10. Show that for any non-square d and any integer M there is a finite (possibly empty) set S = {(xi, y₁)} of solutions to x² − dy² = M such that for any solution (a, b) to x²-dy² = M, there exists (x, y) = S such that a+b√d = ±e" (x₂ + y; √d), where e is the fundamental 1-unit. [Hint: mimic the construction, given in lectures, of a solution to x² - dy² = 1 from infinitely many solutions to r²-dy² = M.]

Q: Let A = B = C= {x|x is a real number). Let f : A → B and g : B → C be defined as follows: f(a) = 6a…

A:

Q: Suppose ū = 2j + k, ū = 2i – 4j + 2k and w = -5i – j+ k. %3D Compute the following values:

A:

Q: 10. Write the sustem of lineor equations in the form Arebuand solve this motrit equat 2と, X, t Xu-9…

A: Given: 2x1  -x2 +x4 =93x2 -x3 -x4 =-9x1 +x3 -3x4 =-2x1 +x2 +2x3 =0 To find : x1x2x3x4 for system of…

Q: sider the following quadratic form on R²:

A: Given the quadratic form on R2:                 Q(x)=-4617x12+2917x22+4017x1x2

Q: Only solve z for both a) & b) on the attached.

A: To compare two proportions, a hypothesis test can be conducted. If there are ,  are the number of…

Q: Prove that there exists a unique real number solution to the equationx +x² – 1 = 0 between x = 2/3…

A:

Q: 2) Let T [ ] = DT [₁2 i] = i) 0 ii) 2T = 1 + (-c-d)x+ (-d)²x² + (a + c)³ x³. Then we have: -7 - =

A:

Q: Use the equation y = x² - 17x + 72 to answer the following question. For what values of x is y ≥ 0?

A:

Q: Given f(x, y, z) = 7x − 7y² + 6z², x = √u + v, y = (u + w) In(v), z = 2 – v + 7w. Find and ди, ду of…

A: Firstly differentiate f with respect to x ,y ,z  and then differentiate  x , y , z with respect to…

Q: Let a ERR. Prove lal = √a². (note: y = √a² iff y²=a and y=-√a² y = 0)

A: Step 1: Step 2: Step 3: Step 4:

Q: A) |2x² + 5x + 1 = 1 B) 12/1 x-1 x+1 C) 5x + | ≥ 1

A: