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Two British and three Indian men and their wives are to be seated around a circular table. If the seating arrangements are defined only by the relative positions of the people, then the total number of ways in which each British man can be seated next to his wife and each Indian man can also be, simultaneously, seated next to his wife is:

81 equally efficient people could complete a piece of work in 15 days. Initially 45 such people started working together. But after they worked together for 6 days their employer wanted that the work should be completed in a total of 21 days. However any new worker who joins is only $60\%$ as efficient as any of the 45 people already working. How many new workers did the employer hire to work alongside the initial 45 people to meet the deadline?

If $f(x) = x - 1$, and $g(f(x)) = 3 + 2x - x^2 + x^3$, which of the options below give a correct expression for $g(x)$?

A bank offers $5\%$ compound interest per annum, calculated on half-yearly basis. A customer deposits $₹3200$ each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is:

A train running between two towns arrives at its destination 10 minutes late when it goes at 60 km per hour and 16 minutes late when it goes at 40 km per hour. Determine the speed at which the train reaches on time.

If $2^{x+y} = 4^{x-y} = \sqrt{32}$, then $xy =$ ___.

The sum of the first 3 terms of an AP is 6 and that of the last 3 terms is 16. If the AP has 13 terms, what is the Arithmetic mean of the three terms that are exactly at the middle of the progression?

Sahil can complete a job in 20 days when working alone. Manoj is twice as fast as Sahil and three times as fast as Ashish in the same job. The collaborative work arrangement involves Sahil and Manoj working on the first day, Sahil and Ashish on the second day, and Manoj and Ashish on the third day. This three-day cycle continues till the work is complete.

What fraction of the total work will then be done by Ashish till the job is completed?

The digit in the units' place of $13^{2003} - 7^{291}$ is ___.

A cylindrical vessel of base radius 42 cm contains water to a certain level. A solid cylinder of height 18 cm is dropped into it and is completely submerged in water. If the level of the water in the vessel rises by 8 cm, what is the total surface area (in cm²) of the solid cylinder? (Use $\pi = \frac{22}{7}$ ) (Assuming no overflow.)

While selling $15$ items, a shopkeeper earned a profit equal to the total cost price of $3$ items. If each item is marked $25\%$ above its actual cost and a discount of $₹2$ is given while selling it, then what is the difference between the total cost price of $20$ items and the total selling price of $12$ items?

In $\Delta XYZ$, length of $XY = 13$ cm. If the length of the median drawn from vertex $Z$ to base $XY$ is $6.5$ cm, then $\Delta XYZ$ is a/an:

The roots of the equation $x^2 + 18x - 703 = 0$ are given as $a$ and $b$, with $a > b$. The roots of the equation $x^2 - 10x - 336 = 0$ are given as $c$ and $d$, with $c > d$. Which of the options below gives the correct relation between $a$, $b$, $c$, and $d$?

A sum of $₹7,740$ is divided between X, Y, and Z such that the ratio of the share of X to the combined share of Y and Z is $4 : 11$ and the ratio of the share of Z to the combined share of X and Y is $1 : 3$. What is the share (in Rs.) of Y?

The sum of the cost prices of three items was $₹24,150$. The person sold the first item at a profit of $72\%$, the second item at a profit of $40\%$, and the third item at a profit of $126\%$. The total profit earned from the sale of these three items was $₹19,432$. If the cost price of the second item was $₹175$ more than what it actually was, then the ratio of the cost prices of the second and the third items, respectively, would have been $4 : 5$. What was the difference (in Rs.) between the profits earned from the sale of the first and the second items?

Find the sum of the smallest value and the greatest value of $x$ that satisfy the inequality $|5x + 5| - 8 \leq 17$.