STUPIDSID
Short Questions
1(a)
Express \( \sqrt{3}-i \) into polar form
1 M
1(b)
Evaluate Δ cos x.
1 M
1(c)
Evaluate \( \lim_{z\rightarrow i}\dfrac{z-i}{z^2+1} \)
1 M
1(d)
Find the radius of convergence for the series \( \sum ^{\infty}_{n=0}Z^n \)
1 M
1(e)
Write formula for Simpson's 3/8 rule.
1 M
1(f)
Find the fixed points of \( w=\dfrac{z-1}{z+1} \)
1 M
1(g)
Give the names of two iterative methods for the solution of system of linear equations.
1 M
1(h)
State the theorem, ' Cauchy's Integral Formula'.
1 M
1(i)
Find the pole and its order for \( f(z)=\dfrac{e^z-1}{z^3} \)
1 M
1(j)
Find the third divided difference with arguments 2, 4, 9, 10 of the
function f(x) = x3 - 2x.
1 M
1(k)
Find Res (f(z), 1) for \( f(z)=\dfrac{1}{z(z-1)} \)
1 M
1(l)
Find the interval for x3 - x ' 11 = 0 in which the root lies.
1 M
1(m)
State DeMoivre's Theorem.
1 M
1(n)
Write iterative formula to find \( \sqrt{7} \) using Newton-Raphson method.
1 M
2(a)
Find all the values of \( \left ( \dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i \right )^{3/4}. \)
3 M
2(b)
Show that the function f(z) = xy + iy is continuous everywhere but is
not analytic.
4 M
Solve any question from Q.2(c) & Q.2(d)
2(c)(i)
If u = ex (x cos y - y sin y), find the analytic function f(z).
3 M
2(c)(ii)
Find the value of \( \int _0^{2+i}(\overline{z})^2 dz, \) along the real axis from 0 to 2 and then vertically from 2 to 2 + i.
4 M
2(d)(i)
If f(z) is a regular function of z, prove that \( \left ( \dfrac{\partial ^2}{\partial x^2}+\dfrac{\partial ^2}{\partial y^2} \right )|f(z)|^2=4|f'(z)|^2. \)
3 M
2(d)(ii
Prove tha \( \sinh^{-1}x=\log \left \{ x+\sqrt{x+1} \right \} \)
4 M
Solve any three question from Q.3(a), Q.3(b), Q.3(c) & Q.3(d), Q.3(e), Q.3(f)
3(a)
Evaluate \( \int _c \dfrac{e^{2z}}{(z+1)^3}dz \), where C : 4x2 + 9y2 = 16 using residue theorem.
3 M
3(b)
Find the bilinear transformation which transforms z = 2, 1, 0 into w= 1, 0, i.
4 M
3(c)
Expand \( \dfrac{1}{z(z^2-3z+2)} \) about z = 0, for the regions (i) 0 < |z| < 1 (ii) 1 < |z| < 2 (iii) |z| > 2.
7 M
3(d)
Evaluate \( \oint _c \dfrac{z-1}{(z+1)^2(z-2)}dz, \) where C is the circle |z-i| = 2.
3 M
3(e)
Find the image of |z ' 3i| = 3 under the mapping \( w=\dfrac{1}{z}. \)
4 M
3(f)
Evaluate \( P.V.\int ^{\infty}_{-\infty}\dfrac{x\cos x}{x^2+9}dx \).
7 M
Solve any three question from Q.4(a), Q.4(b), Q.4(c) & Q.4(d), Q.4(e), Q.4(f)
4(a)
Using Newton's divided difference formula, find a polynomial function satisfying the following data:
| x | -4 | -1 | 0 | 2 | 5 |
| f(x) | 1245 | 33 | 5 | 9 | 1335 |
3 M
4(b)
The table below gives the values of function y=tanx. Obtain the value of tan(0.40) using Newton's backward interpolation.
| x | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 |
| y=tanx | 0.1003 | 0.1511 | 0.2027 | 0.2553 | 0.3093 |
4 M
4(c)
Use the power method to find the largest eigen value and corresponding eigen vector of the matrix \( A=\begin{bmatrix}
1 & 6 & 1\\
1 & 2 & 0\\
0 & 0 & 3
\end{bmatrix} \).
7 M
4(d)
Find the root of the equation x2 - 4x ' 10 = 0 correct to three decimal
places by using bisection method.
3 M
4(e)
Compute f(4) from the tabular value given
Using Lagrange interpolating polynomial.
| x | 2 | 3 | 5 | 7 |
| f(x) | 0.1506 | 0.3001 | 0.4517 | 0.6259 |
Using Lagrange interpolating polynomial.
4 M
4(f)
Solve the following system of equations by Gauss ' Jordan method:
10x + y + z = 12, 2x + 10y + z = 13, x + y + 5z = 7.
10x + y + z = 12, 2x + 10y + z = 13, x + y + 5z = 7.
7 M
Solve any three question from Q.5(a), Q.5(b), Q.5(c) & Q.5(d), Q.5(e), Q.5(f)
5(a)
The velocity v of a particle at distance s from point on its path is given by the following table:
Find the time taken to travel 60 meter, using Simpson's 1/3 rule. (Use \( v=\dfrac{ds}{dt}. \)
| s(meter) | 0 | 10 | 20 | 30 | 40 | 50 | 60 |
| v(meter/Sec) | 47 | 58 | 64 | 65 | 61 | 52 | 38 |
Find the time taken to travel 60 meter, using Simpson's 1/3 rule. (Use \( v=\dfrac{ds}{dt}. \)
3 M
5(b)
Use the method of Regula Falsi to find the root of x = e-x correct to
three decimal places.
4 M
5(c)
Use fourth order Runge Kutta method to find the value of y at x = 1 given that \( y'=\dfrac{y-x}{y+x} \) such that y(0) = 1. (Take h = 0.5)
7 M
5(d)
Use Gauss Seidel method to solve: 83x + 11y ' 4z = 95, 7x + 52y + 13z = 104, 3x + 8y + 29z = 71.
3 M
5(e)
Evaluate the integral \( \int _{-2}^6(1+x^2)^{3/2}dx \) by the Gaussain formula for n = 3.
4 M
5(f)
Using Euler's method solve for y at x = 0.1 from \( \dfrac{dy}{dx}=x+y+xy , \) y(0) = 1, in five steps.
7 M
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