1. Prove that 1 {2" : n E Z} = {; :n e Z}. 2n

Q: 12. Define def M e-$dWg• s=0 For each t > 0, compute (a) E[M¡]. (b) E[M?]

A: Given  Mt=∫s=0te-s dWs      , where W is a Brownian motion.

Q: 5. Prove that So*(y – 0)dydr =

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Q: E. Prove the following: 四- Iz.l 1.

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Q: 12 3. If u lx + my, v = mx-ly, show that ди 4 + m

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Q: (b) En=1 m

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Q: Compute for the value of tanh (1 – i) Iny + 2.13i In 2 + 2.13i 2 In 5 - 4.25i 2 In v2 + 2.13i

A: To Compute: tanh-11-i Concept: 1. tanh-1z=12ln1+z-ln1-z2. lnz=lnr+iθ+2nπwhere,…

Q: 0 <m < n such that am = e. (Hin

A: Subgroup generated by any element in a finite group will have at most n element where n is order of…

Q: 5. Use Taylor's theorem to write down an explicit formula for R, (1). Conclude that R, (1) + 0, and…

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Q: If f(z, y) – e . In(9y) , then fz is given by: O - 9e In(9y) o - 27 e In(9y) O 9 e In(9y) O 27 e…

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Q: alue of the pur C: |z| = (:-1)(z-

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Q: Show Sn+3 diverges by osing 6n +5 the test for divegence 1こV

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Q: 2. If z = x° In 2.xy, find at 1,-

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Q: 6.13 Show that t: = B,/[s//E=, (x;- %3D VI- r in (6.20). |

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Q: 1 О. 15 Show that %3D (D - Q) k e k

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Q: 4x obtain the section of the equation Ex xl with the condition u oon X4

A: Let us consider the given PDE is P(x,y,u)ux+Q(x,y,u)uy=R(x,y,u) The equation is solved as Determine…

Q: S Prove that L(cos 3t) : %3D (s2+9)

A: We know that Lf(t) = ∫0∞e-stf(t) dt Then Lcos 3t = ∫0∞e-st cos 3t dt                             (1)…

Q: 7. Let Te L(V, V). Prove that T2 = 0 if and only if T(V) c n(T). %3!

A: See the attachment

Q: 1. Prove each of the following results: (a) |zw|=|w-z, Vw, z EC (b) ||zw|≤|zw, w z (c) |z| ≤ Re(z) ≤…

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Q: 1)z = e(4+j3) In(6230") 2) z = (-;)(* 2) z = (-)(r)"

A: (1) The  given complex equation is                                                Z=e4+j3ln6∠30°

Q: S Prove that L(cos 3t) (s²+9)

A: Given L(cos3t)

Q: 2) Prove that: d? L„0) = n;(n – 1) %3D dx²

A: Consider the given information: Let Laguerre polynomial is given by, Lnx=∑k=0n(-1)kk!nkxk…

Q: Suppose that C> 1. Prove that if t > Vm ln C+In C, then ()/F) >C.

A: We will use the following Strining's approximation :  n! ≈ √(2πn) e-n nn To prove the given…

Q: S 1.Prove that L (coswt): %| s2 +w2

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Q: Let Sn=Vn. Find an and show that an O but that Sn=ER=1@k diverges. ->

A: Solution:-

Q: prove that var (XY) == 0} 03+ 2} · o3+ 13.03

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Q: eb Let A E M₁ (TR) be given If AAt = On HOME JATOT prove that A = On - Does this hold if A EMn (C)?

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Q: Show that os fj ex* +en+ 22n+1 + (2n+1)n! 25 e*(22n+3) dx – |2 + ... 5(2!) (n+1)!

A: The given integral is ∫02ex2dx. Using Taylor series, the exponential function can be written as:…

Q: ronskian of the DE : xp(x)y' + 3y = 0 = er² et? et?

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Q: Compute 32+n21-3n n=0 If this is divergent, input "O" as your answer If applicable, submit answers…

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Q: 28. (a) Prove that for x e R" and s < t, Ds(x) C D;(x).

A: The notation Dsx represent  an open ball of radius s, centered at x∈ℝn. Let x=x1, x2, …, xn,…

Q: Find e² in the form of u + jv and le²| for z = 2 + 3nj

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Q: Show for Re(z) > 1 ex – 1 dr = T(z)5(2). et – 1

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Q: S 1.Prove that L (coswt) = %3D s2+w2

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Q: solve the following P.D.E: azu ди + u at əx² u(0, t) = u(1,t) = 0 (1, 0<x < 1/2 (0,1/2 < x< 1 %3D…

A: In this question, we solve the partial differential equation by the method of separation of variable…

Q: 1. If x, y, z E R, prove that (a) \x+y\ < \x| + \yl- (b) |x– y| < |x – z|+|z– y|, and (c) ||지| -…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: 2.4.10 Show that det(e4) = etr(A) for all A E Mn(C). D %3D

A: The given problem is to show the given identity. To show: det(eA)=etr(A) for all A belongs to Mn(C).

Q: 1. Prove that L(t)=. 1 s2

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Q: 2 2 2 1+ 2 t3十 Cn)和 9 Prove that

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Q: Prove that (A = B) where (A = ax + ay + az) and (B = bx + by + bz)?

A: Given- A=ax+ay+az and B=bx+by+bz To Prove- That A=B

Q: 1. Prove that L(t)=. s2

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Q: In GL(2, Z,,). det

A:

Q: 3. If f(x,y,z)= In 4 xyz Ite t3y:²s²) , find f,,f, and f..

A: Given that: f(x,y,z)=lnxyz4+e-(3xy2z2)

Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

A:

Q: Suppose A=LU. Find t, where: 1 0. 01 [2 3 4 L = 1 0. U = [0 3 1 5 3 01 0 0 1 3 0 4 and A = 1 8. 10 t…

A: given that A=LU…

Q: Show that for all x, y E F, there exists e with e < Emachine such that fl(x * y) = (x * y)(1+ €) is…

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Q: 12 Find the work done for a force F N from æ =2 to x =6 m. x2

A: Here we solve this by taking the integration with given limits

Q: B. Prove by direct proof general. P: Vn e ZE, (n – 1) (n+ 1) e Z,

A: I have solved B; you just need to expand (n-1)(n+1) and then use the fact that square is always…

Q: 2. Use a theorem to find the ff. Zx if yz? + z = 5xy – 1.

A: Find out the zx when function is given as yz2+z=5xy-1 or yz2+z-5xy+1=0

Q: Prove that e“ p.w. on R. n! n=0 M8

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Q: 13. a) Show that EX=o (-1)*(")=0, and then verify that ) +C) + C) + - - G) +G) + () +- ... = b)…

A: We will use of binomial theorem to solve it

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.