10. a. Show that the roots of the equation \( 2\left(a^{2}+b^{2}\right) x^{2}+2(a+b) x+1=0 \) can never be real unless \( a=b \). b. Prove that the roots of the equation \( x^{2}+(2 k-1) x+k^{2}=0 \) are real roots if \( k \leq \frac{1}{4} \).

\( \frac { 4 x y } { x ^ { 2 } + y ^ { 2 } } \)

Calaula las saguientes determinantes \( \begin{array}{ll}\text { a) }\left|\begin{array}{cc}-8 & 9 \\ -1 & 20\end{array}\right| & \text { b }\left|\begin{array}{ll}2 & \frac{1}{3} \\ \frac{1}{2} & -1\end{array}\right| \text { c) }\left|\begin{array}{cc}5 & -a \\ 2 a & 4 a^{2}\end{array}\right|\end{array} \)

Aplica el método de fracciones parciales a la función: \( \frac{x^2 + 2}{(x-1)(x+3)} \)

If \( 33-a=a+27-5 a \), what is the value of \( 33+3 a ? \)

Сложите почленно неравенства. \( 11<13 \) и \( 5,1>2,7 \)