To find the arithmetic mean (average) of a set of numbers, we sum all the numbers in the set and then divide the sum by the total count of numbers in the set.
A) 43.7 is the correct arithmetic mean calculated by summing all the numbers and dividing by their count.
The mean of 11 numbers is 44. If the mean of the first 6 numbers is 39 and that of the last 6 numbers is 48, then what is the 6th number?
This problem involves calculating the sum of numbers given their mean. We will use the formula for the mean, which is \[\text{Mean} = \frac{\text{Sum of numbers}}{\text{Count of numbers}}\]. We'll find the total sum of 11 numbers, then the sum of the first 6 numbers, and the sum of the last 6 numbers. The 6th number is common to both the first 6 and the last 6 numbers. By adding the sums of the first 6 and last 6 numbers and subtracting the total sum of all 11 numbers, we can find the value of the 6th number.
Calculate the total sum of 11 numbers:
Given that the mean of 11 numbers is 44.
\[ \text{Sum of 11 numbers} = \text{Mean} \times \text{Count} = 44 \times 11 = 484 \]Calculate the sum of the first 6 numbers:
Given that the mean of the first 6 numbers is 39.
\[ \text{Sum of first 6 numbers} = \text{Mean} \times \text{Count} = 39 \times 6 = 234 \]Calculate the sum of the last 6 numbers:
Given that the mean of the last 6 numbers is 48.
\[ \text{Sum of last 6 numbers} = \text{Mean} \times \text{Count} = 48 \times 6 = 288 \]Find the 6th number:
When we sum the first 6 numbers and the last 6 numbers, the 6th number is included twice. The sum of all 11 numbers includes the 6th number only once. Therefore, to find the 6th number, we add the sum of the first 6 numbers and the sum of the last 6 numbers, and then subtract the total sum of all 11 numbers.
\[ \text{6th number} = (\text{Sum of first 6 numbers} + \text{Sum of last 6 numbers}) - \text{Sum of 11 numbers} \] \[ \text{6th number} = (234 + 288) - 484 \] \[ \text{6th number} = 522 - 484 \] \[ \text{6th number} = 38 \]A) 38 is the correct answer, as calculated in the step-by-step solution.
The question asks about the meaning of the letters NPK on a fertilizer pack. This is a common abbreviation used in agriculture to denote the primary macronutrients essential for plant growth. Understanding these abbreviations is crucial for selecting appropriate fertilizers.
B) Nitrogen, Phosphorus, Potassium. This option correctly identifies the three primary macronutrients represented by NPK on a fertilizer pack.
The problem asks us to identify the option that is different from the rest based on a certain pattern. We need to analyze the relationship between the two letters in each option, likely by their positions in the English alphabet.
Correct Option: A) QK is the different option because the difference between the alphabetical positions of its letters (6) does not follow the pattern observed in the other options (17, 15, 13).
The question asks about the Air India logo as it existed until 2023. This requires knowledge of the historical branding of Air India.
Correct Option: D) a flying swan with the wheel of konark inside it
The question asks to identify which option is NOT a direct source of sea energy. Sea energy refers to various forms of renewable energy derived from the ocean. We need to evaluate each option to determine if it originates from the sea.
Correct Option: A) Solar energy is not a direct source of sea energy. While solar radiation heats the ocean and drives weather patterns that create waves, solar energy itself originates from the sun, not the ocean.
The question asks to identify the disease caused by a deficiency of Vitamin C. This requires knowledge of common vitamin deficiency diseases.
Correct Option: D) scurvy — Scurvy is a disease resulting from a severe lack of Vitamin C (ascorbic acid). It is characterized by symptoms such as fatigue, swollen and bleeding gums, joint pain, poor wound healing, and bruising due to impaired collagen synthesis.
The question asks about the sector that contributes the most to India's GDP. This requires knowledge of the sectoral composition of India's economy and how it has evolved over time.
Correct Option: A) Tertiary sector — The tertiary sector, also known as the service sector, has emerged as the dominant sector in India's economy, contributing the largest share to the Gross Domestic Product (GDP).
For x > 0 find the value of \(\left(1+\frac{1}{x+1}\right)\left(1+\frac{1}{x+2}\right)\left(1+\frac{1}{x+3}\right)\left(1+\frac{1}{x+4}\right)\).
\(\frac{x+1}{x+5}\)
\(\left(1+\frac{1}{x+5}\right)\)
\(\frac{1}{x+5}\)
\(\frac{x+5}{x+1}\)
The problem asks us to simplify a product of four terms, each of the form \(\left(1+\frac{1}{A}\right)\). The key to solving this is to simplify each individual term first and then observe if there's a pattern of cancellation when they are multiplied together.
Each term is of the form \(\left(1+\frac{1}{k}\right)\). We can combine the terms in the parenthesis by finding a common denominator:
\[ 1 + \frac{1}{k} = \frac{k}{k} + \frac{1}{k} = \frac{k+1}{k} \]Now, multiply these simplified terms together:
\[ \left(\frac{x+2}{x+1}\right) \left(\frac{x+3}{x+2}\right) \left(\frac{x+4}{x+3} ight) \left(\frac{x+5}{x+4}\right) \]Observe that many terms cancel out diagonally:
\[ \frac{\cancel{(x+2)}}{x+1} \times \frac{\cancel{(x+3)}}{\cancel{(x+2)}} \times \frac{\cancel{(x+4)}}{\cancel{(x+3)}} \times \frac{x+5}{\cancel{(x+4)}} \]After cancellation, the expression simplifies to:
\[ \frac{x+5}{x+1} \]Correct Option: D) \(\frac{x+5}{x+1}\)
The step-by-step simplification of each term and subsequent multiplication with cancellation leads directly to this result.
If θ = 60°, then the value of \(\frac{2\cot^2\theta}{1-\cot^2\theta}\) = ?
The problem requires us to evaluate a trigonometric expression given a specific angle. We need to recall the value of \(\cot\) for the given angle and then substitute it into the expression to simplify.
Identify the given angle \(\theta\):
\[ \theta = 60^\circ \]Recall the value of \(\cot\) for \(\theta = 60^\circ\):
We know that \(\cot\theta = \frac{1}{\tan\theta}\).
The value of \(\tan 60^\circ = \sqrt{3}\).
Therefore,
\[ \cot 60^\circ = \frac{1}{\tan 60^\circ} = \frac{1}{\sqrt{3}} \]Substitute the value of \(\cot 60^\circ\) into the given expression:
The expression is \(\frac{2\cot^2\theta}{1-\cot^2\theta}\).
Substitute \(\theta = 60^\circ\) and \(\cot 60^\circ = \frac{1}{\sqrt{3}}\):
\[ \frac{2\left(\frac{1}{\sqrt{3}}\right)^2}{1-\left(\frac{1}{\sqrt{3}}\right)^2} \]Calculate \(\left(\frac{1}{\sqrt{3}}\right)^2\):
\[ \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1^2}{(\sqrt{3})^2} = \frac{1}{3} \]Substitute this value back into the expression and simplify:
\[ \frac{2\left(\frac{1}{3}\right)}{1-\left(\frac{1}{3}\right)} = \frac{\frac{2}{3}}{1-\frac{1}{3}} \]\[ = \frac{\frac{2}{3}}{\frac{3-1}{3}} = \frac{\frac{2}{3}}{\frac{2}{3}} \]\[ = 1 \]Correct Option: D) 1