44. (a) Let a, b, ceZ with (a, b) = 1 and a | bc. Prove that a c. (b) Provide a counterexample to show why the statement of part (a) doe not hold if (a, b) # 1. С.

Q: 11. If AMNP is reflected across the line y = 2, then translated using the rule (x + y) – (x – 8, y –…

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Q: Show that |qbu)|&x + 7元? the closed on interval [-元,T」 fo1 -π2 2くだ gluen that glven gux) for

A: Given g (x) = x +x² To find the maximum value of g(x). We can differentiate the function and equate…

Q: b) %3D Difleventiating both sides writ sc on dy %3D doc Applying chain rule; dy d. du du doc de U =…

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Q: O Discuss he Singularities and Paks ZE Sinz (--2)(Z²-)}(Z"+ SZ+6)(2+7)(214)

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Q: 2.2 If fe R(a) on [a, b] and if Saf da prove that a must be constant on [a, b]. = 0 for every f…

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Q: Show that : sup {reQ:r<a} =a, for each a e R. %3D

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Q: ) Find the fomily of the equ ation of all lines in 3-space perpen diuler to 2x -y +2u u Given line

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Q: 1. Use the Intermediate Value Theorem to prove that the equation 3x - 4x? = 3 is solvable on the…

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Q: 5. Verify that the. functrun.u=1/ xy2? satis fies the threendimensianal.place cop.x+Uyy.+Uzz =o %3!…

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Q: - check the exi's tence of limol of the Lunetiom 41n at X= 1 %3D 2X+/ こ Xラ-3

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Q: 3) Use Green's Theorem to evaluate o-ydx+xydy where C is the line segment from (-2,0) to (2,0) then…

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Q: 2] Let x,yER Prove that VxVy(10x+21y>21⇒(x>11/10 v y>10/21))

A: Let x, y∈ℝ. Prove the statement ∀x∀y10x+21y&gt;21⇒x&gt;1110∨y&gt;1021. Prove the statement using…

Q: Prove that in the pairs of positive numbers ¤, y for æ² + y° = r the sum a+y is greatest when z = y.…

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Q: Suppose n is an integer and z-z3D =r is a contour traversed n dz %3D once in counter clockwise, then…

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Q: 5. If z = In then determine if the following statement is true or false. Justify your answer. + y…

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Q: ove that 1 cosh(tanh) = -1 -1 < r < 1. V1- 2

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Q: C) = ("; ')-(,",) n Q1. Show that : r

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Q: 7. If (x, y, 2) = 3xz³y – y?, find grad o at the point (1, -2, –1) Ans. - (16î + 9j + 4k)

A: The gradient of a scaler point function φ is obtained by applying the del operator ∇ to the scaler…

Q: 6. Prove: 4 |x – y| 0, then x = y Vx, yER

A: To solve Q.6 Prove: If x-y&lt;ε, ∀ε&gt;0, then x=y ∀x,y∈ℝ If (Given) Then (To prove) ∀ε&gt;0,…

Q: 20. Let -(). -- (). (). u = w = If au + yv = w. (x, y E R), then a - y = (A) 2/3 (В) —3/2 (C) 3/2…

A: To find  The value of x-y

Q: Show that is Reciz) e -ImZ p (ii) Im(iz) = Relz). %3D

A: Solution:- we have to show that  (1)    Re(iZ) = - ImZ (2)    Im(iZ) = Re(Z) where , Z = x + iy

Q: Let f be q functiom that admits contırous second partial denvatives Df (X.Y)=(4x²- 2q²x+ 2q°y ,4y³ +…

A: Given, the second partial derivatives              ∇f(x, y)=4x3-2a2x+2a2y,  4y3+2a2x-2a2ywith a&gt;0

Q: 6. so be it r = 3xi + yi – 2zk and f (r) : -- where |r| = r find all of grad(f) and div (-)

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Q: 3. Suppose that f(1) = 2, f(4) 7, f'(1) 5, f'(4) = 3, and f" is continu- %3D | af" (x) da. ous. Find…

A: The given data is: f1=2f4=7f'1=5f'4=3  f'' is continuous  To find the value of ∫14xf''xdx.

Q: If u, v are given unequal numbers anu (un-1+ Pn-1), vn = V (u –1 Vn-1), where n2 2 , %3D Un %3D 2…

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Q: Suppose v(t) = 6t² - 3t on the interval [3, 5]. What is the displacement on [3, 5]? O 160 O 165 O…

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Q: 1. Is the following always true? If yes, clearly explain why. If no, give a specific counterexample…

A: Note: AS per our guideline we are supposed to answer only one question. Kindly repost the other…

Q: 9:Find the absolute exlemum Verlues of flx) x²-4x+24 over the interNal Cl,41

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Q: in de pndent Variables り if メ,Y,2 and z function which achieves the eguation ャ+3x/2=5 Pind a2 in , Y

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Thene m for Voritly functiom Mean Valne fex) 2. 3X -4 +S the

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Q: Verify that the conclusion of Clairaut's theorem is valid, that is, that uxy = uyx u = x4y3 - y4…

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Q: 3. If u= (x+y +z') , prove that +y

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Q: 4. Sketch the proph of the following funcfions; メーメ+4 (x+4 a) f(x) = (x+4) x b) gex) = %3D

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Q: 2.) Use the chain cule to find dz and d ond guen the fllowng: Yoy must show in work that your the…

A: We have to find partial derivatives.  Concept:

Q: Find the condition for the two lines to be coplanar. a1 + bjx + C7x + d, =0 = az + b2x + C2x + d2 az…

A: The given lines are    a1 +b1x+c1x+d1=0a2+b2x+c2x+d2=0a3+b3x+c3x+d3=0a4+b4x+c4x+d4=0  To examine…

Q: Prove false by creating a counterexample: P(AUB) S 9(A) U 9(B).

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Q: Use the following problem: Given that Z1= Z4, prove that 22 = 23. 3 Write your proof (in any…

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Q: 2) Prove by using the pigeonhole principle that among any 17 points in a square of side one there is…

A: Given: A square of side one contains 17 points. To show: There is at least a pair whose distance is…

Q: f Shoo that the interior operat1on has the tollouni ng Propeities. if E, CE2, then interier (E) C…

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Q: Let S-(xeZ vmod12-8) and T (xe2| 4x). Prove that SCT but TgS La s-(rez| 3reZz, x-12r+7) and T-(xeZ|…

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Q: SU SW. Complete the proof that ASTW ASXU. T. V W Statement ZW ZU SU N SW ZS ZS 4 ASTW ASXU 2.

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Q: (2, Prove that Ž (Pn+ Pn+1) hr 9=U (y+1) 27-²8 (14+ 2x=-1) y

A: What is Legendre Polynomial: Legendre polynomials are a set of complete and orthogonal polynomials…

Q: B a. In the figure, if AX = (3x + 1) cm, BX = 4x - 1 cm, AY = 4x + cm, YC= 5x-1 cm and x = 2, prove…

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Q: 3)suppose fle) isadifferentialle fmehon on the intervall E-7,J Such that Hl-7) =4 and fl)=1.…

A: Intermediate value theorem

Q: l? sc tue suell metuozd to fired tue voune of te Solid genented the by rerolning the plene rgim…

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Q: Ihow do you bnd the imit of as x approaches 0?

A: The limit of x+1-1x as x approaches to 0

Q: Suppose that fo sf,:X-gy and g,-3 YoZ Prove that (Hint : First use ce) to show that gfe g,, and then…

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Q: a,b,x,y, are negative numbers a⁵ + b⁵<= 1 x⁵ + y⁵ <= 1 prove that a²x³ + b²y³ <= 1

A: Given information: The numbers a,b,x and y are negative numbers. The conditions are a5+b5≤1,…