Prove: If a > b > 0, then 0 < 1/a < 1/b.

Q: b. Prove that coth² x-1=csch?x

A: Sinh(x)= (e^x-e^-x)/2. Cosh(x)=(e^x+e^-x)/2

Q: x+y uis = 1 n upi = no • prove that x-+ y-

A:

Q: OLet @ = (0,1)n Q = {x€ @104 x<1}. Prove that %3D 2) 2000m

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Q: 3.) Prove (cA)" = cCA) %3D

A: Let A be a m×n matrix so A=(aij)  1≤i≤m and 1≤j≤n ie A has m rows and n columns. By AT we mean…

Q: d) Using Demoivre's theorem, show that, if z = e“, then i(:-" - ") sin no = ne N+ 2

A:

Q: Prove by mathematical induction that, for all positive integers n, de (e* sin x) = 23"e* sin(x…

A: To Prove →dndxn ex sin x = 212n ex sinx + 14nλBy mathematical induction for all positive integen…

Q: 5. (6 + 15i ) 3

A: this is a problem of complex analysis.

Q: Prove that I(e) = dr, e> 0.

A:

Q: 5. prove thatS" erf (x) dx =o CA

A: ddx(erf(x)) =2 e-x2π   I =∫-aa erf(x) dxI= erf(x) x -∫ ddxerf(x) x dxI = x erf(x) -2π∫ x e-x2 dx-x2…

Q: If I, = [x^tan"x dx, Prove that (n+1), + (n-1),,2 ° п 1 %3D 2 n

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Q: XYZ is given with X(2, 0), Y(0, −2), and Z(−1, 1).  Which of the following can be used to prove that…

A: The answer is given as follows :

Q: Use direct prove to prove that for any two positive numbers x and y, x+y Vxy s 2

A:

Q: Is it possible to find a number  n  such that is ∫0∞  xn dxconvergent?

A:

Q: Q6:-Prove that: y" = 1 if y=;csc (tan"

A:

Q: b? – a? x dx 9. - Proof Prove that 2 a

A: This can be proved using Fundamental theorem of calculus. The second part of the theorem is used.…

Q: Prove by the indirect method: Given: AMPN is not isosceles Prove: PM + PN

A: Concept used: Step of indirect proof: First assume opposite of conclusion of the statement. After…

Q: Prove that 12) d SIAX 13)let y=x, find Ay/Ax

A: Derivative of function

Q: 1-a*** a zi i. La 1-a

A:

Q: OVESTON Prove that A is inveitine if at0 and at b l Gind The pivos 9 A %3D a a

A:

Q: if JERLA.b] and it hex)- J(x). except for finite mumber of Paint in Cab] then that he Rlab] prove.…

A: if g ∈R[a,b] and if h(x) = g(x) except for a finite number of points in [a,b] then prove that h…

Q: Prove by mathematical induetion. Ine I+2 +2+2 +2° = 2.

A:

Q: find the values of c that satisfy Rolle's Theorem

A:

Q: J prove that for 0 <r<R

A:

Q: Given: LM NO; ZLMO = ZNOM Prove: LO NM

A: Using Side angle side test,

Q: 2. (– 13 + 3i)31

A:

Q: Prove that d (sec"x)= 1.

A:

Q: 1 Show that n2 + 1 IN V

A:

Q: Show that if x ~ Nm (u, 2), where N is positive definite, then (x – u)'N-I (x – µ) ~ Xm: 2

A: Given  x~Nmμ,Ω, where Ω is positive definite.

Q: Let x e R,-1. Prove that VNEN (1+x)" >1+ nx

A:

Q: 5. Prove that if x, y ER, then x + y ≤ x + y. Use the following cases: (i.) x, y ≥ 0, (ii.) x, y <…

A: We now prove that if x,y ∈ ℝ , then x+ y≤ x+y with the help of given three cases.

Q: 2.Prove that I (sinwt) : s2+w?

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Q: Let a in F_p, a is non zero then show

A:

Q: Use the mean value theorem to prove that for 0 < a <b we ha < v6 – Va- b-a b-a 2va Use part [a] to…

A:

Q: Let x and y be positive real numbers. Prove that a + y > 2/xy.

A: See solution below

Q: 1. (10 + 7i )®

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Q: 2 (x) 2. prove that s

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Q: 3. ncnt) use PMI to prove that E

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Q: Use Euler-Maclaurin's formula to prove that E n %3D

A:

Q: let X~~ Per(P) Prove ô is unbiare if ô-X let x exp(0) Prave e is unbiar

A: Introduction: One random variable follows Bernoulli with parameter p and other random variable…

Q: b. Let x, y,p E Z, where p is prime. Prove that x2 - p²y - p + 0. (Hint: you may use without proof…

A:

Q: Suppose f" exists and is bounded on Co,0), Prove that if bam fa)=0, then bim fGx)=0.

A:

Q: Prove that S. 8 dx √π 2

A: This question is related to integration using gamma function.

Q: EXs Prove that [* Ju cx) = x'Jv-1 (x)

A: This is a problem of Bessel's function.

Q: Then show that N(A) = R(AT)+ and N(AT) = R(A)+. %3D 11-1 02-3 -1 1-2 N(A) = N(AT) =

A:

Q: R Given: OR L ON Prove: ZROT is complementary to ZNOT Proof:

A:

Q: Q3. Let z, y be real numbers such that ry 1 and z #y. Prove that

A:

Q: se Indirect Proof to prove that if a and b are both positive, and a2 < bi

A:

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.