. Let X and Y be any two random variables and a, b be constants. Prove thatCov(aX, bY ) = ab Cov(X, y).

15. State Central Limit theorem.

2

Page 32: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

PART B

16. (a) The joint probability density function of a random variable (X,Y ) is given byf(x, y) = kxye−(x2+y2), x > 0, y > 0. Find the value of k and prove that X andY are independent. (8)

(b) If the independent random variables X and Y have the variances 36 and 16respectively, find the correlation coefficient, rUV where U = X + Y and V =X − Y . (8)

17. (a) The joint pdf of the random variable is given by

f(x, y) = e−(x+y), for x ≥ 0, y ≥ 0.

Find the pdf of U =X + Y

2. (6)

(b) If the joint probability density function of (X, Y ) is given by

f(x, y) =

{c(x2 + y2), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1

0, otherwise

Find the conditional densities of X given by Y and Y given X. (10)

18. The joint pdf of two random variables X and Y is given by

f(x, y) =

{k[(x + y)− (x2 + y2)], 0 < (x, y) < 1

0 otherwise

Show that X and Y are uncorrelated but not independent. (16)

19. Let X and Y be random variables having joint density function

f(x, y) =

{x + y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1

0, otherwise

Find the correlation coefficient rXY . (16)

20. The probability density function of two random variables X and Y is given by

f(x, y) =

3

2(x2 + y2), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1

0, otherwise

Find the lines of regression of X on Y and Y on X. (16)

RANDOM PROCESSES

PART A

21. Define strict sense and wide sense stationary process.

22. Prove that sum of two independent Poisson process is a Poisson process.

3

Page 33: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

23. A fair dice is tossed repeatedly. If Xn denotes the maximum of the number occurringin the first n tosses, find the transition probability matrix P of the Markov chain{Xn}.

24. Obtain the steady state probabilities for an (M/M/1) : (N/FIFO) queuing model.

25. State Little’s formula for an (M/M/1) : (∞/FIFO) queueing model.

PART B

26. (a) If {N1(t)} and {N2(t)} are two independent Poisson process with parameterλ1 and λ2 respectively, show that

P (N1(t) = k/N1(t) + N2(t) = n) =

(n

k

)pkqn−k

where p =λ1

λ1 + λ2

and q =λ2

λ1 + λ2

. (8)

(b) Let {Xn} be a Markov chain with state space {0, 1, 2} with initial probabilityvector p(0) = (0.7, 0.2, 0.1) and the one step transition probability matrix

P =

0.1 0.5 0.40.6 0.2 0.20.3 0.4 0.3

Compute P (X2 = 3) and P (X3 = 2, X2 = 3, X1 = 3, X0 = 2). (8)

27. (a) Show that the random process X(t) = A cos(ωt+θ) is a Wide Sense Stationaryprocess if A and ω are constants and θ is a uniformly distributed randomvariable in (0, 2π). (8)

(b) Consider a Markov chain with transition probability matrix

P =

0.5 0.4 0.10.3 0.4 0.30.2 0.3 0.5

Find the steady state probabilities of the system. (8)

28. (a) Assume a random process X(t) with four sample functionsx(t, s1) = cos t, x(t, s2) = − cos t, x(t, s3) = sin t, x(t, s4) = − sin t whichare equally likely. Show that is is wide-sense stationary. (10)

(b) If customers arrive at a counter in accordance with a Poisson process with amean rate of 2 per minute, find the probability that the interval between 2consecutive arrivals is (i) more than1 min, (ii) between 1 min and 2 min and(iii) 4 min or less. (6)

29. (a) The probability distribution of the process {X(t)} is given by

P (X(t) = n) =

(at)n−1

(1 + at)n+1, n = 1, 2, 3, · · ·

at

1 + at, n = 0

Show that it is not stationary. (10)

4

Page 34: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

(b) Arrivals at the telephone booth are considered to be Poisson with an averagetime of 10 min. between one arrival and the other. The length of the phone callis assumed to be distributed exponentially with mean 3 min.Find the averagenumber of persons waiting in the system. What is the probability that a personarriving at the booth will have to wait in the queue? (6)

30. In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day.Assuming that the inter-arrival time follows an exponential distribution and theservice time is also exponential with an average of 36 minutes. Find

(a) the mean queue size

(b) the average waiting time in the system

(c) the average number of trains in the queues

(d) the average waiting time in the queue

(e) the probability that the queue size exceeds 10

(16)

RELIABILITY ENGINEERING

PART A

31. Four units are connected with reliabilities 0.97, 0.93, 0.90 and 0.95. Determine thesystem reliability when they are connected (i) in series (ii) in parallel.

32. Reliability of the component is 0.4. Calculate the number of component to beconnected in parallel to get a system reliability of 0.8.

33. The reliability of a component is given by R(t) =

(1− t

t0

)2

, 0 < t < t0, where t0

is the maximum life of the component. Determine the hazard rate function.

34. A fuel pump with an MTTF of 3000 hours is to operate continuously on a 500 hourmission. Determine the reliability.

35. A computer has a constant failure rate of 0.02 per day and a constant repair rate of0.1 per day. Compute the interval availability for the first 30 days and the steadystate availability.

PART B

36. (a) The density function of the time to failure(in years) of a component manufac-

tured by a certain company is given by f(t) =200

(t + 10)3, t ≥ 0.

i. Derive the reliability function and determine the reliability for the firstyear of operation.

ii. Compute the MTTF.

iii. What is the design life for the reliability of 0.95?

5

Page 35: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

(8)

(b) The time to repair a power generator is denoted by its pdf

m(t) =t2

333, 1 ≤ t ≤ 10 hours.

i. Find the probability that the repair will be completed in 6 hours.

ii. What is the MTTR?

iii. Find the repair rate.

(8)

37. (a) Given that R(t) = e−√

0.001t, t ≥ 0,

i. Compute the reliability for a 50 hour mission.

ii. Find the hazard rate function.

iii. Given a 10 hour warranty period, compute the reliability for a 50 hourmission.

iv. What is the average design life for a reliability of 0.95, given a 10 hourwarranty period?

(8)

(b) A critical communication relay has a constant failure rate of 0.1 per day. Onceit has failed the mean time to repair is 2.5 days. What are the point availabilityat the end of 2 days, the interval availability over a 2 day period and the steadystate availability. (8)

38. (a) A component is found to have its life exponentially distributed with a constantfailure rate of 0.03× 10−4 failures per hour

i. What is the probability that the component will survive beyond 10,000hours?

ii. Find the MTTF of the component.

iii. What is the reliability at the MTTF?

iv. How many hours of operation is necessary to get a design life of 0.90?

(8)

(b) Discuss the reliability of a two component redundant system with repair usingMarkov analysis. (8)

39. (a) Find the variance of the time to failure for two identical units, each with a fail-ure rate λ. placed in standby parallel configuration. Compare the results withthe variance of the same two units placed in active parallelconfiguration. (8)

(b) Six identical components with constant failure rates are connected in (i) highlevel redundancy with 3 components in each sub system (b) low level re-dundancy with 2 components in each subsystem. Determine the componentMTTF in each case to provide a system reliability of 0.90 after 100 hours ofoperation. (8)

6

Page 36: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

40. (a) Find the reliability of the system diagrammed below

0.95 0.99

0.7

0.7

0.7

0.75

0.75

0.9

H

O

F

G

C

D

E

A B

(8)

(b) The density function of time to failure of an appliance is f(t) =32

(t + 4)3, t > 0.

Find the reliability function R(t), the failure rate λ(t) and the MTTF. (8)

DESIGN OF EXPERIMENTS AND QUALITY CONTROL

PART A

41. What are the basic principles of experimental design?

42. Describe Latin Square Design.

43. Depict the ANOVA table for two way classification.

44. The data given below are the number of defectives in 10 samples of 100 items each.

Sample No. 1 2 3 4 5 6 7 8 9 10Number of defects 6 16 7 3 8 12 7 11 11 4

Construct a p-chart and comment on the nature of the process.

45. The following data gives the number of defects in 15 pieces of cloth of equal lengthwhen inspected in a textile mill.

Sample No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Number of defects 3 4 2 7 9 6 5 4 8 10 5 8 7 7 5

Construct a c-chart and comment on the nature of the process.

PART B

46. In order to determine whether there is significant difference in the durability of 3makes of computer, samples of size 5 are selected from each make and the frequencyof repair during the first year of purchase is observed. The results are as follows:

7

Page 37: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

A 5 6 8 9 7B 8 10 11 12 4C 7 3 5 4 1

Test whether there is significant difference in the durability of the 3 makes of thecomputers. (16)

47. Three machines A, B, C gave the production of pieces in four days as below:

A 17 16 14 13B 15 12 19 18C 20 8 11 17

Is there a significant difference between machines? (16)

48. Yields of four varieties of paddy in three blocks are given in the following table

Farmers I II IIIA 10 9 8B 7 7 6C 8 5 4D 5 4 4

(a) Is the difference between varieties significant?

(b) Is the difference between blocks significant?

(16)

49. Four farmers each used four types of manures for the crop and obtained the yield(in quitals) as below:

Farmers 1 2 3 4A 22 16 21 12B 23 17 19 13C 21 14 18 11D 22 15 19 10

Is there any significant difference between (i) farmers (ii) manures? (16)

50. Analyze the variance in the Latin square of yields of wheat where P ,Q,R,S representthe different manures used.

S222 P221 R223 Q222Q224 R223 P222 S225P220 Q219 S220 R221R222 S223 Q221 P222

Test whether the different manures used have given significantly different yields.

(16)

8

Page 38: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

MA040 PROBABILITY AND QUEUING THEORY QUESTION BANK

PROBABILITY AND RANDOM VARIABLES

PART-A

1. If A and B are independent events with P(A)= 21 and P(B) = 3

1 , find P( A ∩ B )

and P( A ∩B).

2. In a community, 32% of the population are male smokers; 27% are female smokers. What percentage of the population of this community smoke?

3. A discrete random variable has moment generating function Mx(t) = e 2(et

-1).

Find E(X) and P(X=2).

4. For exponential random variable X, prove that P(X> x+y) / X>x) = P(X>y).

5. If a random variable X has the probability density function f(x)=

⎪⎩

⎪⎨⎧ <

otherwise,,0

2||,41 x

find P(X<1) and P(2X+3>5).

PART- B 6. (i) Let A and B be two independent events. It is known that P(A B)=0.64 and

P(A B)=0.16. Find P(A) and P(B). ∪

(ii) A continuous random variable X has probability density function

f(x)= . Find the value of constant C. Obtain the moment

generating function of the random variable X and hence obtain its mean and variance.

⎩⎨⎧

≤>

0x,00,cxe-2x x

(iii) A random variables X has the probability density function f(x) = e , - < x |x |2− ∝ ∝ . If Y = X , find the probability density function of Y and P (Y < 2).

2

7. (i) A bag contains 3 black and 4 while balls. 2 balls are drawn at random one at a time without replacement.

(a) What is the probability that a second ball drawn is white.

(b) What is the conditional probability that first ball drawn is white if the second ball is known to be white?

(ii) Let X be a exponential random variable with mean 1. Find the probability density function of Y= -loge X and E(Y).

(iii) A random variable X has a mean of 4 and a variance of 2. Use the Chebyshev’s inequality to obtain the upper bound.

8. (i) Three machines A, B and C produce identical items of their respective output 5%, 4% and 3% of the items are faulty. On a certain day A has produced 25%, B has produced 30% and C has produced 45% of the total output. An item selected at random is found to be faulty. What are the chance that it was

Page 39: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

produced by C?

(ii) A test engineer discovered that the cumulative distribution function of the

lifetime of an equipment in years is given by F (x)= .0x,00,1 5

1

⎪⎩

⎪⎨⎧

<≥−

−xe

x

(a) What is the expected lifetime of the equipment?

(b) What is the variance of the lifetime of the equipment?

(c ) Find P(X>5) and P(5< X < 10).

(iii) If a random variable X has the probability density function f(x) =

21 e ,

-∝< x ∝ , find the M.G.F of X and hence obtain its mean.

|x |−

9, (i) Suppose that, for a discrete random variable X, E(X) = 2 and E(X(X-4)) = 5. Find the variance and standard deviation of -4X + 12.

(ii) Let X be a geometric random variable with parameter p.

(a) Determine the moment generating function of X.

(b) Find the mean of X for p = 2/3.

(c) Find P(X>10) for p=2/3.

(iii) Message arrive at a switchboard in a Poisson manner at a average rate of six per hour. Find the probability for each of the following events:

(a) Exactly two messages arrive within one hour.

(b) At least three messages arrive within one hour.

10. (i) A carton 24 hand grenades contains 4 that are defective. If three hand generates are randomly selected from this carton, what is the probability that exactly 2 of them are defective?

(ii) Each sample of water has a 10% chance of containing a particular organic pollutant. Assume that the samples are independent with regard to the presence of the pollutant.

(a) Find the probability that in the next 18 samples, exactly 2 contain the pollutant.

(b) Determine the probability that at least four samples contain the pollutant.

(c) Find the expected number of pollutant.

(iii) Let X be a random variable with probability density function

f(x)=⎪⎩

⎪⎨⎧ ≤≤−

otherwise.,0

11,21 x If Y=Sin

2Xπ , find probability density function of Y.

TWO DIMENSIONAL RANDOM VARIABLES

PART- A

1. If the joint probability density function of (X,Y) is f(x,y) =

⎪⎩

⎪⎨⎧

≤≤≤≤

otherwise,,020,20

,41

yx

find P(X+Y 1). ≤

Page 40: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

2. The joint probability mass function of random variables X and Y is given by

P(X=x, Y=y) = ⎪⎩

⎪⎨⎧ ==+

otherwise.,0

2,1,2,1),2(181 yxyx

Find the marginal probability mass functions of X and Y.

3. Show that if X=Y, then Cov(X,Y)= Var(X) = Var(Y).

4. Prove that the correlation coefficient ρ xy takes value in the range -1 to 1.

5. State the central limit theorem for independent and identically distributed random variables.

PART- B 6. If the joint probability density function of X and Y is given by

g(x,y) = Find P(X>Y). ⎩⎨⎧ ≥≥+−

otherwise.,00,0,y)x( yxe

7. (i) Let X and Y have the joint probability mass function

Y X 0 1 2

0 0.1 0.4 0.1

1 0.2 0.2 0

(a) Find P(X+Y>1).

(b) Find the marginal probability mass function of the random variable X.

(c ) Find P(X=x / Y=0).

(d) Are X and Y independent random variables? Explain.

(ii) (X1 , X2) is a random sample from a population N(0,1). Show that the distribution of X1

2 + X22 and

2

1

XX

are independent and write down the

probability density functions.

8 (i) The joint probability density function of random variables X and Y is given by

f(x,y)= ⎩⎨⎧ ≤≤≤

otherwise.,010,2 yxCxy

(a) Determine the value of C.

(b) Find the marginal probability density functions of X and Y.

(c) Calculate E(X) and E(Y).

(d) Find the conditional probability density function of X given Y=y.

(ii) The joint probability mass function of a bivariate random variables (X,Y) is given by P(X=0, Y=0)=0.45, P(X=0, Y=1)=0.05, P(X=1, Y=0) =0.1, P( X=1, Y=1)=0.4. Find the correlation coefficient of X and Y.

9. (i) The joint probability density function of a bivariate random variable (X, Y)

Page 41: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

is given by f(x,y)= ⎩⎨⎧ <<<<

otherwise,,010,10,Kxy yx

where K is constant.

(a) Find the value of K.

(b) Are X and Y independent?

(c) Find P(X+Y<1) and P(X>Y).

(ii) Let X and Y be positive independent random variable with the identically probability density function f(x)=e-x , x>0. Find the joint probability

density function of U= X+Y and V=YX . Are X and Y independent ?

10. (i) Let the conditional probability density function X given that Y=y be

f(x/y) =⎪⎩

⎪⎨⎧ >>

++ −

otherwise.,0

0,0,1

yx yxey

y

Find

(a) P(X<1/ Y=2).

(b) E(X/Y=2).

(ii) The joint probability density function of random variables X and Y is given as

f(x,y) = ⎩⎨⎧ ≤≤≤

otherwise.,010,2 xy

(a) Calculate the marginal probability density functions of X and Y respectively.

(b) Compute P(X<

21 ), P(X<2Y) and P(X=Y) .

(c) Are X and Y independent random variables ? Explain .

11. (i) Verify the central limit theorem for the following i.i.d random variables:

For i =1,2,3,... Xi =⎪⎩

⎪⎨

− .21y probabilit with,121y probabilit with,1

(ii) The joint probability density function of X and Y is given by

f (x, y) = ⎩⎨⎧ >>

otherwise.,00,0,2e 2y--x yx

Compute

(a) P(X>1, Y<1).

(b) P(X<Y).

(c) P(X< ½).

(d) E(XY) .

(e) Cov (X,Y).

Page 42: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

RANDOM PROCESSES PART-A

1. Distinguish between wide-sense stationary and strict stationary processes.

2. Describe a Binomial process and hence obtain its mean.

3. Let X(t) be a Poisson process with rate λ . Find E(X(t) X (t+τ )).

4. Let X(t) = A cos 2π t, where A is some random variable. Is the process first order stationary? Explain.

5. Let be a renewal process with CDF F(t). Show that }{ 0);( ≥ttN

P (N(t) = n) = F (t) - F (t) where F )( (t) is the n-fold convolution of F(t) with itself.

)(n )1( +n n

PART- B 6.. (i) Consider a random process X(t) defined by X(t) = Y cosω t, t o where ≥ ω

is a constant and Y is a uniform random variable over (0,1).

(a) Describe X(t).

(b) Sketch a few typical sample functions of X(t).

(ii) Show that the time interval between successive events (or inter-arrival times) in a Poisson process X(t) with rate μ are independent identically distributed exponential random variables with parameter μ .

7. (i) Consider a random process X(t) defined by X(t) = Y cos(ω t+φ ) where Y and φ are independence random variables and are uniformly distributed over (-A, A) and (- ππ , ) respectively.

(a) Find E(X(t)).

(b) Find the autocorrelation function Rxx (t, t+τ ) of X(t) .

(c ) Is the process X(t) wide-sense stationary?

(ii) Express the answers to the following questions in terms of probability functions.

(a) State the definition of a Markov process.

(b) State the definition of an independent increment random process.

(c) State the definition of the second order stationary process.

(d) State the definition of the strict-sense stationary process.

8. (i) Obtain the probability generating function of a pure birth and death process with λ and μ as birth and death rates, assuming the initial population size as one.

(ii) Let X(t) be a Poisson process with rate λ .

Find E { 2))()(( sXtX − } for t > s.

9. (i) Define renewal process and renewal density function. Establish the integral equation for the renewal function.

(ii) Consider a random process X(t) defined by X(t) = U cost+ V sint, where U and V are independent random variables each of

which assumes the values -2 and 1 with the probabilities 1/3 and 2/3 ,<∝∝<− t

Page 43: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

respectively. Show that X(t) is wide-sense stationary but not strict-sense stationary.

10. (i) Define Poisson process and obtain the probability distribution for that.

(ii) Consider a renewal process { 0);( ≥ttN0≥

} with an Erlang (2,1) inter-arrival time distribution f(t) = t e , t . Find the renewal function t−

M(t) =E(N(t)) and obtain t Lim →∝t

M(t) .

MARKOV CHAIN AND RELIABILITY PART- A

1. Consider a Markov chain { }0,1,2,...n;Xn = with state space S={1,2} and one-step

transition probability matrix P = . Is state 1 periodic? If so, what is it

period.

⎥⎦

⎤⎢⎣

⎡0110

2. Find the invariant probabilities (stationary probabilities) for the Markov chain with state space S={1,2} and one-step transition probability matrix { 0 n;Xn ≥ }

P = ⎥⎦

⎤⎢⎣

102

12

1.

3. The hazard rate function Z(t) is given as

Z(t) = ⎩⎨⎧ >>>−

otherwise.,00,0,0,1 βααβ β tt

Find the reliability function and the failure time density function.

4. It is known that the cumulative distribution function of a certain system is

F(t) = 1 - e 3t− - e 6

t− + e 2t− where t is in years. Find the reliability function and

the MTTF for the system.

5. A component has MTBF = 100 hours and MTTR = 20 hours with both failure and repair distributions exponential. Find the steady state availability and non-availability of the component.

PART-B 6. (i) Let be a Markov chain with three states 0, 1, 2 and one-step

transition probability matrix { 0; ≥nX n }

P =

⎥⎥⎥⎥

⎢⎢⎢⎢

41

430

41

21

41

041

43

and the initial distribution P(Xo = i) = 1/3, i = 0, 1,2. Find

(a) P(X =2, X1 =1 / X 0 =2). 2

(b) P(X =2, X1 =1 X =2). 2 0

(c) P(X =1, X =2 , X1 =1, X =0). 3 2 0

Page 44: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

(d) Is the chain irreducible Explain .

(ii) Discuss the preventive maintenance of the system and hence obtain MTTSF for a system having n-identical units in series with exponential failure time distribution.

7. (i) Consider the system, shown in the following figure, in which four different electronic device must work in series to produce a given response. The reliability, R, of the various components are shown on the figure. Find the reliability of the system.

(ii) Consider a Markov chain {Xn; n≥0} with state space S = {0, 1} and one-step

transition probability matrix P = .2

12

1⎢⎣

0.9

0.9

0.97 0.95

0.9

0.9

0.9

01⎥⎦

(a) Draw the state transition diagram.

(b) Is the chain irreducible? Explain.

(c) Show that state 0 is ergodic.

(d) Show that state 1 is transient.

8. (i) Discuss the reliability analysis for 2 - unit parallel system with repair.

(ii) Consider a Markov chain {Xn; n 0} with state space S= {1,2} and one-step

transition probability matrix P =

⎥⎥⎦

⎢⎢⎣

21

21

41

43

. Find the invariant (stationary)

probability distribution of the chain. Find P(X2 =1/ Xo =1) also.

9. (i) The life length of a device is exponentially distributed. It is found that the reliability of the device for 100 hour period of operation is 0.90. How many hours of operation is necessary to get a reliability of 0.95?

(ii) Discuss the availability analysis for 2-unit parallel system with repair.

10. (i) A system has n components, the lifetime of each being an exponential random variable with parameterλ . Suppose that the life times of the components are independent random variables and the system fails as soon as any of its components fails. Find the probability density function of the time until the system fails.

(ii) The following circuit operates only if there is a path of functional devices from left to right. The probability that each device functions is shown on the graph. Assume that devices fail independent? What is the reliability of the circuit.

Page 45: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

0.9

0.9

0.95

0.95

0.99 0.9

QUEUING THEORY

PART A

1. In a given M| M| 1 FCFS queue, ρ = 0.5. What is the probability that the queue contains 5 or more customers. Find also the expected number of customers in the system.

2. Define the effective arrival rate for M| M| 1/N FCFS queueing system.

3. Consider an M| M|C FCFS with unlimited capacity queueing system. Find the probability that an arriving customer is forced to join the queue.

4. In an M| D|1 FCFS with infinite capacity queue, the arrival rate 5=λ and the

mean service time E(S) = 81 hour and Var(S)=0. Compute the mean number of

customers Lq in the queue and the mean waiting time Wq in the queue.

5. Using Little’s formula, obtain the mean waiting time in the system for

M|M| 1/N FCFS queueing system.

PART B 6. (i) An average of 10 cars per hour arrive at a single-server drive-in teller.

Assume that the average service time for each customer is 4 minutes, and both inter-arrival times and service times are exponential.

(a) What is the probability that the teller is idle?

(b) What is the average amount of time a driven-in customer spend in the bank parking lot (including time in service)?

(c ) What is the average number of cars waiting in line for teller?

(d) On the average, how many customers per hour will be served by the teller?

(ii) Find the average number of customers in the M| M| 1/N FCFS queuing system.

7. (i) Suppose that the car owners fill up when their tanks are exactly half full. At the present time, an average of 7.5 customers per hour arrive at a single pump gas station. It takes an average of 4 minutes to service a car. Assume that inter-arrival times and service times are both exponential.

(a) Compute the mean number of customers and mean waiting time in the system.

(b) Suppose that a gas shortage occurs and panic buying takes place. So that all car owners now purchase gas when their tanks are exactly three-quarters full. Since each car owner is now putting less gas into the tank during each visit to the station, we assume that the average service time has been

Page 46: Ma131 Mathematics i Question Bank Question Bank Matrices: Part –a

reduced to 3 31 minutes. How has panic buying affected the mean number

of customers in the system and the mean waiting time in the system.

(ii) For the M|M|C FCFS with unlimited capacity queuing system, derive the steady-state system size probabilities. Also obtain the average number of customers in the system.

8. (i) A one-man barber shop has a total of 10 seats. Inter-arrival times are exponentially distributed, and an average of 20 prospective customers arrive each hour at the shop. Those customers who find the shop full do not enter. The barber takes an average of 12 minutes to cut each customer’s hair. Haircut times are exponentially distributed.

(a) On the average how many haircuts per hour will the barber complete?

(b) On the average, how much time will be spent in the shop by a customer who enters?

(ii) Consider an M|G|1 queuing system in which an average of 10 arrivals occur each hour. Suppose that each customer’s service time follows an Erlangian distribution with rate parameter1 customer per minute and shape parameter 4.

(a) Find the expected number oasf customers waiting in line.

(b) Find the expected time that a customer will spend in the system.

(c ) What fraction of the time will the server will be idle?

9. (i) For an M|M|2 queueing system with a waiting room of capacity 5, find the average number of customers in the system, assuming that arrival rate as 4 per hour and mean service time 30 minutes.

(ii) Consider a bank with two tellers. An average of 80 customers per hour arrive at the bank and wait in a single line for an idle teller. The average time it takes to serve a customer is 1.2 minutes. Assume that inter-arrival times and service times are exponential. Determine

(a) The expected number of customers present in the bank

(b) The expected length of time a customer spends in the bank

(c ) The fraction of time that a particular teller is idle.

10. Discuss the M|G|1 FCFS unlimited capacity queueing model and hence obtain P-K formula.


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