Question:

1. If the zeros of the polynomial \({{x}^{3}}-9{{x}^{2}}-54x+c=0\) form a geometric progression, then what is c ?
2. The system of equations
   \[\begin{aligned} & {{x}^{2}}+{{y}^{2}}={{z}^{2}} \\ & {{x}^{2}}=y+z \\ \end{aligned}\]
   has many solutions \(\left( x,y,z \right)\) for positive integers x , y and z . Find 5 such solutions.
3. Two candles of equal length are lighted. On will burn completely in 4 hours while the other needs 5 hours to burn completely. In how many hours after lighting will one be four times the length of the other?
   (where I is the initial length). The condition is
   \[4{{L}_{1}}={{L}_{2}}\,\,\,\Rightarrow \,\,\,\frac{4I\left( 4-t \right)}{4}=\frac{I\left( 5-t \right)}{5}\]
   \[\Rightarrow \,\,\,5\left( 4-t \right)=5-t\,\,\,\Rightarrow \,\,\,\,20-5t=5-t\,\,\,\Rightarrow \,\,\,\,4t=15\,\,\,\Rightarrow \,\,t=3.75\text{ hours}\]
4. If \({{\log }_{b}}N=3\), then what is \({{\log }_{N}}{{b}^{2}}\) ?
5. A station wagon has a seating capacity of 8 persons; 2 in addition to the driver in the front seat; 2 in the middle seat, and 3 in the rear seat. In how many ways can a group of 8 persons be seated in this vehicle, if only 3 can drive and 1 of the others is too tall to occupy the rear seat?

(f) The GCD of two positive integers is 24. Their LCM is 288. If neither of them is 24, then what are the possible values of A and B .

Price: $2.99

Solution: The downloadable solution consists of 4 pages
Deliverable: Word Document