Numerical Methods - Study Mode

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[#41] If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statements is TRUE?

(A) f(a) $$ cdot $$ f(b) = 0

(B) f(a) $$ cdot $$ f(b) < 0

(C) f(a) $$ cdot $$ f(b) > 0

(D) $$frac{{{ ext{f}}left( { ext{a}} ight)}}{{{ ext{f}}left( { ext{b}} ight)}} leqslant 0$$

Correct Answer

(C) f(a) $$ cdot $$ f(b) > 0

[#42] Consider p(s) = s 3 + a 2 s 2 + a 1 s + a 0 with all real coefficients. It is known that it is derivative p'(s) has no real roots. The number of real roots of p(s) is

(A) 0

(B) 1

(C) 2

(D) 3

Correct Answer

(B) 1

[#43] The differential equation $$frac{{{ ext{dx}}}}{{{ ext{dt}}}} = left[ {frac{{1 - { ext{x}}}}{ au }}
ight]$$ xa0 is discretised using Euler's numerical integration method with a time step ΔT > 0. What is the maximum permissible value of ΔT to ensure stability of the solution of the corresponding discrete time equation?

(A) 1

(B) $$frac{ au }{2}$$

(C) $$ au $$

(D) $$2 au $$

Correct Answer

(D) $$2 au $$

[#44] Match the correct pairs Numerical Integration Scheme Order of Fitting Polynomial P. Simpson's 3/8 Rule 1. First Q. Trapezoidal Rule 2. Second R. Simpson's 1/3 Rule 3. Third

(A) P-2, Q-1, R-3

(B) P-3, Q-2, R-1

(C) P-1, Q-2, R-3

(D) P-3, Q-1, R-2

Correct Answer

(D) P-3, Q-1, R-2

[#45] The minimum number of equal length subintervals needed to approximate $$intlimits_1^2 {{ ext{x}}{{ ext{e}}^{ ext{x}}}{ ext{dx}}} $$ xa0to an accuracy of at least $$frac{1}{3} imes {10^{ - 6}}$$ xa0using the trapezoidal rule is

(A) 1000e

(B) 1000

(C) 100e

(D) 100

Correct Answer

(A) 1000e