The sum of the numerator and th denominator of a fraction is 18 . If is added to both its numerator an denominator, the reduced form of th new fraction so obtained is \( \frac{2}{3} \). The the original fraction is ........

(C) If \( a, b, c \) and \( d \) are proportion quantities, then prove that: \( \frac{a+2 c}{b+2 d}=\frac{c-a}{d-b} \) (a) If \( \frac{a}{x+y}=\frac{b}{y-2 x}=\frac{c}{x+4 y} \), find the value of : \( \frac{b+2 c}{2 a+b} \) (b) If \( y=3+a \) and \( a \propto \frac{1}{x} \) and \( y=5 \) when \( x=1 \), find the relation bet \( x \) and \( y \), then find the value of \( y \) when \( x=2 \)

b. \( \sqrt{1^{3}}+2^{1}+3^{3}+\cdots+n^{\overline{3}} \)

Esefcizio 10.7 Sia dato \( S \) il sottoinsieme di \( \mathbb{R}[x] \) definito da \[ S=\{f(x) \in \mathbb{R}[x] \mid f(-1)=f(1)=0, \operatorname{deg} f(x) \leqslant 4\} \cup\{0\} \] sottospazio vettoriale di \( \mathbb{R}[x] \). Determinare la dimensione, la codimensione, una base, equazioni cartesiane, equazioni parametriche ed un complemento per \( S \).

\( 1 \frac{3}{2} \leqslant x<5 / 2 \) et \( \frac{2}{3} \leqslant y \leqslant \frac{4}{3} \) exarder \( x+y, x-y, x+y, \frac{x-3 y}{x} \)

Вычисли дискриминант квадратного уравнения: \( 5 \mathrm{x}-2+11 \mathrm{x}^{2}=0 \) Запиши в поле ответа верное число. \( \mathrm{D}= \)