# Engineering mathematics set 1

 Question 1 [CLICK ON ANY CHOICE TO KNOW THE RIGHT ANSWER]
What's the general solution of $\frac{dy}{dx} = 3y^{2/3}$?
 A $y = (x+C)^3/27$. B $y = (3x+C)^3/27$. C $y = (2x+C)^3/27$. D $y = (x+C)^3/22$.
Question 1 Explanation:
On the left hand side we get, $3y^{\frac{1}{3}}$ and on the right hand side we get, $x+C$ $\Rightarrow 3y^{\frac{1}{3}}=x+C \Rightarrow y = (x+C)^3/27$.
 Question 2 [CLICK ON ANY CHOICE TO KNOW THE RIGHT ANSWER]
Which of the following is true?
 A ${\sqrt 2}^{\sqrt 2}$ can be rational B There exists two irrational numbers $a, b$ such that $a^b$ is rational C Both A and B D None of the above
Question 2 Explanation:
Two irrational numbers $a, b$ such that $\\a^b\\$ is rational Let $\\a = \sqrt{2}^\sqrt{2}\\$ and $\\b = \sqrt{2}\\$, $\\a^b = \sqrt{2}^{\sqrt{2}^2} = \sqrt{2}^2 = 2\\$ which is clearly rational. If $\sqrt{2}^\sqrt{2}$ is irrational, then we have found our example. If $\sqrt{2}^\sqrt{2}$ is however rational, then of course $a=b=\sqrt{2}$ is an example of irrationals $a, b$ such that $a^b$ is rational. Henceforth, there exists two irrational numbers $\\a, b\\$ such that $\\a^b\\$ is rational.
 Question 3 [CLICK ON ANY CHOICE TO KNOW THE RIGHT ANSWER]
If $\frac{1}{a} +\frac{1}{b}+ \frac{1}{c} =\frac{1}{a+b+c}$
 A $(\frac{1}{a} +\frac{1}{b}+ \frac{1}{c})^{2n+1}=\frac{1}{(a+b+c)^{2n+1}}$ B $\left(a^{2n+1}+b^{2n+1}\right)\left(a^{2n+1}+c^{2n+1}\right)\left(b^{2n+1}+c^{2n+1}\right)=0,$ C Both A and B D None of the above
Question 3 Explanation:
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ it's just $(a+b)(a+c)(b+c)=0$ and we need to prove that $$\left(a^{2n+1}+b^{2n+1}\right)\left(a^{2n+1}+c^{2n+1}\right)\left(b^{2n+1}+c^{2n+1}\right)=0,$$ which is true because $a^{2n+1}+b^{2n+1}$ divided by $a+b$.
 Question 4 [CLICK ON ANY CHOICE TO KNOW THE RIGHT ANSWER]
\begin{align} -20 &=-20\\ 16-36 &= 25-45\\ 4^2-4\times 9&=5^2-5\times 9\\ 4^2-4\times 9+81/4&=5^2-5\times 9+81/4\\ 4^2-4\times 9+(9/2)^2&=5^2-5\times 9+(9/2)^2\\ \end{align} Considering the formula $a^2+2ab+b^2=(a-b)^2$, one has \begin{align} (4-9/2)^2&=(5-9/2)^2\\ \sqrt{(4-9/2)^2}&=\sqrt{(5-9/2)^2}\\ 4-9/2&=5-9/2\\ 4&=5\\ 4-4&=5-4\\ 0&=1 \end{align}
 A $\sqrt {a^2}=|a|$ B cannot root the negative integer C Both A and B D None of the above
Question 4 Explanation:
\begin{align} (-2)^2 = 4 &\implies \sqrt{(-2)^2} = \sqrt{2^2} \\ &\implies-2 = 2 \\ &\implies-2 + 2 = 2 +2 \\ &\implies 0 = 4 \end{align} Can you see the mistake?
 Question 5 [CLICK ON ANY CHOICE TO KNOW THE RIGHT ANSWER]
It took Marie 10 minutes to saw a board into 2 pieces. If she works just as fast, how long will it take for her to saw another board into 3 pieces?
 A 20 B 15 C 10 D All the above
Question 5 Explanation:
Cutting something into three pieces requires two cuts. hence 20 minutes

If you cut it differently into three pieces. You can cut it in 15 minutes.

One can cut in 10 minutes too. Saw must look like this



| |
| |
| |
| | <---cutting edges
| |
| |
+--+--+
| <---handle
|


There are 5 questions to complete.