If \( \frac{a^{2}}{b^{2} + c^{2}} = \frac{b^{2}}{c^{2} + a^{2}} = \frac{c^{2}}{a^{2} + b^{2}} \), then what is the value of \( a^{4} + b^{4} + c^{4} \) equal to?
Let \( P = 5^{5} \times 15^{15} \times 25^{25} \times 35^{35} \) and \( Q = 10^{10} \times 20^{20} \times 30^{30} \times 40^{40} \). What is the number of consecutive zeros at the end of the sum \( (P + Q) \)?
The LCM and HCF of two polynomials \( p(x) \) and \( q(x) \) are \( (x + a)(x^{3} - a^{3}) \) and \( (x^{2} - ax + a^{2}) \), respectively. If \( p(x) = x^{4} + a^{2}x^{2} + a^{4} \), then what is \( q(x) \) equal to?
Which of the following expressions can divide both the polynomials \( x^{3} + 2x^{2} - 5x + 2 \) and \( x^{3} + 4x^{2} + x - 6 \) exactly?
I. \( x - 1 \)
II. \( x + 1 \)
III. \( x + 2 \)
Select the correct answer using the code given below:
Eight men and 24 women can finish a piece of work in one day while 12 men and 18 women can also finish it in one day. What is the time taken by 24 men and 72 women to finish the work?
