Prove the following using resolution refutation: 4. Given: (p Aq) -→r, p^ s A~w, (s A t) →q, tv w Prove: r

Q: A natural cubic spline S on (0,2] is defined by Jsin - 1+2-r. S) = if Osr<l. |S,0) = 2+ b - 1) + cư…

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Q: 3. Show that the line integral + 2xy) dx + x² dy + 2xz dz, is independent of path in R3, where C…

A: We will find out the required value.

Q: Evaluate the line integral S 3ydx + 6xdy where C is the straight line path from (1, 2) to (6, 5). So…

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Q: Iff: [a,b] R is bounded and has only a finit number of discontinuous points, then f is Riemann…

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Q: 17. Evaluate the line integral xezds, where C is the line segment from (0,0,0) to (1, 2, 3).

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Q: - Show that the norm tion t= aT+B. Use [a, b] onto [0, 1] (7) = 1, ý(7)= T, V т,

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Q: Prove that cosh-z = In (z + vz? - 1) %3D

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Q: 5.Use thelefinition to Rrave that Jw wh- 2

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Q: Show that the following line integral is path independer (3,5) y² %3D cosh( )(10xdx +6ydy) (0,0) 3

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Q: Prove That:- o sinhz= loy (Z+lz*+l )

A: To prove, sinh-1z=logz+z2+1     Lets consider, y=sinh-1z…

Q: Prove: Given that ll Xllo sl1x|1, <vñ lIXloo for XEL". 2 Show that for

A: The matrix p- norm = operator norm                            ||A||p = Maxx≠0 || Ax||p||x||p…

Q: Define stpes to obtain the general solution of a 2nd order homogeneous ODE with real coincident…

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Q: SVA- 4-y² √8-x²-y2² z² dzdxdy. √x²+y²

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Q: By using the substitution method, show that for any positive numbers x and y ry 1 dt = Sdt

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Q: Given: mTRV = 60° M2TRS = (4x)° %3D Prove: x = 30 Z. 60° R (4x)° W

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Q: Suppose that (Q x Q, +,.) is a ficld where (x, y) + (z,w) (x+z,y + w) and (x, y). (z, w) (xz +2yw,…

A: I have find identity of the field and then inverse

Q: . Compute the line integral of E from B(0,0,0) to A(2,1,3) if E = -4xya, – (4y + 2x2)a, + 8zaz. Use…

A: The line integral of vector field E→ along curve C is ∫C   E→.dr=∫abE→(r→(t).r→'(t)dt  where r(t)→…

Q: 5) Set the iterated double integral ļ) dydx メモービ テ as Jady up

A: Please see the below explanation.

Q: Find the ared of the reyion R bound ed.by and the liney=x+2 2. 2. =y-yand the line y--x ......

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Q: Suppose that h is integrable and that , h(r) dr = 0 and ſ, h(r) dr = 6. What is h(r) dr ?

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Q: Complete a two-column proof. Given: ZRSU VST Prove: ZRSV LUST

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Q: どタ -e -at u Ct) = (a +jwjñ+!

A: To prove that tne-atut↔n!a+jwn+1 It is known that  ut=1 , t&gt;0=0, t&lt;0 Therefore,…

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: Show that the following line integral is path independent (3,2) cosh (0,0) (8xdx+6ydy) 4.

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Q: Find over xlapping area two equatione also gemetrical nepresentahon 2. 4'+v?=n, u?+cV- vn)=1 whore n…

A: Given curves are,…

Q: Suppose that U(x; y) = xe' + ysin z, and that x, y and z can be measured with maximum possible…

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Q: In R', the point (1, 1, 1) does not belong to the cylinder x + y = 2. Select one: True O False

A: Given problem:- In R³, the point (1, 1, 1) does not belong to the cylinder x² + y² = 2.  True  False

Q: Let X and Y be independent and distributed as N(mu, 1). Is T=X-Y sufficient?

A: According to factorization theorem, if the likelihood function Lx,θ can be expressed in the form.…

Q: The scalar path integral 3(16-a y-) ds, where the path C is the straight line from the point (0,0)…

A: This is a problem of line integral. Since the path given for line integral is embedded on the x…

Q: Assume that u = U (x,y) iu differentiible and x =r ca Ô and y=rsin o eapress 2. entinely in termv 0…

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Q: Show that the line integral 22 +2ry) dx + x? dy + 2xz dz, is independent of path in R, where C runs…

A: We will find out the required values.

Q: 7. Use the Fundamental Theorem of Line Integrals to evaluate S(6xy, 3x² + 6y²) · dr where C is any…

A: Line integral theorem

Q: 3. Evaluate JIJ 62² dV where G is bounded by x + 2y + 4z = 8 in the first octant.

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Q: compute J secx dr. I n trying to integrate

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Q: Evaluate the integral. 4 2t 9 -i- j+¬Vt k dt (9+?)*

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Q: prove that tr(aA+bB)=atr(A)+btr(B)for any A,B belongs to Mmxn(F)

A: Let A=aij and traceA=trA=∑i=1maii.. Note that, aA+bB=aaij+bbij. Then,…

Q: Find the Selution of (1-t +3t)4 (t) di - , (2) In which IL) denoter Bessel function

A: Leibniz integral rule : ddx∫bxaxFtdt=∫bxax∂∂xFtdt+Faxddxax-Fbxddxbx   Linear Volterra Equation of…

Q: Evaluate the line integral x²z ds, where C is the line segment from (0,6, 2) to (8, 5, 1).

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Q: -H.. OProve that fcz) =z²,s coutinuous at z=z, O prove that: if Z#20 is dis continnity at Z=Zo,…

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Q: Evaluate the integral fff, xyzdV,where: E = {(x, y, z)|–1 <x< 1,0< ys 2,1 szs 2}

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Q: /show that fo-1x is-Contindals everywhere.

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Q: Evaluate by Cauchy’s integral formula: ∫ c e3z/(z+2)4dz where c: mode z = 4.

A: To find - Evaluate by Cauchy's Integral formula : ∫Ce3zz + 24 dz where C : z = 4     Formula used -…

Q: Suppose that f and h are integrable and that: i fx) = -1, f(x) dx = 5, h (x) dx = 4 ,Find: a. S–2…

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Q: 9) suppose that horx)=1, f,(x)=x ond dz(X) Prove that Ba(f, - where Ba is a Bernyterh Polymomial.

A: Given functions are f0(x)=1, f1(x)=x and f2(x)=x2 Since Bernstein polynomial is…

Q: Evaluate the line integral ∫Cx^3z ds, where C is the line segment from (0,3,8) to (8,4,6).

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Q: flz) = Im z is continuous %3D

A: The objective is to check f(z) =Im Z is continuous Or not.

Q: 2. Compute the line integral of E from B(0,0,0) to A(2,1,3) if E = -4xyax - (4y + 2x2)a, + 8zaz. Use…

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Q: (Compute Sc (-3y+ony Sin(3))dxt (5x²-(ye)du Where C is a Simple Clased cUne in the plane oriented…

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Q: (R, d), where d(r, y) = |r – y|, is separable. %3D

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Q: Prove,J w' Sin wx du %3D COSZ 2

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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.