Q6. Given M = {(x, y), 1 < a² + y² < 4} c R². (a) Determine dM and aM. (b) Is M connected? Pathwise connected? Justify. (c) Let N = {(, y), x² + y² < 4}. Is N = M ? prove your answer.

Q: Find the length s of the path (3 + 10r,9 + 2r312) over the interval 5 < t < 6.

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Q: 4. Evaluate fe fds where f(r, y, z) , ) and 0 <t< 1. = 12r2 and the path C is given by r(t)

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Q: Evaluate I = √x+e²×¯¹dx 2x-1 a) By using Trapezoidal rule with 12 segments b) By using Simpson's one…

A: The given integral is I=∫17x+e2x-1dx. To Evaluate: (a) Using Trapezoidal Rule with 12 segments. (b)…

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Q: (b) If V'u =0 for x2 0 and if u f(y) on x=0, Show that ,y): +(y-5)

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Q: 3)- find ahe pr patikular solustien of theditfrential equatisnx+Cx =CL given the x6)=a Ax 20 at t=o…

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Q: Q4: 16 u. nin N, then 5 bounded fo belongs t and infinum

A: We define a subsequence  an = Sin(pi/2) + (-1)n = 1 + (-1)n . So, a2n = 2 and a2n - 1 = 0.

Q: Q4) Using Green's theorem to evaluate: (12 marks) Sc [ (x3 – xy³)dx + (y³ – 2xy) dy] Where C is the…

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Q: 5.2 Q3: By using Trapezoidal rule find I | In xdx , where n=6.

A: The objective is to find the integration of the given function in the interval [4,5.2] by using…

Q: Given M = {(x, y), 1 < x² + y² < 4}c R². (a) Determine dM and aM. (b) Is M connected? Pathwise…

A: See the attachment

Q: Given w = xysin(z²), x= s-t, y = s², z= t². Find ay

A: Explanation of the answer is as follows

Q: Q4 If f(x, y) = In(x² + y² ) %3D Prove that dy2

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Q: AISO is isosceles with IS = IO. 22 = 2x+24, mZ3 = 6x– 52, %3D %3D nd mZ1. 1/ 3 bX 2 bx -52-180 S x-…

A: We have to solve the problem

Q: find the Lineartransfromation T:Manz (R)R whose matrit Mp --

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Q: Q3. Let A = {(r, y), 4x² + 9y² < 36} C R² Is A connected? pathwise connected? Justify your answer.

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Q: Use the modified Euler method of Lecture 3, with n 10, to find xt1 = -t, (0) = 0

A: Consider the given differential equation.

Q: In Trapezoidal rule method Error Bound: kf"(a) <ES kf"(b), where k (b-a)3 True False О

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Q: Evaluate ſc(x – y) ds over the path C: F(t) = 4tỉ + 3tj for 0 <t< 2.

A: Given: The curve C: r→t=4ti→+3tj→ for 0≤t≤2 To Evaluate: The integral∫Cx-yds

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Q: 3.2 Evaluate (1² + y³)dz where z = r+ iy with y the path on line from z = 2+i to z = 1+ 2i.

A: Topic:- complex number

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Q: 3. Show that S4 has the presentation (x, y|x* = y² = (xy) = 1). %3D

A: To show: S4 has the presentation x,y| x4=y2=(xy)3=1

Q: dw ,s = 1 If w = 2x? ds + y, x = 2 - s, y = 1+ 2s², then by chain rule when %3D సా | 8

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Q: he IVP d's (p² – 121) dp1 d's ds + s 1 = sin(p) dp3 p2 +1 dp s(7) = -994 ds = 19 dp l7 d's = 4 dp2…

A: Given the IVF (p2-121)d4sdp4+p4d3sdp3+1p2+1dsdp+s = sin(p)s(7)=-994[dsdp]7=19[d2sdp2]7=4[d3sdp3]7=10…

Q: s) Given u(r, s) =r2 - 2s2, wheter= and s = y? - r, compute for du

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Q: 2 Evaluate (2² + y²)dz| where z=x + iy with y the path on line from z=2+i to z=1+2i.

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Q: Compute the length of the path r(t)= (t, 4t3/2, 2t3/2) over the interval 0<t<8.

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Q: {(x. y), z? ly > I» ,N { 4y <1}. < 1}. Given Al (2, y), 2 (a) Is M connected? Pathwise connected?…

A: We are given : M=(x,y),x2-4y2&gt;1 Plotting the region M, we have : The Purple region represents M.…

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A: Length of the cadioid is given by below formulaLength = ∫0πaθ3dθ

Q: Q: Examine y (o.4), Using Runge - Kutta Order 4 method if X = 4n + 2

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Q: 9 Give with an not example of T: W≤X →Y, to XT Ext. .R. boundedag. - keše bes

A: Please find the solution innext step this is an non-example of Tietze extension theorem in Topology.

Q: 3.2 Evaluate (2² + y² )dz where z = x + iy with y the path on line from z = 2 +i to z =1+ 2i.

A:

Q: 5) s how that the line of intersection f the Plas and 5x-2y-Z=0 is Parrliel to theline X+2y-22= 5…

A: Given that plane  ,P1:x+2y-2z=5plane , p2:5x-2y-z=0line ,l:x+32=y3=z-14Normal of P1  ,…

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Q: What is the largest interval on which the following IVP sure to have a unique solution: (t…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 5Is there a value of r sotha X = 1 y = 2 and Z=r is asolution to the linear systen ? If there is ,…

A: Putting equations as;23-11-1241-2 xyz=11-712Augmented…

Q: O The madices of cervadere af r= éat any poind Y, is (@) r?VTt (b) rvite? (C) Vitaž (d)

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Q: {(x, y), 4.x2 + 9y² < 36} C R2 ected? pathwise connected? Justif

A: Connected and path connectedness

Q: Q6. Given M = {(x, y), 1 < x² + y² < 4} C R². (a) Determine dM and aM. (b) Is M connected? Pathwise…

A: Given that M={(x,y),1≤x2+y2&lt;4} (a) Given M is a region enclosed between open circle of radius 2…

Q: In the text we obtained the result (Vx F) = F, OF, ду C3 P (Xp, yo, 0) by integrating over a small…

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Q: Compute the length of the curve r(t) = (21, In(?), r²) over the interval 1 < t < 7. (Use symbolic…

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Q: What is t' (x) when t(x) = In( 2)? 2z2+3z+2 Select the correct answer below: O ť(z) = ? + 4z+3…

A: Given: The function is  t(x)=lnx22x2+3x+2   To determine: The derivative t'(x) of the function…

Q: a the diagram, PU||QT, Ì = 42°, RQS = 82°, = v, UPT =r and QPT = x+40°. Prove that PT|QS. %3D P…

A: Given: PU||QT Concept: Pair of interior angles on the same side of the transversal are supplementary

Q: Prove that for the two solutions of BVPS a" (t) + q(t)x(t) = 0, r(0) = 1, x (5) = 4, y" (t) +…

A: Boundary Value Problem: A boundary value problem is a differential equationtogether with a set of…

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Q: {(x, y), 1 < x² + y? < 4} C R². Q6. Given M = (a) Determine dM and aM. (b) Is M connected? Pathwise…

A: Given that M=x,y∈ℝ2:1≤x2+y2&lt;4. a) Given M is a region enclosed between open circle of radius 2…

Q: Which of the following is a general solution to the followin x²y"+xy'+ (x²– 1)y=0 y=c,(x) +cY_,(x) O…

A:

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.