The slant height of a cone is equal to twice the radius of its base. If the cone has a volume of 401π cm³, find the magnitude of the cube of its base radius.
401\(\sqrt{3}\)
412\(\sqrt{3}\)
402\(\sqrt{3}\)
410\(\sqrt{3}\)
To solve this problem, we need to use the formulas for the volume of a cone and the relationship between its radius, height, and slant height. We are given the relationship between the slant height and the radius, and the volume of the cone. We will use these to find the height in terms of the radius, substitute into the volume formula, and then solve for the cube of the base radius.
Correct Option: A) \(401\sqrt{3}\)
To find the number of diagonals of a regular polygon, we first need to determine the number of sides of the polygon. We can do this by using the formula for the sum of interior angles of a polygon. Once the number of sides (n) is known, we can use the formula for the number of diagonals.
Formula for the sum of interior angles:
The sum of the interior angles of a polygon with \(n\) sides is given by the formula:
\[ \text{Sum} = (n - 2) \times 180^\circ \]Find the number of sides (n):
We are given that the sum of the interior angles is \(1260^\circ\). So, we can set up the equation:
\[ (n - 2) \times 180^\circ = 1260^\circ \]Divide both sides by \(180^\circ\):
\[ n - 2 = \frac{1260}{180} \] \[ n - 2 = 7 \]Add 2 to both sides:
\[ n = 7 + 2 \] \[ n = 9 \]So, the polygon has 9 sides (it is a nonagon).
Formula for the number of diagonals:
The number of diagonals in a polygon with \(n\) sides is given by the formula:
\[ \text{Number of diagonals} = \frac{n(n - 3)}{2} \]Calculate the number of diagonals:
Substitute \(n = 9\) into the formula:
\[ \text{Number of diagonals} = \frac{9(9 - 3)}{2} \] \[ \text{Number of diagonals} = \frac{9 \times 6}{2} \] \[ \text{Number of diagonals} = \frac{54}{2} \] \[ \text{Number of diagonals} = 27 \]B) The number of diagonals of the regular polygon is 27.
This problem involves calculating individual weights using given average weights. We will use the formula for average weight, which is the sum of weights divided by the number of items. By setting up equations based on the given averages, we can solve for the unknown weight of B.
B) 39 kg is the weight of B, as calculated from the given average weights.
The question asks about the structural characteristics of alveoli that primarily contribute to efficient gas exchange. Efficient gas exchange requires maximizing the rate at which gases (oxygen and carbon dioxide) can diffuse across the alveolar and capillary membranes. We need to identify the option that best describes features facilitating this process.
D) Large surface area combined with a thin alveolar-capillary barrier — This combination is the primary structural characteristic that allows for highly efficient gas exchange in the alveoli. A large surface area provides ample space for diffusion, and a very thin barrier minimizes the diffusion distance, both of which are crucial for rapid transfer of oxygen and carbon dioxide between the air and blood.
The question asks about the target amount for the second Asset Monetization Plan (2025-30) as per the Union Budget 2025-26, intended for reinvestment in new projects. This is a factual question related to government economic policy announcements.
A) ₹10 lakh crore is the correct amount announced for the second Asset Monetization Plan (2025-30) in the Union Budget 2025-26, with the aim to reinvest in new projects.
38 people are standing in a row facing north. Hari is 11th from the right end while Hema is 14th from the left end. How many people are there between Hari and Hema?
To find the number of people between Hari and Hema, we first need to determine their positions from a common reference point (e.g., the left end of the row). We are given the total number of people and their positions from opposite ends. We will calculate Hari's position from the left end and then subtract the positions of Hari and Hema from the total number of people, adjusting for the people already counted.
The position of a person from the left end can be calculated using the formula:
\[ \text{Position from Left} = \text{Total People} - \text{Position from Right} + 1 \]\[ \text{Hari's position from Left} = 38 - 11 + 1 = 27 + 1 = 28 \]So, Hari is 28th from the left end.Since both positions are now from the left end, we can find the number of people between them. Hari is at position 28 from the left, and Hema is at position 14 from the left. To find the number of people between them, we subtract the smaller position from the larger position and then subtract 1 (because we are looking for people *between* them, not including them).
\[ \text{People between Hari and Hema} = \text{Hari's position from Left} - \text{Hema's position from Left} - 1 \]\[ \text{People between Hari and Hema} = 28 - 14 - 1 = 14 - 1 = 13 \]B) 13 is the correct answer. There are 13 people between Hari and Hema.
This problem involves calculating the rate of simple interest. We are given the principal amount, the simple interest earned, and the time period. We can use the formula for simple interest to find the rate.
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]where \(P\) is the principal, \(R\) is the rate of interest per annum, and \(T\) is the time in years.To find the rate, we can rearrange the formula as:
\[ R = \frac{SI \times 100}{P \times T} \]Now, plug in the values we have:
\[ R = \frac{700 \times 100}{1400 \times 5} \]Perform the multiplication and division:
\[ R = \frac{70000}{7000} \]\[ R = 10 \]The rate of interest is 10% per annum.
Correct Option: A) 10% — The calculation shows that a rate of 10% per annum is required for ₹1,400 to yield ₹700 as simple interest in 5 years.
To find the smallest number that should be added to 13874 to make it completely divisible by 67, we need to perform division. We will divide 13874 by 67 and find the remainder. The difference between the divisor (67) and the remainder will be the smallest number that needs to be added.
207
______
67 | 13874
-134
----
47
-0
--
474
-469
----
5
So, when 13874 is divided by 67, the quotient is 207 and the remainder is 5.
Correct Option: D) 62
This question tests knowledge of recent major film awards, specifically the Academy Awards (Oscars). To answer correctly, one needs to recall the winners from the most recent Oscar ceremony or be aware of upcoming predictions if the year is in the future.
Correct Option: D) Sean Baker
Sean Baker is a strong contender for the Best Director award at the 2025 Oscars due to the critical success of his film Anora, which won the Palme d'Or at the 2024 Cannes Film Festival. This prestigious award often signals a film's potential for Oscar recognition in the following year.
The question asks about Vidarbha's performance in the Ranji Trophy 2024-25. To answer this, one needs to recall or know the recent history of the Ranji Trophy, specifically Vidarbha's achievements in the tournament.
Correct Option: B) They won the title for the third time in seven seasons