Questions in mathematics

Questions in mathematics

📝 Answered - In the sequence 3, 6, 9 ..., which term is 270?

📝 Answered - Look at each simplified expression. Recognizing operations in an expression will guide you in how to simplify it. [tex] \begin{array}{l} \sqrt{5}(-\sqrt{2})=-\sqrt{10} \\ \sqrt{5}(\sqrt{7}-\sqrt{2})-\sqrt{35}-\sqrt{10} \\ (\sqrt{5}+\sqrt{2})(-\sqrt{7})=-\sqrt{35}-\sqrt{14} \\ (3 \sqrt{5})(-7 \sqrt{2})=-21 \sqrt{10} \end{array} [/tex] Choose two of the expressions that require the use of the distributive property to simplify. [tex](\sqrt{5}+\sqrt{2})(-\sqrt{7})[/tex] [tex](3 \sqrt{5})(-7 \sqrt{2})[/tex] [tex]\sqrt{5}(-\sqrt{2})[/tex] [tex]\sqrt{5}(\sqrt{7}-\sqrt{2})[/tex]

📝 Answered - A fraction was multiplied by [tex]$\frac{5}{6}$[/tex] to get [tex]$\frac{25}{48}$[/tex]. What was the original fraction?

📝 Answered - Determine whether the following equation defines y as a function of x. [tex]x+y=29[/tex] Does the equation [tex]x+y=29[/tex] define [tex]y[/tex] as a function of [tex]x[/tex]? Yes No

📝 Answered - Consider the function $f(t)=8 \sec ^2(t)-9 t^3$. Let $F(t)$ be the antiderivative of $f(t)$ with $F(0)=0$. Then $F(t)=$ $\square$

📝 Answered - 1. What is \( \frac{2}{3} - \frac{1}{5} \)? 2. What is \( \frac{3}{4} - \frac{4}{8} \)? 3. What is the value of \( N \) in \( \frac{3}{6} - \frac{1}{5} = -N \)? 4. Find the difference between \( 2 \frac{2}{9} \) and \( \frac{2}{3} \).

📝 Answered - Assume [tex]$\alpha$[/tex] increases monotonically and [tex]$\alpha' \in \mathbb{R}$[/tex] on [tex]$[a, b]$[/tex]. (Here [tex]$\alpha'$[/tex] is the derivative of [tex]$\alpha$[/tex] with respect to [tex]$x$[/tex] in [tex]$[a, b]$[/tex]). Let [tex]$f$[/tex] be a bounded real function that maps [tex]$[a, b]$[/tex] into [tex]$\mathbb{R}^k$[/tex]. Then [tex]$f \in R(\alpha)$[/tex] if and only if [tex]$f\alpha' \in R$[/tex]. In that case, [tex]$\int_a^b f \, d\alpha = \int_a^b f(x) \alpha'(x) \, dx$[/tex].

📝 Answered - (i) Complete the table. | Figure number | Number of white hexagons | Number of grey hexagons | Total number of hexagons | |---|---|---|---| | 1 | 1 | 0 | 1 | | 2 | 1 | 1 | 2 | | 3 | 2 | 1 | 3 | | 4 | 2 | 2 | 4 | | ... | ... | ... | ... | | 12 | 6 | 6 | 12 | (ii) Find an expression, in terms of n, for the number of grey hexagons in Figure n. (iii) Which figure has the same number of white hexagons as Figure 156?

📝 Answered - 1) 2x - a = 4 - 5x; 2) 3(x - a) = 2(4 - x); 3) 2(x + a) = 8 - a + x; 4) 3(x - a) = 4 - 5a + x. 1) 2(a - x) = a + 2 - 4x; 2) 3(x - 2a) = 2x + a - 3; 3) 5(4a - x) = 7(2 - x); 4) 3(2a - 3x) = 2(3 - x). 1) 3(x + a) = 2 - x, x > 2; 2) 5(a - x) = 3 + 2a, x < 1; 3) 3(3a - 2x) = 12a - 2, x < 3; 4) 5(2a + 3x) = 6a + 4, x < 1. 1) 5(10 - a) x^2 - 10x + 6 - a = 0; 2) (a - 3) x^2 - 2(3a - 4) x + 7a - 6 = 0; 3) 5(a + 4) x^2 - 10x + a = 0; 4) a x^2 + 2(a + 1) x - 2a - 1 = 0. 2) (a^2 - 9) x = a^2 + 2a - 3.

📝 Answered - Find by triple integral the volume of the paraboloid of revolution \( z^2 + y^2 = 4z \), cut off by the plane \( z = 4 \).