25. (a) Prove that Re(z) . Under what circumstances are they equal? (b) Prove that |z+ wl|z |lw. Begin by writing |z + w (z w)(z w).

Q: If Osf(x)s g(x) on [a, ∞) and | g(x)dx is divergent, then | F(x)dx is divergent. Select one: O True…

A: We have to state true or false.

Q: 25-let F(xs.2)=x-x22 @ Find VF 6 write the equution of tngent Plame to the at Paint (2,1,1). Find…

A: The given function is,  F(x,y,z)=x2y2-xz2 We have to find,  a) ∇F b) Equation of the tangent plane…

Q: Q2= A) for E= 2xĭ-3Y3+K %3D prove That divc curl È)=0 %3D

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Q: TQ residue of f(z) = e (z – 1) (z + 9) at z = -9 is 10e-9 :Select one True False

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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: Prove that v = In (x- 2)^2+(y-N)^2 is harmonic in every region does not include the point (2,N) and…

A: Given v=lnx-22+y-N2 where N=44. So, v=lnx-22+y-442                                    ........1 To…

Q: Z1 is complementary to ZM 22 is complementary to ZM Z1 = 2 Given: Prove: M R 2

A: Given: ∠1 is complimentary to ∠M. ∠2 is complimentary to ∠M. To prove: ∠1≅∠2 According to the…

Q: 2. Show that c(x, t) = solves 1 ac(x, t) 2 ax ac(x, 1) ar

A:

Q: If h(x) — За — — 1, j(x) k(x) = 21x? – 26x – 11 , show - 7x + 11 , and - - - - h(x) · j(x) = k(x).

A:

Q: If r = (3 cosh 2t)i + (3 sinh 2t)j + 6tk and t = In 2 then اخترأحد الخيارات 8. 16 and T= 867 O K =…

A: Curvature   k=32867=0.0369088811995386.

Q: Q53- X--) (xij-Xi - X.j+X)=Zero 13D 「に )に prove that e to

A: We need to prove the given equation.

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

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Q: Use Fermat's Theorem in your proof that Vn E Z>², 2 is irrational. I

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Q: Prove H'n(x) = 2nHn−1(x), n = 0, 1, . . .

A:

Q: 2. Let X, Y c R. Prove that X U (Y – X) = Y if it is true. If not, give counterexamples to explain.

A: Given statement is false.

Q: For z C, show that the solutions of cosh(z) = 1 are z= i2km, for ke Z.

A: We need to prove that the solutions of the equation: cosh(z)=1 are z=i2kπ, where z∈ℂ, k∈ℤ.

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: In(1+2x) Find the Padé approximation R1,2(x) for f(r) %3D

A:

Q: o prove that knVA=| Test the seuence (ConViordív) of (10) an = (^_-) ) an 15 ら 9n: 3ntl 3n-1 (の+ (7)…

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Q: The evaluation of S V1 + tan2x dx is C In(V2) +1 In(vZ+1) C In(vZ-1) O-In(vZ+1)

A:

Q: HW klx) = 2x+5 ㄱ 1. Find ke-¹(x) 2. Prove by kokorklok

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Q: 2h 1+h )-2 = 1 – if h<<r Prove that ( %3D r

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Q: 2- Prove that: Jzt=27 dz %3D 2in. Whot is the princinle nart for f(z) = ze %3D

A: Since you have asked multiple questions in a single request we would be answering only the first…

Q: if x, y>o such that x Sy then Vã svy Prove by direct method

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Q: (b) Prove that the multiplicative inverse law is false for S. [That is, don't just write down a…

A:

Q: If Σa, is divergent and Eb, is divergent, then Z(a + b) is divergent.

A: The given statement is false as sum of two divergent series may converge. We illustrate this as…

Q: (b) f(z) = tan¯"(5z) - (In(2"))®

A: We have to find derivative

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

A:

Q: 27. Show that कक्या 1 Vn E Z*. T1\2i+1 2i+2. i=0 (2n + 2)!'

A: Consider the given expression. ∏i=0n12i+1·12i+1=12n+2! Consider the left side of the equation.…

Q: Q/Use Definitien 13.4-1 +o prove that f(x,r}>x+y' is difterentialeat (o,0).

A: Given function is  fx,y=x2+y2

Q: 3) Prove the following statement using the Intermediate Value Theorem (IVT): 2ª + 3* = 4* has a…

A:

Q: 7. Prove that if v >-1/2 ; p) = L{xvJ, (; = xı-at-a) H

A:

Q: 7. a) Prove that for r > 0 * (E) + c. Vr2 – r²dx = vr² – x² + -1 sin 2 +C.

A: To prove: ∫r2-x2dx=x2r2-x2+r22sin-1xr+C

Q: 10. Show that S t2 if 0 1, 2(1 - e-) then Ldf %3D

A: Laplace of differential equation.

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

A:

Q: Use Cramer's rule to solve for z. Зх — 4у + z %3D 3 2х + 3у = 8 5y + 8z = 11

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

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Q: Q1 : prove 1 2(1–z) 3(1--z) dydxdz = 1

A:

Q: (b) Let z = x + iy = re?". Find |z| and z² first in terms of x and y and then in terms of r and 0.…

A: Given:

Q: Defn `x→- `xj diverges to minus infinity" VR>O 3N, such that Vj zN *j<-R 8;<- R Hw7 het x; = - 8j…

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Q: 1. A force of F (x) 2x = xe + 6x – 2 acts on an object. What is the work required to move the object…

A:

Q: 15.If the Wronskian of f and g is t cos t − sin t, and if u = f + 3g, v = f − g, find the Wronskian…

A: Consider the provided information, Wf,g=fg'-gf'⇒tcost-sint ...... (1) Find the Wronskian of u and v.…

Q: dv 4.)Prove that f -Incosv + Insinv + C sinvcosv

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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: 2. Give the condition which ensure that |ez| < 1 where z in C.

A:

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

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.