Collecting all Math questions and answers in one place

$\frac{20b}{{\left(4b^3\right)}^3}$

Let $$Z_1$$ and $$Z_2$$ be independent standard normal random variables and $$U_1=Z_1$$ and $$U_2=Z_1+Z_2$$. Are $$U_1$$ and $$U_2$$ independent? Why?

See answers (1)

Find, in radians the general solution of ${\displaystyle \cos 3x=\sin 5x}$
I have said, ${\displaystyle \sin 5x=\cos (\frac{\pi }{2}-5x)}$
so
${\displaystyle \cos 3x=\sin 5x\Rightarrow 3x=2n\pi \pm (\frac{\pi }{2}-5x)}$
When I add ${\displaystyle (\frac{\pi }{2}-5x)\text{ to }2n\pi }$ I get the answer ${\displaystyle x=\frac{\pi }{16}(4n+1)}$, which the book says is correct.
But when I subtract I get a different answer to the book. My working is as follows:
${\displaystyle 3x=2n\pi -\frac{\pi }{2}+5x}$
${\displaystyle 2x=\frac{\pi }{2}-2n\pi }$
${\displaystyle x=\frac{\pi }{4}-n\pi =\frac{\pi }{4}(1-4n)}$

See answers (3)

1 The change in the Gibbs energy of a certain constant-pressure process was found to fit the 
expression ∆𝐺 / 𝐽 = −85.40 + 36.5𝑇 where 𝑇 is the temperature in Kelvins. Find the values of
Δ𝐻 (in 𝐽) and Δ𝑆 (in 𝐽 · 𝐾−1) for the process. 
4. A Carnot cycle uses 1.00 𝑚𝑜𝑙 of a monatomic ideal gas as a working substance. From an 

See answers (1)

What are logarithms?
I have heard of logarithms, and done very little research at all. From that little bit of research I found out its in algebra 2. Sadly to say, I'm going into 9th grade, but yet I'm learning [calculus!?] and I don't know what a logarithm is! I find it in many places now. I deem it important to know what a logarithm is even though I'm jumping the gun in a sense. My understanding of concepts, is just like that of programming. In the mean time, you know its there, and your ITCHING SO HARD to find out what that is, but nope! For now we use it, tomorrow we learn what it does.
I just know that to identify a logarithm at my level, I just look for a log. :P

See answers (2)

What is the z-score of X, if n = 20, ${\displaystyle \mu =20}$, SD =3, and X =31?

See answers (1)

Please, find the derivative of ${\displaystyle \frac{1}{x^2}}$.

See answers (3)

Use trigonometric identities to transform the left side of the equation into the right side.
$\tan \theta \cot \theta =1$

See answers (1)

What angle is subtended at the center of a circle of radius 2 km by an arc of length 9 m?

See answers (2)