Download PTU B-Tech AR-Automation-And-Robotics 2020 Dec 5th Sem 70482 Numerical Methods In Engineering Question Paper

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Roll No. ‘ ‘ ’ ‘ ‘ ‘ ’ ‘ ‘ ’ ‘ ‘ ‘ Total No. of Pages : 02
Total No. of Questions : 18

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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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Answer the following :

1. Define a cubic spline interpolant with natural boundary.
2. What do we mean by unconditionally stable method?
3. Find the condition number of the function f(x) = cos x.
4. Determine the Lagrange interpolating-polynomial passing through the points (2,4) and (5,3).

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5. Out of chopping of numbers and rounding off of numbers, which one introduce less error? Explain suitably.
6. Find the l₁ norm of the vector (1,√6,3)^T.
7. 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.
8. Use the forward-difference formula to approximate the derivative of f(x) = ln x at x₀=1.8 using h =0.01.
9. Compute ∫ x sin x dx using Simpson’s rule.

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10. Explain Lagrange's interpolation.

SECTION-B

11. Use Euler’s method to approximate the solution of the following initial value problem y' = y - t², 1 < t < 2, y(1)=1, h=0.1.
12. Construct a clamped spline S(x) which passes through the points (1,2), (2,3) and (3,5) that has S'(1)=2 and S'(3) = 1.
13. The following data is given :

    x	1.0	1.3	1.6	1.9	2.2
    f(x)	0.7651977	0.6200860	0.4554022	0.2818186	0.1103623

    Use Lagrange’s formula to approximate f(1.5).

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14. Let f(x) = (x cos x — sin x)/(x — sin x). Use four digit rounding arithmetic to evaluate f(0.1). The actual value is f(0.1) =-1.99899998, using this value find the relative error.
15. Use backward-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 u(0,t)=u(1,t)=0, t>0, u(x,0) =sin(πx), 0 < x < 1.

SECTION-C

16. Determine the values of h that will ensure an approximation error of less than 0.00002 when approximating ∫ sin x dx and employing:
    1. Composite trapezoidal rule:
    2. Composite Simpson's rule.

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17. Draw the graph of 4x = tan x. Use Newton's method to find the first two positive roots of 4x = tan x (Note: You can use the graph drawn for selecting your initial guesses.).
18. Use Gauss elimination method with scaled partial pivoting to solve the following linear system of equations 2.11x₁ - 4.21x₂ + 0.921x₃ = 2.01,
    4.01x₁ + 1.02x₂ - 1.12x₃ = -3.09,
    1.09x₁ + 0.987x₂ + 0.832x₃ = 4.21.

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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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