The problem involves converting percentage statements into mathematical equations and then simplifying them to find the relationship between the given expressions. We will first express 'b' in terms of 'a' using the first statement and then substitute this value into the second statement to find its equivalent form.
"20% of a = b" can be written as:
\[ \frac{20}{100} \times a = b \]\[ 0.20 \times a = b \]\[ b = \frac{a}{5} \quad \text{(Equation 1)} \]We need to find the value of "b% of 20". This can be written as:
\[ \frac{b}{100} \times 20 \]Substitute \(b = \frac{a}{5}\) into the expression \(\frac{b}{100} \times 20\):
\[ \frac{\left(\frac{a}{5}\right)}{100} \times 20 \]To express \(\frac{a}{25}\) as a percentage of 'a', we can write it as:
\[ \frac{1}{25} \times a \]\[ \frac{1}{25} \times 100\% \times a \]\[ 4\% \times a \]\[ 4\% \text{ of } a \]C) 4% of a
The question asks to complete an analogy based on the relationship between an instrument and the quantity it measures. We need to identify what a hygrometer measures and then apply the same relationship to a barometer.
Correct Option: C) Pressure — A barometer is an instrument specifically designed to measure atmospheric pressure. This fits the established "Instrument : Measured Quantity" relationship.
We are given the curved surface area and volume of a hemisphere. We need to find its radius. We can use the formulas for the curved surface area and volume of a hemisphere, set up equations, and solve for the radius.
Recall the formulas:
The curved surface area (CSA) of a hemisphere is given by:
\[ \text{CSA} = 2\pi r^2 \]The volume (V) of a hemisphere is given by:
\[ \text{V} = \frac{2}{3}\pi r^3 \]where \(r\) is the radius of the hemisphere.
Set up equations using the given values:
Given CSA = 2772 cm²:
\[ 2\pi r^2 = 2772 \quad \text{(Equation 1)} \]Given V = 19404 cm³:
\[ \frac{2}{3}\pi r^3 = 19404 \quad \text{(Equation 2)} \]Divide Equation 2 by Equation 1 to find \(r\):
Dividing Equation 2 by Equation 1 eliminates \(\pi\) and simplifies the expression for \(r\):
\[ \frac{\frac{2}{3}\pi r^3}{2\pi r^2} = \frac{19404}{2772} \]Simplify the left side:
\[ \frac{1}{3}r = \frac{19404}{2772} \]Calculate the right side:
\[ \frac{19404}{2772} = 7 \]So, we have:
\[ \frac{1}{3}r = 7 \]Solve for \(r\):
\[ r = 7 \times 3 \]\[ r = 21 \text{ cm} \]Verify the result (optional but good practice):
Using \(r = 21\) cm:
CSA \[= 2 \times \frac{22}{7} \times (21)^2 = 2 \times \frac{22}{7} \times 441 = 2 \times 22 \times 63 = 44 \times 63 = 2772 \text{ cm}^2\]. This matches the given CSA.
Volume \[= \frac{2}{3} \times \frac{22}{7} \times (21)^3 = \frac{2}{3} \times \frac{22}{7} \times 9261 = 2 \times 22 \times 441 = 44 \times 441 = 19404 \text{ cm}^3\]. This matches the given volume.
Both values match, confirming our radius calculation.
Correct Option: A) 21 cm
The question asks to identify the substance that is stored and concentrated in the gallbladder. This requires knowledge of the basic functions of the gallbladder in the human digestive system.
Correct Option: D) Bile is the substance produced by the liver, stored, and concentrated in the gallbladder before being released into the small intestine to aid in fat digestion.
The problem requires us to analyze a given statement and determine which of the provided conclusions logically follow from it. We need to evaluate each conclusion based solely on the information given in the statement, without introducing external knowledge or assumptions.
The statement is: "To pass the examination, one must work hard."
Let's analyze each conclusion:
Based on this analysis, only Conclusion III logically follows from the given statement.
A) Only conclusion III follows. This is correct because the statement "To pass the examination, one must work hard" implies that hard work is a necessary condition for passing. Therefore, if one does not work hard, one will not pass.
This problem involves the concept of inverse proportionality between the number of soldiers and the duration the food supply lasts. The total amount of food remains constant. We will first calculate the total 'man-days' of food available and then use this to find out how long the food will last for the reduced number of soldiers.
Calculate the total food supply in terms of 'man-days':
Initially, there are 250 soldiers, and the food supply is sufficient for 30 days.
Total food supply = Number of soldiers \(\times\) Number of days
\[ \text{Total food supply} = 250 \text{ soldiers} \times 30 \text{ days} = 7500 \text{ man-days} \]Calculate the new number of soldiers:
50 soldiers did not attend the camp.
New number of soldiers = Original number of soldiers - Soldiers who did not attend
\[ \text{New number of soldiers} = 250 - 50 = 200 \text{ soldiers} \]Calculate how many days the food will last for the new number of soldiers:
The total food supply (7500 man-days) remains constant.
Number of days = Total food supply / New number of soldiers
\[ \text{Number of days} = \frac{7500 \text{ man-days}}{200 \text{ soldiers}} = 37.5 \text{ days} \]Calculate the extra days the food supply will last:
Extra days = New duration - Original duration
\[ \text{Extra days} = 37.5 \text{ days} - 30 \text{ days} = 7.5 \text{ days} \]Correct Option: B) 7.5 days
If tan² 45° - cos² 60° = x sin 45° cos 45° cot 30°, then find the value of 'x'.
\(\frac{1}{\sqrt{2}}\)
\(\frac{3}{2}\)
\(\frac{\sqrt{3}}{2}\)
\(\frac{2}{\sqrt{3}}\)
To find the value of 'x', we need to substitute the standard trigonometric values for the given angles into the equation and then solve for 'x'.
The given equation is:
\[ \tan^2 45^{\circ} - \cos^2 60^{\circ} = x \sin 45^{\circ} \cos 45^{\circ} \cot 30^{\circ} \]Correct Option: C) The value of 'x' is \(\frac{\sqrt{3}}{2}\).
This is a question based on syllogism. We need to analyze the given statements and determine which of the conclusions logically follow. We can use Venn diagrams to represent the relationships described in the statements.
B) Both conclusions I and II follow
The question asks to identify the individual who inaugurated an advanced electronic surveillance system along a section of the India-Bangladesh border in Assam in 2019. This is a factual question related to a specific event and requires knowledge of political appointments and their responsibilities.
D) Rajnath Singh — As the Union Home Minister in March 2019, Rajnath Singh was responsible for internal security and border management. He inaugurated the "Smart Fence" project, a Comprehensive Integrated Border Management System (CIBMS), along the 61-km stretch of the India-Bangladesh border in Dhubri, Assam, to enhance surveillance and prevent illegal infiltration.
To identify which option is NOT a physical change, we need to understand the difference between physical and chemical changes. A physical change alters the form of a substance but not its chemical composition. A chemical change results in the formation of new substances with different chemical properties.
Correct Option: B) Fermentation of substances