To solve this problem, we need to first identify the even and odd digits in the given number. Then, we apply the specified operations: add 2 to each even digit and subtract 2 from each odd digit. Finally, we sum up all the even digits in the newly formed number.
Thus, the sum of all the even digits in the new number is 18.
Correct Option: D) 18
The problem presents a numerical series, and we need to find the next term. We will analyze the differences between consecutive terms to identify a pattern. If the first-level differences don't reveal a clear pattern, we will look for a pattern in the second-level differences or other mathematical relationships.
135, 123, 109, 93, 75, ?
So, the series of differences is: 12, 14, 16, 18.
The differences (12, 14, 16, 18) form an arithmetic progression where each term increases by 2. This means the next difference in this sequence should be \(18 + 2 = 20\).
The last term given in the original series is 75. To find the next term, we subtract the next difference (20) from 75.
\[ 75 - 20 = 55 \]The next term in the series is 55.
Correct Option: B) 55
The problem asks us to find the missing letter-clusters, # and %, based on a pattern and relationship observed in letter-cluster pairs. We need to analyze the given pair 'PKN' and 'TOR' to identify the rule and then apply it to find the missing clusters.
Correct Option: A) # = KFI, % = YTW
If x + \( \frac{1}{x} \) = \(\sqrt{5}\), x > 0, find the value of x⁴ + \( \frac{1}{x^4} \).
The problem requires us to find the value of \(x^4 + \frac{1}{x^4}\) given \(x + \frac{1}{x} = \sqrt{5}\). We can solve this by repeatedly squaring the given expression. First, we square \(x + \frac{1}{x}\) to find \(x^2 + \frac{1}{x^2}\), and then square the result again to find \(x^4 + \frac{1}{x^4}\).
Given expression:
\[ x + \frac{1}{x} = \sqrt{5} \]Square both sides of the given equation:
\[ \left(x + \frac{1}{x}\right)^2 = (\sqrt{5})^2 \]Using the identity \((a+b)^2 = a^2 + b^2 + 2ab\), we get:
\[ x^2 + \left(\frac{1}{x}\right)^2 + 2 \cdot x \cdot \frac{1}{x} = 5 \] \[ x^2 + \frac{1}{x^2} + 2 = 5 \]Solve for \(x^2 + \frac{1}{x^2}\):
\[ x^2 + \frac{1}{x^2} = 5 - 2 \] \[ x^2 + \frac{1}{x^2} = 3 \]Square both sides of the new equation \(x^2 + \frac{1}{x^2} = 3\):
\[ \left(x^2 + \frac{1}{x^2}\right)^2 = (3)^2 \]Using the identity \((a+b)^2 = a^2 + b^2 + 2ab\) again, where \(a = x^2\) and \(b = \frac{1}{x^2}\):
\[ (x^2)^2 + \left(\frac{1}{x^2}\right)^2 + 2 \cdot x^2 \cdot \frac{1}{x^2} = 9 \] \[ x^4 + \frac{1}{x^4} + 2 = 9 \]Solve for \(x^4 + \frac{1}{x^4}\):
\[ x^4 + \frac{1}{x^4} = 9 - 2 \] \[ x^4 + \frac{1}{x^4} = 7 \]Correct Option: A) The value of \(x^4 + \frac{1}{x^4}\) is 7.
The problem asks us to identify the letter-cluster that does not follow the same pattern as the others, based on the English alphabetical order. We need to find the positional value of each letter in the alphabet and then look for a consistent numerical relationship between the letters within each cluster.
Let's apply this to each option:
Upon analyzing the differences between consecutive letters:
The pattern for options A, C, and D is a sequence of differences (5, 3, 4). Option B has a different sequence of differences (5, 4, 3).
B) EJNQ — This cluster has differences of (5, 4, 3) between consecutive letters, which is different from the (5, 3, 4) pattern observed in the other clusters.
This is a linear arrangement puzzle. We need to deduce the seating order of seven people (A, B, L, M, N, S, T) based on the given clues and then identify the person sitting third from the left end.
Clue 1: Only five people sit between T and A.
Since there are 7 people in total, and 5 people sit between T and A, this means T and A must be at the ends of the row. There are two possibilities:
Clue 2: N sits to the immediate left of A.
Let's re-evaluate Clue 1. "Only five people sit between T and A" means T and A are separated by 5 people. This implies they are at the ends of the row, but not necessarily the absolute ends of the 7-person row. Let's list the 7 positions: 1 2 3 4 5 6 7.
If T is at position 1, A is at position 7. (5 people between them). This is the only way to have 5 people between them in a 7-person row.
So, the arrangement must be: T _ _ _ _ _ A
Clue 2 (revisited): N sits to the immediate left of A.
Given T _ _ _ _ _ A, A is at position 7. N must be at position 6. The arrangement becomes: T _ _ _ _ N A
Clue 3: Only one person sits between N and L.
N is at position 6. If one person sits between N and L, L can be at position 4 (L _ N) or position 8 (N _ L). Since there are only 7 positions, L must be at position 4.
The arrangement becomes: T _ _ L _ N A
Clue 4: M sits at some place to the left of S but at some place to the right of B.
The remaining people are B, M, S. The remaining positions are 2, 3, 5.
Combining these, the order must be B, M, S. The only way to fit B, M, S into the remaining positions (2, 3, 5) is:
The final arrangement is: T B M L S N A
Identify the person sitting third from the left end.
Counting from the left: 1st is T, 2nd is B, 3rd is M.
C) M sits third from the left end of the line.
This is a coding-decoding problem where we need to find the code for a common word by comparing two given coded statements. The strategy is to identify the common word in the statements and then find the common code associated with it.
D) wk is the common code found in both statements for the common word 'down'.
The mode of a set of observations is the value that appears most frequently in the dataset. To find the mode, we need to count the occurrences of each distinct observation and identify the one with the highest frequency.
20, 20, 20, 35, 29, 34, 25, 33, 35, 32, 33, 27, 22, 31, 31
Comparing the frequencies, the number 20 has the highest frequency, appearing 3 times.
Since 20 appears most frequently, it is the mode of the given observations.
Correct Option: A) 20 is the mode because it appears 3 times, which is more than any other number in the given set of observations.
This problem involves concepts of work and time, specifically related to pipes and cisterns. We need to calculate the individual filling rates of the taps, then their combined filling rate without leakage, and finally determine the effective filling rate with leakage to find the leakage rate. Once the leakage rate is known, we can find the time it takes for the leakage alone to empty the tank.
Tap X can fill the tank in 12 hours. So, in 1 hour, tap X fills:
\[ \text{Rate}_X = \frac{1}{12} \text{ of the tank} \]Tap Y can fill the tank in 24 hours. So, in 1 hour, tap Y fills:
\[ \text{Rate}_Y = \frac{1}{24} \text{ of the tank} \]If both taps X and Y are open, their combined filling rate in 1 hour is:
\[ \text{Combined Rate} = \text{Rate}_X + \text{Rate}_Y = \frac{1}{12} + \frac{1}{24} \]\[ \text{Combined Rate} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8} \text{ of the tank} \]If the combined rate is \(\frac{1}{8}\) of the tank per hour, then the time taken to fill the tank without leakage is:
\[ \text{Time (no leakage)} = \frac{1}{\text{Combined Rate}} = \frac{1}{1/8} = 8 \text{ hours} \]Due to leakage, it takes 30 minutes more to fill the tank. 30 minutes is equal to 0.5 hours.
\[ \text{Time (with leakage)} = 8 \text{ hours} + 0.5 \text{ hours} = 8.5 \text{ hours} \]If the tank is filled in 8.5 hours with leakage, the effective filling rate (taps X and Y minus leakage) is:
\[ \text{Effective Rate} = \frac{1}{8.5} = \frac{1}{17/2} = \frac{2}{17} \text{ of the tank per hour} \]The leakage rate is the difference between the combined filling rate (without leakage) and the effective filling rate (with leakage):
\[ \text{Leakage Rate} = \text{Combined Rate} - \text{Effective Rate} \]\[ \text{Leakage Rate} = \frac{1}{8} - \frac{2}{17} \]\[ \text{Leakage Rate} = \frac{17}{8 \times 17} - \frac{2 \times 8}{17 \times 8} \]\[ \text{Leakage Rate} = \frac{17}{136} - \frac{16}{136} = \frac{1}{136} \text{ of the tank per hour} \]If the leakage empties \(\frac{1}{136}\) of the tank per hour, then the time taken for the leakage alone to empty the full tank is:
\[ \text{Time to empty} = \frac{1}{\text{Leakage Rate}} = \frac{1}{1/136} = 136 \text{ hours} \]Correct Option: D) 136 hours
The question asks about the winners of a specific event at the ISSF World Cup 2025, where Indian shooters Rudrankksh Patil and Arya Borse competed. To answer this, one needs to recall or research the results of the 10 m Air Rifle Mixed Team event at that particular competition.
B) A team from Norway — At the ISSF World Cup 2025, the Norwegian team defeated India's Rudrankksh Patil and Arya Borse in the gold medal match of the 10 m Air Rifle Mixed Team event.