JNTU
JNTU B.Tech I Year MATHEMATICS-III Previous Question Paper (May - 2019 - R16)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks.
Part A
Let X denotes the number of heads in a single toss of 4 fair coins. Determine P(1 < X ≤ 3).
Answer
X follows a binomial distribution B(n=4, p=0.5). We need P(1 < X ≤ 3) = P(X=2) + P(X=3). Using the formula P(X=k) = 4Ck(0.5)4, we get P(X=2) = 6/16 and P(X=3) = 4/16. Thus, P(1 < X ≤ 3) = 10/16 = 0.625.
Define moment generating function of a random variable.
Answer
The moment generating function (MGF) of a random variable X is defined as MX(t) = E[etX]. It is the expected value of etX, calculated as ∑etxP(x) for discrete variables or ∫etxf(x)dx for continuous variables, provided the expectation exists.
Define central limit theorem.
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A random sample of size 100 has a standard deviation of 5. What can you say about maximum error with 95% confidence?
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Define Type I and Type II errors.
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Explain one way classification of ANOVA.
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Establish an iterative formula for computing 1/N by Newton Raphson method.
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Construct normal equations for fit a straight line by method of least squares.
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Write Simpsons 1/3rd and 3/8th rule formulas.
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Given y' = xy with y(0)=1. Find y(0.2) with h=0.1 by Euler’s method.
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Part B
A sample of 4 items is selected at random from a box containing 12 items of which 5 are defective. Find the expected number of defective items.
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In a normal distribution, 7% of the items are under 35 and 89% are under 63. Determine the mean and variance of the distribution.
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Let the continuous random variable X have the probability density function f(x) = 3x^2, if 0 < x < 1, and 0 otherwise. Find F(x).
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A discrete random variable X has the mean 6 and variance 2. If it is assumed that the distribution is Binomial find the probability that 5 ≤ X ≤ 7.
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A random sample of size 100 is taken from an infinite population having mean µ=76 and the variance σ^2 = 256. What is the probability that mean of the sample will be between 75 and 78?
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Assuming that σ = 20.0, how large a random sample be taken to assert with probability 0.95 that the sample mean will not differ from the true mean by more than 3.0 points?
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A normal population has a mean of 0.1 and standard deviation of 2.1. Find the probability that mean of a sample of size 900 will be negative.
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Find 95% confidence limits for the mean of a normally distributed population from which the following sample was taken: 15, 17, 10, 18, 16, 9, 7, 11, 13, 14.
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In a random sample of 60 workers, the average time taken by them to get to work is 33.8 minutes with a standard deviation of 6.1 minutes. Can we reject the null hypothesis µ = 32.6 minutes in favour of alternative null hypothesis µ > 32.6 at α = 0.025 level of significance.
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The mean life of a sample of 10 electric bulbs was found to be 1456 hours with S.D. of 423 hours. A second sample of 17 bulbs chosen from a different batch showed a mean life of 1280 hours with S.D. of 398 hours. Is there a significant difference between the means of two batches?
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The following are the number of typing mistakes made in four successive weeks by four typists. Using ANOVA, test at 0.05 level of significance whether the difference among the four sample means can be attributed to chance.
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Find a real root of xe^x - cos(x) = 0 using Newton-Raphson method.
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Fit a least square parabola curve to the given data.
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Find the root of the equation x log10(x) - 7 = 0 which lies between 3.5 and 4 by regula-falsi method.
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Solve the system of equations x1+x2+x3=8, 2x1+2x2+3x3=12, x1+2x2+x3=4 by Gauss-Seidel method.
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Find y(0.1) and y(0.2) using 4th order Runge – Kutta method given that y' = y^2 + xy, y(0) = 1.
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Solve the equation y' = x^2 + y subject to the condition y(0) = 1 by Picard’s method.
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