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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: 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
Q10: 5. Suggest an algorithm for a search engine that lets you plan travel connections between cities. Its input is a list of possible connections, where each connection is given as a combination of a starting time, an ending time, a starting location and a destination location. (For example, one such connection could be "a train departing from London at 08:48, arriving in Manchester 10:55".) 1. Describe an algorithm for finding a fastest journey from one city to another, with a given starting time. Note that a journey can consist of many connections. [15 marks] 2. Explain how to adjust your algorithm to accommodate minimum change overtimes, e.g., when switching from one train to another at least a 20-minute margin should be given, and when switching from one plane to another a margin of at least one hour is required. [10 marks]See Answer
Q11: 1. Illustrate the Kruskal's algorithm on the graph I. In addition show how Union-Find data structure changes throughout the algorithm. [15marks) 2. Using Kruskal's algoritm, determine how many minimum spanning trees (MSTS) the graph I'. [10 marks] See Answer
Q12: Exercise 9. Let (X, d) be a connected metric space and let A be a connected subset of X. Assume that the complement of A is the union of two separated sets B and C. Prove that AUB and AUC are connected. Prove also that if A is closed,then so are AUB and AUC.See Answer
Q13: 7. The First Isomorphism Theorem has two important corollaries: the Second Isomorphism Theorem andthe Third Isomorphism Theorem. For this exam, we will investigate the Third Isomorphism Theoremfor rings: Theorem 0.1. (Third Isomorphism Theorem for rings) Let I and J be ideals of ring R, with I C J.Then I is an ideal of J, and (R / I) /(J / I) \simeq R / J Note that this theorem allows us to greatly simplify cases where we would construct a factor ring out ofanother factor ring. For example, (Z/18Z)/(9Z/18Z) = Z/9Z This question will walk you through the steps for the proof of the Third Isomorphism Theorem. Considerthe homomorphism o : R/I → ŘĮJ with 6(a+I) = a+J. (I will allow you to assume o is a well-definedhomomorphism on this exam). Prove I is an ideal of J. (This allows us to define J/I) \text { Prove that ker } \phi=J / I \text {. } \text { s) Prove } \phi(R / I)=R / J \text {, i.e. prove } \phi \text { is onto. } d) (4 points) Use the First Isomorphism Theorem on o to prove the Third Isomorphism Theorem.(Using parts (a),(b), and (c), you are able to write part (d) with only one or two lines of proof).See Answer
Q14: When exam scores are low, students often ask the teacher whether he or she is going to "curve" the grades. The hope is that by curving a low score on the exam, the students will wind up getting a higher letter grade than might otherwise be expected. The term curving grades, or grading on a curve, comes from the bell curve of the normal distribution. If we assume that scores for a large number of students are distributed normally (as with SAT scores) and we also assume that the class average should be a"C," then a teacher might award grades as listed in the table below. A1.5 standard deviations above the mean or higher 0.5 to 1.5 standard deviations above the mean within 0.5 standard deviation of the mean D 0.5 to 1.5 standard deviations below the mean F1.5 standard deviations below the mean or lower Suppose a teacher curved grades using the bell curve as in the table above and the grades were indeed normally distributed.What percent of students would get a grade of "F"? Round your answer as a percentage to one decimal place. What percent of students would get a grade of "B"? Round your answer as a percentage to one decimal place. Suggestion: To find the percentage of students getting a grade of "B," subtract the percentage of students 0.5 standard deviation or less above the mean from the percentage of students 1.5 standard deviations or less above the mean.%See Answer
Q15: What is the linear combination of gcd(25,5)? \square \operatorname{gcd}(25,5)=4 \cdot 1+5 \cdot 5 \operatorname{gcd}(25,5)=5 \cdot 4-1 \cdot 25 \square \operatorname{gcd}(25,5)=-4 \cdot 5+1 \cdot 25 \square \operatorname{gcd}(25,5)=25 \cdot 0+5 \cdot 1 Greatest common divisor gcd(124,24) is_(1)– Linear combination of greatest common divisor of 124 and 24 is Find the linear combination of gcd(345,50) is (Using recursive formula) Therefore, linear combination of gcd(345,50) is 345 \cdot-(5)-50 \cdot-(6) The linear combination of gcd(546,234) is 546 \cdot-(1)_{-}+234 *-(2) See Answer
Q16: Prove that the below function is surjective: f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=x^{2}-2 x-2 Show your work, and give a counterexample if you find that the function is not surjective.See Answer
Q17: 12.Find the equation of the following polynomial function: See Answer
Q18: (a) In the network in Figure 3, it is known that there is a flow of 12. Explain how thatflow can be obtained. State whether that flow is a maximum flow. (b) Calculate \vec{D}\left(S_{2} \bar{S}\right) (c) Explain the difference between a s-t cut and the capacity of a s-t cut. (d) Determine whether the following sets of edges are s-t cut sets for the network inFigure 3. If any set is an s-t cut set then write its capacity. -\{(\mathrm{a}, \mathrm{c}),(\mathrm{s}, \mathrm{c}),(\mathrm{s}, \mathrm{d})\} \{(\mathrm{s}, \mathrm{a}),(\mathrm{s}, \mathrm{c}),(\mathrm{s}, \mathrm{d})\} \text { - }\{(\mathrm{s}, \mathrm{a}),(\mathrm{s}, \mathrm{c}),(\mathrm{d}, \mathrm{t})\} Figure 3: Q4(a),(d): The capacity of the edges (b) and the current flow (a) in each edge isgiven (alb) for each edge. (e) State whether the network in Figure 4 satisfies the two constraints in the net workflow problem. Explain your answer. Figure 4: Q4(e): The capacity of edges (b) and the current flow (a) in each edge is given(alb) for each edge State whether the network in Figure 5 has a perfect matching. Explain your answer. ) The network of Figure 6 (overleaf)has source S, sink T, a capacity on each edge anda capacity on certain vertices (denoted by circled numbers). Convert the network toa basic network with only edge capacities and then use the Ford-Fulkerson algorithm to find the maximum flow from S to T. See Answer
Q19: . Determine approximate solutions for each equation in the interval x = [0, 2pi], to the nearest hundredth of a radian. \text { a) } \sin x-\frac{1}{4}=0 \text { b) } \cos x+0.75=0 \text { d) } \sec x-4=0See Answer
Q20:8. Suppose that the manufacturer of a gas clothes dryer has found that when the unit price is p dollars, the revenue R (in dollars) is R(p) = -4p² +8,000p. (a) At what prices p is revenue zero? (b) For what range of prices will revenue exceed $1,000,000?See Answer

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