Download SGBAU B-Tech 4th Sem Applied Mathematics Question Paper

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B.Tech. Fourth Semester (Chem. / Poly / Food / Pulp & Paper / Qil & Paint / Petro) (Old)

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Pages :

AW - 3553

Applied Mathematics — II : 4 SCT 1

Time : Three Hours Max. Marks :

Notes: 1. Assume suitable data wherever necessary.

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2. Use of calculator normal distribution table significance table is permitted.

3. Use of pen Blue/Black ink/refill only for writing the answer book.

1. A tightly stretched string with fixed end point x = 0 and x = L, is initially in a position given by y(x, 0) = y₀ sin² (πx/L). If it is released from rest from this position, find the displacement y at any distance x from one end and at any time t.

   OR

2. 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 at B is reduced suddenly to 0°C and kept so, while that of A is maintained. Find the temperature u(x, t) at a distance x from A and at any time t.

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1. 1. Show that the function u=x³ —3xy² is harmonic. Find its harmonic conjugate function and corresponding analytic function in terms of z.
   2. Prove that cosh²z—sinh²z = 1.
   3. Find all values of z, which satisfies the equation z⁴ +1=0.

   OR

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2. 1. If F(z) is regular function, prove that |(∂²/∂x²) + (∂²/∂y²)| |f(z)|²=4|F'(z)|²
   2. Prove that (x +iy)^m/n +(x —iy)^m/n = 2(x² +y²)^m/2n cos((m/n)tan^-1(y/x))
   3. Prove that log(1 +itana) = logseca +ia .

1. 1. Find the root of the equation X² +X—1=0 by iterative method.

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   2. Evaluate ∫₀¹ x³ dx by using Trapezoidal rule, with five sub intervals.

   OR

2. 1. If y(0)=-3,y(3)=9, y(4)=30,y(6)=132 ; find the Lagrange's interpolation polynomial that takes the same values as y at the given point.
   2. Given that

      x	1.0	1.1	1.2	1.3	1.4	1.5	1.6
      y	7.99	8.40	8.78	9.13	9.45	9.75	10.03

      Find dy/dx at x=1.1.

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1. 1. Solve the LPP graphically

      Maximize Z = 2x₁ + X₂,

      subject to X₁+2x₂ ≤ 10

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      X₁ +X₂ ≤ 6

      X₁—X₂ ≤ 2

      X₁ , X₂ ≥ 0

   2. Solve the LPP by using simplex method :

      Maximize Z = 3x₁ +4x₂

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      subject to 2X₁+3x₂ ≤ 9

      4x₁ +3X₂ ≤ 12

      with x₁ ≥ 0, x₂ ≥ 0

   OR

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2. 1. Prove that "the feasible region of LPP is convex"
   2. Solve the following LPP using simplex method and comment on the solution

      Minimize f = -3x₁ - 2x₂,

      subject to X₁—-x₃ =1,

      3X₁ + 2x₃ ≤ 6

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      with x₁ ≥ 0, x₂ ≥ 0

1. 1. For the following distribution. compute the mean and S.D. of 100 students.

      Marks	60-62	63-65	66-68	69-71	72-74
      No. of students	5	18	42	27	8

   2. The mean and variance of a binomial distribution are 4 and 2 respectively, find P(x = 1).

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   OR

2. 1. If the probability that an individual suffers a bad reaction from a certain injection is 0.001, determine the probability that out of 2000 individuals -
      1. exactly 3
      2. atleast 2
      3. none will suffer a bad reaction.

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   2. Students of a class were given an aptitude test. Their marks were found to be normally distributed with mean 60 and S.D. of 5. What is the percentage of students scored more than 6C marks ?

1. 1. Using samples of sizes 10 and 16 with variances S₁² =50 and S₂² =30 ; assuming normality of the population. test the hypothesis H₀ :σ₁² = σ₂² against the alternative σ₁² > σ₂². Choose α=5%.
   2. Find the regression line of y on x for the data.

      x	1	2	3	4	5
      y	2	5	3	1	4

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   OR

2. 1. Distinguish between -
      1. Null and Alternative hypothesis.
      2. Type - 1 and Type - II errors.

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   2. In a sample of 600 men from a certain city, 450 are found smokers. In another sample of 900 men from another city 450 are smokers. Do the data indicate that the cities are significantly different with respect to the habit of smoking among men ?

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