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Q1:We consider the following first order differential equation x' + A(t)r - B(t)r* = 0 with k an integer greater or equal to 2. a) Setting y = 2¹-k prove that (1) is transformed into y' + (1 − k)A(t)y − (1-k)B(t) = 0 b) Use this result to solve r x = -x³. (1)See Answer
Q2: (2) If ABCD is a parallelogram such that ABD = 30, Show that DM >AM where M is the point of the intersection of the two diagonals of the parallelogram.See Answer
Q3: (1) If ABCD is a quadrilateral such that AC BD. Show that the quadri-lateral PQRS is a rhombus where P,Q, R and S are the midpoints of the sides of the quadrilateral ABCD.See Answer
Q4: Determine the points of intersection between the circle, x2 +y2 = 3, and the hyperbola, xy = 1. As shown in the plot, there are four roots. However, it is enough to find only one root because the others can be deduced from symmetry. a) Solve the system of equations using the Successive Substitution method,starting with the initial guess xo = 0.5 and yo = 1.5. Show two complete iterations. b) Solve the system of equations using Newton-Raphson method, starting with the initial guess xo= -0.5 and yo =-1.5. Show two complete iterations. Evaluate &s for the second iteration. c) Solve the system of nonlinear equations by calling the MATLAB function newtmult. Yoursolution should achieve an accuracy of 6 significant figures. Report the solution, number ofiterations and errors. See Answer
Q5: Use eigenvalues and eigenvectors to find the general solution of the following system of differential equations \mathrm{d} / \mathrm{dt} \times(\mathrm{t})=-\mathrm{x}(\mathrm{t}) \quad \mathrm{d} / \mathrm{dt} \mathrm{y}(\mathrm{t})=-4 \mathrm{x}(\mathrm{t})+4 \mathrm{y}(\mathrm{t}) \text { First find the eigenvalues } \lambda_{1}, \lambda_{2} \text { and the corresponding eigenvectors } V_{-} 1, V_{-} 2 \text { of the matrix of coefficients. } \text { Write the eigenvalues in ascending order (that is, } \lambda_{1} \leq \lambda_{2} \text { ): } Write the eigenvectors in their simplest form, without simplifying any fractions that might appear and one of components is 11 or –1–1:See Answer
Q6: a) The gcd(-12, 22) is. b) The gcd(188, -56) is c) The gcd(-546, - 428) is . Are 13, 67, 124 pairwise relative prime? O Yes O No Are 3, 52, 31 pairwise relative prime? O Yes O NoSee Answer
Q7: 10. (15 points total)Small hard particles are found in the molten glass from which glass bottles are made. On average 20 particles are found in 100 kg of molten glass. The distribution of these particle sis Poisson. If a bottle made of this glass contains one or more such particles it has to be discarded. Bottles of mass 1 kg are made using this glass. What is the probability that a randomly chosen bottle needs to be discarded? (b) (5 points) Using the expression above for the probability that a randomly chosen bottle is defective, determine the probability that in a randomly chosen batch of 10 bottles, 4out of the 10 are defective? (c) (5 points) Suppose you randomly sample the bottles one by one, what is the probability that the first defective bottle found is the 5th sample?See Answer
Q8: \text { We consider the matrix } A=\begin{array}{ccc} -1 & 0 & 0 \\ -1 & 2 & 0 \\ 0 & 4 & B \end{array} \text { Write the eigenvalues of } A \text { in ascending order (that is, } \lambda_{1} \leq \lambda_{2} \leq \lambda_{3} \text { ): } \text { (ii) Write the corresponding eigenvectors }\left(\vec{\nabla}_{1} \text { corresponds to } \lambda_{1}, \vec{V}_{2} \text { corresponds to } \lambda_{2}, \vec{\nabla}_{3} \text { corresponds to } \lambda_{3}\right. \text { ) } in their simplest form, such as the components indicated below are 1. Do not simplify any fractions that mightappear in your answers. (iii) Write the diagonalisation transformation X such that X^{-1} A X=\begin{array}{ccc} \lambda_{1} & 0 & 0 \\ 0 & \lambda_{2} & 0 \\ 0 & 0 & \lambda_{3} \end{array} and such that X has the following components equal to 1, x21 = X22=X33=1:See Answer
Q9: 9. Find the radius of convergence of each of the following power series: \text { (a) } \sum n^{3} z^{n}, \text { (b) } \sum \frac{2^{n}}{n !} z^{n} \text { (c) } \sum \frac{2^{n}}{n^{2}} z^{n} \text {, } \text { (d) } \sum \frac{n^{3}}{3^{n}} z^{n}See Answer
Q10: D. Suppose that the coefficients of the power series sigma a z are integers, infinitely many of which are distinct from zero. Prove that the radius of convergence is at most 1.See Answer
Q11: 1)do you think provide the more valid data? With reference to the sample preparation methods employed in the practical, and how the portable XRF instrument works why do Present a basic statistical analysis on the XRF data. Which of the particle size fractions you think this might be?See Answer
Q12: This practical includes a number methods that you would expect an accredited lab to employ as part of their quality control system. Give two examples. Describe how they give confidence to the validity of the data.See Answer
Q13: From the ICP data which of the two wavelengths measured for iron do you think give the most valid results and why?See Answer
Q14: Exercise 7. Give an example of a set Ø + ACQ that that is both open and closed-in Q. Justify your answer.See Answer
Q15: Describe one additional QC methodology and explain how it might be used here. An example might be that demonstrated in validating the ICP data from the Birmingham canal dredging study.See Answer
Q16: Assess the ICP data. Describe the main methodological controls on the values reported.See Answer
Q17: Exercise 10. Let (X,d) be a metric space and let A, B be two closed subsets of X such that AUB and A n B are connected. Prove that A is connected.See Answer
Q18: 3. Find the exponential Fourier series of the waveform and draw the frequency spectrum See Answer
Q19: 4. Plot amplitude and phase spectra for figure in question 2.See Answer
Q20: 5. Find the trigonometric and exponential FS of waveform given below See Answer

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