When \(3^{333}\) is divided by \(11\), the remainder is

If \(x\) and \(y\) satisfy the equations \(|x| + x + y = 15\) and \(x + |y| - y = 20\), then \((x - y)\) equals

If \(m\) and \(n\) are natural numbers such that \(n > 1\) and \(m^n = 2^{25} \times 3^{40}\), then \(m - n\) equals

A function \( f \) maps the set of natural numbers to whole numbers such that \( f(xy)=f(x)f(y)+f(x)+f(y) \) for all \( x,y \) and \( f(p)=1 \) for every prime number \( p \). Then the value of \( f(160000) \) is

\(ABCD\) is a trapezium in which \(AB\) is parallel to \(CD\). The sides \(AD\) and \(BC\) when extended, intersect at point \(E\). If \(AB=2\text{ cm}\), \(CD=1\text{ cm}\), and the perimeter of \(ABCD\) is \(6\text{ cm}\), then the perimeter, in cm, of \(\triangle AEB\) is

Amal and Vimal together can complete a task in 150 days, while Vimal and Sunil together can complete the same task in 100 days. Amal starts working on the task and works for 75 days, then Vimal takes over and works for 135 days. Finally, Sunil takes over and completes the remaining task in 45 days. If Amal had started the task alone and worked on all days, Vimal had worked on every second day, and Sunil had worked on every third day, then the number of days required to complete the task would have been

Bina incurs 19% loss when she sells a product at Rs. 4860 to Shyam, who in turn sells this product to Hari. If Bina would have sold this product to Shyam at the purchase price of Hari, she would have obtained 17% profit. Then, the profit, in rupees, made by Shyam is

The sum of the infinite series

\( \frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right) +\left(\frac{1}{5}\right)^2\left(\left(\frac{1}{5}\right)^2-\left(\frac{1}{7}\right)^2\right) \)

\( +\left(\frac{1}{5}\right)^3\left(\left(\frac{1}{5}\right)^3-\left(\frac{1}{7}\right)^3\right)+\cdots \) is equal to

A. \( \dfrac{5}{408} \), B. \( \dfrac{7}{816} \), C. \( \dfrac{7}{408} \), D. \( \dfrac{5}{816} \)

If \(a\), \(b\), and \(c\) are positive real numbers such that \(a>10\ge b\ge c\) and \(\dfrac{\log_{8}(a+b)}{\log_{2} c}+\dfrac{\log_{27}(a-b)}{\log_{3} c}=\dfrac{2}{3}\), then the greatest possible integer value of \(a\) is

A company has 40 employees whose names are listed in a certain order. In the year 2022, the average bonus of the first 30 employees was Rs. 40000, of the last 30 employees was Rs. 60000, and of the first 10 and last 10 employees together was Rs. 50000. Next year, the average bonus of the first 10 employees increased by 100%, of the last 10 employees increased by 200% and of the remaining employees was unchanged. Then, the average bonus, in rupees, of all the 40 employees together in the year 2023 was