7806714 Anna University Maths Question Bank

MA131 MATHEMATICS I QUESTION BANK Question Bank MATRICES: PART –A ⎛4 1⎞ ⎟. ⎟ ⎝ 3 2⎠ ⎛0 a⎞ ⎟. 2. Find the eigen values and eigen vectors of A = ⎜ ⎜ ⎟ ⎝a 0⎠ ⎛ 8 −6 2 ⎞ ⎜ ⎟ 3. Find the sum of the eigen values of 2A if A = ⎜ − 6 7 − 4⎟ . ⎜ 2 −4 3 ⎟ ⎝ ⎠ ⎛ 5 3⎞ 4. Verify Cayley- Hamilton theorem for A = ⎜ ⎟ ⎜ 1 3⎟ . ⎠ ⎝ 1. Find the eigen values of 2A+I given that A = ⎜ ⎜ 5. Find the matrix of the quadratic form x + y + z + 2 xz + 4 2 yz . 2 2 2 PART - B 6. (i) Using Cayley-Hamilton theorem, find the inverse of A where 3⎞ ⎛1 1 ⎟ ⎜ A = ⎜ 1 3 − 3⎟ ⎜ 2 − 4 − 4⎟ ⎠ ⎝ (ii) Using diagonalization, find A6 given that A = ⎜ ⎜ (8 Marks) ⎛5 4⎞ ⎟. 1 2⎟ ⎝ ⎠ (8 Marks) 7. Find an orthogonal transformation which reduces the quadratic form 2 xy + 2 yz + 2 xz to a canonical form. Also find its nature, index, signature and rank. (16 Marks) 7. Reduce the quadratic form 2 x + y + z + 2 xy − 4 yz − 2 xz to a canonical form by an orthogonal transformation. Also find its nature, index, signature and rank. (16 Marks) 2 2 2 8. (i) Using Cayley-Hamilton theorem, find An given that A = ⎜ ⎜ find A3. ⎛1 4⎞ ⎟ . Hence 2 3⎟ ⎝ ⎠ (8 Marks)

⎛ 2 0 4⎞ ⎟ ⎜ (ii) Diagonalize A by an orthogonal transformation where A = ⎜ 0 6 0 ⎟ . ⎜ 4 0 2⎟ ⎠ ⎝ (8 Marks) 9. Find the characteristic equation of the matrix A and hence find the matrix respresented by A − 5 A + 7 A − 3 A + A − 5 A + 8 A − 2 A + I where 8 7 6 5 4 3 2 ⎛2 1 1⎞ ⎜ ⎟ A = ⎜0 1 0⎟. ⎜ 1 1 2⎟ ⎝ ⎠ ANALYTICAL GEOMETRY: (16 Marks) 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 2 x − y + z + 7 = 0 and x + y + 2 z − 11 = 0 . 3. Find x −1 y − 3 z +1 x +1 y +1 z = = = & = are perpendicular. k −k −1 5 3 −4 the value of k if the lines 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 x 2 + y 2 + z 2 − 2 x + 4 y + 6 z − 12 = 0 . 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 x −1 y − 2 z − 3 x−3 y +2 z +4 & = = = = . 1 2 3 1 2 3 (8 Marks) 7. (i) Find the length of the shortest distance line between the lines x − 2 y +1 z = = ; 2 x + 3 y − 5 z − 6 = 0 = 3x − 2 y − z + 3 . 2 3 4 (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 x − 13 y + 8 z − 31 = = . 5 −8 1 (8 Marks) 2

8. (i) 2 Show 2 2 that the plane equation 2 x − 2 y + z + 12 = 0 of the sphere touches having the the sphere circle x + y + z − 2 x − 4 y + 2 z = 3 and find also the point of contact. (8 Marks) (ii) Find x 2 + y 2 + z 2 + 10 y − 4 z − 8 = 0; x + y + z = 3 as a great circle. (8 Marks) 9. (i) Find the equation of the sphere that passes through the circle x 2 + y 2 + z 2 + 2 x + 3 y + z − 2 = 0; 2 x − y − 3z − 1 = 0 orthogonally the sphere x + y + z − 3x + y − 2 = 0 . 2 2 2 and cuts (8 Marks) circle given by (ii) 2 Find 2 2 the centre and radius of the x + y + z + 2 x − 2 y − 4 z − 19 = 0; x + 2 y + 2 z + 7 = 0 . x +1 y − 2 z = = ; 3x + 2 y − 5 z − 6 = 0 = 2 x − 3 y + z − 3 . 3 2 1

(8 Marks) (i) Find the length and equations of the shortest distance line between the lines (8 Marks) (ii) Find the equation of the sphere passing through the circle given by x 2 + y 2 + z 2 + 3x + y + 4 z − 3 = 0 and x 2 + y 2 + z 2 + 2 x + 3 y + 6 = 0 and the point (1,-2,3). GEOMETRICAL APPLICATIONS OF DIFFERENTIAL CALCULUS PART-A 1. Find the curvature of the curve given by y = c tan x 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 (8 Marks) 1 p2 = 1 a2 + 1 b2 − r 2 a 2b 2 . 4. Find the envelope of the family of lines y = mx + a , where m is a parameter. m 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 on it. (ii) Find the centre of curvature at x y + = 1 at any point (x,y) a b x = 2 cos t + cos 2t , y = 2 sin t + sin 2t . θ =π 2 (8 Marks) on the curve (8 Marks) 3

7. (i) Find the equation of the circle of curvature of the parabola y = 12 x at the point (3,6). (ii) Find the radius of curvature of the curve r = a(1 + cosθ ) at the point 2 8. (i) If ϕ be the angle which the radius vector of the curve r = f (θ ) makes with θ =π 2. (8 Marks) ⎛ dϕ ⎞ = sin ϕ ⎜1 + ⎟ , where ρ is the radius of ρ dθ ⎠ ⎝ curvature. Apply this result to show that ρ = a/2 for the circle r = a cosθ . the tangent, then prove that r (ii) 9. (i) x = a(θ + sinθ ); y = a(1 − cosθ ) . Find the Find the radius of curvature at of the origin the y sin t + x cos t = a + a cos t log tan(t / 2) . Find the radius of envelope (8 Marks) for the cycloid (8 Marks) straight lines (8 Marks) point θ on (8 Marks) (8 Marks) (ii) x = 3a cosθ − a cos 3θ , y = 3a sinθ curvature at − a sin 3θ . 3 the

10. (i) Find the evolute of the astroid x = a cos θ , y = a sin 3 θ . (ii) Show that the envelope of a circle whose centre lies on the parabola y 2 = 4ax and which passes through its vertex is the cissoid (8 Marks) y 2 ( 2a + x ) + x 3 = 0 . FUNCTIONS OF SEVERAL VARIABLES: PART-A 1. Evaluate ∂z ∂z & if x + y + z = log z . ∂x ∂y −t 2. If u = log( x + y + z ) & x = e , y = sin t , z = cos t , then find 3. Find du . dt du 2 2 2 2 , when u = sin( x + y ) & x + 4 y = 9 . dx 4. Write down the Maclaurin’s series for sin(x + y) y2 x2 ∂ ( x, y ) 5. If u = ,v = , then find . x y ∂ (u , v) 4

PART-B 6. (i) Expand x y + 3 y − 2 in powers of (x-1) and (y+2) using Taylor’s expansion. (8 Marks) 2 (ii) Find the maximum and minimum values of x − xy + y − 2 x + y . (8 Marks) 7. (i) Find the volume of the greatest rectangular parallelepiped that can be inscribed 2 2 + + = 1. (8 Marks) a2 b2 c2 ∞ −x e (ii) Prove that ∫ (1 − e − ax )dx = log(1 + a ) where a>-1 using differentiation 0x in the ellipsoid under the integral sign. 8. (i) If z = f(u,v) where (8 Marks) x2 y2 z2 u = x 2 − y 2 , v = 2 xy , then show that (8 Marks) z xx + z yy = 4( x 2 + y 2 )( zuu + zvv ). variables x and y in the equation z xx − z yy = 0 to u and v. 9. (i) If x + y + z − 2 xyz = 1, show that 2 2 2 (ii) By using the transformations u = x + y, v = x − y , change the independent (8 Marks) dx 1− x x 2 + dy 1− y 2 + dz 1− z 2 = 0. (8 Marks) (ii) Expand e cos y in powers of x and y as far as the terms of third degree. (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 e log(1 + y ) in powers of x and y as far as the terms of third degree. (8 Marks) DIFFERENTIAL EQUATIONS: PART-A x 1. Solve ( D − 2) y = e 2

2x . 2 −x 2. Find the Particular Integral of ( D + 2) y = e 3. Solve ( D + 4) y = sin 2 x . 2 cos x . 5

4. Solve d 4x dt 4 = n4 x . 2 −2 x 5. Find the Particular Integral of ( D − 3) y = xe PART-B 6. (i) Solve . d2y dx 2 +4 dy + 4 y = e − 2 x + e 3 x sin x . dx d2y dx 2 (8 Marks) (ii) Solve by the method of variation of parameters + 4 y = 4 tan 2 x . (8 Marks) 7. (i) Solve the simultaneous equations x = 2, y = 0 when t = 0. dx dy + y = sin t ; + x = cos t given that dt dt (8 Marks) (ii) Solve d2y dx 2 + y = sec x by the method of variation of parameters.(8 Marks) dx dy + 2 x − 3 y = 5t ; − 3 x + 2 y = 2e 2 t . (16 Marks) dt dt dy d2y 9. Solve the equation + (1 − cot x) − y cot x = sin 2 x by the method of 2 dx dx 8. Solve reduction of order. 10. Solve x 2 (16 Marks) dy dx 2 2 +x dy + y = 4 sin(log x) . dx --End-(16 Marks) 6

MA132 MATHEMATICS – II QUESTION BANK MULTIPLE INTEGRALS: PART- A 1. Evaluate ∫ ∫ 11 ab dxdy . xy 2. Evaluate ∫ π a sin θ 0 0 ∫ r dr dθ . 0 1− x 2 0 0 3. Change the order of integration in ∫ ∫ f ( x, y )dx dy . 4. Define Gamma function and Beta function. 5. Show that β ( m, n) = β ( n, m) . PART-B 6. (i) Prove that β (m, n) = (ii) Evaluate ∫ e 0 ∞ −x 7 Γ ( m )Γ ( n ) . Γ ( m + n) x dx . 2 2 7. (i) Find the volume bounded by the cylinder x + y = 4 and the planes y + z = 4 and z = 0 . (ii) Evaluate 2 2 ∫∫ x y dx dy over the positive quadrant of the circle x + y = 1. aa 8. (i) Change the order of integration I = ∫ ∫ a a2 − x2 0 x dx dy and then evaluate. x2 + y2 0y (ii) Evaluate I = 9. (i) Evaluate I = −a ∫ 2 2 ∫ ( x + y ) dx dy . π /2 0 6 7 ∫ sin θ cos θ dθ . (ii) Evaluate I = π /2 0 ∫ cot θ dθ . − x2

10. (i) Evaluate I = ∫ e 0 ∞ ⎛1⎞ dx and prove Γ⎜ ⎟ = π . ⎝ 2⎠ (ii) Prove that Γ( n + 1) = n! , when n is a positive integer.

VECTOR ANALYSIS: PART – A 1. Find a unit normal to the surface x + y = z at (1, 2, 5) . 2 2 2. Find directional directive of x + y + z at (1,0,1) in the direction 2i + 3k . 2 2 2 3. Prove that curl ( gradφ ) = 0 . 4. Evaluate ∇ (log r ) . 2 5. If ∇φ = yz i + zx j + xy k , find φ . PART – B 6. (i) Show that F = (6 xy + z ) i + (3 x − z ) j + (3 xz − y ) k is irrotational. Find 3 2 2 φ such that F = ∇φ . (ii) Find the work done when a force F = ( x − y + x ) i − ( 2 xy + y ) j moves a 2 2 particle in the xy plane from (0, 0) to (1,1) along the curve y = x . 2 7. (i) Prove that the area bounded by a simple closed curve C is given by 1 ∫ ( x dy − y dx). Hence find the area of the ellipse. 2C (ii) Find the area between y = 4 x and x = 4 y by using Green’s theorem. 2 2 8. Verify Gauss’s divergence theorem for F = x i + y j + z k taken over the cube bounded by planes x = 0, x = 1, y = 0, y = 1, z = 0 and z = 1. 2 2 2 9. Verify Green’s theorem for ∫ [( xy + y ) dx + x dy ]where C is the boundary of the 2 2 common area between y = x and y = x. 10. Verify Stoke’s theorem for a vector field F = ( x − y ) i + 2 xy j in the rectangular region of the plane z = 0 bounded by the lines x = 0, x = a, y = 0 and y = b. 2 2 C 2

ANALYTIC FUNCTIONS: PART – A 1. Find the invariant points of the transformation w = 3z − 5 . z +1 2. Prove that an analytic function with constant real part is constant. 3. Find the Bilinear transformation which maps w = 2, −1,3 respectively. 4. Test whether the function f ( z ) = 5. Define: Conformal mapping. PART-B z = 0,1, 2 into the points 1 is analytic or not. z 6. (i) Verify that x is harmonic and find u such that x + y2 w = u + iv is analytic. Express w as a function of z . (ii) Find the bi-linear transformation which maps 1, i, −1 in z -plane to 0,1,∞ of the w -plane. v = x2 − y 2 + 2 2 7. (i) Under the transformation w = z , obtain the map in the with vertices (0, 0), ( 2, 0),2, 2) and (0, 2) in z -plane. (ii) Under the transformation w -plane of the square w= z − 2i = 2 . 1 find the image of the circle z + 1 = 1and z 2 2 ⎫ 2 ⎧∂ ⎫ ⎧∂ f ( z) ⎬ + ⎨ f ( z ) ⎬ = f ′( z ) 8. If f (z ) is an analytic function of z , prove ⎨ ⎩ ∂x ⎭ ⎩ ∂y ⎭ 9. If f ( z ) = u + iv is an analytic function find f (z ) and v if u = 10. If f ( z ) = u + iv is an analytic function find f (z ) given that sin 2 x . cos 2 x + cosh 2 y u+v= sin 2 x . cos h 2 y − cos 2 x

COMPLEX INTEGRATION: PART – A 1. Evaluate ∫ z + 2 dz where C is c z z = 3. 2. Find the singular points of f ( z ) = z 1 . z sin z 3. Expand f ( z ) = e in a Taylor’s series about z = 0 . 4. Find the residues at the isolated singularities of the functions 5. Define essential singularity with an example. PART – B z . ( z + 1)( z − 2) 6. Evaluate (i) c ∫ z 2 + 2 z + 4 , where C is z + 1 + i = 2 . ∫ ( z 2 + 9) 3 , where C is c ( z + 1)dz the following integrals, using Cauchy’s residue theorem (ii) dz z − i = 3. 7. Find the Laurent’s series of f ( z ) = 1 valid in the region z (1 − z ) (i) z + 1 < 1 , (ii) 1 < z + 1 < 2 and (iii) z + 1 > 2 8. Use Cauchy’s integral

formula to evaluate (i) sin π z 2 + cos π z 2 ∫ ( z − 2)( z − 3) dz , where C is the circle z = 4 . C (ii) x2 y2 dz , where c is the ellipse + = 1. ∫2 4 1 C z − 3z − 4 2π 7z −1 9. (i) Evaluate ∫ 2 + cosθ 0 dθ using contour integration. ∞ (ii) ) Evaluate ∞ dx using contour integration (1 + x 2 ) 2 0 ∫ x2 − x + 2 10. Evaluate ∫ 4 dx using contour integration. 2 −∞ x + 10 x + 9

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 χ 2 distribution. PART-B 6. (i) Find the coefficient of correlation between X and Y using the following data X: Y: 5 10 15 20 25 16 19 23 26 30 (ii) A study of prices of rice at Chennai and Madurai gave the following data: Chennai Mean S.D. 19.5 1.75 Madurai 17.75 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 Sample I Sample II 8 7 Mean 1234 hours 1036 hours S.D. 36 hours 40 hours Is the difference in the means sufficient to warrant superior to type II bulbs? (ii) Two samples of sizes of squares of deviations from their respective means respectively. Can they be regarded as drawn from the

that type I bulbs are nine and eight gave the sums equal to 160 and 91 same normal population?

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

MA231 MATHEMATICS-III QUESTION BANK PARTIAL DIFFERENTIAL EQUATIONS PART-A 1. Find the partial differential equation from (x − a)2 + (y − b)2 + z 2 = r2 by eliminating the arbitrary constants a and b. 2. Find the partial differential equation from z = x + y + f (xy) by eliminating the arbitrary function f. √ √ 3. Find the complete integral of p + q = 1. 4. Find the general solution of the Lagrange’s equation px2 + qy 2 = z 2 . 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 + q 2 . (ii) Solve z 2 (p2 + q 2 ) = x + y. 7.(i) Solve the equation x2 (y − z)p + y 2 (z − x)q = z 2 (x − y). (ii) Form the partial differential equation by eliminating f from f (z − xy, x2 + y 2 ) = 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 − D2 D − 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<1 . 2−x 1<x<2 1

  1 + 2x ,   l 7.(i) Obtain the Fourier series of the function given by f (x) =    1 − 2x , l −l ≤ x ≤ 0 . 0≤x
9. A rod of length l has its ends A and B kept at 0◦ C and 100◦ C respectively until steady state conditions 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 state conditions prevail. If the temperature of A is suddenly raised to 40◦ C while that the other end B is 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) = (ii) Find the Laplace transform of the periodic function f (t) = and f (t + 2a) = f (t). 7.(i) Find the inverse Laplace transform of s2 3s + 7 . − 2s − 3 a, 0≤t3 (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 = t2 e2t , y(0) = 2 and y (0) = 6, using Laplace transform. 9.(i) Prove that L{f (t)} = s2 F (s) − sf (0) − f (0) where F (s) = L{f (t)}. (ii) Find the inverse Laplace transform of 2 . (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) = 3. Find the Fourier sine transform of f (x) = e−ax (a > 0). 1, 0, |x| < a . |x| > a 3

4. If Fs (s) is the Fourier sine transform of f (x), prove that the Fourier cosine transform of f (t) is 2 FC {f (t)} = sFs (s) − f (0). π 5. Show that FC {f (t) cos at} = of f (x). 1 [Fc (s + a) + Fc (s − a)], where Fc (s) is the Fourier cosine transform 2 PART-B 6.(i) Find the Fourier integral representation of f (x) = (ii) Find the Fourier transform of f (x) = e−a|x| , a > 0. 7. Find the Fourier transform of f (x) = ∞ 0, x<0 e−x , x > 0 1 − x2 , 0, |x| < 1 . Hence evaluate |x| > 1 ∞ 0 x cos x − sin x x3 x cos dx. 2 8.(i) Evaluate 0 dx (x2 + a2 )(x2 + b2 ) using transform methods. (ii) Show that (1) Fs [xf (x)] = − d d Fc (s) and (2) Fc [xf (x)] = Fs (s). ds ds a − |x|, 0, |x| < a and hence evaluate |x| > a > 0 ∞ 0 9. Find the Fourier transform of f (x) = sin x x 2 dx. 10. Find the Fourier sine and cosine transform of xn−1 . −End− 4

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. 3 1 2 4. If A and B ate events such that P ( A ∪ B ) = , P ( A ∩ B ) = , P A = . Find P A | B 4 4 3 () ( ) 5. Suppose that for a RV X, E ⎡ X n ⎤ = 2n , n = 1, 2,3....Calculate its moment generating ⎣⎦ function. PART B 6. (i) Let X be a continuous RV with pdf f X ( x ) = 2 ,1 < x < 2. Find E [ log X ] x2 (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 ( ⋅3et + ⋅7 ) , what is the MGF of Y=3X+2. Also find the mean 2 and variance of X. (6) 7. (i) If the continuous RV X has pdf of Y=X2. ⎧2 ⎪ ( x + 1) , −1 < x < 2 , fX ( x) = ⎨9 ⎪ 0 otherwise ⎩ find the pdf (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)

(iii) The cumulative distribution function FX ( x ) = 1 − e −3 x , x ≥ 0 , find Var(3X+2). for a (4) RV X is given by 8. (i) Let X be an exponential RV with parameter λ = 1 . Use Chebyshev’s inequality, (6) to find P {−1 ≤ X ≤ 3} . Also, find the actual probability. (ii) Let X be a continuous RV with pdf and P {2 X + 1 > 2} 1 f X ( x ) = , − 2 < x < 2 .Find P { X > 1} 4 (6) (iii) Let X be a RV with the pdf f X ( x ) = 1 − ∞ < x < ∞. Find the pdf of π (1 + x 2 ) (4) Z = tan −1 X . 9. (i) The time that it takes for a computer system to fail is exponential with mean 700 hours. If a lab has 20 such computer systems what is the probability that atleast two fail before 1700 hours of use ? (6) (ii) The Pap test makes a correct diagnosis with probability 95%. Given that the test is 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) (iii) Experience has shown that while walking in a certain park, the time X, in minutes, between seeing two people smoking has a density function of the form f X ( x ) = λ xe − x , x > 0. Calculate the value of λ . Find the cumulative distribution function of X.What is 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) 10. (i) Let X be a Gamma RV with parameters n and λ . Find the moment generating function of X and use it to find E[X] and Var(X). (6) (ii) Suppose that, on an average, a post office handles 10,000 letters a day with a variance of 2000. What can be said about the probability that this post office will handle between 8000 and 12000 letters tomorrow? (6) (iii) Peter and Xavier play a series of backgammon games until one of them wins five games. Suppose that the games are independent and the probability that Peter wins a game is 0.58. (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. (4)

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 0. Find the joint pdf of U=X+Y and V=eX. (8) (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 f ( x, y ) = (ii) If X y − xy e , x > 0, 0 < y < 2 find E[e X |2 |Y = 1] 2 (8) pdf (8) and Y are continuous RVs with joint 3 f ( x, y ) = 2 xy + y 2 , 0 < x < 1, 0 < y < 1 , find the conditional pdfs X and Y. 2

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 σ 2 . 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 6.(i) Consider the random process x(t ) = cos(t + φ ), φ is a RV with pdf 1 π π fφ ( x) = , − < φ < check whether x(t) is first order stationary. (8) 2 2 π (ii) A stochastic process is described by x(t ) = A sin t + B cos t 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 + τ > 0) ; t>0 , τ > 0 . ii) Let { x(t ), t ≥ 0} be a Poisson process with parameter λ . Suppose each arrival is registered with probability p independent of other arrivals. Let { y (t ), t ≥ 0} be the process of registered arrivals. Prove that Y(t) is a Poisson process with parameter λ p . (6) 8.(i) Prove that the process x(t)=8+A with f A (α ) = 1;0 < α < 1 is (a) first order stationary (b) Secondorder 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 e −4 4 K . Classify the above process. P (k transitions ) = (8) K! 9. Show that the random process x(t ) = cos(t + φ ) where φ is uniformly distributed in (16) ( 0, 2π ) is (a) first order stationary (b) stationary in wide sense (c) Ergodic.

10.(i) Let { x(t ), t ≥ 0} be a random process with stationary independent increments, where (8) and assume that x(0) = 0 . Show that cov ( x(t ), x( s ) ) = σ 12 min(t , s ) , σ 12 = Var ( x(1) ) . (ii) Classify the random process x(t ) = A cos ωt where A and ω are RVs with joint 1 pdf f Aω (α , β ) = , 0 < α < 2, 8 < β < 12. (8) 8 CORRELATION FUNCTION: PART A 1. Statistically independent zero-mean random processes x(t) and y(t) have −τ and Ryy (τ ) = cos 2πτ respectively. Find the autocorrelation functions Rxx (τ ) = e autocorrelation function of the sum Z(t)=x(t)+y(t). 2. Suppose that a random process is wide sense stationary with autocorrelation function Rxx (τ ) = e − τ 2 Find the second moment of the random variable x(5) –x(3) 3. Show that Rxx ( 0 ) = x 2 . 4. Show that the autocorrelation function Rxx (τ ) is maximum at τ = 0 . 5. If x(t) is a stationary random process having mean value E[x(t)]=3 and autocorrelation function Rxx (τ ) = 10 + 3e − τ . Find the variance of x(t). PART B 6. Given the random process x(t ) = ⎧1 0 < t < b where g (t ) = ⎨ , Ai ' s n =−∞ ⎩0 otherwise 1 1 and φ are independent RVs with density functions f A (α ) = δ (α − 1) + δ (α + 1) and 2 2 1 fφ (α ) = , 0 < α < b. The joint probability mass function of Ai is given by b ∑ A g (t − nb + φ ) n ∞ Ai +1 Ai -1 +1 -1 +1 0.2 0.3 0.3 0.2

Find the autocorrelation function Rxx (τ ) in 0 < τ < b for the above process. 7. Given x(n) = ∑ X k δ ( n − k ) where the joint mass function for the RVs X k is given k below:X k +1 Xk 0 1 0 1 0.2 0.3 0.3 0.2 Rxx ( k ) Find the autocorrelation function k = 0,1, 2,3. 8. Consider the random process x(n) = by the joint mass function and covariance function Lxx (k ) for k =−∞ ∑ X δ (n − k ) k ∞ where the X k ' s are characterized X k +1 Xk 0 1 0 1 0.5 0.1 0.1 0.3 Find the autocorrelation function Rxx ( k ) and covariance function for k = 0,1, 2,3. for the above process. 9. Given the random process x(t ) = ⎧1 0 < t < b where g (t ) = ⎨ and n =−∞ ⎩0 otherwise 1 the An ' s and φ are independent RVs with density functions fφ (α ) = , 0 < α < b. and b ∑ A g (t − nb + φ ) n

∞

1 1 f Ai (α ) = δ (α − 1) + δ (α + 1) Evaluate the autocorrelation function Rxx (τ ) in 2 2 0 < τ < b for the above process. Also the joint probability mass function for the Ai ' s is given by Ai +1 Ai -1 +1 -1 1 4 1 4 +1 1 4 1 4 ∞ ⎧1 0 < t < b 10. Given the random process x(t ) = ∑ An g (t (t ) = ⎨ An ' s −∞ ⎩0 otherwise 1 1 and φ are independent RVs functions f Ai (α ) = δ (α − 1) + δ (α + 1) 2 2 1 and fφ (α ) Evaluate the autocorrelation function Rxx (τ ) in 0 < τ < b . probability mass function of Ai is given by Ai +1 Ai -1 +1 -1 1 3 1 6 +1 1 6 1 3

− nb + φ ) where g with density = , 0 < α < b. b The joint

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 Rxx (τ ) = 3 + 2e −4τ . Find the 2 power spectral density of x(t). 3. A widesense stationary noise process N(t) has an autocorrelation function RNN (τ ) = Pe −ϑ τ , − ∞ < 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) a b+ f 2 (ii) a f −b 2 PART B 6. (i) Consider two independent zero-mean random processes x(t) and y(t) with power spectral densities S xx ( jw) and S yy ( jw) respectively. Define new random processes z(t)=x(t)+y(t), x(t)= x(t)-y(t) and ω (t ) =x(t)y(t). Find formulas for, S zz ( jw) , Suu ( jw) and Sωω ( jw) . (8) (ii) Given the power spectral density power in the process x(t). 1+ ω2 . Use residue theory to find the average ω 4 + 4ω 2 + 4 (8) 7.(i) Two random processes x(t) and y(t) are given by x(t ) = A cos(ωt + θ ) and y (t ) = A sin(ωt + θ ) where A and ω are constants and θ is a uniform RV over ( 0, 2π ) . Find the cross-spectral density functions S xy (ω ) and S yx (ω ) and verify S xy (ω ) = S yx ( −ω ) . (ii) Find the cross-correlation function corresponding to the cross-power spectrum 6 . S xy (ω ) = 2 ( 9 + ω ) + ( 3 + jω )2

8.(i) The autocorrelation function of a signal is e spectral density and average power. − τ2 2k 2 where k is a constant. Find the 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) (ii) A random process x(t) is given by x(t ) = b cos(ωt + 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 1 by S xx (ω ) = . Find its autocorrelation and average power. 22 (4 + ω ) (8) --End--

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 ( ¬P ∧ ( ¬Q ∧ R )) ∨ (Q ∧ R ) ∨ (P ∧ R ) ⇔ R. 7. Without constructing truth table verify whether the formula Q ∨ (P ∧ ¬Q ) ∨ ( ¬P ∧ ¬Q ) is a contradiction or tautology. 8. Without constructing truth table obtain PCNF of (P → (Q ∧ R))∧ (¬P → (¬Q ∧ ¬R)) and hence find its PDNF. 9. Use rule CP to show that (∀x )(P( x ) → Q( x)), (∀x )(R( x) → ¬Q( x )) ⇒ (∀x )(R( x ) → ¬P( x)) . 10. Use indirect method of proof to show that (∃z )Q( z) is not valid conclusion from the premises (∀x )(P( x ) → Q( x )) and (∃y )Q( y ) . COMBINATORICS PART-A 1. 2. 3. 4. Use mathematical induction to prove that State and prove Pigeon hole principle. How many positive integers not exceeding 100 that is divisible by 5? 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 , where n > 1.

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. 8. Write the recurrence relation for Fibonacci numbers and hence solve it. 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 PARTA 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. 3. 4. Show that the inverse of an identity element in a group G, ∗ is itself.. If G,∗ is an abelian group show that for all a, b in G, (a ∗ b )n = an ∗ bn . 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 a ∗ b −1 ∈ H for all a, b ∈ H. 10. Show that the Kernel of a group homomorphism is a normal subgroup of a group.

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 , x ∨ (y ∧ ( x ∨ z)) = ( x ∨ y ) ∧ ( x ∨ z) . 10. Which of the following lattices given by the Hasse diagrams are complemented, distributive and modular? 30 35 9 15 d 1 e c e 15 6 10 d c b a 3 1 2 3 5 5 a 0 b 1 (a) (b) (c) (d)

GRAPHS PART-A 1. 2. How many edges are there in a graph with 10 vertices each of degree 5? If the simple graph G has n vertices and m edges, how many edges does have? 3. 4. 5. 6. 7. 8. Define regular graph and a complete graph. What is meant by isomorphism of graphs? Define Euler and Hamilton paths. PART-B If G is a simple graph with n vertices with minimum degree that G is connected. Show that if g is a self complementary simple graph with then . Verify the following graphs are isomorphic. , show vertices, 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--

MA039 PROBABILITY AND STATISTICS QUESTION 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 exactly one of them will solve it. 2. A continuous random variable X has the following probability density function 2 x −1 < x < 2 f (x) = 3 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) = the standard deviation of X. 4. For a binomial distribution with mean 6 and standard deviation two terms of the distribution. √ 3 , obtain 3−t 2, find the first 2 5. If X is a Poisson random variable such that P (X = 2) = 3 P (X = 1), find P (X = 0). PART B 6. (a) The function of an infinite distribution is given by 1 P (X = j) = j for j = 1, 2, · · · , ∞. Verify if it is a legitimate 2 probability mass function and also find P (Xis even), P (X ≥ 5) and P (Xis divisible by 3). (8) probability (b) If a random variable X has a pdf   1 , −1 < x < 2 f (x) = 3 0, otherwise find the moment generating function of X. variance of X. Hence find the mean and (8) 7. (a) Find the first four moments about the origin for a random variable X having 4x(9 − x2 ) the pdf f (x) = , 0 ≤ x ≤ 3. (8) 81 (b) In a bolt factory machines A, B, C manufacture respectively 25,35 and 40 percent of the total. Of their output 5,4 and 2 percent are defective bolts respectively. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machine A? (8) 1

8. (a) Find the moment generating function of a Poisson random variable and hence find its mean and variance. (8) (b) Suppose that a trainee soldier shoots a target in an independent fashion. If the 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 and hence find its mean and variance. (8) (b) The time required to repair a machine is exponentially distributed with parameter 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 hours given 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 functions fX (x) = 2e−2x , x ≥ 0 and fY (y) = 3e−3y , y ≥ 0, find the density function of U =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) = xy 2 + P (X < Y ). x2 , 0 ≤ x ≤ 2, 0 ≤ y ≤ 1. Find 8 13. Let (X, Y ) be a two dimensional non-negative continuous random variable having the joint density function f (x, y) = xy 36 0 x, y = 1, 2, 3 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 that Cov(aX, bY ) = ab Cov(X, y). 15. State Central Limit theorem. 2

PART B 16. (a) The joint probability density function of a random variable (X, Y ) is given by 2 2 f (x, y) = kxye−(x +y ) , x > 0, y > 0. Find the value of k and prove that X and Y are independent. (8) (b) If the independent random variables X and Y have the variances 36 and 16 respectively, find the correlation coefficient, rU V 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. X +Y . 2 (b) If the joint probability density function of (X, Y ) is given by Find the pdf of U = f (x, y) = c(x2 + y 2 ), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 0, otherwise (10) (6) Find the conditional densities of X given by Y and Y given X. 18. The joint pdf of two random variables X and Y is given by f (x, y) = k[(x + y) − (x2 + y 2 )], 0 < (x, y) < 1 0 otherwise Show that X and Y are uncorrelated but not independent. 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 (16) Find the correlation coefficient rXY . 20. The probability density function of two random variables X and Y is given by   3 (x2 + y 2 ), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 f (x, y) = 2 0, otherwise Find the lines of regression of X on Y and Y on X. 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 (16) (16)

23. A fair dice is tossed repeatedly. If Xn denotes the maximum of the number occurring in 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/F IF O) queuing model. 25. State Little’s formula for an (M/M/1) : (∞/F IF O) queueing model. PART B 26. (a) If {N1 (t)} and {N2 (t)} are two independent Poisson process with parameter λ1 and λ2 respectively, show that P (N1 (t) = k/N1 (t) + N2 (t) = n) = where p = n k n−k pq k λ1 λ2 and q = . space {0, 1, 2} step transition Compute P (X2 =

(8) λ1 + λ2 λ1 + λ2 (b) Let {Xn } be with initial probability vector p(0) probability matrix   0.1 0.5 0.4 P 3) and P (X3 = 2, X2 = 3, X1 = 3, X0

a = = =

Markov chain with state (0.7, 0.2, 0.1) and the one 0.6 0.2 0.2 0.3 0.4 0.3 2). (8)

27. (a) Show that the random process X(t) = A cos(ωt+θ) is a Wide Sense Stationary process if A and ω are constants and θ is a uniformly distributed random variable in (0, 2π). (8) (b) Consider a Markov chain with transition probability matrix   0.5 0.4 0.1 P = 0.3 0.4 0.3 0.2 0.3 0.5 Find the steady state probabilities of the system. (8) 28. (a) Assume a random process X(t) with four sample functions x(t, s1 ) = cos t, x(t, s2 ) = − cos t, x(t, s3 ) = sin t, x(t, s4 ) = − sin t which are equally likely. Show that is is wide-sense stationary. (10) (b) If customers arrive at a counter in accordance with a Poisson process with a mean rate of 2 per minute, find the probability that the interval between 2 consecutive 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  n−1  (at)  , n = 1, 2, 3, · · · n+1 P (X(t) = n) = (1 + at)  at  ,n = 0 1 + at Show that it is not stationary. 4 (10)

(b) Arrivals at the telephone booth are considered to be Poisson with an average time of 10 min. between one arrival and the other. The length of the phone call is assumed to be distributed exponentially with mean 3 min.Find the average number of persons waiting in the system. What is the probability that a person arriving 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 interarrival time follows an exponential distribution and the service 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 the system 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 be connected in parallel to get a system reliability of 0.8. 33. The reliability of a component is given by R(t) = t , 0 < t < t0 , where t0 t0 is the maximum life of the component. Determine the hazard rate function. 1− 2 34. A fuel pump with an MTTF of 3000 hours is to operate continuously on a 500 hour mission. Determine the reliability. 35. A computer has a constant failure rate of 0.02 per day and a constant repair rate of 0.1 per day. Compute the interval availability for the first 30 days and the steady state availability. PART B 36. (a) The density function of the time to failure(in years) of a component manufac200 tured by a certain company is given by f (t) = , t ≥ 0. (t + 10)3 i. Derive the reliability function and determine the reliability for the first year of operation. ii. Compute the MTTF. iii. What is the design life for the reliability of 0.95? 5

(8) (b) The time to repair a power generator is denoted by its pdf t2 , 1 ≤ t ≤ 10 hours. 333 i. Find the probability that the repair will be completed in 6 hours. ii. What is the MTTR? iii. Find the repair rate. m(t) = (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 hour mission. iv. What is the average design life for a reliability of 0.95, given a 10 hour warranty period? (8) (b) A critical communication relay has a constant failure rate of 0.1 per day. Once it has failed the mean time to repair is 2.5 days. What are the point availability at the end of 2 days, the interval availability over a 2 day period and the steady state availability. (8) 38. (a) A component is found to have its life exponentially distributed with a constant failure rate of 0.03 × 10−4 failures per hour i. What is the probability that the component will survive beyond 10,000 hours? 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 using Markov analysis. (8) 39. (a) Find the variance of the time to failure for two identical units, each with a failure rate λ. placed in standby parallel configuration. Compare the results with the variance of the same two units placed in active parallel configuration. (8) (b) Six identical components with constant failure rates are connected in (i) high level redundancy with 3 components in each sub system (b) low level redundancy with 2 components in each subsystem. Determine the component MTTF in each case to provide a system reliability of 0.90 after 100 hours of operation. (8) 6

40. (a) Find the reliability of the system diagrammed below C F 0.7 0.75 A 0.95 B 0.99 D 0.7 H 0.9 O E 0.7 G 0.75 (8) (b) The density function of time to failure of an appliance is f (t) = 32 , t > 0. (t + 4)3 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 of experimental design? 42. Describe Latin Square Design. 43. table for two way classification. 44. The data given below are defectives in 10 samples of 100 items each. Sample No. Number 345 738 678 12 7 11 9 10 11 4

the basic principles Depict the ANOVA the number of of defects 12 6 16

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 length when inspected in a textile mill. Sample No. Number of defects 123 342 4567 7965 8 9 10 4 8 10 11 12 58 13 14 77 15 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 3 makes of computer, samples of size 5 are selected from each make and the frequency of repair during the first year of purchase is observed. The results are as follows: 7

A B C 568 8 10 11 735 97 12 4 41 Test whether there is significant difference in the durability of the 3 makes of the computers. (16) 47. Three machines A, B, C gave the production of pieces in four days as below: A 17 B 15 C 20 16 14 12 19 8 11 13 18 17 (16) Is there a significant difference between machines? 48. Yields of four varieties of paddy in three blocks are given in the following table Farmers A B C D I II 10 9 77 85 54 III 8 6 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 A B C D 1 22 23 21 22 2 16 17 14 15 3 21 19 18 19 4 12 13 11 10 (16) Is there any significant difference between (i) farmers (ii) manures? 50. Analyze the variance in the Latin square of yields of wheat where P ,Q,R,S represent the different manures used. S222 Q224 P220 R222 P221 R223 Q219 S223 R223 P222 S220 Q221 Q222 S225 R221 P222 Test whether the different manures used have given significantly different yields. (16) 8

MA040 PROBABILITY AND QUEUING THEORY QUESTION BANK PROBABILITY AND RANDOM VARIABLES PART-A 1. If A and B are independent events with P(A)= 1 and P( A ∩ B). 2. 3. In a community, 32% of the population are male smokers; 27% are female smokers. What percentage of the population of this community smoke? A discrete random variable has moment generating function Mx(t) = e 2(e Find E(X) and P(X=2). 4. 5. If a random variable X has the probability density function 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). A continuous random variable X has probability density function -2x ⎧cxe , x > 0 f(x)= ⎨ . Find the value of constant C. Obtain the moment 0, x≤0 ⎩ generating function of the random variable X and hence obtain its mean and variance. A random variables X has the probability density function −2| x | 2 f(x) = e , - ∝ < x ∝ . If Y = X , find the probability density function of Y and P (Y < 2). A bag contains 3 black and 4 while balls. 2 balls are drawn at random one at a time without replacement. What is the probability that a second ball drawn is white. What is the conditional probability that first ball drawn is white if the second ball is known to be white? Let X be a exponential random variable with mean 1. Find the probability density function of Y= -loge X and E(Y). A random variable X has a mean of 4 and a variance of 2. Use the Chebyshev’s inequality to obtain the upper bound. 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 For exponential random variable X, prove that P(X> x+y) / X>x) = P(X>y). ⎧1 ⎪, | x |< 2 f(x)= ⎨ 4 ⎪ 0, otherwise, ⎩ t 2 and P(B) = 1 , find P( A ∩ B ) 3 -1) . (ii) (iii) 7. (i) (a) (b) (ii) (iii) 8. (i)

produced by C? (ii) A test engineer discovered that the cumulative distribution function of the 1 −x ⎧ ⎪1 − e 5 , x ≥ 0 lifetime of an equipment in years is given by F (x)= ⎨ . ⎪ 0, x<0 ⎩ (a) (b) (c ) (iii) What is the expected lifetime of the equipment? What is the variance of the lifetime of the equipment? Find P(X>5) and P(5< X < 10). If a random variable X has the probability density function f(x) = - ∝ < x ∝ , find the M.G.F of X and hence obtain its mean. 9, (i) (ii) (a) (b) (c) (iii) (a) (b) 10. (i) 1 −| x | e, 2 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. Let X be a geometric random variable with parameter p. Determine the moment generating function of X. Find the mean of X for p = 2/3. Find P(X>10) for p=2/3. 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: Exactly two messages arrive within one hour. At least three messages arrive within one hour. 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? 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. Find the probability that in the next 18 samples, exactly 2 contain the pollutant. Determine the probability that at least four samples contain the pollutant. Find the expected number of pollutant. Let X be a random variable with probability density function ⎧1 πX ⎪ , −1 ≤ x ≤ 1 , find probability density function of Y. If Y=Sin f(x)= ⎨ 2 2 ⎪ 0, otherwise. ⎩ (ii) (a) (b) (c) (iii) TWO DIMENSIONAL RANDOM VARIABLES PART- A 1. 0 ≤ x ≤ 2, ⎧1 ⎪, If the joint probability density function of (X,Y) is f(x,y) = ⎨ 4 0≤ y≤2 ⎪ 0, otherwise, ⎩ find P(X+Y ≤ 1).

2. The joint probability mass function of random variables X and Y is given by ⎧1 ⎪ ( 2 x + y ), x = 1,2, y = 1,2 P(X=x, Y=y) = ⎨18 ⎪ 0, otherwise. ⎩ Find the marginal probability mass functions of X and Y. 3. 4. 5. Show that if X=Y, then Cov(X,Y)= Var(X) = Var(Y). Prove that the correlation coefficient ρ xy takes value in the range -1 to 1. State the central limit theorem for independent and identically distributed random variables. PART- B 6. the joint probability density function of X and Y is given by − ( x + y) ⎧e , x ≥ 0, y ≥ 0 g(x,y) = ⎨ Find P(X>Y). otherwise. ⎩ 0, (i) Let X and Y have the joint probability mass function YX 0 1 (a) (b) (c ) (d) (ii) Find P(X+Y>1). Find the marginal probability mass function of the random variable X. Find P(X=x / Y=0). Are X and Y independent random variables? Explain. (X1 , X2) is a random sample from a population N(0,1). Show that the distribution of X12 + X22 and probability density functions. X1 X2 If 7. 0 0.1 0.2 1 0.4 0.2 2 0.1 0 are independent and write down the 8 (i) The joint probability density function of random variables X and Y is given by ⎧Cxy 2 , 0 ≤ x ≤ y ≤ 1 f(x,y)= ⎨ otherwise. ⎩ 0, (a) (b) (c) (d) (ii) Determine the value of C. Find the marginal probability density functions of X and Y. Calculate E(X) and E(Y). Find the conditional probability density function of X given Y=y. 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. The joint probability density function of a bivariate random variable (X, Y) 9. (i)

⎧ Kxy, 0 < x < 1,0 < y < 1 is given by f(x,y)= ⎨ otherwise, ⎩ 0, where K is constant. (a) (b) (c) (ii) Find the value of K. Are X and Y independent? Find P(X+Y<1) and P(X>Y). 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 X density function of U= X+Y and V= . Are X and Y independent ? Y Let the conditional probability density function X given that Y=y be ⎧x + y −y e , x > 0, y > 0 ⎪ f(x/y) = ⎨ 1 + y ⎪ 0, otherwise. ⎩ 10. (i) Find (a) (b) (ii) P(X<1/ Y=2). E(X/Y=2). The joint probability density function of random variables X and Y is given as ⎧2, 0 ≤ y ≤ x ≤ 1 f(x,y) = ⎨ ⎩0, otherwise. (a) (b) (c) 11. (i) Calculate the marginal probability density functions of X and Y respectively. Compute P(X< 1 ), P(X<2Y) and P(X=Y) . 2 Are X and Y independent random variables ? Explain . Verify the central limit theorem for the following i.i.d random variables: 1 ⎧ ⎪ 1, with probability 2 For i =1,2,3,... Xi = ⎨ 1 ⎪− 1, with probability . 2 ⎩ (ii) The joint probability density function of X and Y is given by ⎧2e -x-2y , x > 0, y > 0 f (x, y) = ⎨ otherwise. ⎩ 0, Compute (a) (b) (c) (d) (e) P(X>1, Y<1). P(X
RANDOM PROCESSES PART-A 1. 2. 3. 4. 5. Distinguish between wide-sense stationary and strict stationary processes. Describe a Binomial process and hence obtain its mean. Let X(t) be a Poisson process with rate λ . Find E(X(t) X (t+τ )). Let X(t) = A cos 2π t, where A is some random variable. Is the process first order stationary? Explain. Let {N (t ); t ≥ 0 } be a renewal process with CDF F(t). Show that P (N(t) = n) = F (n ) (t) - F ( n +1) (t) where F ( n ) (t) is the n-fold convolution of F(t) with itself. PART- B 6.. (i) (a) (b) (ii) 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). Describe X(t). Sketch a few typical sample functions of X(t). 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 . 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 (- π , π ) respectively. Find E(X(t)). Find the autocorrelation function Rxx (t, t+τ ) of X(t) . Is the process X(t) wide-sense stationary? Express the answers to the following questions in terms of probability functions. State the definition of a Markov process. State the definition of an independent increment random process. State the definition of the second order stationary process. State the definition of the strict-sense stationary process. 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. Let X(t) be a Poisson process with rate λ . ( Find E { X (t ) − X ( s )) 2 7. (i) (a) (b) (c ) (ii) (a) (b) (c) (d) 8. (i) (ii) } for t > s. Establish the 9. (i) (ii) Define renewal process and renewal density function. integral equation for the renewal function.

Consider a random process X(t) defined by X(t) = U cost+ V sint, − ∝< t <∝, 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

respectively. Show that X(t) is wide-sense stationary but not strict-sense stationary. 10. (i) (ii) Define Poisson process and obtain the probability distribution for that. Consider a renewal process {N (t ); t ≥ 0 } with an Erlang (2,1) inter-arrival time distribution f(t) = t e − t , t ≥ 0 . Find the renewal function M(t) =E(N(t)) and obtain Lim t →∝ M(t) . t MARKOV CHAIN AND RELIABILITY PART- A 1. Consider a Markov chain {X n ; n = 0,1,2,...} with state space S={1,2} and onestep ⎡0 1⎤ transition probability matrix P = ⎢ ⎥ . Is state 1 periodic? If so, what is it ⎣1 0⎦ period. Find the invariant probabilities (stationary probabilities) for the Markov chain {X n ; n ≥ 0} with state space S={1,2} and one-step transition probability matrix ⎡1 P =⎢ 2 ⎣0 2. 1⎤ 2⎥ . 1⎦ 3. The hazard rate function Z(t) is given as ⎧αβt β −1 , t > 0, α > 0, β > 0 Z(t) = ⎨ otherwise. ⎩ 0, 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 3 - e 6 + e the MTTF for the system. 5. −t −t −t 2 where t is in years. Find the reliability function and A component has MTBF = 100 hours and MTTR = 20 hours with both failure and repair distributions exponential. Find the steady state availability and nonavailability of the component. PART-B 6. (i) Let {X n ; n ≥ 0} be a Markov chain with three states 0, 1, 2 and one-step transition probability matrix ⎡3 ⎢4 P = ⎢1 4 ⎢ ⎢0 ⎣ 1 1 3 4 2 4 0⎤ ⎥ 1⎥ 4 1⎥ 4⎥ ⎦

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

(d) (ii) Is the chain irreducible Explain . 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. 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. 0.9 0.9 7. (i) 0.9 0.97 0.95 0.9 0.9 (ii) Consider a Markov chain {Xn; n ≥ 0} with state space S = {0, 1} and one-step 0⎤ ⎡1 transition probability matrix P = ⎢ 1 1 ⎥. 2⎦ ⎣2 Draw the state transition diagram. Is the chain irreducible? Explain. Show that state 0 is ergodic. Show that state 1 is transient. Discuss the reliability analysis for 2 - unit parallel system with repair. Consider a Markov chain {Xn; n ≥ 0} with state space S= {1,2} and one-step 1⎤ ⎡3 4 ⎥ . Find the invariant (stationary) transition probability matrix P = ⎢ 4 1 1⎥ ⎢2 2⎦ ⎣ probability distribution of the chain. Find P(X2 =1/ Xo =1) also. 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? Discuss the availability analysis for 2-unit parallel system with repair. A system has n components, the lifetime of each being an exponential random variable with parameter λ . 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. 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. (a) (b) (c) (d) 8. (i) (ii) 9. (i) (ii) 10. (i) (ii)

0.9 0.95 0.99 0.9 0.95 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. Define the effective arrival rate for M| M| 1/N FCFS queueing system. Consider an M| M|C FCFS with unlimited capacity queueing system. Find the probability that an arriving customer is forced to join the queue. In an M| D|1 FCFS with infinite capacity queue, the arrival rate λ = 5 and the 1 mean service time E(S) = hour and Var(S)=0. Compute the mean number of 8 customers Lq in the queue and the mean waiting time Wq in the queue. Using Little’s formula, obtain the mean waiting time in the system for M|M| 1/N FCFS queueing system. PART B 2. 3. 4. 5. 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 interarrival times and service times are exponential. What is the probability that the teller is idle? What is the average amount of time a driven-in customer spend in the bank parking lot (including time in service)? What is the average number of cars waiting in line for teller? On the average, how many customers per hour will be served by the teller? Find the average number of customers in the M| M| 1/N FCFS queuing system. 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. Compute the mean number of customers and mean waiting time in the system. 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 (a) (b) (c ) (d) (ii) 7. (i) (a) (b)

minutes. How has panic buying affected the mean number 3 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. A oneman 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. On the average how many haircuts per hour will the barber complete? On the average, how much time will be spent in the shop by a customer who enters? 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. Find the expected number oasf customers waiting in line. Find the expected time that a customer will spend in the system. What fraction of the time will the server will be idle? 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. 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 The expected number of customers present in the bank The expected length of time a customer spends in the bank The fraction of time that a particular teller is idle. Discuss the M|G|1 FCFS unlimited capacity queueing model and hence obtain P-K formula. reduced to 3 1 8. (i) (a) (b) (ii) (a) (b) (c ) 9. (i) (ii) (a) (b) (c ) 10.