F(x, y) = (x2+xy)\(\hat{a}_{x}\)+(y2+xy)\(\hat{a}_{x}\). It’s line integral over the straight line from (x, y) = (0, 2)to (x, y) = (2, 0) evaluates to ________.
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Solution
The matrix [A] =\(\begin{bmatrix} 2 & 1\\ 4 &-1 \end{bmatrix}\) is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix[U]. The properly decomposed [L] and [U] matrices respectively are
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Solution
Consider the differential equation
\(\frac{d^{2}y(t)}{dt^{2}}+2\frac{dy(t)}{dt}+y(t)=\delta (t)\)with \(y(t)\mid_{t=0^{-}}=-2\: and\: \frac{dy}{dt}\mid _{t=0}=0\)
The numerical value of \(y(t)\mid_{t=0^{+}}\) is__________ .
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Solution
The function f(x) = ex – 1 is to be solved using Newton-Raphs on method. If the initial value of x0 is taken as 1.0,then the absolute error observed at 2nd iteration is
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Solution
A fair coin is tossed till head appears for the first time. The probability that the number of required tosses is odd is________.
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Solution
Given
\(f(z)=\frac{1}{z+1}-\frac{2}{z+3}\). If C is a counterclockwise path in the z-plane such that|z + 1|= 1, the value of \(\frac{1}{2\pi j}\oint_{c} f(z)dz\) is-
Solution
The analytic function f(z)=\(\frac{z-1}{z^{2} +1}\)has singularities at
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Solution
With initial condition x(1) = 0.5, the solution for the differential equation \(t \frac{dx}{dt} + x = t\) is
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Solution
The minimum value of the function f(x) = x3– 3x2 – 24x + 100 in the interval [– 3, 3] is _______ .
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Solution
f'(x) = 3x2– 6x – 24
f'(x) =0
3x2– 6x –24 = 0
x2– 2x – 8= 0 Þx = – 2, 4 Þx = – 2∈[– 3,3]
f(– 3) = (– 3)2– 3(–3)2– 24 (–3) + 100 = 118
f(– 2)= (–2)2– 3(–2)2– 24 (–2) + 100 = 128
f(– 3) = (3)2– 3(3)2– 24 (3) + 100 = 28
Hence, minimum value of f(x) at x = 3 is 28.
Given a system of equations:
x +2y + 2z = b1
5x + y + 3z = b2
Which of the following is true regarding its solutions?
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Solution
\(\begin{vmatrix} 1 &2 & 2 &: & b_{1} \\ 5 &1 & 3 &: & b_{2} \end{vmatrix}\)
Here, R(A:B) = R(B) = n(no. of unknown)Hence, the system will have infinitely many solutions for any given b1 and b2.