Exercise

Let $a$ and $b$ be two positive integers such that $a=p^3{q}^4$ and $b=p^2{q}^3$, where $p$ and $q$ are prime numbers. If $\mathrm{HCF}\left(a, b\right)=p^m{q}^n$ and $\mathrm{LCM}\left(a, b\right)=p^r{q}^s$, then $\left(m+n\right)\left(r+s\right)=$

Shown below are the graphs of two cubic polynomials, $f\left(x\right)$ and $g\left(x\right)$. Both the polynomials have the zeroes $-1, 0$ and $1$

Anya said, " Both the graphs represent the same polynomial, $f\left(x\right)=g\left(x\right)=\left(x+1\right)\left(x-0\right)\left(x-1\right)$ as they have the same zeroes.

Pranit said," Both the graphs represent two different polynomial, $f\left(x\right)=\left(x+1\right)\left(x-0\right)\left(x-1\right)$ and $g\left(x\right)=-\left(x+1\right)\left(x-0\right)\left(x-1\right)$ and only two such polynomial exist that can have the zeroes $-1, 0$ and $1$

Aadar said, " Both the graphs represent two different polynomial and infintely many such polynomials exist that have zeroes $-1, 0$ and $1$.

Who is right and justify your answer?

Find the distance between the zeroes of the polynomial $f\left(x\right)=2x^2-x-6$. Show your steps.

$p\left(x\right)=2x^2-6x-3$. The two zeroes are of the form: $\frac{3\pm \sqrt{k}}{2}$; where $k$ is a real number.

Use the relationship between the zeroes and the coefficients of a polynomial to find the value of $k$. Show your steps.

At the Bengaluru marathon, the $42$ km route is designed such that marathoners run in a straight line for $21$ km, and return back along the same path in opposite direction. 

A marathoner, running against the wind, covered the first half of the marathon in $2$ hrs. Then, he covered the second half, running with the wind, in $1.5$ hours. Assume that the marathoner ran at a constant speed and that the wind speed and direction did not change throughout the marathon.

Calculate the speed (in km/h) of the marathoner and the wind. Show your steps. 

Ananya had red, blue and yellow marbles in the ratio $4:5:3.$ She gave all her red marbles and some blue marbles to Neha. The ratio of the number of blue marbles and yellow marbles left with Ananya was $7:9.$

If Ananya gave $20$ marbles to Neha, how many of them are red marbles$?$ Show your work.

The break-even point for a new flavour ice cream is the number of units sold at which the total cost price of the ice cream is the same as the total selling price. The parlour launches another new flavour whose fixed cost is Rs$10000,$ variable cost is Rs $40$ and the selling price of each unit Rs $60.$

Find the number of units at the break-even point of the other new flavour. Show your work. 

A company has a locker in which valuable documents are kept. The passcode is a four digit number of the form $xyyx.$ The Chief Executive Officer (CEO) and the Vice President (VP) of the company have each been given one clue. On solving BOTH clues, the passcode that opens the locker can be found.

CEO's clue: When twice the ones digit is subtracted from the tens digit, the result is $1.$

VP's clue: Three more than the tens digit is thrice the ones digit.

Find the passcode that opens the locker. Show your work.