13. APPLICATIONI LINEARI \[ F(x-1)=(1,2,0) \quad F\left(x^{2}+1\right)=(1,2,0) \] Studiare poi la suriettività e Iiniettivita di \( F \). Svolamento I vettori in partemza ano indipendenti ma pochi. Allora esistono infinite applicarioni con quelle proprieta date. In particolare esacre non saranno iniettive poichê mandano quei due vettori nella stesa immagine ( \( 1,2,0 \) ). Siccome in partenza e in arrivo la dimensione è 3 , se \( F \) non è iniettiva, non sarà nemmeno suriettiva. Si ponga anche per semplicita \( F(1)=(1,2,0) \). Una possibile applicarione lineare eे \[ F\left(a x^{2}+b x+c\right)=(2 b+c, 4 b+2 c, 0) \]

A certain quadratic pattern has the following characteristics: - \( T_{1}=p \) - \( T_{2}=18 \) - \( T_{4}=4 T_{1} \) - \( T_{3}-T_{2}=10 \) Determine the value of \( p \).

(a) 3 (b) -3 2. If \( (3-x, x-1) \) lies in the fourth quadrant where \( x \in \mathbb{Z} \), then \( x= \) \( \begin{array}{ll}\text { (a) } 4 & \text { (b) } 3\end{array} \) (c) -1 (Aswan 24) (d) 1 (a) 4 (b) 3 3 If \( (125, \sqrt{y})=\left(x^{3}, 4\right) \), then \( x+y=\ldots \ldots \ldots \ldots \) (a) 15 (c) 2 (Itmailia 17) (d) zero (a) 15 (b) 21 (E1-Shartia 2.3) 4 If \( (2 a, b)=(3 b, 2) \), then \( \frac{a}{b}= \) (c) 7 (d) 10 (a) \( \frac{3}{2} \) (b) \( \frac{2}{3} \) (c) \( \frac{1}{2} \) (Luxar 24) 5. If \( \mathrm{n}(\mathrm{X})=2, \mathrm{n}(\mathrm{X} \times \mathrm{Y})=6 \), then \( \mathrm{n}\left(\mathrm{Y}^{2}\right)= \) \( \qquad \) (Giza 23) (a) 4 (b) 9 (c) 16 (d) 12 16 If \( X=\{3\} \), then \( X^{2}= \) \( \qquad \) (El-Shurkia i7) (3) 9 (b) \( (3,3) \) (c) \( \{9\} \) (d) \( \{(3,3)\} \) If \( X=\{1,2\} \quad, \quad Y=\{3,4\} \), then \( (3,4) \in \) (Qema 23) (a) \( X \times Y \) (b) \( \mathrm{Y} \times \mathrm{X} \) (c) \( \mathrm{X}^{2} \) (d) \( \mathrm{Y}^{2} \) 단 If \( \mathrm{X} \times \mathrm{Y}=\{(2,3)\} \), then \( \mathrm{X}^{2}=\ldots \ldots \) (El-Sharkia /S) (a) \( \{(4,9)\} \) (b) \( \{(4,3)\} \) (c) \( \{(2,2)\} \) (d) \( \{(2,9)\} \) 93 If \( f(x)=4 x+b, f(3)=15 \), then \( \mathrm{b}= \) (Et-Katyoubia 18) (a) 156 (b) 3 (c) 4 (d) -3 (10) If \( f: f(x)=\mathrm{n} x^{2}+2 x^{n}-3 \), then the set of possible values of \( n \) that make \( f \) a function of the second degree is \( \qquad \) (El-Dakailia 101 (a) \( \{2,3\} \) (b) \( \{1,-1\} \) (c) \( \{2,1,0\} \) (d) \( \{2,1\} \) \( 517: f(x)=5 \) is represented by a straight line parallel to the \( x \)-axis and passing through the point (Ismuilia 16) (a) \( (0,5) \) (b) \( (5,0) \) (c) \( (5,-5) \) (d) \( (0,0) \)

d. \( \frac{x-1}{x+2} \div \frac{x+3}{x^{2}+6 x+5} \)

First Multiple choice questions 1. If the point \( (3, b-1) \) lies on the \( x \)-axis, the \( b= \) (Aswan 24) (a) 3 (b) -3 (f) -1 (d) 1 VI If \( (3-x, x-1) \) lies in the fourth quadrant where \( x \in \mathbb{Z} \), then \( x= \) \( \qquad \) (Ismailia (7) (a) 4 g/ 3 (c) 2 (d) zero 3. If \( (125, \sqrt{y})=\left(x^{3}, 4\right) \), then \( x+y= \) (b) 21 (c) 7 (d) 10 4. If \( (2 a, b)=(3 b, 2) \), then \( \frac{a}{b}= \) \( \qquad \) (Luxor 24) (1) \( \frac{3}{2} \) \( 4 \frac{2}{3} \) (c) \( \frac{1}{2} \) (d) 2 If If \( \mathrm{n}(\mathrm{X})=2, \mathrm{n}(\mathrm{X} \times \mathrm{Y})=6 \), then \( \mathrm{n}\left(\mathrm{Y}^{2}\right)= \) (Giza 23) (a) 4 (1) 9 (c) 16 93. If \( X=\{3\} \), then \( X^{2}= \) \( \qquad \) (El-Shurkia 17) (a) 9 (b) \( (3,3) \) (c) \( \{9\} \) If \( X=\{1,2\}, Y=\{3,4\} \), then \( (3,4) \in \) \( \qquad \) (Qena 23) (a) \( \mathrm{X} \times \mathrm{Y} \) (b) \( \mathrm{Y} \times \mathrm{X} \) \( C \times x^{2} \) (d) \( \mathrm{Y}^{2} \) 31. If \( \mathrm{X} \times \mathrm{Y}=\{(2,3)\} \), then \( \mathrm{X}^{2}= \) (E1-Sharkia 18) (a) \( \{(4,9)\} \) (b) \( \{(4,3)\} \) (c) \( \{(2,2)\} \) (d) \( \{(2,9)\} \) (9). If \( f(x)=4 x+b, f(3)=15 \), then \( b= \) \( \qquad \) (E1-Kalyoubia 18) (a) 156 (b) 3 (c) 4 (d) -3 [10. If \( f \) : \( f(x)=n x^{2}+2 x^{n}-3 \), then the set of possible values of \( n \) that make \( f \) a function of the second degree is \( \qquad \) (El-Dakahlia 16) (a) \( \{2,3\} \) (b) \( \{1,-1\} \) (c) \( \{2,1,0\} \) (d) \( \{2,1\} \) \( 131 f: f(x)=5 \) is represented by a straight line parallel to the \( x \)-axis and passing through the point \( \qquad \) (Isnailia 16) (a) \( (0,5) \) (b) \( (5,0) \) (c) \( (5,-5) \) (d) \( (0,0) \)

\( \begin{array}{ll}\text { 2.1 } & x y-5 x y^{2}+10 x^{3} y^{3} \\ \text { 2.2. } & m(r+8)-p(r+8) \\ \text { 2.3 } & r(t-2)-5(2-t) \\ 2.4 & k^{2}(3 x+4)+(-4-3 x)\end{array} \)