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  • Q1:(3) Define g(z) - Inr+i0 <0<T. where z=rei and r>0 and (i) Sketch the domain D of the function g(z) and show that g(z) is analytic on this domain and find g'(z). (ii) Show that the composite function G(z) = g(2²+1) is analytic on the half-plane H = {z=z+iy : x>0}, with derivative 2z 2² +1 G(z)= (4) (i) Define what it means for a subset of C to be open. (ii) Define what it means for a subset of C to be a domain. (iii) Suppose f(2)= u(x, y) +iu(x, y), z=1+iy, x,y ER, for z in a domain D. Prove that if f is analytic on this domain D then f'(2)=0 for all z in D, and hence f is constant on D.See Answer
  • Q2: Show how to use the "slide method" to determine the GCF and LCM of 3250 and 4000.See Answer
  • Q3: Solve the IVP. 4 y^{\prime \prime}+12 y^{\prime}+9 y=0 \quad y(0)=1 \quad y^{\prime}(0)=2See Answer
  • Q4: (a) Determine which of the following maps are linear (giving reasons for your answers). f: R_{3}-\rightarrow R_{2} \quad\left[x_{1, X 2, X 3}\right\} 7-\rightarrow[x 2+1, x 3-1\} f: \mathrm{R} 3-\rightarrow \mathrm{R} \quad[x 1, X 2, X 3\} 7-\rightarrow X 1 \times 2 \times 3 f: \mathrm{R} 3-\rightarrow \mathrm{R} 3 \quad[X 1, X 2, X 3\} 7-\rightarrow[X 1+X 2+2 X 3,0, X 2\}See Answer
  • Q5: \text { What is the Fourier transform of the function } 4 \cos (5 t) \text { on the interval }[-\pi, \pi] ? \text { a. } \cos (2 t)+\cos (4 t) b. none of the others c. 2 cos (t) +7 sin(2t) d. 4 cos(5t) 7 sin(2t) + 4 cos(5t)e.See Answer
  • Q6: \text { or the continuous odd function } f(t) \text { if } \int_{0}^{a} f(t) d t=4, a>0 \text { , then } \int_{-a}^{a} f(t) d t=? a. 4 b. all of the others c. 2 d. 8 e. OSee Answer
  • Q7: \text { What is the Fourier transform of the function } 2 \sin (3 t) \text { on the interval }[-\pi, \pi] \text { ? } \text { a. } \sin (t)+\sin (2 t) \text { b. } 3 \sin (2 t)+4 \sin (3 t) \text { c. } 2 \sin (3 t) \text { d. } 4 \sin (t)+2 \sin (2 t) e. none of the othersSee Answer
  • Q8: Using (13) sketch or graph a figure (similar to Fig. 291 in Sec. 12.3) of the deflection u(x, t) of a vibrating string(length L = 1, ends fixed, c = 1) starting with initial velocity 0 and initial deflection (k small, say, k = 0.01). \text { 7. } f(x)=k \sin 2 \pi x See Answer
  • Q9: Find all trigonometric functions of the acute angle 0 in the right triangle below: See Answer
  • Q10: \text { What is the fundamental period of the function } f(t)=\cos (3 t) ? \text { a. } \frac{2 \pi}{3} b. none of the others \text { c. } \frac{\pi}{2} \text { d. } 4 \pi \text { e. } 8 \piSee Answer
  • Q11: . Consider the eigenvalue problem y^{\prime \prime}+\lambda y=0 ; \quad y^{\prime}(0)=0, \quad y(\pi)=0 .See Answer
  • Q12: O Find the values of x for which the following functions have stationary values and using your results, sketch a graph of each function.bed tomoidelngog oIT y=e^{2 x}-2 x y=4 e^{x}-2 x-3See Answer
  • Q13: A nonlinear dynamic system is described by the following three state variable equations: \dot{q}_{1}=q_{2}-q_{3} \dot{q}_{2}=u+q_{2}+\frac{1}{2}\left|q_{1}\right| q_{1} \dot{q}_{3}=3 q_{1}+6 q_{3} Assume that 73 = 2. What is the value of 71?See Answer
  • Q14: 1. CAS PROJECT. Effect of Damping. Consider a vibrating system of your choice modeled by y^{\prime \prime}+c y^{\prime}+k y=\delta(t) (b) What happens if e is kept constant and k iscontinuously increased, starting from 0? (c) Extend your results to a system with two8-functions on the right, acting at different times.See Answer
  • Q15: Here are yesterday's high temperatures (in Fahrenheit) in 11 U.S. cities. 47, 49, 62, 65, 68, 68, 71, 72, 75, 78, 82 Notice that the temperatures are ordered from least to greatest.Give the five-number summary and the interquartile range for the data set. Five-number summary Minimum:_____ Lower quartile: _______ Median:________ Upper quartile: ______ Maximum:_______ Interquartile range:_______ See Answer
  • Q16: 5. Use appropriate tests to determine the convergence or divergence of the above series \text { (a) } \sum_{n=1}^{\infty} \frac{n^{2}+2 n+3}{4 n^{5}+n^{2}+3} \text { (b) } \sum^{\infty} \frac{\sqrt{n+c}}{\sqrt{n^{3}}+1} \text { where } c>3 \text { is a constant } \text { (c) } \sum_{n=1}^{\infty} \frac{n^{2}+3 n+1}{3 n^{2}+4 n-1} \text { (d) } \sum_{n=1}^{\infty} \frac{4+2 \sin (n)}{3 n^{c}+4} \text { where } c>1 \text { is a constant } \text { (e) } \sum^{\infty} \frac{4+(-1)^{n}}{3 n^{c}+4} \text { where } 0<c<1 \text { is a constant } \text { (f) } \sum_{n=2}^{\infty} \frac{1}{n \ln n} \text { (g) } \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{c}} \text { where } c>1 \text { is a constant } \text { (h) } \sum_{n=1}^{\infty} \frac{(-1)^{n} n^{2}}{3^{n}+2} \text { (i) } \sum_{n=1}^{\infty} \frac{c^{n}}{n^{4}+3} \text { where } c>1 \text { is a constant } \text { (j) } \sum_{n=1}^{\infty} \frac{n^{2}+2}{(-1)^{n} \cdot n !} (k) \sum_{n=1}^{\infty} \frac{n !}{n^{2}} \text { (1) } \sum_{n=1}^{\infty}\left(-\frac{1}{3}+\frac{1}{n}\right)^{n} \text { (m) } \sum_{n=1}^{\infty}\left(c+\frac{1}{n}\right)^{n} \text { where } c>1 \text { is a constant }See Answer
  • Q17: Given following differential equation: ÿ + 10y + 16y = 0 It has initial conditions y(0) = 1 and y(0) = 2 Solve it by the trial solution method. True or false, the solution is: y(t)=-\frac{2}{3} e^{-2 t}+\frac{5}{3} e^{-8 t}See Answer
  • Q18: How many fractions are there in the partial fractions decomposition of the function f(x)=\frac{3 x+8}{(x-7)(x+1)} ? А. 3 В. 1 C. None of these. D. 4 Е. 2See Answer
  • Q19: . We want to solve the partial differential equation (PDE) \frac{\partial^{2} U(x, y)}{\partial x^{2}}+\frac{\partial^{2} U(x, y)}{\partial y^{2}}=3 x+2 y subject to Dirichlet boundary conditions with g1(x, y) = Y, 92(x, y) = e®+y, and g3(x, y)figure below; h = k = 1/4).= x (see a. Discretise the partial differential equation using a centred difference scheme with error O(h2)in each dimension. Explain what a stencil is and present it for the given problem.[8] b. Use the stencil derived in (a) to present the system of equations to solve. Find the solutions for the problem at P1, P2 and P3.[9] c. Explain how to compute the numerical solution if the condition on g1 (left hand side of the triangle) changes to a Neumann boundary condition.[8]See Answer
  • Q20: If the region under the curve y = 5x is rotated about the x-axis, then the cross-sections of the resulting solid of revolution are disks whose area is A(x) = pie (5x2)See Answer

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