There are three pillars X, Y and Z of different heights. Three spiders A, B and C start to climb on these pillars simultaneously. In one chance, A climbs on X by 6 cm but slips down 1 cm. B climbs on Y by 7 cm but slips down 3 cm. C climbs on Z by 6.5 cm but slips down 2 cm. If each of them requires 40 chances to reach the top of the pillars, what is the height of the shortest pillar?
UPSC-2017
The problem describes three spiders climbing pillars with different rates of climbing and slipping. They all take 40 chances to reach the top. We need to find the height of the shortest pillar. The key insight is that in the last chance, the spider will climb the full distance without slipping.
Since the spiders reach the top in 40 chances, they will have climbed and slipped for 39 chances, and in the 40th chance, they will make the final climb without slipping.
The heights of the pillars are:
Comparing these heights, the shortest pillar is Y, with a height of 163 cm.
B) 163 cm is the height of pillar Y, which is the shortest among the three pillars.
X and Y are natural numbers other than 1, and Y is greater than X, which of the following represents the largest number?
UPSC-2018
The problem asks us to identify the largest number among the given options, where \(X\) and \(Y\) are natural numbers greater than 1, and \(Y > X\). We can solve this by choosing specific values for \(X\) and \(Y\) that satisfy the given conditions and then evaluating each option.
A) XY is the largest number. When \(X=2\) and \(Y=3\), \(XY = 6\), which is the maximum among all calculated values.
A number consists of three digits of which the middle one is zero and their sums is 4. If the number formed by interchanging the first and last digits is greater than the number itself by 198, then the difference between the first and last digits is
UPSC-2018
We are given a three-digit number with its middle digit as zero. We are also given the sum of its digits and a condition about the number formed by interchanging its first and last digits. We need to find the difference between the first and last digits.
Let the three-digit number be \(100x + 10y + z\). According to the problem, the middle digit is zero, so \(y = 0\). Thus, the number is \(100x + z\).
The sum of its digits is 4. So, \(x + y + z = 4\). Since \(y = 0\), we have \(x + z = 4\).
The number formed by interchanging the first and last digits would be \(100z + 10y + x\). Since \(y = 0\), this number is \(100z + x\).
We are given that the new number is greater than the original number by 198. So,
\[ (100z + x) - (100x + z) = 198 \]Simplify the equation:
\[ 100z + x - 100x - z = 198 \]\[ 99z - 99x = 198 \]Divide by 99:
\[ z - x = \frac{198}{99} \]\[ z - x = 2 \]We have two equations:
Add the two equations:
\[ (x + z) + (z - x) = 4 + 2 \]\[ 2z = 6 \]\[ z = 3 \]Substitute \(z = 3\) into the first equation \(x + z = 4\):
\[ x + 3 = 4 \]\[ x = 1 \]The first digit is \(x = 1\) and the last digit is \(z = 3\). The difference between the first and last digits is \(|z - x|\) or \(|x - z|\).
\[ |3 - 1| = 2 \]B) The difference between the first and last digits is 2.
A bookseller sold 'a' number of Geography textbooks at the rate of ₹ x per book, 'a+2' number of History textbooks at the rate of ₹ (x+2) per book and 'a-2' number of Mathematics textbooks at the rate of ₹ (x-2) per book. What is his total sale in ₹?
UPSC-2018
To find the total sale, we need to calculate the revenue from each type of textbook and then sum them up. The revenue from each type of book is the product of the number of books sold and the rate per book.
B) \(3ax+8\) is the correct expression for the total sale, as derived from the step-by-step calculation.
Joseph visits the club on every 5th day, Harsh visits on every 24th day, while Sumit visits on every 9th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
UPSC-2021
To find out when all three individuals will meet again, we need to find the least common multiple (LCM) of the number of days each person takes to visit the club. Once we have the LCM, we can determine the day of the week by counting forward from the initial meeting day.
Therefore, all three of them will meet again on a Wednesday.
Correct Option: B) Wednesday
A 2-digit number is reversed. The larger of the two numbers is divided by the smaller one. What is the largest possible remainder?
UPSC-2017
