Explanation

u = x² + y²  

First take ∂u/∂y = 0 + 2y = 2y  

Now take ∂/∂x of 2y = 0, because 2y has no x

So ∂²u/∂x∂y = 0

Last verified on 05-06-2026

Explanation

RHS all 0 → homogeneous system  

Find det of 3 2 1; 2 1 4; 1 4 1 = -34  

det ≠ 0 → only x1=x2=x3=0 works

det = 0 would mean infinite solutions. Here it’s not 0.

Last verified on 05-06-2026

Explanation

Parabola: y = x² - 4  

Circle: x² + y² = 9  

Put y: x² + (x² - 4)² = 9  

x⁴ - 7x² + 7 = 0  

Let t = x²: t² - 7t + 7 = 0  

D = 49 - 28 = 21 > 0  

Both t roots positive => 2 x values each  

Total intersections = 4

Last verified on 05-06-2026

Explanation

Real polynomial => complex roots come in conjugate pairs  

So roots = 2+i, 2-i, 1-i, 1+i  

Pair them:  

(x - 2 - i)(x - 2 + i) = (x-2)^2 - i^2 = x^2 - 4x + 5  

(x - 1 - i)(x - 1 + i) = (x-1)^2 - i^2 = x^2 - 2x + 2  

Multiply:  

(x^2 - 4x + 5)(x^2 - 2x + 2) = x^4 - 6x^3 + 15x^2 - 18x + 10  

Last verified on 05-06-2026

Explanation

y^2 - (x^2 / a^2) = 1  

=> (y^2 / 1) - (x^2 / a^2) = 1  

For vertical hyperbola, c^2 = 1 + a^2  

Foci = (0, +c) and (0, -c)

Given focus (0, sqrt2) => c = sqrt2 => c^2 = 2  

So 2 = 1 + a^2  

a^2 = 1  

a = 1

Last verified on 05-06-2026