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January 2025
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Discrete Mathematics
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January 6, 2025
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January 6 of 2025
answers to lowest terms. Simplify improper fracti 3. \( \begin{array}{r}\frac{8}{9} \\ -\frac{5}{9} \\ \hline\end{array} \) 5. \( \begin{array}{r}\frac{1}{4} \\ +\frac{2}{3} \\ \hline\end{array} \) 7. \( \begin{array}{r}\frac{9}{10} \\ -\frac{3}{5} \\ \hline\end{array} \) - \( \frac{7}{12}…
4 a . Optimize the function: \( f(x, y)=26 x-3 x^{2}+5 x y-6 y^{2}+12 y \) Subject to: \( 3 x+y=170 \). Hence, determine the marginal impact on the objective function caused by a small change in the constant of the constraint. ( 13 Marks). \( 4 b \). Given the profit function \( \pi=160 x-3…
000000000000000000000 haver \( ^{10} \) MATHS ASSIGNMENT Q1. Simplify (a) \( 3^{2} \times 10^{3} \) (b) \( (-2) 4 \times(-3) 3 \) Q2. Express each in power notation (a) 256625 (b) 121243 Q3. Show the given rational numbers on the number line (a) -185 (b) 78 Q4. Solve the equation (a) \( 4…
7. A set of 5 similar coins is tossed 320 times and the result is : \begin{tabular}{|c|c|c|c|c|c|c|} \hline\( x: \) & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline\( y: \) & 6 & 27 & 72 & 112 & 71 & 32 \\ \hline \end{tabular} Test the hypothesis that the data follows a Binomial distribution,
Let \( B(z) \) be the Blaschke product with zeroes at the points \( \alpha_{n}=1-\frac{1}{n^{2}} \), for \( n \in \mathbb{N} \). Show that \( B(z) \) defines a holomorphic function on the unit disk. Prove that \( \lim _{r \rightarrow 1} B(r)=0 \), where \( 0<r<1 \). Hint: Show the…
6. Use the definition of factorials to simplify: \( \frac{\left(y^{2}-4\right)!}{(y-2)\left(y^{2}-5\right)!} \) 7. Use geometric series to convert \( 3.567567567 \cdots \) in to a fraction. 8. Use the principle of mathematical induction to show that: \[ \sum_{t=1}^{n-1}…
9.Find the value of \( x \) and \( y \) in (i)
21.Consider the following the open statements with sets of all real numbers as the universe \( p(x): x \geq 0, q(x): x^{2} \geq 0, r(x): x^{2}-3 x-4=0, s(x): x^{2}-3>0 \). Determine the truth values of the following statements (i) \( \exists x p(x) \wedge q(x) \) (ii) \( \forall x, p(x)…