BTech 3rd Semester Mathematics 3 Past Paper PDF 2021-2022

# 3rd Semester Regular / Back Examination: 2021-22 ## MATHEMATICS - III **BRANCH(S):** AIE, AERO, AG, AUTO, BIOMED, BIOTECH, C&EE, CHEM, CIVIL, CSE, CSEAIME, CST, ECE, EEE, EIE, ELECTRICAL, ELECTRICAL & C.E, ELECTRONICS & C.E, ENV, ETC, IT, MANUTECH, MECH, METTA, MINERAL, MINING, MME, MMEAM, PE, PL...

# 3rd Semester Regular / Back Examination: 2021-22 ## MATHEMATICS - III **BRANCH(S):** AIE, AERO, AG, AUTO, BIOMED, BIOTECH, C&EE, CHEM, CIVIL, CSE, CSEAIME, CST, ECE, EEE, EIE, ELECTRICAL, ELECTRICAL & C.E, ELECTRONICS & C.E, ENV, ETC, IT, MANUTECH, MECH, METTA, MINERAL, MINING, MME, MMEAM, PE, PLASTIC, PT **Time: 3 Hour** **Max Marks: 100** **Q.Code: OF575** **Registration No:** **Total Number of Pages : 02** **B.Tech** **RMA3A001** ## Answer Question No.1 (Part-1) which is compulsory, any eight from Part-II and any two from Part- M. ## The figures in the right hand margin indicate marks. ## Part-I ### Q1 **Answer the following questions:** **(2×10)** a) Rate of convergence of the Newton-Raphson method is generally. b) Using Newton's Forward formula, find sin(0.1604) from the following table. | X | f(x) | |---|---| | 0.160 | 0.1593182066 | | 0.161 | 0.1603053541 | | 0.162 | 0.1612923412 | c) Rewrite dy/dx +2y=1.3exy(0)=5 In dy/dx=f(x,y) y(0)=yo form. d) Given that x=2y+4 and y=kx +6 are the lines of 'regression of x 'on y and y on x respective. Find the value of k if r is 0.5. e) A coin is tossed three times. What is the probability of three heads? f) What is the meaning of probability in statistics? g) What will be the variance of the Bernoulli trials, if the probability of success of the Bernoulli trial is 0.3. h) Explain the primary philosophical difference between the parameters of the probability distribution function (PDF) and the cumulative distribution function - (CDF). i) The mean of hypergeometric distribution is. j) A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random from the bag. The probability that all of them are red, is: ### Q2 ## Part-II ### Only Focused-Short Answer Type Questions- (Answer Any Eight out of Twelve) **(6×8)** a) Using Lagrange's interpolation formula find y(10) from the following table: | X | y | |---|---| | 5 | 12 | | 6 | 13 | | 9 | 14 | | 11 | 16 | b) Find the cube root of 12 using the Newton Raphson method assuming x0 = 2.5. c) Baby boys have a mean weight of 6.4 kg, with a standard deviation of 0.7. Baby girls have a mean weight of 5.9 kg, with a standard deviation of 0.7. The weights of 3-month old babies are normally distributed. What is the probability that a 3-month old baby boy weighs more than 7.3 kg? d) For an integral Ja f(x) dx derive the one-point Gauss quadrature rule. e) valuate [02 Xx2 dx using the Trapezoidal Rule, with n = 2. f) Find the value of k1 by Runge-Kutta method of fourth order if dy/dx = 2x + 3y2 and y(0.1) = 1.1165, h = 0.1 g) A die is rolled twice and two numbers are obtained, let X be the outcome of first role and Y be the outcome of the second roll. Given that X+Y=5, what is the probability of X=4 or Y=4? h) It is estimated that 50% of emails are spam emails. Some software has been applied to filter these spam emails before they reach your inbox. A certain brand of software claims that it can detect 99% of spam emails, and the probability for a false positive (a non-spam email detected as spam) is 5%. Now if an email is detected as spam, then what is the probability that it is in fact a non-spam email? i) The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then prove that P (A') + P (B) = 2-2p + q. j) Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If X denotes the number of red ball drawn, find the probability distribution of X. k) A binomial probability experiment is conducted with the given parameters. Compute the probability of xx successes in the nn independent trials of the experiment n=10, p=0.75, x=8 l) Let's return to the example in which X1,X2,..., Xn are normal random variables with mean u and variance o². What are the method of moments estimators of the mean u and variance σ²? m) Suppose that X is a discrete random variable with the following probability mass function: where 0 ≤ 0 ≤ 1 is a parameter. The following 10 independent observations were taken from such a distribution: (3,0,2,1,3,2,1,0,2,1). What is the maximum likelihood estimate of 0. https://www.bputonline.com ## Part-III ### Only Long Answer Type Questions (Answer Any Two out of Four) **(8)** **Q3** a) Find the cube root of 12 using the Newton Raphson method assuming xo = 2.5. **(8)** b) Solve Equations 2x+5y=16,3x+y=11 using Gauss Seidel method **(8)** **Q4** a) Find an approximate value of 56x³dx using Euler's method of solving an ordinary differential equation. Use a step size of h = 1.5. **(8)** b) A pair of dice is thrown and let X be the random variable which represents the sum of the numbers that appear on the two dice. Find the mean of X. **(8)** **Q5** a) Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. [2ox√(x²+1)dx n =4. **(8)** b) Let X and Y be two independent Uniform (0,1) random variables. Let also Z = max (X,Y) and W = min (X,Y). Find Cov (Z,W). **(8)** **Q6** a) Calculate the regression coefficient and obtain the lines of regression for the following data | X | Y | |---|---| | 1 | 9 | | 2 | 8 | | 3 | 10 | | 4 | 12 | | 5 | 11 | | 6 | 13 | | 7 | 14 | **(8)** b) X is a normally distributed variable with mean u = 30 and standard deviation σ = 4. Find: a) P(x 40) b) P(x > 21) c) P(30 < x < 35)