Mathematics Questions and Answers May 01, 2023

A check digit is a way of verifying that a particular sequence of numbers has been entered correctly. They are found in many places for example, bar codes, credit card account numbers, and Australian tax file numbers. In this question we will study the example of 13digit ISBNS, which are used to identify books. 9788175 257665 EAN Group (a) (1)

Let A Mn, n (R). Prove that if A has a zero column, then A is singular. E

Let A E Mn, n (R) and c E R. Prove that CA is nonsingular c0 and A is nonsingular.

Let A (a) (b) 3 2 1 1 102 2 which is known to be nonsingular. Find A1. Express A as a product of finitely many 3 x 3 elementary matrices.

Explain why there is no power series t ant. n0

2. For the matrix B given below 1120 0111 1 1121 2 2131 a. Find the basis for Row B, Col B, and Nul B. PMC 4.2.5 points b. Find the dimension of Row B, Col B, and Nul B. 1,1, 1 points) c. Find the rank of B. 1 point D,2. For the matrix B given below 120 0111 1 1 1 2 1 2 2131 T a. Find the basis for Row B. Col B, and Nul B. 14.2.5 points b. Find

1. Let U, A, and B be sets such that AC U and BCU, where U is the relative universal set. The symmetric difference between A and B is the set of elements that are either in A or in B but not in both, and it is denoted by . (a) Write AAB in terms of A, B, A, B and their unions and inter sections. (b) Draw the Venn Diagrams representing the

Question 1. Let 7 be the class of subsets of N consisting of and all subsets of N of the form E, n, n1, n2,... with ne N. (i) Show that r is a topology on N. (ii) If A 2, 6), find A. (iii) If A 5, 11, 31), find A.

Obtain the temperature distribution as a function of (x, y, t) T 1 T ax dy a t Where at x 0 y 0 T To, and at x L T dy 0, y W T T Show all solution steps and simplify all equations Show all values of the eigen values T dx and the initial condition, T(x, y, t 0) f(x,y) 0 Apply the nonhomogenous initial condition to determine the

Use the substitution, t tan Evaluate to find I se secx dx. 4x5 6x4 2x 3x 5x 7 S x3 2x x dx.

Compute sin (2ln(v)) V d v. Do not forget to add C (upper case) to your answer. This question accepts numbers or formulas. Help I Switch to Equation Editor Preview

Compute f cos cos (t) e5 sin(t) dt. Do not forget to add C (upper case) to your answer. This question accepts numbers or formulas. Help I Switch to Equation Editor Preview

An object is originally located atz 3. Beginning at time zero, the object moves along the axis with velocity given by 5 t 22 Determine the exact location of the object at time 3. If the answer is a fraction do not enter its decimal approximation but the exact value like 13, not 0.3333. This question accepts numbers or formulas Help I Switch to

A manufacturer has found that when m employees are working, the number of units of product produced per day is q 10m 4900 700 The demand equation for the product is 8qp 19,300 0 where p is the selling price when the demand for the product is q units per day. (a) Determine the manufacturers marginalrevenue product when m 240. (b) Find the

Compute S sin(3t) cos (3t) dt. Do not forget to add C (upper case) to your answer. This question accepts numbers or formulas. Help I Switch to Equation Editor Preview

Evaluate the integral 224x2x dz Reminder type sqrt (2) to enter 2. For example, to enter 2 21 type 2 (34) sqrt (21). (Use preview to check for syntax correctness before grading.) This question accepts formulas in Maple syntax. Plot I Help I Preview

The meanreverting OrnsteinUhlenbeck process is the solution X of the stochastic differential equation dXt (m Xt)dt od Bt where m, o are real constants, B E R. b) Find EX and VarX E(Xt EX).

Prove by induction that 11 8 is a multiple of 3 for all n E N. (7) Prove that the function given by x f(x) 2 3 x 2 is 11 on R and find a formula for the inverse of f. (6) O Consider the functions f(x) 2x 1 and g(x) x 1. Indicate the domain of definition of each of the following functions f, g, fog, gof. (6)

Find the solution u(x, t) of the secondorder equation Utt 2 Uxt 3Uxx 0, xER, t 0. Show that the function 1 3 B(t) L (23 0(2,1) 2 u(x, 1)) da E ut (x) t) dx 0 is an energy function for any solution of () that vanishes whenever x 0, 1. ()

(i) Let f(a) be a polynomial such that f(A) Onxn. Convince me that m(a) f(a). (ii) Up to similarity, classify all 5 x 5 matrices such that A 5A 615. Hint Let f(a) a 5a 6. Hence, by hypothesis, f(A) 05x5. By (i), mA(a) f(a). Hence m(a) a 2 OR mA(a) a3 OR m(a) f(a). To 0 0 1 (iii) Let A be a 4 x 4 such that A is similar to H 00 1 0 0 0

a) Given a stochastic matrix A with nonnegative entries whose row sums are equal to 1, which also satisfies the PerronFrobenius assumptions. Its invariant distribution is its unique left eigenvector. Find the unique left eigenvector and then compute its derivatives with respect to a. 0.6 a 0.4 a 0.2 0.8 a 20 b) Given A is any stochastic matrix of

QUESTION 1. Assume A, B are similar n x n matrices, say A QBQ. (i) We know (a) CB(a). Prove m(a) mB(a). (ii) Assume that a is an eigenvalue of A and v, v2, ..., Uk is a basis for Ea (A). Prove that Qu, Qu2,..., QUk is a basis for Ea (B). Hint Observe that BQ QA and since Q is invertible, Qw On iff w On

(a) (b) Let k, L 0 and consider the nonhomogeneous heat equation ut(x, t) kuxx (x, t) H(x, t) over the interval 0, L subject to general initial and Dirichlet conditions u(x,0) f(x), u(0, t) a(t), u(L, t) b(t). Show that at most one solution exists for any given choice of a, b, , H. Derive a formula for the solution of the problem in part (a)

Suppose that a decision tree model has been built for a twoclass classification problem in a twodimensional space. The decision boundaries as shown in the figure below, where x and y represent the two axes of the space, and x, y 0. Suppose that five samples, as shown in the table below, are used to testify the performance of the classifier. YA 3

We say that a function f A R is Lipschitz on A if there is some fixed k 0 such that for all x, y A, we have f(x) f(y) kxy. (5 points) Let A be connected. Suppose f A R is differentiable. Prove that if f has bounded derivative on A, then f is Lipschitz on A. Indeed, the converse is also true but this is not necessary for this problem. (5

Given curve is a boundary of domain r. y ER y 4 Ar0 Ay0 oriented counterclockwise. (a) Compute circulation of f(x,y) (ry, y2y) over the curve. (b) Suggest another approach of the computation (just one equality). For a vector function f(x, y) (In(x y), 1) compute the circulation over a curve (a) y R y 1 oriented counterclockwise. (b) R2 (1)

The algebraic system reads Axb, where A 2 0 2 0 2 1 0 1 2 1 01 2 b 2 (a) Make a Maple procedure for Successive OverRelaxation (SOR). (b) Perform 15 iterations of Jacobi and GaussSeidel methods, starting from xo 0,0,0,0. (c) Try to find the best w with which the SOR method converges fastest during the first 15 iterations. (d) Find the spectral

Make a maple procedure for Gaussian elimination with partial pivoting and solve the below linear equation system

21x2x3 1 21 224x3 2 2x1x22x3 3 Compare the result with Gaussian elimination without pivoting.

Multiply below natives together

1 et 1 4et 2et 1 e 2t 2t e 2e2t t et. (et2t1) 5 (3t1)et1 5

Use the method of separation of variables to find the product solution of the PDES (a) u Uy3u (b) xu 2yuy

What is T(n)'s order of growth? a. T(n) = 2T(n-1)+4n^2+3n+2 b. T(n)= 2T(n/2)+ 4n^2+3n+2 Solve these two by using master theorem

PDF Given the initial value problem y(3) 3y 4y 12y 0 y(0) a, y(0) 0, y(0) 0 1. Solve the initial value problem by hand. 2. Solve the same problem using the Matlab or Mathematica and find the value of constants using the given initial conditions. 3. Plot the graph of the solution for a 3, a 2, a 1, a 1, a 2, a 3 in single box.

Suppose you want to estimate ln(1.1). The nth Taylor polynomial Tn(x) of the function f(x) ln(1 x), centered around c 0, can be used to find the approximate value of ln(1.1) by setting x 0.1 in Tn(2). Use the formula for the error term En(x) in Proposition 2.9 to de termine the smallest value of n so that Tn (0.1) approximates In(1.1) to

Let P SCU o be the stereographic projection introduced in our course. If y a, b C is a continuously differentiable simple curve, that is, 7 is injective, then answer the followings (a) Find the formula for the length of the curve (a, b) C C. Give the reason for your answer clearly. Hint Use the notion of the chordal distance d(z, z) 2zz 1 21 22

Let c, R > 0 and suppose that , : R R are smooth functions such that (x) = (x) = 0 whenever |x| R. Let u(x, t) be the solution of u_tt c^2u_xx = 0, u(x, 0) = (x), u_t(x, 0) = (x), where xR and t0. Show that there exists some t_0 >0 such that

2 10 4 14 (x, t) dx c u(x, uz(x, t) dx for all t to.

In 2009, the average price of a new home sold in a country was $294,000. In 2004, the average price of a new home in the same country was $297,000. Let y be the average price of a new home in the year x, where x=0 represents the year 2004 a. Write a linear equation that models the average price of a new home in terms of the year x [Hint: The line

Find the value of c: x^(2)+10x=5(c+4x)

Jermaine spent $204 on 21 shirts. Large shirts were $12 and small shirts were $8. How many larges did he buy?

If -1-2 is is one zero of f(x)=x^(3)+6x^(2)+13x+20, what is the other complex zero?

The below sequence is given in recursive form. a_(1)=8 a_(n)=2a_(n-1)+5, for n>=2 What is the value of the third term of this sequence?

Divide the rational expressions. Write the answer in its fully factored form. (((7y-35)/(15y^(2)-10y)))/(((7y-56)/(5y^(2)+15y)))

In the question ODE is given, where y y(x) 2yy 4x x 5 0 At x 0, the value of y is given as 1. Solve the equation for the interval of x 0 1. (1) Find by Forward Eulers method with a stepsize of 0.5. (II) Find by Modified Euler (Heun) method with a stepsize of 0.5.

Student tickets to the school football game cost $5.80 each and general admission tickets cost $7.40 each. The school sells 380 tickets for a total of $2372. Which system of equations models this situation?

1.During a two-hour period, the temperature in a city dropped from a high of 57F to a low of 19F. What was the range of the temperatures during this period? F 2. On an examination given to 8,400 students, Hal's score of 405 was higher than the scores of 3,752 of the students who took the examination. What is the percentile for Hal's score? (Round

1.On a reading test, Shaylen's score of 455 was higher than the scores of 4288 of the 7238 students who took the test. Find the percentile, rounded to the nearest percent, for Shaylen's score. th percentile 2. On a placement examination, Rick scored lower than 1176 of the 12,700 students who took the exam. Find the percentile, rounded to the

A cyclist rode the first 26mile portion of his workout at a constant speed. For the 16mile cooldown portion of his workout, he reduced his speed by 5 miles per hour. Each portion of the workout took the same time. Find the cyclists speed during the first portion and find his speed during the cooldown portion. The cyclists speed during the first

a. Use the formulas shown below to write the company's profit function, P, from producing and selling x radios b. Find the company's profit if 19,000 radios are produced and sold. C(x)=126,000+43x and R(x)=50x

for the given below vector use the expansion method to find their cross product. A=2i-3j+4k and B=1+2j+5k

Determine for what value of a will the expression x^(2)+2x-a divided by x+4 give remainder 3?

Factor the below polynomial given that it has a zero at 3 with multiplicity 2 . x^(4)-8x^(3)+13x^(2)+30x-72

In an arithmetic sequence, the nth term an is given by the formula a a (n 1)d, where a is the first term and d is the common difference. Similarly, in a geometric sequence, the nth term is given by a a r1. Here r is the common ratio. Use these formulas to n1 determine the indicated term in the given sequence. The 30th term of 1, 3, 5, 7,...

Use the remainder theorem to find P (3) for P(x)x2x3x9. Specifically, give the quotient and the remainder for the associated division and the value of P (3). Quotient 11 Remainder P (3) 0 X

Give an example of a symmetric 3 x 3 matrix A such that the corresponding quadratic form Q(z) A is negative semidefinite and has no cross product terms the range of the corresponding linear transformation T(7) Az is 2dimensional and one of the eigenvalues of A has algebraic multiplicity equal to 2. A Find the minimum value of Q(z) 4, on the

The outcomes of an experiment and the probability of each outcome are given in the table below. Outcome 5 6 7 8 9 10 Probability 0.4 0.34 0.1 0.08 0.07 0.01 Compute the expectation for this experiment.

The table below shows the population of a herd of elk between 2000 and 2011. Complete parts (a) through (c) below. Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 Population 509 480 459 443 424 406 390 383 362 347 334 324 REITS a. Use a calculator to find an exponential model that fits the data. Let y be the population and let x

Find the quadratic function that is the best fit for f(x) defined by the table below. X 2 4 6 f(x) 4.48 19.04 78.24 173.12 www f(x) (Use integers or decimals for any numbers in the expression. Round to the nearest hundredth as needed.)

Solve the equation. Check your solution(s). 26 213242 13 Question 2 Solve the equation. Check your solution(s). 122 2

920 42 1616 225 Simplify. 8a16 Use ControlCommandSpace Bar and search superscript to insert any exponents Question 2 216 Simplify. 3x1824x 27x4 Use ControlCommandSpace Bar and search superscript to insert any exponents 1 pts

what would be the answer for it?

134080011 2 Which number is equivalent to A 2 B 9 81 D 729 3

In the table below, the mathematical steps to factorize the trinomial have been provide Provide an explanation for each step. Factorize 6a 12a18a Step 1 2 2.2 3 4 5 6 7 Mathematical steps 6(a2a 3a) 6a(a 2a 3) 6a(a3aa3) 6a(a3a) (a 3) 6aa(a 3) (a 3) 6a(a 3)(a 1) 6a(a 3)(a 1) 87 Explanation Factor out 6 which is common. 2 The final

Solve the following for X Jx10X2

As an intern for an office you are tasked with gathering lunch orders and bringing back the receipt for reimbursement. You take orders for a local deli that sells turkey sandwiches, paninis, and salads. On Monday, you place an order for 6 turkey sandwiches, 9 paninis, and 9 salads that totaled to $184.14. On Wednesday, you place an order for 2

The Anderson Creamery TM offers two types of ice cream available commercially in quart packaging. Their Victorious Vanilla flavor requires 3 cup(s) of cream and 4 egg(s) for each quart, whereas their Magical Mocha requires 4 cup(s) of cream and 3 egg(s)( for each quart. If the Creamery has 2,170 eggs and 2,310 cups of cream on hand, How many of

Find the interest rate r if the interest is compounded continuously and no withdrawals or further deposits are made. Initial Amount: $1,500; Amount in 5 years: $2,000

We can use methods from linear algebra to find fits for various functions through data points, and learn about the nature of such fits. Use the digits of your stu dent number to construct five numbers in the following way if the digits are dd2d3d4d5d6d7, write down the twodigit numbers y d2d3, y2 d3d4, y3 d4d5, y4d5d6 and y5d6d7. So, for

Rearrange the below formula and find S? 0.44 = (6764)/ Sqrt [35 + S^2]

The formula APe describes the accumulated value, A, of a sum of money, P, the principal, after t years at annual percentage rate r (in decimal form) compounded continuously. Complete the table for a savings account subject to continuous compounding. Amount Invested Annual Interest Rate 9500 9 Accumulated Amount Time t in Years Double the amount

Use common logarithms or natural logarithms and a calculator to evaluate the expression. log 0.5 18.7 Use the change of base property to rewrite the given expression. log 0.5 18.7 (Do not evaluate.) Evaluate the expression. log 0.5 18.7 (Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places as

K Write the equation in its equivalent exponential form. Then solve for x. log 3(x1) 4 log 3(x1) 4 in its equivalent exponential form is X (Do not simplify.)

Let f be a function, defined and continuous on the entire real line, with the property that for every real number C, there is another real number D such that x D f(x) C. (An informal way to say this is f(x) as x o. For example, the function f(x) r has this property given C, we may take D to C but the function g(x) does not.) be Apply the

In image editing software such as Photoshop generally express positions with respect to an (x, y) coordinate system with some origin (0, 0). Rotation about (0, 0) can be thought of as a linear operation on the coordinates. In a 300 400 image with bottom-left corner at the origin, the bottom right corner is rotated to (288, 84) and the top left

need to express a part of x r 12x sin t answer as a convolution integral. x(0) 1, r(0) 0

For the function f(x) X1 X3 find f(x), the third derivative of f, and f(4) (x), the fourth derivative of f. Begin by finding the first and second derivatives. What is the first derivative f(x) and f(x)

2 For the function f(x) x 100, find f(x). Then find f(0) and f(9).

1. 2. 3. Given A Given X 3 4 1 0 7 3 and B and Y 1 5 1 Given the matrix D 2 3 3 2 Determine AB and BA 1 9 Show that (XY)(X Y) X Y 1 1 3 4 2 2 , find the inverse matrix of D. (3) (10) (8)

(a) On the grid, draw the image of (i) triangle A after a reflection in the yaxis, (ii) triangle A after a translation by the vector (3) (b) Describe fully the single transformation that maps triangle A onto triangle B.,5 2 2 3 B 8 9 10

Determine without converting the form and leave your answer in the form abi, where a, be R. (0) (ii) 2 (12)(13) (12i) (31) 2i (2) (4) Simplify the following complex number, without changing the polar form and leave your answer in polar form 3 (24) (344) 423 (6)

Given Nitt that 1 t 2 Show that 5x, e 99x 2 333x 35 7x 2 8 1 16 and t 2 3 e 1 , 141000 2 31 3 tici 3 3819 x and write down the rading of convergence at the series.

Assume that investors want to plan what should be a fraction x of their assets j 1,2,..., n. To help with the planning they use historical data that show what was the annual return a 20 for the asset j in the years i1,2,..., m. The risk aversive investor tries to maximize the total annual return by using the data aij as follows In this year, the

In a 4D Euclidean space with metric ds dx dy dz dw, we define the 3D surface x y w R2, with a positive constant R. This surface is called a 3sphere. We introduce the periodic coordinates (x. 0, 9) on the 3sphere, defined through x Y 2 Rsin x sin cos p. Rsin x sin sin p Rsin x cos 0, such that wRxy22 R cos x. Show that the line element of

(a) The transformation from Schwarzschild coordinates (t, r, 0, 4) to outgoing Eddington Finkelstein coordinates (u, r, 0, ) is given by U 0 tr. T 0 where r. satisfies dr.dr (12Mr). Use the rule for the coordinate transformation of metric components to obtain the line element in the (u, r, 0, ) coordinates. (b) Show that radial worldlines with

An employee at Best Beans Expresso is earning 13.70 per hour. If the employee is paid time and a half for working overtime (any hours over 40 hours in a week,) write an expression for the weekly earnings of the employee (E) as a function of the number of hours in a week the employee works (h). Be sure your function is defined for 0 to 60 hours of

1 Given that S 3 a) b) 5 1 p 2. 63 If S5, find value of p. If p12, find S by using adjoint matrix or cofactor method. (5 marks) (10 marks)

Amount of Sales to be reported in T2125 Should Advertising expense be claimed ? If yes, How Much Should meals and entertainment expense be claimed? If Yes. how much Should Senators tickets be expensed? Yes or No Should Red blacks tickets be expensed? Yes or No Should storgae locker fee be claimed? If yes, how much Should Office Furniture be added

According to Newton's law of cooling, the temperature u ( t ) of an object satisfies the differential equation \f

Consider the Cauchy problem for the autonomous system : x' = f(x), r(to) = To , in which f : $2 -> R" is a Lipschitz function and $ is an open set of R". a) Show that if f is also bounded in ? then the problem has an unique so- lution define in some open set Ia(to) for any (to, To) E R" x Rn

Hey! I'm having a trouble with my calc homework. Please help me! A 17,500-cubic-foot room has 300 smoke particles per cubic foot. A ventilation system is turned on that each minute brings in 700 cubic feet of smoke-free air, while an equal volume of air leaves the room. Also, during each minute, smokers in the room add a total of 10,000 particles

answer to this problem Write the exponential equation: 6 y = x in logarithmic form. o log 6 x = y O log 6 y = X O log y x = 6 Ology 6 =X

CodeLab - Web Based Programming Learning System Write the definition of a function named quadratic that receives three double parameters a, b, c. If the value of a is 0 then the function prints the message "no solution for a=0" and returns. If the value of "b squared" 4ac is negative, then the code prints out the message "no real solutions" and

Please help 4. (Strang 2.7.9) The product of two permutation matrices, Pi and Pi, is also a permutation matrix. (a) Give an example of two permutation matrices such that P1P2 = P2P1, where Pi * P2. (b) Give an example of two permutation matrices such that P3PA # PAP3

t) Find the values of a and b that make the following function differentiable for all x f(2) = ax +b , if x > -1 ba2 - 3 , if 2

de Moivre's Theorem - ( r (cos + i sin )) n = r n (cos( n ) + i sin( n )). Use de Moivre's theorem to find the two square roots of z = 04i.

I need help 1) Solve the following system by matrix row operations: x + V - 3=-3 2x - 3y + 4: =23 -3x + v - 2: - -15

According to CTIA.org, there were 327.6 million cell phones in the U.S. in 2014. The U.S. population was approximately 317.9 million at that time. How many cell phones per capita were there in the U.5. in 2014 and what does this value tell us

Could you explain the problem? (15 points). Suppose A is a (4 x 4) matrix. (a) If det A = 1, then what is det (2A)? det (2A) = (b) If the characteristic polynomial of A is p()) = () + 1)2(1 - 1)2, then what is the characteristic polynomial of 2A

Keys are made by cutting incisions of various depths in a number of positions in a blank key. Suppose there are eight possible depths of incision. We need to be able to make one million different key types: for simplicity of calculation assume that we need at least 2^20 different keys (which is just a little over a million). How many different

In 1980, a Domino's large pizza cost $4.99. What would be the cost of that pizza in 1982 dollars? Express your answer rounded to the nearest cent. $

5. Jane can paint the office by herself in 7 hours. Working with an associate, she can paint the office In 3 hours. How long would it take her associate to do it working alone

Which of the following techniques yields a simple random sample? a. Choosing volunteers from an introductory psychology class to participate b. Listing the individuals by ethnic group and choosing a proportion from within each ethnic group at random. c. Numbering all the elements of a sampling frame and then using a random number table to pick

Assume that the figures shown to the right are similar. Given the lengths of sides and measures of angles in the left figure, what information is known about the right figure? 38 The measure of angle I is 38 0 60 48 The length of side is