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Other Questions from Jan 07,2025

Browse the Other Q&A Archive for Jan 07,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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12. Find the square roots of the complex number \( 3-4 i \), leaving your answer in rectangular form 11. Find the square roots of the complex number \( 2+2 \sqrt{3} i \), leaving your answer in rectangular form. (Hint: Solve \( z^{2}=2+2 \sqrt{3} i \), where \( z=a+b i \) and \( a \) and \( b \) are real numbers.) \( \begin{array}{l}\text { Direction: Identify the following question. Write your answer on your notebook. } \\ (1-6) \\ \text { 1. Defined as an arrangement to have something (room, table or } \\ \text { seat) held for your use at a later time. } \\ \text { 2. Also known as 'kitchen less kitchen'. } \\ \text { 3. A growing trend in reserving a seat in a restaurant. } \\ \text { 4. This system is usually used in cafeteria, restaurants, small } \\ \text { hospital and school canteens. } \\ \text { 5. Use of the internet through a website, where all the necessary } \\ \text { information needed for a reservation. } \\ \text { 6. This kind of system food is prepared in one place then } \\ \text { transported to satellite kitchens. }\end{array} \) Suppose the firm uses two inputs to produce its output: capital (K) and labor (L). The firm's production function is \( q=L^{1 / 2} K^{1 / 3} \). Suppose that in the short run capital (K) is fixed at 27. If the wage is \( \$ 4.5 \) and the rental rate on capital is \( \$ 4 \), 1. Does the production function display decreasing, constant, or increasing returns to scale? Explain 2. What is the short run total cost equation (function with \( q \) )? 3. What is the level of output (q) that minimizes the total cost? Suppose the firm uses two inputs to produce its output: capital (K) and labor (L). The firm's production function is \( q=L^{1 / 2} K^{1 / 3} \). Suppose that in the short run capital (K) is fixed at 27. If the wage is \( \$ 4.5 \) and the rental rate on capital is \( \$ 4 \), 1. Does the production function display decreasing, constant, or increasing returns to scale? Explain 2. What is the short run total cost equation (function with \( q) \) ? 3. What is the level of output (q) that minimizes the total cost? - Calculate the components and the magnitude of the vectors \( \overrightarrow{V_{1}}=\vec{A}+\vec{B}, \overrightarrow{V_{2}}=\vec{A}-\vec{B} \) and \( \overrightarrow{V_{3}}=2 \vec{A}-3 \vec{B} \). - Determine the unit vectors carried by the veetors \( \overrightarrow{V_{1}}, \vec{V}_{2} \) and \( \overrightarrow{V_{3}} \). - calculate the dot product and cross product of vectors \( \vec{A} \) et \( \vec{B} \). Fxercice 08: we consider three vectors: \( \vec{V}_{1}\left(\begin{array}{l}x \\ 3 \\ z\end{array}\right) \), \[ \overrightarrow{V_{2}}\left(\begin{array}{l} 1 \\ 2 \\ 2 \end{array}\right) \text { and } \overrightarrow{V_{3}}\left(\begin{array}{c} 2 \\ -2 \\ 1 \end{array}\right) \] 1- Show that the vectors \( \vec{V}_{2} \) and \( \vec{V}_{3} \) are orthogonal. 2- Détermine \( x \) and \( z \) so that the vectors \( \vec{V}_{1} \) and \( \vec{V}_{2} \) are parallel. 3- Determine the direction (cosines) of the vector \( \vec{V}_{3} \). 4. We suppose that \( x=2 \) and \( z=2 \). Calculate the area of the parallelogram constructed by the vectors \( \overrightarrow{V_{1}} \) and \( \overrightarrow{V_{3}} \). 10. If \( \mathrm{A}=\{2,4,8,16\} \) and \( A \cup B=\{1,2,4,8,16) \), find one element in set B . II. If \( \mathrm{A}=\{ \) all even numbers less than 10\( \} \) and \( \mathrm{B}= \) (all multiples of 3 less than 10\( \} \), list the elements in A and B and draw a \( V \) enn diagram showing the relationship B. 12. If \( \mathrm{A}= \) \{all positive numbers less than 8\( \} \) and \( \mathrm{B}=\{ \) all multiples of 3 less than 10\( \} \), list the elenents in the sets (i) \( \mathrm{A} \cap B \) (ii) \( A \cup B \). e) Given that \( V=\left[V_{1}, V_{2}, V_{3}, V_{4}, V_{5}\right] \) \[ V_{1}=\left(\begin{array}{r}1 \\ -1 \\ -3 \\ 2\end{array}\right), V_{2}=\left(\begin{array}{r}-1 \\ 1 \\ 1 \\ -1\end{array}\right), V_{3}=\left(\begin{array}{r}2 \\ 2 \\ -2 \\ -1\end{array}\right) \cdot V_{4}=\left(\begin{array}{r}-2 \\ -6 \\ 1 \\ 3\end{array}\right), V_{5}=\left(\begin{array}{r}1 \\ -2 \\ -2 \\ 1\end{array}\right) \text { in } \] Determine : (i) the rank of A (ii) the null space of A (iii) if the vectors \( V_{1}, V_{2}, V_{3}, V_{4} \) and \( V_{5} \) span \( \mathbb{R}^{4} \) (e) Given that \( V=\left[V_{1}, V_{2}, V_{3}, V_{4}, V_{5}\right] \) \[ V_{1}=\left(\begin{array}{r}1 \\ -1 \\ -3 \\ 2\end{array}\right), V_{2}=\left(\begin{array}{r}-1 \\ 1 \\ 1 \\ -1\end{array}\right), V_{3}=\left(\begin{array}{r}2 \\ 2 \\ -2 \\ -1\end{array}\right) \cdot V_{4}=\left(\begin{array}{r}-2 \\ -6 \\ 1 \\ 3\end{array}\right), V_{5}=\left(\begin{array}{r}1 \\ -2 \\ -2 \\ 1\end{array}\right) \text { in } \] Determine : (i) the rank of A (ii) the null space of A (iii) if the vectors \( V_{1}, V_{2}, V_{3}, V_{4} \) and \( V_{5} \) span \( \mathbb{R}^{4} \) \[ A=\left(\begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) \] EsERcizio 1 Provare che le seguenti applicazioni non sono forme bilineari: (X) \( b: \mathbb{R}^{2} \times \mathbb{R}^{2} \rightarrow \mathbb{R} \) tale che \( \left.b\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right)=-x x^{\prime}-x y^{\prime}+3 ;\right) \mathbb{N} \) (L) \( b: \mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R} \) tale che \( b\left((x, y, z),\left(x^{\prime}, y^{\prime}, z^{\prime}\right)\right)=e^{x x^{\prime}}-y y^{\prime} \); (Ni) \( b: \mathcal{M}_{2}(\mathbb{R}) \times \mathcal{M}_{2}(\mathbb{R}) \rightarrow \mathbb{R} \) tale che \( b(A, B)=\operatorname{det}(A+B) \) (iv) \( b: \mathbb{R}_{\leqslant 1}[x] \times \mathbb{R}_{\leqslant 1}[x] \rightarrow \mathbb{R} \) tale che \( b\left(a x+\alpha, a^{\prime} x+\alpha^{\prime}\right)=a a^{\prime}+\alpha \); (v) \( b: \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R} \) tale che \( b\left(\left(x_{1}, \ldots, x_{n}\right),\left(y_{1}, \ldots, y_{n}\right)\right)=\left|x_{1} y_{1}+\cdots+x_{n} y_{n}\right| \). SVOLGIMENTO (i) Si trova \( b(\mathbf{0}, \mathbf{0})=3 \), pertanto \( b \) non può essere una forma bilineare. (i) Abbiamo \( b(\mathbf{0}, \mathbf{0})=1 \). (if) La funzione determinante non ci permette di portare fuori le costanti moltiplicative. Per esempio sia \( \lambda=2 \). Allora \[ b\left(2 I_{2}, \mathbf{0}_{2}\right)=\operatorname{det}\left(2 I_{2}\right)=4 \] mentre \[ 2 b\left(I_{2}, \mathbf{0}_{2}\right)=2 \operatorname{det}\left(I_{2}\right)=2 \] (iv) Abbiamo che \( b(1,0)=1 \).
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