To maximize the largest integer, we need to minimize the values of the other integers while adhering to the given conditions: the average of the 6 positive integers is 55, and the difference between the largest and smallest is 24.
Let the 6 positive integers be \(n_1, n_2, n_3, n_4, n_5, n_6\) in ascending order.
The average of these 6 integers is 55. So, their sum is:
\[ \text{Sum} = 6 \times 55 = 330 \]Let the smallest integer be \(S\) and the largest integer be \(L\). We are given that \(L - S = 24\), which implies \(S = L - 24\).
To maximize \(L\), we must minimize the other four integers (\(n_2, n_3, n_4, n_5\)). Since all integers must be positive, the smallest possible value for any integer is 1. However, we also have the condition that \(S = L - 24\).
The integers are in ascending order: \(S \le n_2 \le n_3 \le n_4 \le n_5 \le L\).
To minimize \(n_2, n_3, n_4, n_5\), we set them to be as small as possible, which means they should be equal to the smallest integer \(S\). So, we can set:
This configuration ensures that the first five numbers are as small as possible, allowing \(L\) to be as large as possible. All numbers must be positive, so \(S \ge 1\).
Now, substitute these values into the sum equation:
\[ S + S + S + S + S + L = 330 \] \[ 5S + L = 330 \]Substitute \(S = L - 24\) into the equation:
\[ 5(L - 24) + L = 330 \] \[ 5L - 120 + L = 330 \] \[ 6L = 330 + 120 \] \[ 6L = 450 \] \[ L = \frac{450}{6} \] \[ L = 75 \]With \(L = 75\), the smallest integer \(S = L - 24 = 75 - 24 = 51\). All integers are positive (51, 51, 51, 51, 51, 75). This set satisfies all conditions.
Correct Option: B) 75
The question asks for the full form of 'HITEC' as in 'HITEC City' which refers to Hyderabad. This is a general knowledge question related to the IT sector and urban development in India.
A) Hyderabad Information Technology and Engineering Consultancy — This is the correct and official full form of HITEC as it refers to the IT hub in Hyderabad.
The question asks to identify the Governor of Maharashtra as of March 2021. This requires knowledge of current affairs and political appointments during that specific period.
Correct Option: B) Bhagat Singh Koshyari was indeed the Governor of Maharashtra as of March 2021. He held the office from September 2019 to February 2023.
The problem requires us to identify which two numbers, when interchanged in the given equation, will make the equation mathematically correct. We will test each option by performing the interchange and then evaluating the expression using the order of operations (BODMAS/PEMDAS).
The original equation is: \(29 - 19 + 25 \times 87 \div 44 = 100\)
Let's evaluate the original equation first to see if it's already correct:
\[ 29 - 19 + 25 \times 87 \div 44 \]\[ = 29 - 19 + 2175 \div 44 \]\[ = 29 - 19 + 49.43 \]\[ = 10 + 49.43 = 59.43 \]Since \(59.43 \neq 100\), the original equation is incorrect.C) 29 and 44
Interchange 29 and 44 in the equation:
\[ 44 - 19 + 25 \times 87 \div 29 \]Now, evaluate the expression:The question asks to identify the date observed as World Blood Donor Day. This is a factual question related to important international observances.
Correct Option: B) 14 June is observed as World Blood Donor Day.
The question asks to identify which of the listed devices/interfaces was invented by Douglas Engelbart. This requires knowledge of the history of computing and key inventions.
Correct Option: D) Computer mouse
Douglas Engelbart is credited with inventing the computer mouse in the 1960s while working at the Stanford Research Institute (SRI). He demonstrated it publicly in 1968 during what is now known as "The Mother of All Demos." His original mouse was a wooden shell with two metal wheels.
To find the missing number in the series, we need to identify the pattern or rule that governs the sequence of numbers. We will analyze the differences or relationships between consecutive terms.
Thus, the next number in the series is 17.
Correct Option: B) 17
24 students are sitting in a row. F is 17th from the right end and R is 19th from the left end. How many students are between F and R?
This problem involves finding the number of students between two individuals in a row. We need to determine the positions of F and R from a common reference point (e.g., the left end) and then calculate the difference, considering potential overlaps.
To determine which latitude is most distant from India, we need to understand India's latitudinal extent and then compare the distances to the given options. The Earth is a sphere, and distances along meridians (lines of longitude) can be approximated by differences in latitude. India is located entirely in the Northern Hemisphere, predominantly between approximately \(8^\circ\text{N}\) and \(37^\circ\text{N}\) latitude.
A) South Pole (90°S) is the most distant latitude from India. The angular distance from India's southernmost point (\(\approx 8^\circ\text{N}\)) to the South Pole (\(90^\circ\text{S}\)) is approximately \(90^\circ + 8^\circ = 98^\circ\).
This problem involves work and time, specifically comparing the work rates of men and women. We will first find the individual work rate of one man and one woman, then determine the combined work rate of 6 men and 8 women, and finally calculate the total time required to complete the task.
Calculate the total work units for men:
If 14 men complete a task in 9 days, the total work done by men can be expressed as:
Calculate the total work units for women:
If 16 women complete the same task in 7 days, the total work done by women can be expressed as:
Equate the total work to find the relationship between man-days and woman-days:
Since the task is the same, the total work units are equal:
Find the ratio of work done by one man to one woman:
From the equality above, we can find the equivalent work of men in terms of women, or vice versa. Let's find how many women are equivalent to one man:
Convert the work of 6 men into equivalent woman-days:
Now, we need to find the work done by 6 men in terms of woman-days:
Calculate the total equivalent woman-days for 6 men and 8 women:
The combined team consists of 6 men and 8 women. In terms of woman-days, this is:
Calculate the number of days to complete the task:
We know that 112 woman-days are required to complete the task. If the combined team works at a rate of \(\frac{40}{3}\) woman-days per day, the number of days required is:
Correct Option: B) 8.4