Solve any two: i) d^2y/dx^2 - 4dy/dx + 4y = sin2x ii) d^2y/dx^2 + y = sec(x) by method of variation of parameters. iii) x^2 d^2y/dx^2 - 3x dy/dx + 5y = sin(logx)

Solve: dx/(y-z) = dy/(z-x) = dz/(y-z)

Solve any two: i) d^2y/dx^2 - 4dy/dx + 4y = x^2 cos x ii) d^2y/dx^2 + y = e^x sin x by method of variation of parameters. iii) (1+x)^2 d^2y/dx^2 + (1+x) dy/dx + y = 4cos[log(1+x)]

Solve: dx/dt - y = e^t, dy/dt - x = e^-t

Find the fourier integral representation of the function f(x) = 1 if |x|<1, f(x)=0 if |x|>1 and hence evaluate integral from 0 to infinity sin(lambda) cos(lambda x) / lambda d(lambda)

Solve any one: i) Find the z-transform and its ROC of 2k, k >= 0 ii) Find Z^-1 {z^2 / ((z-1/2)(z-1/3)); 1/3 < |z| < 1/2}

Obtain f(k) , given that 12 f (k+2) – 7 f (k + 1) + f (k) = 0; k > 0, f (0) = 0, f (1) = 3.

Solve any one: i) Find z{f (k)} if f (k) = 4ksin (2k + 3), k >= 0 ii) Obtain Z^-1{f (k) } by use of the inversion integral method when F (z) is given by F (z) = z / ((z-1)(z-2))

Find the fourier cosine integral representation for the function f(x) = 2 if 0<x<a, f(x)=0 if x>a

Solve the integral eqn integral 0 to infinity f(x) cos(lambda x) dx = 1-lambda if 0<=lambda<=1, 0 if lambda>1 and hence show that integral 0 to infinity (sin^2(z) / z^2) dz = pi/2 where z = lambda/2

Solve any TWO. i) (D2 + 6 D + 9) y = e-3x/x3 ii) Solve by the variation of parameters method d2y/dx2 + y = cosec x iii) x2(d2y/dx2) - 5x(dy/dx) + 4y = x6

Solve dx/(y-z) = dy/(z-x) = dz/(x-y)

Solve any TWO. i) (D2 – 1) y = x sin x ii) Solve by the variation of parameters method d2y/dx2 + y = x2 e-x iii) (x+2)2(d2y/dx2) - (x+2)(dy/dx) + y = 3x+4

Solve the simultaneous linear differential equations with given conditions. du/dx + v = sin x, dv/dx + u = cos x Given that when x = 0, then u = 1 and v = 0.

Find the Fourier transform of f(x) = 2, |x| <= a; = 0, |x| > a

Using inverse Fourier sine transform, find f (x), given Fs(λ) = e-aλ/λ

Solve any one i) Find Z transform of f(k) = 2k/k, k >= 1. ii) Find inverse Z transform of z/(z-5), |z| > 5.

Solve any one i) Find Z transform of f(k) = 5k, k >= 0 ii) Find inverse Z transform of z/(z-1)(z-2), |z| >= 2

Obtain f (k), given f(k+2) - 4f(k) = 0, f(0) = 0, f(1) = 2

Solve the following integral equation: Integral from 0 to infinity of f(x)cos(λx)dx = e-λ, λ > 0

(D2 – 4D + 4)y = ex + sin2x + x

(D2 + 1)y = cotx [Use variation of parameter method)

(1+x)^2 d^2y/dx^2 + (1+x) dy/dx + y = cos[log(1+x)]

The deflection of a strut of length 'l' with one end (x = 0) built-in and the other supported, subjected to end thrust 'p' satisfies the differential equation d^2y/dx^2 + a^2y = R/P(l-x), prove that the deflection curve is y = R/P [sin ax/a - x + (l - sin al/a cos al) cos ax]

(1+2tanx) d^2y/dx^2 - 2 dy/dx + (1+2tanx)y = e^x sec^2x

d^2y/dx^2 + 4y = sin^2x

dx/(y-z) = dy/(z-x) = dz/(x-y)

A horizontal beam is uniformly loaded. It's one end fixed and other end is subjected to a tensile force 'P'. The deflection of the beam is given by d^2y/dx^2 - n^2y = -W/EI, given that dy/dx = 0, y = 0 at x = 0 show that the deflection of a beam for a given x is y = W/2P [e^nx + e^-nx - 2] where n^2 = P/EI

Use Gauss-elimination method with Partial Pivoting to solve following system of equations: 8y + 2z = –7; 3x + 5y + 2z = 8; 6x + 2y + 8z = 26

Use the Runge-Kutta fourth order method to solve dy/dx = x + y, y(1) = 1.5 at x = 1.1 with h = 0.1

Solve following system of equations by using Cholesky-method: 4x1 + 6x2 + 8x3 = 0; 6x1 + 34x2 + 52x3 = –160; 8x1 + 52x2 + 129x3 = –452

Apply Gauss-Seidel method to solve the equations: 27x1 + 6x2 – x3 = 85; 6x1 + 15x2 + 2x3 = 72; x1 + x2 + 54x3 = 110

Using modified Euler's method find an approximate value of y when x = 0.1 given that dy/dx = x + y; y(0) = 1, by taking h = 0.1

Numerical Solution of the differential Equation dy/dx = y - x^2/5 is tabulated as x: 4, 4.1, 4.2, 4.3; y: 1.0, 1.0049, 1.0097, 1.0143. Use Milne's predictor-Corrector method to find y at x = 4.4 by taking h = 0.1

Solve any two i) d2y/dx2 + 4y = sin 2x ii) (D2 - 6D + 9)y = x e^3x (Use method of variation of parameters) iii) x^2 d2y/dx2 - 2x dy/dx - 4y = x^2

Solve : dx/dt = x-y, dy/dt = x+z, dz/dt = x+y

Solve any two i) (D2 + 2D + 1)y = 4sin 2x ii) (D2 + 4)y = sec 2x (Use method of variation of parameters) iii) x^2 d2y/dx2 + (x-a) dy/dx - 4(x+a) y = x^6

Solve the following simultaneous equations : dx/dt + 5x - 2y = 0, dy/dt + 2x + y = 0

By using Fourier integral representation show that integral_0^infinity (cos lx)/(1+l^2) dl = (pi/2) e^-x, x > 0

Attempt any one : i) Find inverse z-transform of F(z) = z / ((z-2)(z-4)), |z| > 4 ii) Find z-transform of f(k) if f(k) = e^k, k > 0

Solve difference equation f(k+1) + 4f(k) = 4^k, k > 0 given f(0) = 0.

Solve any one : i) Find z-transform of f(k) = 3^k sin (2k + 3), k > 0 ii) Find inverse z-transform of F(z) = z / ((z-4)(z-5)) by inversion integral method.

Find Fourier sine transform of f(x) = sin x, 0 < x < a; 0, x > a

Solve integral equation integral_0^infinity f(x) sin lx dx = (1-l), 0 <= l <= 1; 0, l > 1

Solve the following differential equations (Any two) i) (D – 4)3y = e4x + 3x where D = d/dx ii) (d2y/dx2) - 6(dy/dx) + 4y = x iii) (D2 + 3D + 2)y = sin ex [Use variation of parameter method]

A light horizontal strut AB of length ‘l’ is freely pinned at A & B and is under the action of equal and opposite compressive forces ‘P’ at each of its ends with load ‘W’ at its centre governed by the differential equation (d2y/dx2) + n^2y = Wx/EI for x = 0, y = 0, for x=l/2, dy/dx=0 show that the deflection at the centre is (W/2Pn^2)(tan(nl/2) - (nl/2)) where n^2 = P/EI

Solve the following differential equations (Any two) i) (D2 – 6D + 9)y = e3x/x ii) (D2 – 4D + 3)y = ex. cos 2x iii) dx/yz = dy/xz = dz/xy

Find the elastic curve of a uniform cantilever beam of length ‘l’ having a constant weight ‘W’ kg per unit length and determine the deflection at the free end.

Solve following system of equations by using Gauss-elimination method: 10x + 2y + z = 9; 2x + 20y – 2z = –44; –2x + 3y + 10z = 22

Use the Runge - Kutta fourth order method to solve dy/dx = x2 + y2 ; y(0) = 1 at x = 0.1 with h = 0.1

Solve the following system of equations by using Cholesky-method: 9x1 + 6x2 + 12x3 = 17.4; 6x1 + 13x2 + 11x3 = 23.6; 12x1 + 11x2 + 26x3 = 30.8

Solve by Jacobi Iteration method: 10x + y – z = 11.19; x + 10y + z = 28.08; –x + y + 10z = 35.61. Correct to two decimal places.

Use Euler’s modified method to find the value of satisfying the equation dy/dx = log(x + y) ; y(1) = 2. Find y for x = 1.2 by taking h = 0.2.

Numerical solution of the differential equation dy/dx = xy + y2 is tabulated as x: 0, 0.1, 0.2, 0.3; y: 1.0000, 1.1169, 1.2773, 1.5049. Find y at x = 0.4 by Milne’s predictor - corrector method by taking h = 0.1.

Solve any Two i) (D2 + 1) y = 2 sinx sin 2x ii) (D2 – 2D + 2) y = ex tanx (Use method of variation of parameters) iii) d^2y/dx^2 - 5dy/dx + 4y = 6x^2

Solve dx/(y-z) = dy/(z-x) = dz/(x-y)

Solve any TWO i) (D2 – 4D + 4) y = ex cos2x ii) (D^2 - 6D + 9) y = e^3x/x^2 (Use method of variation of parameters) iii) (x+2)^2 d^2y/dx^2 - 3(x+2)dy/dx + 4y = cos ln(x+2)

Solve dx/dt - 4x + y = 0, dy/dt - x - y = 0

Find the Fourier cosine integral representation for the function f(x) = {2, 0 < x < a ; 0, x > a}

Solve the integral equation integral from 0 to infinity f(x) sin(lambda x) dx = {1, 0 <= lambda < 1 ; 0, lambda >= 1}

Attempt the following (Any One) : i) Find Z-transform of f(k) = (1/4)^k for all k. ii) Find Z^-1 [z / ((z-3)(z-2))], 2 < |z| < 3

Attempt the following (Any One) : i) Find Z-transform of f(k) = 4^k sin(2k+3), k >= 0. ii) Find Z^-1 [(z^2+z) / ((z-1)(z^2+z+1))], 1 < |z| < 2

Solve the difference equation f(k + 2) + 3 f(k + 1) + 2 f(k) = 0, f(0) = 0, f(1) = 1, using z-transform.

By Considering Fourier cosine integrals of e–mx, (m > 0), prove that integral from 0 to infinity (cos(lambda x) / (lambda^2 + m^2)) d(lambda) = (pi / 2m) e^(-mx), m > 0, x > 0

Solve the following differential equations (any two) : i) d2y/dx2 - 3dy/dx + 2y = e^x.cosh(x) ii) d2y/dx2 + y = x^2 iii) d2y/dx2 + 4y = sec(2x) (By Variation of parameter method)

The differential equation satisfied by a beam, uniformly loaded with one end fixed and second subjected to a tensile force 'p' is given by E.I. d2y/dx2 = w/P(x^2/2) - P*y where n^2 = P/EI. Show that the elastic curve for the beam under conditions y = 0, dy/dx = 0 for x = 0 is given by y = W/P*n^2 * [e^(nx) + e^(-nx)/2 - 1 - (nx)^2/2].

Solve the following differential equations (any two) : i) (3x+1)^2 d2y/dx2 + 3(3x+1) dy/dx + 9y = 0 ii) d2y/dx2 + 4dy/dx + 3y = e^x iii) dx/(y^2) = dy/(x^2) = dz/(z^2)

Find the elastic curve of a uniform cantilever beam of length 'l' having a constant weight 'w' kg per unit length and determine the deflection at the free end.

Solve the following system of equations by using Gauss-elimination method: 4x + y + z = 4, x + 4y – 2z = 4, 3x + 2y – 4z = 6

Using Runge-Kutta method of fourth order to solve dy/dx = xy + y^2, y(1) = 2, at x = 1.2 with h = 0.2.

Solve the following system of equations by using Cholesky-method. 4x1 + 2x2 + 14x3 = 14, 2x1 + 17x2 – 5x3 = –101, 14x1 – 5x2 + 83x3 = 155

Apply Gauss Seidel method to solve the equations: 10x1 + x2 + x3 = 12, 2x1 + 10x2 + x3 = 13, 2x1 + 2x2 + 10x3 = 14

Solve the equation dy/dx = x^2 + y, y(0) = 1 to find y at x = 0.05, using Euler's modified method by taking h = 0.05.

Numerical Solution of the differential equation dy/dx = xy + x^2 is tabulated as follows : x: 1.0, 1.2, 1.4, 1.6; y: 1.0, 1.6, 2.2771, 3.0342. Find y at x = 1.8 by Milne's predictor-corrector method by taking h = 0.2.

Solve any two: i) x^2(d^2y/dx^2) - x(dy/dx) + y = log x ii) (d^2y/dx^2) + y = sin^2 x by applying the method of variation of parameters. iii) (D^2 - 5D + 6)y = e^x + x^2

The deflection of a strut of length l with one end (x = 0) built in and the other supported and subjected to end thrust P satisfies the equation (d^2y/dx^2) + a^2y = (R/P)(l-x) [5] Prove that the deflection curve is y = (R/P) [(1/a^2)(sin ax/sin al - x/l) + (x-l)/a^2] where al = tan al.

Solve any two: i) dx/(x^2-y^2-z) = dy/(xy) = dz/(xz) ii) dx/dt = 3x+8y, dy/dt = -x-3y if x(0)=6 and y(0)=-2 iii) (D^2 + 3D + 2)y = xe^x + cos x

The differential equation for the elastic curve of a beam is (d^2y/dx^2) + n^2y = -(W/EI)(l/2-x) where E, I, W and P are constants. Assume the beam to be positioned horizontally with one end at x = 0 and the other end at x = l. With y(0) = 0 and dy/dx = 0 at x=l/2, show that the deflection at the centre is (W/2Pn^2)(1 - 1/cos(nl/2)) where n^2 = P/EI

Solve the following system by Cholesky method: 2x - y = 3, -x + 2y - z = 0, -y + 2z = 1

Use Runge Kutta fourth order Method to solve dy/dx = x+y^2 to find y at x = 0.4, given y(0) = 1, take h = 0.2.

Solve by Jacobi’s Iteration method system of equations: 20x + y - 2z = 17, 3x + 20y - z = -18, 2x - 3y + 20z = 25

Solve the following system using Gauss Seidel Method: 20x + y - 2z = 17, 3x + 20y - z = -18, 2x - 3y + 20z = 25

Use modified Euler’s Method to solve dy/dx = x+log y; y(1) = 2 for x = 1.2 and x = 1.4 by taking h = 0.2.

Solve the equation dy/dx = 1+y^2 is tabulated as: x: 0, 0.2, 0.4, 0.6; y: 0, 0.2027, 0.4228, 0.6841. Use Adam-Moulton method to find y at x = 0.8 taking h = 0.2.

Write the correct option for the following multiple choice questions : a) X is normally distributed. The mean of X is 15 and standard deviation 3. Given that for Z = 1, A = 0.3413 then P(X > 12) is given by : i) 0.3413 ii) 0.8413 iii) 0.1587 iv) 0.6587 b) Among 64 off springs of a certain cross between guinea pig 34 were red, 10 were black and 20 were white. According to genetic model, these number should be in the ratio 9 : 3 : 4. Expected frequencies in the order are i) 36, 12, 16 ii) 32, 8, 24 iii) 36, 16, 12 iv) 34, 10, 20 c) Using Newton - Raphson method, the first approximation to a root x1 of the equation x3 + 2x – 5 = 0 in (1, 2) if initial approximation x0 = 2 is ____ i) 0 ii) 3 iii) 1.5 iv) 4 d) If Lagrange's polynomial passes through x: 0, 1; y: –4, –4 then dy/dx at x = 1 is given by i) 0 ii) 2 iii) 1 iv) 1/2 e) The first central moment of a distribution about the mean is i) 1 ii) always positive iii) 0 iv) –1 f) If f(x) is continuous on [a, b] and f(a) f(b) < 0 then to find a root of f(x) = 0, initial approximation x0 by bisection method is ____ i) (a-b)/2 ii) (f(a)+f(b))/2 iii) (a+b)/2 iv) (a-b)/(a+b)

The first four moments about the working mean 30.2 of a distribution are 0.255, 6.222, 30.211, 400.25. Calculate the first four central moments about the mean.

Obtain regression line of x on y for the following data : x: 2, 3, 5, 7, 9, 10, 12, 15; y: 2, 5, 8, 10, 12, 14, 15, 16

Fit a linear curve y = ax + b to the data : x: 0, 2, 4, 6, 8, 12, 20; y: 10, 12, 18, 22, 20, 30, 30

Calculate the coefficient of correlation from the information n = 10, Σx = 40, Σx2 = 190, Σy2 = 200, Σxy = 150, Σy = 40

Fit a curve y = axb for the data x: 2000, 3000, 4000, 5000, 6000; y: 15, 15.5, 16, 17, 18

If regression line of x on y is 9x + y =  and the regression line of y on x is 4x + y =  where means of x and y are 2 and –3 respectively. Find the values of  and  and the coefficient of correlation between x any y.

Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that they are both Queens if : i) the first card drawn is replaced ii) the first card drawn is not replaced

A series of five one-day matches is to be played between India and Australia. Assuming that the result of all the five matches is independent and the probability of India's win in each match is 0.6, find the probability that India wins the series.

A life time of a certain component has a normal distribution with mean of 400 hours and standard deviation of 50 hours. Assuming a normal sample of 1000 components, find number of components whose life time lies between 340 to 465 hours. [Given : A (z = 1.2) = 0.3849, A(z = 1.3) = 0.4032]

The mean and variance of a binomial distribution are 4 and 2 respectively. Find P(r < 2).

Number of road accidents on a high-way during a month follows a Poisson distribution with mean 5. Find the probability that in a certain month number of accidents on the highway will be i) less that 3 ii) more than 3

A die is tossed 300 times gave the following result. Score: 1, 2, 3, 4, 5, 6; Frequency: 43, 49, 56, 45, 66, 41. Is the data consistent at 5% level of significance with hypothesis that the die is unbiased? (Given : χ2 5,0.05 = 11.07)

Using metnod of bisection, find the cube root of 69. (five iterations)

Find the root of the equation x – e–x = 0 that lies between 0.5 and 1 by Newton Raphson method correct up to four decimal places.

Solve by Gauss - Seidel method, the following system of equations. 8x1 + 3x2 + 2x3 = 13; x1 + 5x2 + x3 = 7; 2x1 + x2 + 6x3 = 9

Solve the following system by Gauss elimination method. 2x1 + x2 + x3 = 10; 3x1 + 2x2 + 3x3 = 18; x1 + 4x2 + 9x3 = 16

Solve the following system of equations by Jacobi's iteration method. 20x1 + x2 – 2x3 = 17; 3x1 + 20x2 – x3 = 18; 2x1 – 3x2 + 20x3 = 25

Solve the equation f (x) = x – e–x by Regula-Falsi method with the initial approximations 0.5 and 1 correct up to three decimal places.

Using Newton's backward difference formula find the value of y at x = 3.5 for following data : x: 0, 1, 2, 3, 4; y: 3, 2, 3, 6, 11

Use simpson's 1/3 rd rule to find the value of ∫ 1 to 2 (1/x) dx. Take h = 0.25. Correct the solution upto fourth decimal place.

Use Euler's method to solve the equation dy/dx = 1 + xy with y(0) = 1 and tabulate the solution for x = 0 to x = 0.4. Take h = 0.1 and correct the solution upto fourth decimal place.

Use Runge-Kutta method of fourth order to solve dy/dx = (x2 + y2), y(1) = 1.5 in the interval (1, 1.1) with h = 0.1 and correct the solution upto fourth decimal place.

Given dy/dx = x2 + y, y(0) = 1 determine using modified Euler's method the value of y when x = 0.05. Take h = 0.05 and correct the solution upto fourth decimal place. Use two iterations only.

Find the value of y for x = 0.5 using Newton's forward difference formula for following data : x: 0, 1, 2, 3, 4; y: 1, 5, 25, 100, 250

Attempt the following : a) If 190, 4, 4, 10, 1.732, 2 x y xy x y n Σ = = = = σ= σ= then correlation coefficient r(x y) is i) 0.91287 ii) 0.8660 iii) 0.7548 iv) 0.5324 b) The mean and standard deviation of binomial probability distribution are 36 and 3 respectively. Number of trials n is i) 42 ii) 36 iii) 48 iv) 24 c) The curl of vector field 2 2 F x y j xyz j z yk = + + at point (0, 1, 2) is i) 4 2 2 i j k - + ii) 4 2 2 i j k + + iii) 4 2 i k + iv) 2 4 i k + d) rn is _______. i) nrn–1 ii) 1 1 nr r n + + iii) 2 3 nr r - iv) 2 n n r r - e) r  is i) r ii) 3 iii) r r iv) 0 f) Line of regression y on x is i) ( ) y x y y r x x σ - = - σ ii) ( ) y x y y r x x σ + = + σ iii) ( ) x y x x r y y σ - = - σ iv) ( ) y x x x r y y σ - = - σ

The first four moments of a distribution about the value 2 are 2, 10, 20 and 25. Find first four moments about mean, coefficient of skewness and kurtosis.

Calculate the coefficient of correlation for the following data.

An unbiased coin is thrown 10 times. Find probability of getting. i) Exactly 6 heads. ii) At least 6 heads

Obtain regression line y on x for the following data

On an average 180 cars per hour pass a specified point on a particular road. Using poission distribution. Find the probability that at least two cars pass the point in any one minute.

In a sample of 1000 cases, the mean score in a certain examination is 14 and standard deviation is 2.5. Assuming the distribution to be normal, find the expected number of students scoring between 12 and 15 (both inclusive). (Given : Z1 = 0.4, A1 = 0.1554 ; Z2 = 0.8, A2 = 0.2881)

Find the angle between the tangent to the curve ( ) ( ) ( ) 3 2 2 4 5 2 6 r t i t j t t k = + + - + - at t = 0 and t = 1.

Find the directional derivative of 3 2 2 4 3 xz x y z φ= - at point (2, –1, 2) in the direction 2 3 6 i j k - + .

Given ( ) 2 2 3 2 u xyzi x z y x j xz k = + - + 2 xy yz z φ= + + at point (1,0,–1) Find i) u  ii) u  iii) ( ) u φ

If ( ) 2 3 2 r t t i tj t k = + - then evaluate 1 2 2 0 d r r dt dt   .

Show that vector field ( ) ( ) ( ) 2 2 2 F x yz i y zx j z xy k = - + - + - is irrotational. Find the scalar potential φ such that F =φ.

For constant vector a . Show that i) ( ) a r a   = ii) ( ) 2 a r a   =

Evaluate the line integral C F dr   around the parabolic are y2 = x joining points (0, 0) and (1, 1).

Evaluate S ˆ F n ds   where ‘s’ is the surface of cylinder x2 + y2 = 4, z = 0, z = 3 and 2 2 F 4 2 xi y j z k = - + .

Evaluate ( ) S ˆ F nds    for F 3 yi j xk = + + where ‘s’ is the surface of the paraboloid z = 1 – x2 – y2, z > 0.

Using Green’s theorem evaluate ( ) C cos 1 sin y dx x y dy + -  where ‘c’ is the ellipse 2 2 1 25 9 x y + = , z = 0.

Find the workdone in moving a particle once round the ellipse 2 2 1, 0 16 4 x y z + = = under the force field given by ( ) ( ) ( ) 2 F 2 3 2 4 x y z i x y z j x y z k = - + + + - + - +

Evaluate S ˆ F n ds   where 3 3 3 F x i y j z k = + + and ‘S’ is the surface of the sphere x2 + y2 + z2 = a2 in the first octant.

Solve 2 2 2 2 2 C y y t x   =   with boundary conditions i) ( ) 0, 0 y t = ii) ( ) , 0 y l t = iii) 0 0 t y t    =      iv) ( ) ( ) 2 ,0 ; 0 y x k lx x x l = -  

Solve 2 2 2 u u c t x   =   , u(x, t) satisfies the conditions i) u is finite  t ii) u(0, t) = 0 iii) u(l, t) = 0 iv) u(x, 0) = u0 for 0 < x < l

Solve 2 2 u u t x   =   subject to conditions i) u is finite for all ‘t’ ii) u (0, t) = 0 iii) u (l, t) = 0 iv) ( ) 3 , x u x t l = for 0 < x < l

Solve 2 2 2 2 0 u u x y   + =   subject to the conditions i) u (x, ) = 0, y  ii) u (0, y) = 0 iii) u (10, y) = 0 iv) ( ) ,0 100.sin ;0 10 10 x u x x π =  

Write the correct option for the following multiple choice questions : i) Coefficient of correlation between the variables x and y is 0.8 and their covariance is 20, the variance of x is 16. The standard deviation of y is ________. a) 6.75 b) 6.25 c) 7.5 d) 8.25 ii) The mean and variance of binomial probability distribution are 5/4 and 15/16 respectively. Probability of success in a single trial p is equal to a) 1/2 b) 15/16 c) 1/4 d) 3/4 iii) Using Gauss elimination method the solution of system of equations x+y+z=1, 3x+y+z=5, 4x+4y+z=19 is _______. a) x = 1, y = 2, z = 3 b) x=1/2, y=1, z=1/2 c) x=2, y=1/2, z=1/2 d) x=1, y=1/2, z=-1/2 iv) If Langrange's interpolation polynomial passes through the points (x, y) = (0, 1), (2, 3), (3, 2) then the value of y at x = 1 is a) 2/3 b) 5/3 c) 8/3 d) 5/4 v) Given equation is dy/dx = f(x, y) with initial conditions x = x0, y = y0 and h is step size Euler's formula to calculate y1 at x = x0 + h is given by _______. a) y1 = y0 + hf (x0, y0) b) y1 = y0 + hf (x1, y1) c) y1 = y1 + hf (x0, y0) d) y1 = hf (x0, y0) vi) Coefficient of kurtosis beta2 is given by a) m4/m3^2 b) m4/m2^2 c) m3^2/m2^3 d) m4/m2^3

The first four moments of a distribution about the value 3.5 are 0.0375, 0.4546, 0.0609 and 0.5074. Find the first four central moments about the mean.

Obtain regression line of x on y for the following data : x (6, 2, 10, 4, 8), y (9, 11, 5, 8, 7)

Fit a linear curve y = ap + b by using least square criteria. p (100, 120, 140, 160, 180, 200), y (0.90, 1.10, 1.20, 1.40, 1.60, 1.70)

Calculate the coefficient of correlation from the information n = 5, sum(x) = 100, sum(x^2) = 230000, sum(y^2) = 80, sum(xy) = 500, sum(y) = 2.

Fit y = ax^2 + bx + c to the given data where a, b, c are constants. x (-3, -2, -1, 0, 1, 2, 3), y (12, 4, 1, 2, 7, 15, 30)

The line of regression of y on x is 8x – 10y + 66 = 0 and the line of regression of x on y is 40x – 18y = 214. Find i) the mean values of x and y ii) correlation coefficient between x & y

A throw is made with two dice. Find the probability that : i) the sum is 7 or less ii) the sum is a perfect square

The mean and variance of a binomial distribution are 4 and 2 respectively, find p(r < 2).

Assuming that the diameters of 1000 brass plugs taken consecutively from machine form a normal distribution with mean 0.7515 cm and standard deviation 0.0020 cm. How many of the plugs are likely to be approved if the acceptable diameter is 0.752 + 0.004 cm? [Given A(2.25) = 0.4878, A(1.75) = 0.4599]

An unbiased coin is thrown 10 times. Find the probability of getting i) exactly 8 Heads ii) at least 8 heads

The number of breakdowns of a computer in a week is a Poisson variable with z = np = 0.3. What is the probability that the computer will operate i) with no breakdown ii) at most one breakdown

Demand for a particular spare part in a factory was found to vary from day to day. In a sample study the following information was obtained. Days (Mon, Tue, Wed, Thurs, Fri, Sat), Parts demanded (1124, 1125, 1110, 1120, 1126, 1115). Test the hypothesis that the number of parts demanded does not depend on the day of the week. (Given : chi^2 5,0.05 = 11.07)

Using the bisection method, find an approximate root of the equation xsinx – 1, that lies between x = 1 and x = 1.5 (measured in radians). Perform six iterations.

Obtain the root of the equation x^3 – 4x – 9 = 0 correct to four decimal places by using Newton-Raphson method.

Solve by Gauss - Seidel method, the following system of equations. 20x1 + x2 – 2x3 = 17, 3x1 + 20x2 – x3 = –18, 2x1 – 3x2 + 20x3 = 25

Solve the following system by Gauss - elimination method : 4x1 + x2 + x3 = 4, x1 + 4x2 – 2x3 = 4, 3x1 + 2x2 – 4x3 = 6

Solve the following system of equations by Jacobi - iteration method : 6x1 + 2x2 – x3 = 4, x1 + 5x2 + x3 = 3, 2x1 + x2 + 4x3 = 27

Use method of false position to find the root of the equation x^4 – 32 = 0 correct to three decimal places.

Find the value of y at x = 1.5 for the following data using Newton's forward difference formula. x (0, 2, 4, 6, 8), y (5, 29, 125, 341, 725)

Find the value of integral(dx/(1+x)) from 0 to 3 by using Simpson's 3/8th rule. Take h = 0.5 and correct the solution upto four decimal places.

Use Euler's method to solve the equation dy/dx = xy + 1 with y(0) = 1 and tabulate the solution for x = 0 to x = 0.4. Take h = 0.1 and correct the solution upto fourth decimal place.

Use Runge - Kutta method of fourth order to solve dy/dx = y/(x+y), y(0) = 1 in the interval (0, 0.2) with h = 0.2 correct the solution upto fourth decimal place.

Given dy/dx = x^2 / (y + 1), y(0) = 1 determine using modified Euler's method the value of y when x = 0.05. Take h = 0.05 and correct the solution upto fourth decimal place. Use two iterations only.

Using Newton's backward difference formula find the value of y at x = 3.5 for the following data. x (0, 1, 2, 3, 4), y (3, 2, 3, 6, 11)

Attempt the following : i) Standard deviation of three numbers 9, 10 and 11 is a) 2/3 b) 1/3 c) 2/3 d) 2 ii) If a = 2i + j + k and b = 2i + j + k then angle between a & b is ________. a) cos^-1(2/3) b) cos^-1(1/2) c) cos^-1(1/2) d) cos^-1(1/6) iii) For F = x^2i + xy j the value of integral C F dr for curve y^2 = x joining points (0, 0) and (1, 1) is - a) 1/12 b) 7/12 c) 5/12 d) 2/3 iv) Two dimentional heat flow equation in steady state condition is a) du/dt = c^2(d^2u/dx^2) b) du/dt = c^2(d^2u/dx^2 + d^2u/dy^2) c) du/dt = c^2(d^2u/dx^2 + d^2u/dy^2) d) d^2u/dx^2 + d^2u/dy^2 = 0 v) The vector product of two vectors is a - a) Vector b) Scalar c) Neither vector non scalar d) none of these vi) A card is drawn from a well shuffled a pack of 52 cards, probability of getting a club card is - a) 1/4 b) 3/4 c) 1/3 d) 1/2

If Σf = 27, Σfx = 91, Σfx^2 = 359, Σfx^3 = 1567, Σfx^4 = 7343. Find first four moments about origin also find coefficient of skewness and kurtosis.

From a record of analysis of correlation data the following results are available variance of x is 9 and lines of regression are 8x – 10y + 66 = 0, 40x – 18y = 214, Find (i) mean values of x & y series (ii) coefficient of correlation between x & y series (iii) standard deviation of y series.

If ten percent of articles from a certain machine are defective. What is probability that then shall be 6 defective in a sample of 25?

Obtain the regression line y on x for following data.

Find probability that at most 5 defective fuses will be found in a box of 200 fuses if 2% of such fuses are defective.

In a sample of 1000 cases the mean of a certain test is 14 and standard deviation is 2.5. Assuming that the distribution is normal find (i) How many students score between 12 and 15? (ii) How many score above 18? [Given :- A(0.8) = 0.2881, A(0.4) = 0.1554, A(1.6) = 0.4452)

For the curve x = e^tcost, y = e^tsint, z = e^t. Find the velocity and acceleration of the particle moving on the curve at t = 0.

Find the directional derivative of Φ = xy^2 + yz^3 at (1, –1, 1) along the direction normal to the surface x^2 + y^2 + z^2 = 4 at (1, 2, 2)

Show that the vector field F = (3x^2y + yz)i + (x^3 + xz)j + (xy)k is irrotational. Find scalar potential Φ such that F = grad Φ.

If the vector field F = (x + 2y + az)i + (bx - 3y - z)j + (4x + cy + 2z)k is irrotational. Find a, b, c and determine Φ such that F = grad Φ

Attempt any one : i) div(r cross a) = 0 ii) div(r^n r) = (n+3)r^n

Find directional derivative of xy^2 + yz^2 at (2, –1, 1) along the line 2(x – 2) = (y + 1) = (z – 1)

Use Green’s lemma to evaluate the integral integral (x^2 - y^2)dx + (x^2 + y^2)dy where C is the curve bounding y > 0 and x^2 + y^2 < 1

Evaluate integral integral (curl A dS) where S is the surface of the cone z^2 = x^2 + y^2 above the xy plane and A = (x^2 + z^2)i + (x + yz)j + (xy)k.

Evaluate the surface integral integral curl F dS by transforming it into a line integral, S being that part of the paraboloid z = 1 – x^2 – y^2 for which z > 0 and F = yi + zj + xk.

Find the value of integral F dr where C is part of the spiral r = a*theta, theta = 0 to 2π and where F = r^2i

Obtain the equation of stream lines in case of steady motion of fluid defined by velocity q = (2x + 2y)i + (xy)j + (x + y + z)k

Using Gauss Divergence theorem show that integral integral div(r/r^6) dS = 0 where S is a smooth closed surface in the three dimensional space which contains a region of space whose numerical volume is V0.

A tightly stretched string of length l is initially in equilibrium position is set vibrating by giving to each of its points, the velocity dy/dt (at t=0) = V0 sin(πx/l) find y(x, t) if d^2y/dt^2 = c^2(d^2y/dx^2)

Solve du/dt = k(d^2u/dx^2) if, i) u(0, t) = 0 ii) u(l, t) = 0, iii) u(x, t) is bounded iv) u(x,0) = u0 for 0 < x < l

If d^2y/dt^2 = c^2(d^2y/dx^2) represents vibrations of string of length l, fixed at both ends, find the solution if, i) y(0, t) = 0 ii) y(l, t) = 0 iii) dy/dt (at t=0) = 0 iv) y(x, 0) = k(lx – x^2) 0 < x < l

Solve (d^2u/dx^2) + (d^2u/dy^2) = 0, subject to conditions i) u = 0 as y → ∞ ∀x ii) u = 0 if x = 0 ∀y iii) u = 0 if x = l ∀y iv) u = u0 sin(πx/l) if y = 0 for 0 < x < l

Write the correct option for the following multiple choice questions. a) The first three moments of a distribution about the value 5 are 2,20 and 40. Third moment about the mean is i) – 64 ii) 64 iii) 32 iv) – 32 b) If probability density function f (x) of a continuous random variable x is defined by f(x) = (x/4) for 0 <= x <= 2, else 0, then P(x <= 1) is i) 1/4 ii) 1/2 iii) 1/3 iv) 3/4 c) Using secant method, the first approximation to the root x2 of the equation x3 – 5x – 7 = 0, if the initial approximations are given as x0 = 2.5 and x1 = 3 is i) 2.7183 ii) 3 iii) 2 iv) 0 d) If Lagrange’s polynomial passes through x: 0, 1; y: –4, –4 then dy/dx at x = 1 is given by i) 0 ii) 2 iii) 1 iv) 1/2 e) To compare the variability of two or more than two series, coefficient of variation is obtained using ( x is arithmetic mean and  is standard deviation) i) (σ/x)*100 ii) (x/σ)*100 iii) (σ/x)*100*something iv) (σ^2/x)*100 f) If x0 is initial approximation to the root of the equation f(x) = 0 by Newton - Raphson method, first approximation x1 is given by i) x1 = (x0+x0)/2 ii) x1 = x0 - f(x0)/f'(x0) iii) x1 = x0 + f(x0)/f'(x0) iv) x1 = x0 + f'(x0)/f(x0)

Find arithmetic mean and coefficient of variation for x if the data is, x: 1 2 3 4; f: 9 6 5 3

Fit a straight line of the form y = ax + b for the data x: -2, -1, 0, 1, 2; y: 5, 3, 1, -1, -3

Given the information: x_bar=8.2, y_bar=12.4, σx=6.2, σy=6.2, r=0.9. Find line of regression of x on y. Estimate x for y = 10

The first four moments of a distribution about the value 2 are 2, 10 , 20 and 25. Find first four moments about mean, coefficient of skewness and kurtosis.

Fit a parabola of the type y = ax2 + bx + c for the data x: -1, 0, 1, 2; y: 3, 1, 3, 9

Find the coefficient of correlation for following distribution, x: 5, 7, 9, 11, 13; y: 9, 6, 12, 3, 15

A box contains 6 red balls, 4 white balls and 5 blue balls. Three balls are drawn successively from the box. Find the probability that they are drawn in the order red, white and blue if each ball is not replaced.

A coin is so biased that appearence of head is twice likely as that of tail. If a throw is made 6 times, using Binomial distribution, find the probalility that at least two heads will appear.

In a distribution, exactly normal, 7% of the items are under 35 and 89% are under 63. Find the mean and standard deviation of the distribution. [Given A(z = 1.48) = 0.43, A(z = 1.23) = 0.39]

The average number of misprints per page of a book is 1.5. Assuming the distribution of number of misprints to be poisson, find the number of pages containing more than one misprint if the book contains 900 pages.

A random sample of 200 screws is drawn from a population which represents the size of screws. If a sample is distributed normally with mean 3.15 cm and standard deviation 0.025cm, find expected number of screws whose size falls between 3.12 cm and 3.2 cm. [Given A(z = 1.2) = 0.3849, A(z = 2) = 0.4772]

A nationalised bank utilizes four teller windows to render fast service to the customers. On a particular day, 800 customers were observed. They were given service at the different windows as follows. Window number: 1, 2, 3, 4; Expected no.of customers: 150, 250, 170, 230. Test whether the customers are uniformly distributed over the windows at 5% level of significance. [Given χ2(3,0.05) = 7.815]

Using the Bisection method up to fifth iteration, find a real root of the equation x3 – 4x – 9 = 0.

Find the real root of the equation 2x3 – 2x – 5 = 0 by applying Newton - Raphson method at the end of fourth iteration.

Solve by Gauss - Seidel method, the system of equations: 45x1 + 2x2 + 3x3 = 58; –3x1 + 22x2 + 2x3 = 47; 5x1 + x2 + 20x3 = 67

Solve the following system by Cholesky’s method: 4x1 + 2x2 + 14x3 = 14; 2x1 + 17x2 – 5x3 = –101; 14x1 – 5x2 + 83x3 = 155

Solve the following system by Gauss elimination method: 2x1 – 2x2 + 3x3 = 2; x1 + 2x2 – x3 = 3; 3x1 – x2 + 2x3 = 1

Use method of false position to find the fourth root of 32 correct to three decimal places.

Using Newton’s forward interpolation formula, find the polynomial satisfying the data. x: 0, 1, 2, 3, 4; y: –4, –4, 0, 14, 44

Use simpson’s 1/3 rd rule to obtain integral from 1 to 2 of (1/x) dx dividing the interval into four parts.

Use Euler’s method to solve dy/dx = x+y, y(0)=1. Tabulate values of y for x = 0 to x = 2. Take h = 0.5.

Use Runge - Kutta method of fourth order to solve dy/dx = x^2 + y^2, y(0)=1, x=1.5 to find y at x = 1.1 taking h = 0.1

Using modified Euler’s method, find y(0.1) given that dy/dx = x+y, y(0) = 1 and h = 0.1. Consider accuracy to four decimal places.

Using Newton’s backward interpolation formula, find the polynomial satisfying the data. Also, find y when x = 4.5. x: 1, 2, 3, 4, 5; y: 14, 30, 62, 116, 198

Write the correct option for the following multiple choice questions : a) Given bxy = 0.85, byx = 0.89 and x = 6 then the values of r(x,y) and y are i) r = 0.87, y = 6.14 ii) r = –0.87, y = 0.61 iii) r = 0.75, y = 6.14 iv) r = 0.89, y = 4.64 b) The divergence of a vector field F = xzi + yzj + xyk at a point (1, 1, 1) is i) 4 ii) 6 iii) 5 iv) 3 c) If F = yi + zj + xk, then by using divergence theorem,  F.ds (where S is closed surface bounded by volume V) i) 3 ii) 0 iii) 2 iv) 5 d) The finite and bounded general solution of heat equation u/t = 5 ²u/x² is i) u(x,t) = (c1 cos mx + c2 sin mx)(c3 cos5mt + c4 sin5mt) ii) u(x,t) = (c1 cos mx + c2 sin mx)e^-25mt iii) u(x,t) = (c1 cos mx + c2 sin mx)e^-5mt iv) u(x,t) = (c1e^-mt + c2e^mt)(c3 cos mx + c4 sin mx) e) If x is a poisson random variable with mean value 3 then standard deviation of poisson distribution is i) 1 ii) 2 iii) 3 iv) √3 f) ×r =

First four moments of the distribution are 1, 4, 10 and 46. Compute the first four central moments. Also find 1 and 2.

Find the Co-efficient of correlation for the following data. x: 152, 158, 169, 182, 160, 166, 182; y: 198, 178, 167, 152, 180, 170, 162

With the usual notations, find the probability of the binomial distribution (p) if 9 P(X = 4) = P(X = 2) where X be the random variable.

If x=8.2, y=12.4, σx=6.2, σy=20, r(x,y)=0.9 then find the line of regression y on x. Also find the value of y for x = 10.

If the random variable X follows the poisson distribution such that P(X = 1) = 2P (X = 2), find the i) mean of distribution ii) P(X = 3)

From the past experience the labor contractor knows that per hour wages of skilled labor on an average is Rs. 100 with standard deviation Rs. 2. What percentage of labors will have wages between Rs. 98 and Rs. 102, assuming that the wages are normally distributed . (Given : (1) = (–1) = 0.3413)

Find the angle between velocity and acceleration vectors to the curve x=2sin3t, y=2cos3t, z=8t at t = 0.

Find the directional derivative of the function (x, y, z) = 4 e^2x–y–z at (1, 1, 1) in the direction tangent to the curve x = e^–t cost, y = 2sin t + 1, z = t – cos t at t = 0.

Show that vector field F = (x+2y+4z)i + (2x-3y-z)j + (4x-y+2z)k is irrotational. Find scalar potential  such that F =.

Find the angle between surfaces x² – y² + 2z² = 3 and x² + y² + z² = 16 at (1, 2, 2).

Determine f (r), such that F = f(r) r is solenoidal.

Attempt any one. i) Prove that : (r^n) = n r^(n-2) r ii) Prove that : ²() = ² + 2 + ²

Evaluate  C F·dr for F = (x²+y²)i + (xz-y)j + zk along the curve x=t, y=t², z=t³/4 from t = 0, t = 1.

Evaluate  S (×F)·ds where F = 3yi - xzj + yz²k and S is the surface of the paraboloid x²+y²=z bounded by the plans z = 2.

Evaluate  S F·ds by using Gauss-divergence theorem, where F = (x²+yz)i + (y²+xz)j + (z²+xy)k and S is the surface of the sphere x²+y²+z²=9.

Using Green’s theorem, evaluate  C F·dr, where F = (sin y + x)i + (x cos y + y)j and C is the boundary of an ellipse x²/4 + y²/9 = 1; z = 0.

Using Stoke’s theorem, evaluate  C F·dr, where F = yzi + zxj + xyk and S is the upper part of the sphere x² + y² + z² = 1 above XOY plane.

Evaluate  S (x³i + y³j + z³k)·ds where S is the surface of the sphere x² + y² + z² = a².

Solve one dimensional heat flow equation u/t = ²u/x² subject to the conditions. i) u(0,t) = 0 ii) u(π,t) = 0 iii) u(x,0) = x(π-x), 0 < x < π

Solve wave equation ²y/t² = c² ²y/x² subject to conditions i) y(0,t) = 0 ii) y(L,t) = 0 iii) y/t(x,0) = 0 iv) y(x,0) = y0, 0 < x < L where y0 is constant

Solve Laplace equation ²v/x² + ²v/y² = 0 subject to conditions : i) v(0,y) = 0 ii) v(1,y) = 0 iii) v(x,∞) = 0, 0 < x < 1 iv) v(x,0) = 10, 0 < x < 1

Solve wave equation ²y/t² = c² ²y/x² subject to conditions i) y(0,t) = 0 ii) y(π,t) = 0 iii) y/t(x,0) = 0 iv) y(x,0) = 2sin x, 0 < x < π

If the first four central moments of a distribution are 0, 2.5, 0.7 and 18.75 then the coefficient of Kurtosis 2 is _____. i) 0 ii) 1 iii) 2 iv) 3

The probability distribution of x is x 1 2 3 4 P(x) 1/2 1/4 1/8 1/8 The mathematical expectation E(x) is ____ i) 11/8 ii) 13/8 iii) 15/8 iv) 9/8

A root of the equation x3– 4x – 9 = 0 using bisection method lies between____ i) 0 and 1 ii) 1 and 2 iii) 2 and 3 iv) 3 and 4

If Lagrange’s polynomial passes through x 0 2 y –3 1 then 2 0 y dx is equal to______ i) –1 ii) –2 iii) 1 iv) 2

If x is arithmetic mean, N=f and the data is presented in the form of frequency distribution then the standard deviation  is given by ___ i) (1/N) * sum(f(x-x_bar)^2) ii) (1/N) * sum(f(x-x_bar)^2) iii) sum(fx)/N iv) (1/N) * sum|f(x-x_bar)|

Given equation is dy/dx = f(x,y) with initial condition x=xo, y=yo and h is step size. Euler’s formula to calculate y1 at x=xo+h is given by _____ i) y1 = yo + hf(xo,yo) ii) y1 = yo + hf(x1,y1) iii) y1 = yo + hf(xo,y1) iv) y1 = hf(xo,yo)

The first four moments of a distribution about the value 5 are 2,20,40 and 50. Find the first four central moments about the mean.

Obtain regression line of x on y for the following data. x: 6, 2, 10, 4, 8; y: 9, 11, 5, 8, 7

Fit a linear curve of the type y = ax + b to the data using method of least squares. x: 0, 1, 2, 3, 4, 5, 6, 7; y: –5, –3, –1, 1, 3, 5, 7, 9

Calculate the coefficient of correlation from the information n=10, x=40, x2=190, y2=200, xy=150, y=40

Fit a curve y=axb for the data x: 2000, 3000, 4000, 5000, 6000; y: 15, 15.5, 16, 17, 18

The two regression equations of the variables x and y are x = 19.13 – 0.87y and y = 11.64 – 0.50x. Find mean of x and mean of y and coefficient of correlation.

A mathematics problem is given to the three students A, B, C whose chances of solving it are 1/2, 1/3 and 1/4 respectively. What is the probability that the problem will be solved?

The mean and variance of a binomial distribution are 6 and 2 respectively. Find P (r  1)

A fair coin is tossed 64 times. Using normal distribution with mean 32 and standard deviation 4, find the probability of getting. i) Number of heads between 28 to 40 ii) Number of heads less than 28. [Given: A(1) = 0.3413, A(2) = 0.4772]

On an average a box containing 10 articles is likely to have 2 defectives. If we consider a consignment of 100 boxes, how many of them are expected to have three or less defectives?

Let 10% of the rivets produced by a machine are defective. Using Poisson distribution find the probability that out of 5 rivets chosen at random, at least two will be defective.

A nationalized bank utilizes four teller windows to render fast service to the customers. On a particular day, 800 customers were observed. They were given service at the different windows as follows: Window number: 1, 2, 3, 4; No. of customers observed: 150, 250, 170, 230. Test whether the customers are uniformly distributed over the windows. (Given : χ^2 (3, 0.05) = 7.815)

Use secant method to find a root of the equation f(x) = x^3 – 5x – 7 = 0 correct to three decimal places.

Obtain a root of the equation 3x – cosx –1= 0 (measured in radians), correct to four decimal places, using Newton-Raphson method.

Solve by Gauss-Seidel method, the following system of equations: 10x1+x2+x3=12, x1+10x2+x3=13, x1+x2+10x3=14

Solve the following system by Gauss elimination method: x1+x2+x3=4, 2x1+5x2-6x3=-12, x1+2x2-3x4=4

Solve the following system of equations by Jacobi-iteration method: 20x1+x2-2x3=17, 3x1+20x2-x3=-18, 2x1-3x2+20x3=25

Find a real root of the equation x3–2x–5=0 by the method of false position at the end of fifth iteration.

Find value of y for x=0.5 using Newton’s forward difference formula for following data: x: 0, 1, 2, 3, 4; y: 1, 5, 25, 100, 250

Use Simpson’s 1/3rd rule to find the value of integral (1 to 2) (1/x) dx. Take h = 0.25 correct solution upto fourth decimal place.

Use Euler’s method to solve the equation dy/dx = x^2+y with y(0)=1 and tabulate the solution for x = 0 to x = 0.3 take h = 0.1.

Use Runge-Kutta method of fourth order to solve dy/dx = x^2+y^2, y(1)=1.5 in the interval (1, 1.1) with h=0.1 and correct upto four decimal places.

Given dy/dx = x^2+y, y(0)=1, determine y(0.02) by using modified Euler’s method correct upto forth decimal places. Take h=0.02 (Two iterations only)

Find the value of f (4.5) using Newton’s backward difference formula correct upto 4 decimal places for following data: x: 1, 2, 3, 4, 5; y=f(x): 14, 30, 62, 116, 198

Attempt the following. a) The first four moments of distribution about mean one 0, 16, –64 and 162, then standard deviation of a distribution is____. [2] i) 21 ii) 12 iii) 16 iv) 4 b) The value of ∇^2r is_____ [2] i) 2/r ii) 2r iii) 1/r iv) 0 c) For F= (3xz^2)i + (2y)j - (zk), the value of ∫F.dr along straight line joining points (0,0,0) and (2,1,3) is _____. [2] i) 15 ii) 14 iii) 16 iv) 8 d) The most general solution of PDE ∂^2u/∂t^2 = ∂^2u/∂x^2 is ____. [2] i) u(x,t) = (c1 cos mx+c2 sin mx) e–m2t ii) u(x,t) = (c1 cos mx+c2 sin mx) (c3 cos mt +c4 sin mt) iii) u(x,y) = (c1 cos mx+c2 sin mx) (c3 emy+c4 e–my) iv) u(x,y) = (c1emx+c2 e–mx) (c3 cos my + c4 sin my) e) A throw is made with two dice. The probability getting a score of 10 is ____. [1] i) 1/12 ii) 1/6 iii) 1/5 iv) 2/3 f) The cross product of a & b is defined as a x b = [1] i) ab cos θ ii) n^ sin ab θ iii) ab sin θ iv) b^ cos ab θ

a) Calculate the first four moments about mean of the given distribution also find β1 & β2. [5] b) Find coefficient of correlation from given data. n = 25, Σx = 75, Σy = 100, Σx2 = 250, Σy2=500, Σxy = 325. [5] c) An unbiased coin is thrown 10 times. Find probability of getting. i) Exactly 6 heads ii) At least 6 heads [5]

a) Find lines of regression for the following data. [5] b) One percent of articles from a certain machine are defective. What is the probability of i) No defective ii) One defective [5] c) Assuming that the diagram of 1000 brass plugs taken consecutively from machine form a normal distribution with mean 0.7515 cm and standard deviation 0.0020 cm. How many of the plugs are likely to be approved if the acceptable diagram is 0.752 ± 0.004 cm. [Given A (2.25) = 0.4878, A (+1.75) = 0.4599] [5]

a) Find the angle between velocity and acceleration vectors at t=0 for r = e^t i + (log(t+1))j + (tan t)k. [5] b) In what direction from the point (1, 0, 1) is the directional derivative of φ = x^2 y z^3 a maximum? What is the magnitude of this maximum? [5] c) Show that F=(2xz^2)i + (6yz - 4x)j + (3xz^2)k is irrotational. Find scalar φ such that F= ∇φ. [5]

a) If directional derivative of φ = ax^2 y + by^2 z + cz^2x at (1,1,1) has maximum magnitude 15 in the direction parallel to (x-1)/1 = (y-2)/(-1) = (z-1)/(-2). Hence find the values of a, b, c. [5] b) Attempt any one: i) ∇.(r^2 ∇(1/r^2)) = 0 ii) ∇.(a x r) = 0 [5] c) Show that F = [1/r^2 (a x r) + r(a.r)] is irrotational. [5]

a) Evaluate ∫∇F.dr where F=(x+y)i + (x+y^2)j where C is the square formed by y = ± 1 and x = ±1. r = xi + yj + zk. [5] b) Evaluate ∬f.n ds where f = yzi + zxj + xyk and S is the sphere x^2 + y^2 + z^2 = 1 in the first octant. [5] c) Apply Gauss divergence theorem to evaluate ∬f.n ds where f = 4xi - 2y^2j + z^2k, S being the closed cylinder x^2 + y^2 = 4 bounded by z = 0 and z = 3. [5]

a) Using Green’s lemma for F = (3x^2 - 8y^2)i + (4y - 6xy)j and the curve C bounding the region R formed by x = 0, y = 0 and x + y = 1, evaluate ∬(∇x F).k dxdy. [5] b) Using Gauss divergence theorem evaluate ∬(F.n) ds where F = x^2z i - yj + xz^2k where S is the boundary of the region bounded by the surfaces z = x^2 + y^2 and z = 4y. [5] c) A liquid mass is rotating with a constant angular velocity ω about a vertical axis (positive z-axis) under the action of gravity. Find the pressure at any point of the liquid, if the motion is steady. Use the equation ∇p = ρ(F - dq/dt). [5]

a) If ∂^2y/∂t^2 = c^2 ∂^2y/∂x^2 represents the vibrations of the string of length l fixed at both ends. Find the solution it. i) y(0,t) = 0 ii) y(l, t) = 0 iii) ∂y/∂t = 0 at t=0 iv) y(x,0) = lx – x^2 0 < x < l [8] b) Solve the following one-dimensional heat flow equation, ∂u/∂t = c^2 ∂^2u/∂x^2 subject to conditions. i) u(0,t) = 0, ∀t ii) u(l, t) = 0, ∀t iii) u(x,0) = x 0 < x < l iv) u(x, t) is bounded. [7]

a) If the wave equation of vibration of string is given by, ∂^2y/∂t^2 = c^2 ∂^2y/∂x^2. Find the solution y(x, t), if, i) y(0,t) = 0 ∀t ii) y(l, t) = 0 ∀t iii) y(x,0) = 0 ∀x iv) ∂y/∂t = ax(l-x) at t=0 [8] b) Solve, ∂u/∂t = ∂^2u/∂x^2 if, i) u is finite for all t ii) u(0, t) = 0 iii) u(π,t) = 0 iv) u(x, 0) = πx – x^2 0 ≤ x ≤ π [7]

Write the correct option for the following multiple choice questions. a) If the two regression coefficiens are -8/15 and -5/6 then the correlation coefficient is i) -2/3 ii) 2/3 iii) -1/2 iv) 1/2 b) A and B are independent events such that P(A) = 1/2, P(B) = 1/3 then P(A ∪ B). i) 3/5 ii) 2/3 iii) 1/6 iv) 1/3 c) Using Gauss elimination method the solution of system of equations x + 2y + z = 4, –3y + 2z = –3, –7y – 2z = –6 is i) x=43/16, y=15/8, z=-9/16 ii) x=47/20, y=9/10, z=3/20 iii) x=4/3, y=3/8, z=5/6 iv) x=16/5, y=8/43, z=-9/x d) If a curve passing through (0,0), (2,4), (4,8) is given by y = y0 + u Δ y0 then y at x = 1 is given by (Note : x = x0 + uh) i) 1 ii) 0 iii) –1 iv) 2 e) The range of correlation coefficient ‘r’ for a bivariate data is i) 0 < r < ∞ ii) -∞ < r < ∞ iii) -1 ≤ r ≤ 1 iv) 0 ≤ r ≤ 1 f) If x0, x1 are two initial approximations to the root of f(x) = 0, by secant method next approximation x2 is given by i) x2 = x1 - (x1-x0)/(f(x1)-f(x0)) * f(x1) ii) x2 = (x0+x1)/2 iii) x2 = x1 - f(x1)/f'(x1) iv) x2 = x1 + (x1-x0)/(f(x1)+f(x0)) * f(x1)

The first four moments of distribution abut the value 4 are –1.5, 17, –30 and 108 respectively. Obtain the first four central moments about mean, β1 and β2.

Firt a straight line of the form y = a + bx using least squares method to the following data. x: 0, 1, 2, 3, 4; y: -2, 1, 4, 7, 10

The two regression lines of a bivariate data are 3x + 2y = 26 and 6x + y = 31. Find the mean values of x and y. Also, determine the correlation coefficient between x and y.

Calculate the coefficient of variation for the data given as follows. 36, 15, 25, 10 and 14.

Fit a second degree parabola of the form y = a + bx + cx2 using least squares method to the folowing data x: 0, 1, 2, 3; y: 2, 1, 6, 17

Find the correlation coefficient between the variables population density (x) and death rates (y) as given in the following data. x: 200, 400, 500, 700, 300; y: 12, 18, 16, 21, 10

Find the expected value of the sum of the faces obtained when two fair dice are tossed simultaneously.

An unbiased coin is tossed five times. Find the probability of observing at least four heads.

In a sample of 1,000 cases, the mean score in a certain examination is 14 and standard deviation is 2.5. Assuming the distribution to be normal, find the expected number of students scoring between 12 and 15 (both inclusive). [Given : Z1 = 0.4, A1 = 0.1554; Z2 = 0.8, A2 = 0.2881]

A riddle is given to three students to solve independently. The individual probabilities of the riddle being solved by the three students are 0.3, 0.4 and 0.5 respectively. Find the probability that the riddle gets solved.

On an average, there are two printing mistakes on a page of a book. Using Poision distribution, find the probability that a randomly selected page from the book has at least one printing mistake.

In a mouse breeding experiment, a geneticist has obtained 172 brown mice with pink eyes, 60 brown mice with brown eyes, 62 white mice with pink eyes and 26 white mice with brown eyes. Theory predicts that these types of mice should be obtained in the ratios 9 : 3 : 3 : 1. Test the compatibility of the data with theory, using 5% level of significance. [Given χ2 tab = 7.815]

Find a root of the equation x4 + 2x3 – x – 1 = 0, lying in the interval [0, 1] using the bisection method at the end of fifth iteration.

Obtain the real root of the quation x3 – 4x – 9 = 0 by applying Newton Raphson method at the end of third iteration.

Solve by Gauss - Seidel method, the system of equations : 10x1 + x2 + x3 = 12; 2x1 + 10x2 + x3 = 13; 2x1 + 2x2 + 10x3 = 14

Solve by Gauss elimination method, the system of equations : 2x1 + x2 + x3 = 10; 3x1 + 2x2 + 3x3 = 18; x1 + 4x2 + 9x3 = 16

Solve by Jacobi's iteration method, the system of equations : 20x1 + x2 – 2x3 = 17; 3x1 + 20x2 – x3 = –18; 2x1 – 3x2 + 20x3 = 25

Find a real root of the equation x3 – 2x –5 = 0 by the method of false position at the end of fourth iteration.

Using Newton's forward interpolation formula, find y at x = 8 from the data : x: 0, 5, 10, 15, 20, 25; y: 7, 11, 14, 18, 24, 32

Evaluate ∫ 1/(1+x^2) dx from 0 to 1 using Simpson's 1/3 rd rule, (Take h = 0.25)

Use Euler's method to solve dy/dx = x+y, y(0) = 1. Tabulate values of y for x = 0 to x = 0.3 (Take h = 0.1)

Use Runge-Kutta method of fourth order to solve dy/dx = x^2+y, y(0) = 1 at x = 0.1 with h = 0.1.

Use modified Euler's method to find y(0.1), given dy/dx = xy+y, y(0) = 1 and h = 0.1. (up to two iterations)

Using Newton's backward difference formula, find the value of 155 from the data : x: 150, 152, 154, 156; y: 12.247, 12.329, 12.410, 12.490

Attempt the following. a) If xy = 2638, x = 14, y = 17, n = 10, then cov (x, y) is i) 24.2 ii) 25.8 iii) 23.9 iv) 20.5 b) If F = r2r then F is i) Constant ii) Conservative iii) Solenoidal iv) None of these c) For F= x2 ˆi + xy ˆj , the value of F c dr   for curve y2 = x joining points (0, 0) and (1, 1) is i) 1 ii) 1 3 iii) 3 2 iv) 2 3 d) General solution of PDE 2 2 2 2 4 u u t x      is i) u(x, t) = (C4 cos mx + C5 sin mx) 2 4m t e ii) u(x, t) = (C1 cos mx + C2 sin mx) (C3 cos2mt + C4sin2mt) iii) u(x, y) = (C1emx + C2 mx e ) (C3 cos my + C4 sin my) iv) u(x, y) = (C1 cos mx + C2sin mx) (C3emy + C4 my e ) e) Coefficient of kurtosis 2 is given by i) 4 3   ii) 4 2 2   iii) 3 2 2   iv) 4 3 2   f) The dot product of two vectors a & b is defined as . a b =_______. i) ab cos ii) ab sin iii) ab sinˆn iv) ab

The first four moments of a distribution about value 5 are 2, 20, 40 and 50. From given information find first four central moments. Also find coefficient of skewness and kurtosis.

Find the coefficient of correlation for the following data. x: 10, 14, 18, 22, 26, 30; y: 18, 12, 24, 6, 30, 36

Between 2. p.m and 3.p.m the average number of phone calls per minute coming into company are 2. Find probability that during one particular minute there will be 2 or less calls.

Given the following information. Variable x: Arithmetic mean 8.2, Standard deviation 6.2; Variable y: Arithmetic mean 12.4, Standard deviation 20. Coefficient of correlation between x & y is 0.9. Find the libean regression estimate of x, given y = 10.

On an average a box containing 10 articles is likely to have 2 detectives. If we consider a consignment of 100 boxes, how many of them are expected to have three or less detectives?

In a normal distribution 10% of items are under 40 and 5% are over 80. Find mean and standard deviation of distribution. [Given : A(1.29) = 0.4, A(1.65) = 0.45]

Find the angle between tangents to the curve r = (t2) i + (4t-2) j + (2t2-6) k at t = 0 and t = 2.

Find the directional derivative of  = x2y + xyz + z3 at (1, 2, –1) along normal to the surface x2 + y2 + z2 = 9 at the point (1, 2, 0).

Show that F = (e^xy cos z) i + (e^xy cos z) j - (e^xy sin z) k is irrotational. Find corresponding scalar  such that F = 

If the directional derivative of  = a(x + y) + b(y + z)+ c(x + z) has maximum value 12 in the direction parallel to y axis. Find a, b and c.

Attempt any one. i) (a x r) = - (a + r) ... ii) (e^r/r^4) = ...

Show that the vector field f(r)r is always irrotational and determine f(r) such that the field is solenoidal.

Let F = (xy + y2) i + x2 j. Is the work done along y = x and y = x2 from the common starting point (0, 0) to the common and point (1, 1), the same or different?

Evaluate  F . n dS where F = axi + byj + czk and S in the surface of the sphere x2 + y2 + z2 = r2.

Apply stokes theorem to evaluate  [(x+y) dx + (x+z) dy + (y+z) dz] where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6).

Evaluate  [(x-y) dx + (x+y) dy] applying Green’s lemma where C is the curve x2 + y2 = a2. Is the work done the same along the curves C1 and C2 where C1 is the arc of C from (0, –1) to (0, 1) clockwise and C2 is the arc of C from (0, –1) to (0, 1) anti clockwise.

Let S be the surface of the sphere (z + 3)2 + x2 + y2 = 42 cut off by the plane z = –2. Evaluate  ( x F) . dS where F = (x+y) i + (y+z) j + (z+x) k

Find the surface of equi pressure in case of steady motion of a liquid which has velocity potential  = log (xyz) and is under the action of force F = yzi + zxj + xyk. Use the equation ...

Solve the equation, ^2y/t^2 = c^2 (^2y/x^2), where y(x, t) satisfies the following conditions: i) y(0, t) = 0, ii) y(L, t) = 0, iii) (y/t) = 0 at t=0, iv) y(x,0) = sin(x/L)

Solve the Laplace equation ^2u/x^2 + ^2u/y^2 = 0, with conditions: i) u=0 as y->inf, ii) u=0 at x=0, iii) u=0 at x=, iv) u=u0 at y=0, 0<x<

A tightly stretched string with fixed ends x = 0 and x = l is initially at rest in its equilibrium position. If it is set vibrating giving each point a velocity 3x(l – x) for each 0 < x < l. Find the displacement y (x, t).

Solve, u/t = ^2u/x^2 if, i) u is finite for all l, ii) u (0, t) = 0, iii) u (l, t) = 0, iv) u (x, 0) = 3x

Write the correct option for the following multiple choice questions. a) For a given set of bivariate data, x = 2, y = 3. The regression coefficient of y on x is – 4. Using the regression equation of y on x, the most probable value of y when x =1 is ______. i) –1 ii) 1 iii) –2 iv) 2

If probability density function f (x) of a continuous random variable x is f(x) = x/8 for 0 ≤ x ≤ 4, then p (x ≤ 3) = _____. i) 0 ii) 3/4 iii) 9/16 iv) 1

Lagrange’s polynomial through the points x: 0, 1, 2; y: 4, 5, 12 is given by ______. i) y = 4x2–3x + 4 ii) y = x2+ 4 iii) y = 2x2–x + 4 iv) y = 3x2–2x + 4

Using Gauss elimination method, the solution of system of equations x+y+z=1, x+2y+3z=4, x+3y+4z=5 is ______. i) x = -1, y = 1/2, z = 1/2 ii) x = 1/2, y = 1, z = 1/2 iii) x = 2, y = 1/2, z = 1/2 iv) x = 1, y = 2, z = 3

The first four central moments of a distribution are 0, 0.453, 0.06 and 0.502. The coefficient of Kurtosis β2 is _____. i) 0.0387 ii) 2.4463 iii) 25.8221 iv) 0.4088

If f (x) is a continuous function on [a,b]and f (a) f (b) < 0, then to find a root of f (x) = 0, initial approximation x0 by bisection method is _____.

The first four moments of a distribution about the value 5 are 2, 20, 40 and 50. Obtain the first four central moments, β1 and β2.

Fit a straight line of the form y = a + bx to the following data by the least square method. x: -2, 1, 3, 6, 8, 9; y: 17, 14, 12, 9, 7, 6

For a bivariate data, the regression equation of y on x is 8x–10y = –66 and the regression equation of x on y is 40x – 18y = 214. Find the mean values of x and y. Also, find the correlation coefficient between x and y.

Following are the runs scored by two batsmen in 5 cricket matches. Which batsman is more consistent in scoring runs? Batsman A (x): 38, 47, 34, 18, 33. Batsman B (y): 37, 35, 41, 27, 35

Fit a parabola of the form y = a + bx + cx2. Using the least square method to the following data. x: -2, -1, 0, 1, 2; y: -2, 5, 8, 7, 2

Find the correlation coefficient between age in years (x) and glucose level (y) from the data of 5 people as follows. x: 43, 22, 25, 42, 58; y: 99, 65, 79, 75, 87

A fair die is tossed once. Random variable x denote the digit that appears as top face. Find the expectation E(x).

The number of breakdowns of a computer in a week is a poisson variable with λ = np = 0.3. What is the probability that the computer will operate. i) With no breakdown ii) At most one breakdown in a week.

In a certain city 4000 lamps are installed. If the lamps have average life of 1500 burning hours. Assuming normal distribution. i) How many lamps will fail in first 1400 hours? ii) How many lamps will last beyond 1600 hours? [Given : z = 1, A = 0.3413]

Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that they are both kings if i) The first card drawn is replaced ii) The first card drawn is not replaced

A certain factory turning cotter pins knows that 2% of its product is defective. If it sells cotter pins and gurantees that not more than 5 pins will be defective in a box of 100 pins. Find the approximate probability that a box will fail to meet the guranteed quality.

A bank utilizes four windows to render fast service to the customers on a particular day 800 customers were observed. They were given service at the different windows as follows: Window 1: 150, Window 2: 250, Window 3: 170, Window 4: 230. Test whether the customers are uniformly distributed over the windows. [Given : χ2(3,0.05) = 7.815]

Find the root of the equation x3 –4x + 1 = 0 lying in the interval [0, 0.5] by Bisection method correct upto 3 decimal places (Five iterations only)

Find the root of the equation x2 – 12 = 0 lying between (3, 4) by Newton-Raphson method correct upto 3 decimal places.

Solve by Gauss-Seidel method the system of equations. 5x – y = 9, –x + 5y –z = 4, –y + 5z = –6. Take initial solution as (9/5, 4/5, 6/5) write numerical calculations correct upto three decimal places.

Solve by Gauss elimination method, 2x + y + z = 10, 3x + 2y + 3z = 18, x + 4y + 9z = 16

Solve by Jacobi’s iteration method, 20x1 + x2 –2x3 = 17, 3x1 + 20x2 – x3 = –18, 2x1 – 3x2 + 20x3 = 25. Write numerical calculations correct upto 3 decimal places.

Use Regula-Falsi method to find real root of the equation ex – 4x = 0 lying between [0, 0.5], correct to three decimal places.

Using Newton’s forward interpolation formula, find the population in the year 1895 from given data, Year: 1891, 1901, 1911, 1921, 1931; Pop: 46, 66, 81, 93, 101

Evaluate, integral from 0 to 1 of x*e^x dx, using Simpson’s 1/3rd rule (h = 0.2).

Use Euler’s method to solve dy/dx = x+y^2, y(0) =1, h = 0.1 Tabulate values of y for x = 0.1 to x = 0.4.

Use Runge-Kutta method of 4th order to solve dy/dx = y-x, y(0) = 1 at x = 0.2 with h = 0.2.

Using modified Euler’s method find y (0.1), given dy/dx = 1+xy, y(0) = 1, h = 0.1 (Two iterations only).

Using Newton’s backward difference formula, find y at x = 3.5 from following data, x: 0, 1, 2, 3, 4, 5; y: 5.2, 8, 10.4, 12.4, 14, 15.2

Write the correct option for the following multiple choice questions. a) For a given set of bivariate data, x=2, y=3. The regression coefficient of x on y is –0.11. By using the regression equation of x on y , the most probable value of x when y=0 is ______. i) 0.57 ii) 0.87 iii) 0.77 iv) 1.77 b) If Probability density function f(x) of a continuous random variable x is defined by f(x) = x/2 for -2 <= x <= 4, 0 otherwise, then P(x <= 1) is ______. i) 1/4 ii) 1/2 iii) 1/3 iv) 3/4 c) Lagrange’s polynomial through the points x: 0, 1, 2; y: 4, 0, 6 is given by ________. i) y = 5x2–3x+4 ii) y = 5x3+3x+4 iii) y = 5x2–9x+4 iv) y = x2–9x+4 d) Using Gauss elimination method, the solution of system of equations x+y+z=15, y+z=9, z=5 is ______ i) x=1, y=2, z=3 ii) x=1/2, y=1, z=2 iii) x=1, y=2, z=1/2 iv) x=1, y=1/2, z=-1/2 e) The first four central moments of a distribution are 0,16,–64 and 162. The coefficient of Kurtosis  is ______. i) 1.20 ii) 0.6328 iii) 1 iv) 0.3286 f) If f(x) is continuous on [a,b] and f(a)f(b)<0. then to find a root of f(x)=0, initial approximation x0 by bisection method is ______ i) x0 = (a-b)/2 ii) x0 = f(a)+f(b)/2 iii) x0 = (a+b)/2 iv) x0 = (a-b)/(a+b)

a) If marks scored by five students in statistics test of 100 marks, are given in following table: Student 1 2 3 4 5; Marks(/100)x 46 34 52 78 65. Find standard deviation and arithmetic mean x. b) Fit a law of the form y=ap+b by least square method for the data, p: 100, 120, 140, 160, 180, 200; y: 0.9, 1.1, 1.2, 1.4, 1.6, 1.7. c) If the two lines of regression are 9x+y–=0 and 4x+y= and the means of x & y are 2 & –3 respectively. Find values of , and correlation coefficient between x & y.

a) The first four moments of a distribution about 5 are 2,20,40 and 50. Find first four moments about mean, and 1,2. b) Fit a parabola y=ax2 + bx + c, by using least square method to the following data, x: 0, 1, 2, 3; y: 2, 2, 4, 8. c) Calculate the coefficient of correlation from the following information, n=10, x=40, x2=190, y2=200, xy=150, y=40.

a) Bag 1 contains 2 white and 3 red balls. Bag 2 contains 4 white and 5 red balls. One ball is drawn randomly from bag 1 and is placed in bag2. Later, one ball is drawn randomly from bag2. Find the probability that it is red. b) The expected number of matches those will be won by India in a series of five one day matches between India and England is three. If the probability of India’s win in each match remains the same and the results of all the five matches are independent of each other, find the probability that India wins the series, using Binomial distribution. Assume that each match ends with a result. c) The lifetime of an article has a normal distribution with mean 400 hours and standard deviation 50 hours. Find the expected number of articles out of 2,000 whose lifetime lies between 335 hours to 465 hours. (Given : Z=1.3,A=0.4032)

a) Find the expected value of the number of heads obtained when three fair coins are tossed simultaneously. b) On an average, 180 cars per hour pass a specified point on a particular road. Using Poisson distribution, find the probability that at least two cars pass the point in any one minute. c) The proportions of blood types O,A,B and AB in the general population of a country are known to be in the ratio 49:38:9:4 respectively. A research team observed the frequencies of the blood types as 88,80,22 and 10 respectively in a community of that country. Test the hypothesis at 5% level of significance that the proportions for this community are in accordance with the general population of that country. (Given :  tab=7.815)

a) Find the root of the equation x4+2x3–x–1=0, lying in the interval [0,1] using the bisection method at the end of fifth iteration. b) Find a real root of the equation x3+2x–5=0 by applying Newton-Raphson method at the end of fifth iteration. c) Solve by Gauss-Seidel method, the system of equations: 20x1 + x2 –2x3 = 17; 3x1 + 20x2 –x3 = –18; 2x1 – 3x2 + 20x3 =25

a) Solve by Gauss elimination method, the system of equations: 2x1 + x2 +x3 = 10; 3x1 + 2x2 +3x3 = 18; x1 +4x2 + 9x3 =16. b) Solve by Jacobi’s iteration method, the system of equations: 4x1 + 2x2 +x3 = 14; x1 + 5x2 –x3 = 10; x1 + x2 + 8x3 =20. c) Use Regula-Falsi method to find a real root of the equation ex–4x=0 correct to three decimal places.

a) Using Newton’s forward interpolation formula, find y at x=8 from the following data. x: 0, 5, 10, 15, 20, 25; y: 7, 11, 14, 18, 24, 32. b) Evaluate integral from 0 to 1 of 1/(1+x) dx using Simpson’s 1/3 rule. (Take h=0.2). c) Use Euler’s method, to solve dy/dx = x+y, y(0)=1. Tabulate values of y for x=0 to x=0.3 (Take h=0.1)

a) Use Runge-Kutta method of 4th order, to solve dy/dx = xy+y, y(1)=2 at x=1.2 with h=0.2. b) Using Modified Euler’s method, find y(0.2), given dy/dx = x+y^2, y(0)=0. Take h=0.2 (Two iterations only). c) Using Newton’s backward difference formula, find the value of sqrt(155) from the following data: x: 150, 152, 154, 156; y=sqrt(x): 12.247, 12.329, 12.410, 12.490

The pair of regression Linens are L1 : 8x –10y + 66 = 0 and L2 : 40x –18y = 214 i) L1 is the regression Line y on x. ii) L1 is the regression line x on y. iii) L2 is regression line y or x. iv) L1 and L2 is regression line x on y.

Vector along the direction of the line 1 2 3 2 1 5 x y z      is i) 2 3 14 i j k   ii) 2 5 30 i j k   iii) 2 5 30 i j k   iv) 2 5 30 i j k  

Let (7,1/3) X B = be the Binomial distribution with parameters n = 7 and p = 1/3. Then ( ) ( ) 2 5 p x p x = + = is i) 81/28 ii) 28/81 iii) 7/81 iv) 10/81

If vector field ( ) ( ) ( ) F 3 2 x y i y z j x mz k = + + - + + is solenoidal the value of m is i) –2 ii) 3 iii) 2 iv) 0

Using Stoke’s theorem F c dr    where F xyi yzj zk = + + over the cube whose side is a and it’s face in XOY - plane is missing is equal to i) 0 ii) R y dxdy  iii) R 2 x dxdy  iv) R x dxdy  

Most general solution of 2 2 u u t x   =   is i) ( ) ( ) ( ) 1 2 3 4 , cos sin cos sin u x t c mx c mx c cmt c cm = + + ii) ( ) ( ) 2 4 5 , cos sin m t u x t c mx c mx e = + iii) ( ) ( ) ( ) 1 2 1 2 , cos sin mx mx u x t c e c e c my c my  = + + iv) ( ) ( ) ( ) 1 2 3 4 , cos sin my my u x t c mx c mx c e c e  = + +

A computer while calculating carrelation coefficient between two variables X and Y from 25 pairs of observations obtained the following results : n = 25, X 125,  = 2 X 650,  = Y 100,  = 2 Y 460,  = XY 508.  = Later it was discovered that the values (X, Y) = (8, 12) was copied as (6, 14) and the value (8, 6) was copied as (6, 8). Obtain the correct value of the correlation coefficient.

In a normal distribution 31% of the items are under 45 and 8% are above 64. Find the mean and standard deviation of the distribution. Take Area (0 < z <1.4) = 0.42 and Area (0 < z < 0.5) = 0.19 where z is the standard normal variate.

Verify at 5% level of significance and 4 degrees of freedom if the distribution can be assumed to be poisson given: # defects : 0 1 2 3 4 5 Frequency : 6 13 13 8 4 3 Take 2 0.135. e = in the calculations round off the frequencies to the immediate higher integral value. Take 2 5,0.05 11.07  =

Two examiners A and B award marks to seven students as follows: Roll No. : R1 R2 R3 R4 R5 R6 R7 Marks (A) : 40 44 28 30 44 36 30 Marks (B) : 32 39 26 30 28 34 28 Find the coefficient of correlation.

Assume the mean height of soldiers to be 68.22 inches with a variance of 10.8 inches square. How many soldiers in a regiment of 10,000 would you expect to be over 6 feet? Assume area (0 < z < 1.15) = 0.3749 where z is the standard normal variate.

Among 64 off springs of a certain cross between European horses 34 were red, 10 were black and 20 were white. According to a genetic model these numbers should be in the ratio 9:3:4. Is the data consistent with the model at 5% level of significance? Take 2 2;0.05 5.991  =

Find the angle between the tangents to the curve 2 3 , , x t y t z t = = = at t =1 and t = –1

If ( ) ( ) ( ) 1 F y z i z x j x y k = + + + + + and ( ) ( ) ( ) 2 2 2 2 2 F x yz i y zx j z xy k = - + - + - then show that 1 2 F F  is solenoidal.

If the directional derivative of axy byz czx  = + + at (1, 1, 1) has maximum magnitude 4 in a direction of x-axis. Find a, b and c.

Find the directional derivative of 2 xy y z  = + at the point (1, –1, 1) to wards point (2, 1, 2).

Prove the following identities (any one) i) ( ) 2 a r a   = ii) . a r a  =

Show that ( ) ( ) ( ) 2 2 2 2 2 2 F i xy xz yx yz j zx zy k = + + + + + is irrotational. Find scalar  such that F .  = 

Evaluate F c dr   along the straight line joining points (0, 0, 0) and (2, 1, 3) where ( ) 2 = F = 3 2 x i xz y j zk + - +

Evaluate ( ) S xi yj zk ds + +   over the surface of sphere 2 2 2 1 x y z + + =

Evaluate using Stoke’s theorem ( ) F S ds    where 2 F yi zj xyk = + + and S is surface of paraboloid ( ) 2 2 4 0 z x y z = - -  .

Use Green’s theorem to evaluate ( ) ( ) 2 2 2 2 2 c x y dx x y dy - + +  where ‘C’ is boundary of area enclosed by the axis and circle 2 2 16, 0. x y z + = =

Apply Stoke’s theorem to evaluate F c dr   where F yzi zxj xyk = + + and S is upper part of sphere 2 2 2 1 x y z + + = above XOY plane.

Evaluate ( ) 2 . s xi yj z k ds + +   Where S is the surface of cylinder 2 2 4 x y + = bounded by planes z = 0 and z = 2.

A string stretched and fastened between two points L a part. Motion is started by displacing the string in the form y = a sin L x  from which it is released at time t = 0. Find the displacement ( ) , y x t .

Solve the one dimensional heat equation 2 2 y u k t x   =   subject to conditions. i) u is finite .t  ii) u (0, t) = 0, iii) u(, t) = 0, iv) u(x, 0) = x – x2 0 . x   

A tightly stretched string with fixed ends x = 0 and x = l is initially at rest in its equilibrium position is set to vibration by giving each point a velocity 3x(l – x) for 0 < x < l. Find the displacement y(x, t) at any time t.

An infinitely long uniform metal plate is enclosed between lines y = 0, and y = l for x > 0. The temperature is zero along the edges y = 0, y = l, and at infinity. If edge x = 0 is kept at a constant temperature 0 , v Find the temperature distribution ( ) , v x y .

Write the correct option for the following multiple choice questions : i) y : 1 2 3 x : 1 5 9 The least square fit of the form x = ay + b to the above data is ________. ii) For two events A and B, P(A)=2/3, P(B)=3/8 and P(A∩B)=1/4 then the events A and B are __________. iii) Using Gauss elimination method, the solution of system of equations 5x+y+z=5, x+3y+2z=13 and x+2y+3z=19 is _________. iv) If Lagrange's polynomial passes through (0,1), (1,2) x y then ∫01 ydx = _________. v) If n=5, Σx=10, Σy=17, Σx2=26, Σxy=14 then cov(x,y) = _________. vi) If x0, x1 are two initial approximations to the root of f(x) = 0, by secant method the next approximation x2 is given by ________.

The first four moments of a distribution about 4 are –1.4, 17, –30 and 108. Obtain the first four central moments and coefficient of skewness & kurtosis.

Fit a linear curve of the type y = ax + b, to following data, x: 10 15 20 25 30 y: 0.75 0.935 1.1 1.2 1.3

Find the correlation coefficient for the following data, Population density: 200 500 400 700 800 Death rate: 12 18 16 21 10

Find coefficient of variability for following data, C.I.: 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Freq. (f): 4 7 8 12 25 18 10

Fit a linear curve y = ax + b, by least square method to the data, x: 100 120 140 160 180 200 y: 0.9 1.1 1.2 1.4 1.6 1.7

The regression equations are 8x – 10y + 66 = 0 and 40x – 18y = 214. The value of variance of x is 9. Find i) the mean values of x and y ii) the correlation x and y and iii) the standard deviation of y

Three factories A, B and C produce light bulbs. 20%, 50% and 30% of the bulbs are available in the market by factories A, B and C respectively. Among these, 2%, 1% and 3% of the bulbs produced by factories A, B and C are defective. A bulb is selected at random in the market and found to be defective. Find the probability that this bulb was produced by factory B.

On an average, 20% of the computers in a firm are virus infected. If 10 computers are chosen at random from this firm, find the probability that at least one computer is virus infected, using Binomial distribution.

The height of a student in a school follows a normal distribution with mean 190 cm and variance 80 cm2. Among the 1,000 students from the school, how many are expected to have height above 200 cm? (Given : z = 1.118, A = 0.3686)

A die is tampered in such a way that the probability of observing an even number is twice as likely to observe an odd number. Find the expected value of the upper most face obtained after rolling the die.

The number of industrial injuries per working week in a factory is known to follow a Poisson distribution with mean 0.5. Find the probability that during a particular week, at least two accidents will take place.

A pea cultivating experiment was performed. 219 round yellow peas, 81 round green peas, 61 wrinkled yellow peas and 31 wrinkled green peas were noted. Theory predicts that these phenotypes should be obtained in the ratios 9:3:3:1. Test the compatibility of the data with theory, using 5% level of significance. (Given : χ2 tab = 7.815)

Obtain the root of the equation x3 – 4x – 9 = 0 that lies between 2 and 3 by Newton-Raphson method correct to four decimal places.

Solve 2x – cosx – 3 = 0 by using the method of successive approximations correct of three decimal places.

Solve by Gauss - Seidel method, the system of equations : 2x1 + x2 + 6x3 = 9 8x1 + 3x2 + 2x3 = 13 x1 + 5x2 + x3 = 7

Solve by Gauss elimination method, the system of equations : 4x1 + x2 + x3 = 4 x1 + 4x2 – 2x3 = 4 3x1 + 2x2 – 4x3 = 6

Solve by Jacobi's iteration method, the system of equations : 20x1 + x2 – 2x3 = 17 3x1 + 20x2 – x3 = –18 2x1 – 3x2 + 20x3 = 25

Find a real root of the equation x3 – 2x – 5 = 0 by the method of false position at the end of fifth iteration.

Using Newton's backward difference formula, find y at x = 4.5 for the following data. x: 1 2 3 4 5 y: 3.47 6.92 11.25 16.75 22.94

Use Simpson's 3/8th rule, to estimate ∫17 f(x) dx from the following data. x: 1 2 3 4 5 6 7 f(x): 81 75 80 83 78 70 60

Use Euler's method to solve dy/dx = x+y, y(0)=1. Tabulate values of y for x = 0 to x = 0.3. (Take h = 0.1)

Use Runge-Kutta method of 4th order to solve dy/dx = (y-x)/(y+x), y(0)=1 at x = 0.2 with h = 0.2.

Using modified Euler's method, find y(1.1). Given dy/dx = xy+y2, y(1)=1. Take h = 0.1. (Two iterations only)

Determine the value of √151 using Newton's forward difference formula, from the following data. x: 150 152 154 156 y=√x: 12.247 12.329 12.410 12.490