7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

Q: Solve for r: In(x) + In(x + 2) = 3

A: Use the properties of lograthim and simplify.

Q: Prove that Σ (2n+1) P(x) no (1-2hx+h²) ¾/2₂

A:

Q: Show that MnXn and Rn^2 are isomorphic.

A:

Q: Find 3²y of z = In(xy)

A:

Q: Q2/ fincd lachre trans form for at X- 3x+2X=lee

A: given equationx"-3x'+2x=1-e2tby assuming that initial condition is zero.x"=s2X(s)and x'=sX(s)and…

Q: 5. Find the on the interval [0, 4]

A:

Q: Find the first piece S1 (x) of a natural cubic spline that interpolates the set of points (0,1),…

A: Formula for cubic spline interpolation is…

Q: Show that 1/2 ( Tn + Mn ) = T 2n

A:

Q: A natural cubic spline is defined by Sa,(x-1)' +b,(x-1)² +c, (x - 1) + d, S(x) =- if 0sxs1 (x- 1)' +…

A: Step:-1 Since, we know for cubic spline S(x)=S0x=a0x3+b0x2+c0x3+d0…

Q: Prove using the IVT. For all positive integers k, cos x = xk has a solution.

A: Intermediate value theorem: Let f be a function continuous on a closed interval a,b. Let k be number…

Q: 27. Verify ldt ip|1| =x|x]. Hint: Consider x > 0 and x <0 separately.

A:

Q: 6. Find a²z дгу of z = In(xy)

A:

Q: Suppose x,y,z are mutually coprime positive integers that solve x2+y2=z2. a) Show why exactly one…

A: Basic properties of modular arithmetic and gcd will be used in solving this problem. The detailed…

Q: 1 If z = 2 V3 -i, then find ()*

A: Given complex number is: z=-1-3i2 Find the conjugate of complex number: z¯=-1+3i2 Find z¯4?

Q: Prove that the equation ! + = 1 has no solutions in positive integers æ, y, z.

A:

Q: Find the infimum and supremum of the set X = {3 – Vztn ine N}. -

A:

Q: Show that the solution (if exists) is unique:

A: Given, u =6x + 4y +2 on {x² + 4y² &lt;1} u(x, y) = x+2y+y²-4xy²-8y³ on {²+4y²=1} Using maximum…

Q: 2. Find three positive integers x, y, and z if their product is 27 and the sum is a minimum.

A: Given three positive numbers are  x, y, and z Product = xyz = 27 And sum is minimum

Q: 1 Show that xn-m)}-2X(2). assu me. Zera 1c

A: As per the company's rule, we are supposed to solve one problem or three subparts. As there are two…

Q: 4: verify that wy = Wyx . ریاضيات المرحلة الثانية a w = In (2r + 3y) %3D

A:

Q: Find ²2 of z = In(xy) 8²y

A: Given: z=lnxy   As we know that; The Power law; ddxxn=nxn-1 The Chain Rule: fgx'=f'gx·g'x And…

Q: az Find if yz – In z = x + y ax 3.

A: We have              yz-lnZ=x+y

Q: ² = 4. Maximize Q = 4xy, where x and y are positive numbers such that x+y %3D y: (Type an exact…

A: topic - application of derivatives

Q: 13 3. Use the IVT to prove that y = r Inr has at least one zero on 2 2

A: Above function is continuous function in given interval. So we can apply IVT

Q: 9. Show that 4x - 251 + 10x? – 15x + 20 is irreducible over Q.

A: Answer

Q: Theorem = 7 K 51 M 25

A:

Q: Is 1z = 1 for all z ∈ C?

A:

Q: show that x'-15x +C= O has at most root on E2,2] one

A:

Q: Prove that the equation * + + = 1 has no solutions in positive integers x , Y, Z.

A: x/y + y/z + z/x = 1 if x , y , z are positive integers then case 1 :  If all three numbers are…

Q: Q) prove that (1) -2P xSinx 2.

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

Q: 5x 60(x, +x,) 3 b) Y = In 60(x2 . 3,

A: The given differential equation:                                                 Y=ln5x1360(x2+x1)32…

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

A:

Q: Show that 1 is the only solution of x25 - 1 = 0 in Z37.

A:

Q: Find the clamped cubic spline that interpolates (0,0),(1,1),(2,2) and satisfies the condition: s’(0)…

A: We have to find clamped cubic spline that interpolates (0,0),(1,1),(2,2) and satisfies the…

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

A:

Q: Find all triples of integers (x, y, z) that satisfy x² + 2y² = z2.

A:

Q: Compute Projsw given by |2 -1 – 2 and S is defined by 8 [2 – 12]+t[-1 – 13], s,t e R. x = s ||

A:

Q: 3. Show that à = (6xy + z³)î + (3x² – z)ĵ + (3xz2 – y)k is irrotational. Find p such that A Vọ.

A:

Q: Prove that there are no solutions in integers x and y to the equation 2x2 + 5y² = 14.

A: Two different proofs are given for the given problem as follows :

Q: 4. Prove that x – 15x³ + 10x² – 21x + 2021 is irreducible in Q[x]. -

A: Given :-  Prove that x5 -15x3 +10x2 -21x +2021 is irreducible in ℚx.

Q: 4. Show that if ki = (X; - X)/[Σ(X; – X)2], then (a) Σki = 0; = (b) Σk? = 1/[Σ(X; − x)2]; (c) Σκ;(X;…

A:

Q: D. Let Find 2,-

A: We will find out the required value.

Q: 4. Prove that d x" = nx"-1 ミクー! de

A: To prove that ddx(xn) = nxn - 1

Q: 7.1) 19) Evaluate the integral.

A:

Q: E Show that x'+ 3x+ 2 has 4 zerosin Ze

A:

Q: 16) * Find y' if y In (x2-8), when x 4

A: