Engineering Mathematics Miscellaneous
 The root of the function f(x) = x^{3} + x – 1 obtained after first iteration on application of Newton Raphson scheme using an initial guess of x_{0} = 1 is

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f (x) = x^{3} + x–1
f (1) = 1+1 – 1 = 1
f’(x) = 3x^{2} + 1
f’ (1) = 3+1=4x_{1} = x_{0}  f(x_{0}) = 1  1 f'(x_{0}) 4
= 1 – 0.25
x_{1} = 0.75Correct Option: C
f (x) = x^{3} + x–1
f (1) = 1+1 – 1 = 1
f’(x) = 3x^{2} + 1
f’ (1) = 3+1=4x_{1} = x_{0}  f(x_{0}) = 1  1 f'(x_{0}) 4
= 1 – 0.25
x_{1} = 0.75
 Solve the equation x = 10 cos(x) using the NewtonRaphson method. The initial guess is x = π / 4. The value of the predicted root after the first iteration, up to second decimal, is _____.

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Correct Option: A
 NewtonRaphson method is used to find the roots of the equation, x^{3} + 2x^{2} + 3x – 1 = 0. If the initial guess is x_{0} = 1, then the value of x after 2nd iteration is ________.

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By NewtonRaphson Method,
1^{st} iteration, x^{1} = x^{0} – f^{(0)} f'^{(0)} = 1  f(1) = 1  5 = 1 f'(1) 10 2
Where f(x) = x^{3} + 2x^{2} + 3x – 1 ⇒ f(1) = 5
f'(x) = 3x^{2} + 4x + 3 ⇒ f'(1) = 102^{nd} iteration, x_{2} = x_{1} f(x_{1}) = 0.5  f(0.5) = 0.3043 f'(x_{1}) f'(0.5)
Correct Option: A
By NewtonRaphson Method,
1^{st} iteration, x^{1} = x^{0} – f^{(0)} f'^{(0)} = 1  f(1) = 1  5 = 1 f'(1) 10 2
Where f(x) = x^{3} + 2x^{2} + 3x – 1 ⇒ f(1) = 5
f'(x) = 3x^{2} + 4x + 3 ⇒ f'(1) = 102^{nd} iteration, x_{2} = x_{1} f(x_{1}) = 0.5  f(0.5) = 0.3043 f'(x_{1}) f'(0.5)
 The real root of the equation 5x – 2 cosx – 1 = 0 (up to two decimal accuracy) is _______.

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Let f(x) =5x – 2 cos x – 1
⇒ f'(x) = 5 + 2 sin x
f(0) = – 3; f(1) = 2.9
By intermediate value theorem roots lie between 0 and 1.
Let x_{0} = 1 rad = 57.32º
By Newton Raphson method,X_{n+1} = X_{n}  f(x_{n}) f'(x_{n}) ⇒ x_{n+1} 2x_{n}sin x_{n} + 2 cos x_{n} + 1 5 + 2Sin x_{n}
⇒ x_{1} = 0.5632
⇒ x_{2} = 0.5425
⇒ x_{3} = 0.5424Correct Option: A
Let f(x) =5x – 2 cos x – 1
⇒ f'(x) = 5 + 2 sin x
f(0) = – 3; f(1) = 2.9
By intermediate value theorem roots lie between 0 and 1.
Let x_{0} = 1 rad = 57.32º
By Newton Raphson method,X_{n+1} = X_{n}  f(x_{n}) f'(x_{n}) ⇒ x_{n+1} 2x_{n}sin x_{n} + 2 cos x_{n} + 1 5 + 2Sin x_{n}
⇒ x_{1} = 0.5632
⇒ x_{2} = 0.5425
⇒ x_{3} = 0.5424
 Let X and Y be two independent random variables. Which one of the relations between expectation (n), variance (Var) and covariance (Cov) given below is FALSE?

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X and Y are independent
∴ (a), (b), (c) are true
Only (d) is odd oneCorrect Option: D
X and Y are independent
∴ (a), (b), (c) are true
Only (d) is odd one