PTU B.Tech Automation Robotics Engineering ECE 5th Semester May 2019 70482 NUMERICAL METHODS IN ENGINEERING Question Papers

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Roll No. [ | [ [T TTT] Total No. of Pages : 03
Total No. of Questions : 09

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students have to attempt ANY FOUR questions.
3. SECTION-C contains THREE questions carrying TEN marks each and students have to attempt ANY TWO questions.

SECTION-A

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1. Write briefly :
   1. Write Newton’s formula for interpolation.
   2. Find the condition number of the function f'(x) = sin x.
   3. Define a cubic spline interpolant with clamped boundary.
   4. Determine the Lagrange interpolating polynomial passing through the points (1, 1), (2,4) and (3, 9).
   5. Find the L, norm of the vector (1, 5, 9)^T.

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   6. Explain least square curve fitting.
   7. Compute ∫xe^xdx using Simpson’s rule.
   8. Use the forward-difference formula to approximate the derivative of f(x) = ln x at x₀=1.8 using h =0.1.
   9. What is the order of convergence when Newton Raphson’s method is applied to the equation x² — 6x + 9 = 0 to find its multiple root.
   10. Out of chopping of numbers and rounding off of numbers, which one introduce less error?

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SECTION-B

1. Use forward-difference method with steps sizes h = 0.1 and k = 0.01 to approximate the solution to the heat equation :
   ∂u/∂t (x,t) = ∂²u/∂x² (x,t) = 0, 0 < x < 1, t > 0,
   with boundary conditions
   

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   u(0,t)=u(1,t)=0, t>0,
   and initial condition
   u(x,0)=sin(πx), 0 < x < 1.
2. Apply Taylor’s method of order 2 with N = 10 to initial value problem
   y'=y-t²+1, 0
3. The following data is given
   

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   1.0	1.3	1.6	1.9	2.2
   0.76519771	0.6200860	0.4554022	0.2818186	0.1103623

4. Use Lagrange’s formula to approximate f'(1.5).
5. Use the data points (0, 1), (1, e), (2, e²) and (3, e³) to form a natural spline S(x) that approximates f (x) = e^x.
6. Find the largest interval in which p* must lie to approximate p with relative error at most 10^-4 for p= (√17).

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SECTION-C

1. Derive Secant’s formula for solving the equation f (x) = 0 (specifying the assumptions made). Use the secant method to solve the equation x = cos x starting with an initial guesses 0.5 and 1.
2. Use Gauss elimination method with partial pivoting to solve the following linear system of equations.
   √3x₁+√5x₂—x₃+x₄=0,
   x₁ — x₂ +x₃+2x₄=1,
   

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   x₁+x₂—√5x₃+x₄=2,
   -x₁ —x₂+x₃—√5x₄=3.
3. Determine the values of h that will ensure an approximation error of less than 10^-6 when approximating I= ∫e^-xsin3xdx and employing :
   a) Composite trapezoidal rule.
   b) Composite Simpson’s rule.

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NOTE : Disclosure of identity by writing mobile number or making passing request on any page of Answer sheet will lead to UMC case against the Student.

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