The problem asks us to find the median of a data set given its mean and mode, using the empirical formula. The empirical formula for the relationship between mean, median, and mode for a moderately skewed distribution is given by:
\[ \text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean} \]C) 35.97 is the correct answer because, by applying the empirical formula for mean, median, and mode and performing the necessary calculations, the median is found to be approximately 35.97.
The question asks about the power output of a laser weapon system developed by DRDO in early 2025. This is a factual question related to recent developments in defence technology in India. To answer correctly, one needs to recall the specific power output announced for this system.
Correct Option: C) 30 kilowatts
This problem involves calculating the work rate of pipes filling and emptying a cistern. We will determine the individual filling rates of the inlet pipes and the combined filling/emptying rate when all three pipes are open. From these, we can find the emptying rate of the outlet pipe and then calculate the time it takes to empty the cistern alone.
Calculate the filling rate of the first pipe:
The first pipe fills the cistern in 35 min. So, in 1 minute, it fills \(\frac{1}{35}\) of the cistern.
Calculate the filling rate of the second pipe:
The second pipe fills the cistern in 15 min. So, in 1 minute, it fills \(\frac{1}{15}\) of the cistern.
Calculate the combined filling rate of the two inlet pipes:
In 1 minute, the two pipes together fill:
Calculate the net filling rate when all three pipes are open:
When all three pipes (two inlet and one outlet) are open, the cistern fills in 42 min. So, in 1 minute, the net filling is \(\frac{1}{42}\) of the cistern.
Determine the emptying rate of the outlet pipe:
Let the emptying rate of the outlet pipe be \(\frac{1}{x}\) per minute. The net filling rate is the combined filling rate minus the emptying rate:
Calculate the time taken by the outlet pipe to empty the cistern:
Since the outlet pipe empties \(\frac{1}{14}\) of the cistern in 1 minute, it will take 14 minutes to empty the completely filled cistern.
Correct Option: A) 14 min
The question asks for the estimated capital expenditure of the Ministry of Road Transport and Highways (MoRTH) for the financial year 2024-25. This is a factual question related to the Indian Union Budget announcements.
B) The estimated capital expenditure for the Ministry of Road Transport and Highways in India for the financial year 2024-25 was ₹2.72 lakh crore, as announced in the Union Budget.
The question asks to identify the statement that does NOT describe a role of the Himalayas in shaping India's climate. We need to evaluate each option based on the known climatic influences of the Himalayan mountain range.
Comparing the options, the creation of convection currents is not a primary or unique role of the Himalayas in shaping India's climate, unlike their barrier effects on monsoon winds and cold winds, and their role in snowfall.
Correct Option: C) Creating convection currents
The question asks to identify the recipient of the Lifetime Achievement Award for research in Diabetes in July 2024. This requires knowledge of recent awards and recognitions in the medical field, specifically related to diabetes research.
Correct Option: C) Dr. Jitendra Singh was awarded the Lifetime Achievement Award for research in Diabetes in July 2024. He is a renowned diabetologist and Union Minister of State (Independent Charge) for Science & Technology, among other portfolios.
When two quadratic equations have both roots common, their corresponding coefficients must be proportional. We will use this property to find the value of \(k\).
Given the two quadratic equations:
Equation 1: \( 2x^{2} - 5x + 3 = 0 \)
Equation 2: \( 4x^{2} - (k+1)x + 6 = 0 \)
If two quadratic equations, \( a_1x^2 + b_1x + c_1 = 0 \) and \( a_2x^2 + b_2x + c_2 = 0 \), have both roots common, then the ratio of their corresponding coefficients must be equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]Applying this property to the given equations:
\[ \frac{2}{4} = \frac{-5}{-(k+1)} = \frac{3}{6} \]Simplify the known ratios:
\[ \frac{2}{4} = \frac{1}{2} \] \[ \frac{3}{6} = \frac{1}{2} \]Now, equate the ratio involving \(k\) with the simplified ratio:
\[ \frac{-5}{-(k+1)} = \frac{1}{2} \] \[ \frac{5}{k+1} = \frac{1}{2} \]Cross-multiply to solve for \(k\):
\[ 5 \times 2 = 1 \times (k+1) \] \[ 10 = k+1 \]Isolate \(k\):
\[ k = 10 - 1 \] \[ k = 9 \]A) 9 is the correct value for \(k\) as derived from the property of common roots.
This question tests knowledge of paste options in Microsoft PowerPoint. When pasting content, PowerPoint offers several options to control how the pasted text integrates with the destination slide's formatting. Understanding these options is crucial for effective document creation.
Correct Option: B) The text retains its original formatting from the source.
Find the area of a regular hexagon whose side measures 16\(\sqrt{2}\) cm.
876\(\sqrt{3}\) cm²
786\(\sqrt{3}\) cm²
867\(\sqrt{3}\) cm²
768\(\sqrt{3}\) cm²
To find the area of a regular hexagon, we can use the formula that relates its area to the length of its side. A regular hexagon can be divided into six equilateral triangles. The area of one equilateral triangle can be calculated, and then multiplied by six to get the total area of the hexagon.
Correct Option: D) 768\(\sqrt{3}\) cm²
This problem involves concepts of relative speed and distance, specifically for trains crossing each other and crossing a bridge. We will first find the lengths of Train A and Train B using the information about them crossing each other. Then, using the length of Train B and the time it takes to cross a bridge, we can determine the length of the bridge.
Since the trains are running in opposite directions, their speeds add up.
\[ \text{Relative Speed} = \text{Speed of Train A} + \text{Speed of Train B} \]\[ \text{Relative Speed} = 32 \text{ km/hr} + 28 \text{ km/hr} = 60 \text{ km/hr} \]To convert km/hr to m/s, multiply by \(\frac{5}{18}\).
\[ \text{Relative Speed} = 60 \times \frac{5}{18} \text{ m/s} = \frac{300}{18} \text{ m/s} = \frac{50}{3} \text{ m/s} \]When two trains cross each other, the total distance covered is the sum of their lengths.
\[ \text{Total Distance} = \text{Length of Train A} + \text{Length of Train B} \]Let \(L_A\) be the length of Train A and \(L_B\) be the length of Train B. We are given \(L_B = 1.5 L_A\).
\[ \text{Total Distance} = L_A + 1.5 L_A = 2.5 L_A \]We know that \(\text{Distance} = \text{Speed} \times \text{Time}\). The time taken to cross is 33 seconds.
\[ 2.5 L_A = \text{Relative Speed} \times \text{Time} \]\[ 2.5 L_A = \frac{50}{3} \text{ m/s} \times 33 \text{ s} \]\[ 2.5 L_A = 50 \times 11 \text{ m} \]\[ 2.5 L_A = 550 \text{ m} \]\[ L_A = \frac{550}{2.5} \text{ m} = \frac{5500}{25} \text{ m} = 220 \text{ m} \]When Train B crosses a bridge, the total distance covered is the sum of the length of Train B and the length of the bridge. Let \(L_{\text{bridge}}\) be the length of the bridge.
\[ \text{Distance} = \text{Length of Train B} + \text{Length of Bridge} \]\[ \text{Distance} = L_B + L_{\text{bridge}} \]\[ \text{Distance} = 330 \text{ m} + L_{\text{bridge}} \]\[ \text{Time} = 54 \text{ s} \]\[ \text{Speed of Train B} = \frac{70}{9} \text{ m/s} \]\[ \text{Distance} = \text{Speed} \times \text{Time} \]\[ 330 + L_{\text{bridge}} = \frac{70}{9} \times 54 \]\[ 330 + L_{\text{bridge}} = 70 \times 6 \]\[ 330 + L_{\text{bridge}} = 420 \]\[ L_{\text{bridge}} = 420 - 330 \]\[ L_{\text{bridge}} = 90 \text{ m} \]D) 90 — The length of the bridge is 90 m, as calculated by first finding the lengths of the trains and then applying the distance-speed-time formula for Train B crossing the bridge.