What is the missing number 'X' of the series 7, X, 21, 31, 43?
UPSC - 2015
To find the missing number in a series, we need to identify the pattern or rule governing the sequence of numbers. This often involves looking at the differences between consecutive terms, or other mathematical relationships.
\(7, X, 21, 31, 43\)
The differences are 10 and 12. This suggests that the differences between consecutive terms are increasing by 2 each time. If this pattern holds, the differences would be \(\dots, 6, 8, 10, 12\).
Using \(X - 7 = 6\):
\[ X = 6 + 7 \] \[ X = 13 \]Using \(21 - X = 8\):
\[ X = 21 - 8 \] \[ X = 13 \]Both calculations yield \(X = 13\).
\(7, 13, 21, 31, 43\)
Differences:
The differences are \(6, 8, 10, 12\), which confirms the pattern of increasing by 2.
Correct Option: C) 13
Consider the following three-dimensional figure: 
How many triangles does the above figure have?
UPSC - 2018
To determine the number of triangles in the given three-dimensional figure, we need to carefully count all possible triangles formed by the intersecting lines and edges.
Correct Option: B) 20
Examine the following three figure in which the numbers follow a specific pattern: 
The missing number (?) in the third figure above is
UPSC - 2013
To solve this problem, we need to identify the pattern followed by the numbers in the given figures. Each figure consists of three numbers, and we must determine the relationship between these numbers to find the missing value.
Correct Option: B) 16
In a queue, Mr. X is fourteenth from the front and Mr. Y is seventeenth from the end, while Mr. Z is exactly in between Mr. X and Mr. Y. If Mr. X is ahead of Mr. Y and there are 48 persons in the queue, how many persons are there between Mr. X and Mr. Z?
UPSC - 2011
We need to determine the number of people between Mr. X and Mr. Z. First, we'll find the position of Mr. Y from the front. Then, we'll calculate the number of people between Mr. X and Mr. Y. Since Mr. Z is exactly in the middle of Mr. X and Mr. Y, we can find the number of people between Mr. X and Mr. Z.
Total number of persons in the queue:
\( \text{Total persons} = 48 \)
Position of Mr. X from the front:
\( \text{X's position from front} = 14^{\text{th}} \)
Position of Mr. Y from the end:
\( \text{Y's position from end} = 17^{\text{th}} \)
Calculate Mr. Y's position from the front:
To find Mr. Y's position from the front, we use the formula:
\[ \text{Position from front} = \text{Total persons} - \text{Position from end} + 1 \]\[ \text{Y's position from front} = 48 - 17 + 1 = 31 + 1 = 32^{\text{nd}} \]Calculate the number of persons between Mr. X and Mr. Y:
Since Mr. X is ahead of Mr. Y, Mr. X is at position 14 from the front and Mr. Y is at position 32 from the front.
Number of persons between X and Y \( = (\text{Y's position from front}) - (\text{X's position from front}) - 1 \)
\( = 32 - 14 - 1 \)
\( = 18 - 1 \)
\( = 17 \text{ persons} \)
Calculate the number of persons between Mr. X and Mr. Z:
Mr. Z is exactly in between Mr. X and Mr. Y. This means the number of persons between X and Z is equal to the number of persons between Z and Y.
Let \( N_{XY} \) be the number of persons between X and Y. We found \( N_{XY} = 17 \).
If Mr. Z is exactly in between, then the number of persons between X and Z is half of \( N_{XY} \), if \( N_{XY} \) is even. If \( N_{XY} \) is odd, then Z is the middle person, and the number of persons between X and Z is \( (N_{XY} - 1) / 2 \).
In this case, \( N_{XY} = 17 \). So, Mr. Z is the \( (17+1)/2 = 9^{\text{th}} \) person from Mr. X (not counting Mr. X) and also the \( 9^{\text{th}} \) person from Mr. Y (not counting Mr. Y).
The number of persons between Mr. X and Mr. Z is \( (17 - 1) / 2 \).
\( = 16 / 2 \)
\( = 8 \text{ persons} \)
C) 8
Examine the following statements:
1. Rama scored more than Rani.
2. Rani scored less than Ratna.
3. Ratna scored more than Rama.
4. Padma scored more than Rama but less than Ratna.
Who scored the highest?
UPSC - 2012
This problem involves comparing the scores of five individuals based on a series of relational statements. The best approach is to represent the relationships using inequalities or by arranging the names in descending order of scores as we deduce them from each statement.
Correct Option: D) Ratna scored the highest.
In five flats, one above the other, live five professionals. The professor has to go up to meet his IAS officer friend. The doctor is equally friendly to all, and has to go up as frequently as go down. The engineer has to go up to meet his MLA friend above whose flat lives the professor's friend. From the ground floor to the top floor in what order do the five professionals live?
UPSC - 2012
Engineer, Professor, Doctor, IAS officer, MLA
Professor, Engineer, Doctor, IAS officer, MLA
IAS officer, Engineer, Doctor, professor, MLA
Professor, Engineer, Doctor, MLA, IAS officer
This is a logical deduction problem. We need to determine the order of five professionals living in five flats, one above the other, based on the given clues. We will assign numbers to the floors from 1 (ground floor) to 5 (top floor) and place the professionals step-by-step.
Clue 1: "The doctor is equally friendly to all, and has to go up as frequently as go down."
This implies the doctor lives in the middle flat. Since there are five flats, the middle flat is the 3rd floor.
Clue 2: "The professor has to go up to meet his IAS officer friend."
This means the IAS officer lives above the professor. So, Professor < IAS officer.
Clue 3: "The engineer has to go up to meet his MLA friend above whose flat lives the professor's friend."
Let's break this down:
Combining these, we get the sequence: Engineer < MLA < IAS officer.
Now, let's combine all the deductions:
Consider the possible placements for Engineer, MLA, and IAS officer, keeping in mind the Doctor is on Flat 3:
Let's test the first possibility: Engineer (Flat 1), MLA (Flat 2), IAS officer (Flat 4).
Now, we need to place the Professor. We know Professor < IAS officer. The only remaining flat is Flat 5. If Professor is on Flat 5, then Professor is above IAS officer, which contradicts Professor < IAS officer. So, this arrangement is incorrect.
Let's test the second possibility: Engineer (Flat 2), MLA (Flat 4), IAS officer (Flat 5).
Now, we need to place the Professor. We know Professor < IAS officer. The only remaining flat is Flat 1. If Professor is on Flat 1, then Professor is below IAS officer, which is consistent with the clue. This arrangement works.
Final arrangement from ground floor (Flat 1) to top floor (Flat 5):
D) Professor, Engineer, Doctor, MLA, IAS officer
