# Four easy numerical questions

$7^5 \cdot 3^2 \cdot 5 + 3$, is it a composite number?

Find a quadratic polynomial whose zeroes are $3 + \sqrt{5}$ and $3 - \sqrt{5}$

Solve for x and y $\frac x a + \frac y b = 2$ ; $ax - by = a^2 - b^2$

Prove that $sec^2\theta + cosec^2\theta = sec^2\theta*cosec^2\theta$

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Please give me an explaination instead of hints, my exams are tomorrow and i am sitting on heaps of practice questions... –  aayush Sep 23 '12 at 15:14
For the first question, the number is divisible by $3$ but greater than $3$, so it is composite. –  André Nicolas Sep 23 '12 at 15:17
(1) The number is sum of two numbers,so is even and $> 2,$ so can not be prime. –  lab bhattacharjee Sep 23 '12 at 15:18
@labbhattacharjee, I think you meant to say that the number is a sum of two odd numbers. –  Joel Reyes Noche Apr 21 at 5:28
@JoelReyesNoche, yes. –  lab bhattacharjee Apr 21 at 5:42
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## 1 Answer

Too many questions in too few days, and you show no insights, self effort..., Here are some hints:

$$1)\;\;\; 7^5\cdot 3^2\cdot 5+3=3\,(7^5\cdot 3\cdot 5+1)$$

$$(2)\;\;\;(x-(3+\sqrt 5))(x-(3-\sqrt 5))\;--\text{(Note that this polynomial is rational)}$$

$$(3)\;\;\;\frac{x}{a}+\frac{y}{b}=2\Longrightarrow bx+ay=2ab\Longrightarrow\,\text{we have the linear system:}$$

\begin{align*}ax-by=&a^2-b^2\\bx+ay=&2ab\end{align*}

Since from the given data $\,a,b\neq 0\,$ (why?), we get above $\,x=a\,$

Multiply now the first eq. by $\,a\,$ and the second one by $\,b\,$ and get:

\begin{align*}a^2x-aby=&a^3-ab^2\\b^2x+aby=&2ab^2\end{align*}

Now sum both eq's and solve for $\,x\,$ :

$$(a^2+b^2)x=a^3+ab^2=a(a^2+b^2)$$.

Since from the given data $\,a,b\neq 0\,$ (why?), we get above $\,x=a\,$ . Substitute now in either equation and get $\,y\,$ .

$$\sec^2t+\csc^2t=\frac{1}{\cos^2t}+\frac{1}{\sin^2 t}=\frac{\sin^2t+\cos^2t}{\sin^2t\,\cos^2t}$$

Now just remember the trigonometric Pythagoras Theorem $\,\cos^2x+\sin^2x=1\,$ and you get that the above equals what you want.

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I know you may think badly of me, but i am kind of in alot of stress. I havent slept in 3 days, and the biggest exam of my life is in 2 hours from now. I keep pouring practise question over question, and hoping to get all the ones i cant do solved. –  aayush Sep 24 '12 at 0:35
It's not a matter of thinking bad or not. Hopefully you understand now that it is not a good idea to leave all aside until one week or so before the exam. You must work through this stuff much sooner as to be able to assimilate ideas, tricks, exercises, etc. –  DonAntonio Sep 24 '12 at 2:46
Its not just Math sadly..Its several exams and they are just one after the other. Theres a heck load of material, too much for any one person to process in this much time... There are many topics that are all equally as challanging, and one after the other. These include topics like Math, Physics, Chemistry, Biology and other things like sanskrit(A extremely ancient language) along with hindi, english and even stuff like General knowledge and moral science. That packed with testing on other levels, such as the activities we chose, i chose chess. On top of that 1/4 of your grade is on assignmen –  aayush Sep 24 '12 at 9:34
And EVERY subject(total 15) give us a difficult assignment. All this in half a year, every assignment is too time consuming, and a major portion of the time goes to this. Then all the other things and then the exams...And after this starts the 2nd term which is even more challanging than the first. And after all this is done you get a piece of paper, that is your entire academic career. Those who get a bad score on this marksheet might aswell jump off a building, because no college will ever take them in and they will never get a respectable job... –  aayush Sep 24 '12 at 9:41
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