5. Find real numbers a and b such that (1 + 5i) · (a + bi) = 1.

Q: Use De Moivre's theorem to find the following: 4 i. [1+ v3i]" -3 ii. [-1+ v3 i]

A: Given that: i. [1+3i]4 ii. [-1+3i]-3

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

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

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Q: 4. Let z = -4 + 3i and zw = -7 – 2i. By showing clear steps, find w in the form of a+bi.

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Q: Given z = -5+ 3i and z2 = 3 – 6i, express in the form a + bi %3D i. 4z1* – Z2 ii. Z2

A: Complex Cartesian form

Q: 1) Prove that there are no positive integer solutions to the equation x2-y?=1

A:

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: 17. Given a = 4 – 4i and az = 4 + 3i, express %3D :11 in the form a + bi. a, az CLONE Dec 19

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

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Q: B = and C: .Prove that A = B² =C? = 1. 3. Given A = %3D

A: Given: A=0110, B=0-ii0, C=100-1 We know that for a 2×2 matrixP=abcd, P2 is given by…

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

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Q: у" + Зу" + Зу' +у%3D0

A: We have to solve the differential equations

Q: b) By using De Moivre theorem, calculate in the form of a + bi: 4. i) (VZ – iv2)* | ii) (1– i)6

A: Given (2-i2)4 &   (1-i)6

Q: (2) Find all the numbers z such that elz = 10i.

A: Consider the given. eiz=10i

Q: 1 1 dr = In a. find the exact value of a. %3D V16+9x? 19+4x*

A: Here we will solve for 'a' step by step answer given below,

Q: ez = -2

A:

Q: Prove that: sec a + sec(120° +a)sec(120° – a) = –3 sec 3a

A: To prove that  secα+sec(120∘+α)sec(120∘-α)=-3sec3α

Q: %3D Bi'

A:

Q: Find the imaginary component of In(2z + w) if z = 11 + 9i and w=10+4i .

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: 1 Assume that 1 < a < b_and equation is solvable. + x = Show that this * – b'

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Q: If x,y,z are real numbers satisfying the equation 25 (9x* +y° )+9z²-15(5xy+yz+ 3zx )=0 then prove…

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Q: 6. Show that (1) Cou(B1, g) = 0 (2) Cov(3o, B1) %3D

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Q: 6. Express in a + bi form: 2i(i + 3)

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Q: If Z1 = 4 + 5i and Z 2 = -1 + I,  express in the form a + bi  (i) Z1 + Z2   (ii) Z1 – Z2

A: Formula used  sum of two complex number is given by  (a1+b1i)+(a2+b2i)=(a1+a2)+i(b1+b2)

Q: Find all the values of x in the interval [0,2pi], that satisfy the question.  10cos(x)+5sin(2x)=0

A: Given equation is Solve the given equation for x

Q: (-1+i/3)°, by using deMoivre 's theorem

A:

Q: 28. IfI:zxzr = 0, ris a. positive b. negative с. zero d. impossible to determine from the…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: show that the Pio Diophant ine equation 2x -5y= | has no integer Selutions .

A: Given, 2x²=1+5y² Here 5y2 is a multiple of 5 ending in digit 5 or 0. 2x2 can end in 1 or 6 But no…

Q: 1. Do not use a theorem to find the ff. Z if yz? + z = 5xy – 1. %3D

A: In the multivariable function, if we have to find the partial derivative of one variable with…

Q: Find the indicated power using De Moivre's Theorem. (Express your fully simplified answer in the…

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Q: •1-i Evaluate 4z dz . J3i

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Q: 10. f(x) =-x2 +3x+ m, y 4-IOBI = 10CI %3D A(ABC) = ? B.

A: As per guidelines, we will solve first question poste. 10) Given fx=-x2+3x+m 4OB=OC

Q: Question is attached, Thankyou

A: Let us take z = x + iy.

Q: #8a using Descartes's rule of sign

A: Prove in the following that if all the coefficients of an equation are positive and the equation…

Q: If z = a+bi for some real numbers a and b, show that |2| = |5|.

A: In the regular geometry on R3, we can compute the distance between                 x,y∈R2 and the…

Q: 2.2.12 Prove that if x and y positive real numbers, then Vx+y # Vx+ Vy. Argue by contradiction. are

A: Let x and y are positive real numbers. Also let x+y=x+y Now squaring on both side,we get…

Q: Use DeMoivre's Theorem to find the following. Write your answer in standard form. V2 cis- 20 O…

A: Given (√2cos(9π/20))5

Q: If z = 3 – 3i, express 5/z  in the form a + bi

A: The given value of z is z=3-3i.

Q: A (real or complex) scalar u is said to be an nth rootof unity if un = 1. Let ωn = e 2πi n . Use…

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Q: IRI- 2. <| sinz|< -e 2

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Q: G-a+bi:

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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: By expanding both sides of the identity (1+ x)"+4 = (1+x)"(1+x)* prove that ("**) - (") + «(,",) +…

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

A: