Download AKTU B-Tech 4th Sem 2016-17 NAS401 Engineering Mathematics Iii Question Paper

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B. TECH.

THEORY EXAMINATION (SEM-IV) 2016-17

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ENGINEERING MATHEMATICS-III

Time: 3 Hours

Max. Marks : 100

SECTION-A

1. Attempt all parts of the following question: (2 x 10 = 20)
   1. Evaluate ?_C (ez / (z+1)) dz, where C is the circle |z| = 2

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   2. Prove that f(z) = sinh z is analytic
   3. Prove that Modulation theorem F{f(x) cos ax} = ½[f(s+a)+f(s-a)]
   4. Solve the Z-transform: Y_k+2 + Y_k+1-2y_k =0, y₀ =4 y₁ = 0
   5. What is the meaning of Skewness?
   6. Write Normal equation of y = a + b/x

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   7. Prove that ? = E - 1
   8. Find first approximation value of (17)^1/3 by using Newton Raphson method
   9. Using Picard's method, find the solution of dy/dx = 1+ xy upto the third approximation when x(0) = 0
   10. Find y(0.1) using Euler's method given that dy/dx = log(x + y), y(0) =1.0

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SECTION – B

2. Attempt any five parts of the following question: (5 x 10 = 50)
   1. Prove that the function f(z) defined by f(z) = { [x³(1+i)-y³ (1-i)] / (x² + y²) , z?0 , 0 , z=0 } is continuous and the C.R. equations are satisfied at the origin, yet f'(0) does not exist.
   2. Using Cauchy Integral formula to evaluate ?_C (e^2z / (z+1)⁴) dz, where C is the circle |z| = 3.
   3. Find the Fourier cosine transform of 1 / (1 + x²) and then find Fourier sine transform of x / (1 + x²)
   4. Find the multiple linear regression of X₁ on X₂ and X₃ from the data relating to three variables:
      

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      X1	7	12	17	20
      X2	4	7	9	12
      X3	2	5	8	11

   5. Find the root of the equation using Regular Falsi method correct to four decimal places.
   6. Apply Crout's method and solve the system of equations
      2x+3y+z=9
      x+2y+3z = 6
      

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      3x+y+2z = 8
   7. Find the value y(1.1) using Runge-Kutta method of fourth order, given that dy/dx = y² + xy, y(1) =1.0,take h=0.05

SECTION-C

3. Attempt any two questions of the following: (2 x 15 = 30)
   1. Show that the function defined by f(z) = vxy is not regular at the origin, although Cauchy-Riemann equations are satisfied

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   2. Evaluate: ?₀^2p (d? / (a+bsin?)) if a > b
   3. Solve by Z-transform: Y_k+2-4yk+1+3yk = 5^k
4. 1. Using the convolution theorem, evaluate Z^-1 {z² / ((z-1)(z-3))}
   2. If ? is the acute angle between the two regression lines in the case of two variables x and y, show that tan ?= | (1-r²) / (r * (s_x² + s_y²)) | * s_xs_y ,where r,s_x,s_y have their usual meanings. Explain the significance of the formula when r=0 and r=±1

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   3. By using x²-test, find out whether there is any association between income level and type of schooling
      

      Social status	Health	Poor	Rich	Total
      Below Normal	130	20	150
      Normal	102	108	210
      Above Normal	24	96	120
      Total	256	224	480

5. 1. Find the missing figure in the following table
      

      X	2	3	4	5	6
      f(x)	45	49.2	54.1	?	67.4

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   2. Find a cubic polynomial which approximates the data:
      

      X	y(x)
      -2	-12
      -1	-8
      2	3
      3	5

   3. Find an approximate value of the log 5 by calculating to four decimal places by Simpson's rule, given ?₄⁵ (dx / x)

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