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Questions in mathematics

[Done] Find the coordinates of the focus and equation of the directrix for the parabola given by $y^2=-4 x$. The general formula for this parabola is $y^2=4 p x$. Therefore, the value of $p$ is $\square$ . The coordinates of the focus are $\square$ . The equation of the directrix is $\square$ .

[Done] What is the simplified form of [tex]$\sqrt{400 x^{100}}$[/tex] ? A. [tex]$200 x^{10}$[/tex] B. [tex]$200 x^{50}$[/tex] C. [tex]$20 x^{10}$[/tex] D. [tex]$20 x^{50}$[/tex]

[Done] Select the correct answer. What is this expression in simplified form? [tex]$\frac{\sqrt{32}}{\sqrt{2}}$[/tex] A. 16 B. 2 C. [tex]$\sqrt{30}$[/tex] D. 4

[Done] Select the correct answer. What is the justification for step 4 in the solution process? Step 1 : [tex]\frac{9}{2} b+11-\frac{5}{6} b=b+2[/tex] Step 2 : Step 3 : Step 4 : [tex]\frac{22}{6} b+11=b+2[/tex] [tex]\frac{8}{3} b+11=2[/tex] [tex]\begin{aligned}\frac{8}{3} b & =-9 \\b & =-\frac{27}{8}\end{aligned}[/tex] A. the multiplication property of equality B. combining like terms C. the addition property of equality D. the subtraction property of equality

[Done] Find the value of the expression: $\frac{3^7 \cdot 27}{\left(3^4\right)^3}$

[Done] What is the solution to the following equation? $x^2+5 x+7=0$ A. $x=\frac{-3 \pm \sqrt{-7}}{2}$ B. $x=\frac{3 \pm \sqrt{25}}{2}$ C. $x=\frac{-5 \pm \sqrt{53}}{2}$ D. $x=\frac{-5 \pm \sqrt{-3}}{2}$

[Done] Two percent of mobile phones produced at a factory are defective. Which of the following is a binomial experiment? A. Selecting phones randomly until a non-defective phone is chosen B. Selecting phones randomly until 200 defective phones are chosen C. Selecting 200 phones randomly and recording whether each phone is defective D. Selecting a group of phones randomly and recording the number of defective phones in the group

[Done] Perform the indicated operations, and write the result in the form $a+b i$. (Simplify your answers completely.) $i^{44}$

[Done] Evaluate the following limits: 38. Evaluate [tex]$\lim _{x \rightarrow 1^{-}} \frac{|x-1|}{x-1}$[/tex]. A. 2 B. 1 C. 0 D. -1 39. Evaluate [tex]$\lim _{x \rightarrow \infty} \frac{4-x^2}{4 x^2-x-2}$[/tex]. A. -2 B. 1 C. 2 D. -[tex]$\frac{1}{4}$[/tex]. 40. Evaluate [tex]$\lim _{x \rightarrow 0} \frac{\sin ^2 x}{x}$[/tex]. A. 0 B. [tex]$\infty$[/tex] C. 1 D. -1 41. If [tex]$\lim _{x \rightarrow 0} \frac{1-\cos x}{3 x \sin x}=\frac{1}{k}$[/tex], find the value of [tex]$k$[/tex]. A. 3 B. -1 C. 6 D. -3 42. Evaluate [tex]$\lim _{x \rightarrow 0} x^2 \cos \left(\frac{1}{x}\right)$[/tex]. A. i B. [tex]$\infty$[/tex] C. 0 D. Undefined.

[Done] The derivation for the equation of a parabola with a vertex at the origin is started below. [tex]\sqrt{(x-0)^2+(y-p)^2}=\sqrt{(x-x)^2+(y-(-p))^2}[/tex] 1. [tex](x)^2+(y-p)^2=(0)^2+(y+p)^2[/tex] 2. [tex]x^2+y^2-2 p y+p^2=y^2+2 p y+p^2[/tex] If the equation is further simplified, which equation for a parabola does it become? A. [tex]x^2=4 p y[/tex] B. [tex]x^2=2 y^2+2 p^2[/tex] C. [tex]y^2=4 p x[/tex] D. [tex]y^2=4 p y[/tex]