The total surface area of a solid hemisphere is 42 cm². Its radius (in cm) is:
(Take π = \( \frac{22}{7} \))
\(\frac{7}{\sqrt{11}}\)
\(\frac{7\sqrt{2}}{\sqrt{22}}\)
\(\frac{7\sqrt{3}}{\sqrt{11}}\)
\(\frac{7}{\sqrt{22}}\)
To find the radius of a solid hemisphere given its total surface area, we use the formula for the total surface area of a solid hemisphere and solve for the radius. The total surface area of a solid hemisphere is the sum of its curved surface area and the area of its base.
Recall the formula for the total surface area of a solid hemisphere. A solid hemisphere has a curved surface area of \(2\pi r^2\) and a circular base area of \(\pi r^2\). Therefore, its total surface area (TSA) is:
\[ \text{TSA} = 2\pi r^2 + \pi r^2 = 3\pi r^2 \]We are given that the total surface area is \(42 \text{ cm}^2\). So, we set up the equation:
\[ 3\pi r^2 = 42 \]Substitute the value of \(\pi = \frac{22}{7}\) into the equation:
\[ 3 \times \frac{22}{7} \times r^2 = 42 \]Now, solve for \(r^2\):
\[ \frac{66}{7} r^2 = 42 \]\[ r^2 = \frac{42 \times 7}{66} \]\[ r^2 = \frac{6 \times 7 \times 7}{6 \times 11} \]\[ r^2 = \frac{49}{11} \]Finally, find \(r\) by taking the square root of both sides:
\[ r = \sqrt{\frac{49}{11}} \]\[ r = \frac{\sqrt{49}}{\sqrt{11}} \]\[ r = \frac{7}{\sqrt{11}} \]The radius of the hemisphere is \(\frac{7}{\sqrt{11}} \text{ cm}\).
A) \(\frac{7}{\sqrt{11}}\) is the correct radius calculated from the given total surface area.
The problem presents a series of number pairs and asks us to find the missing number in the last pair based on the relationship observed in the previous pairs. We need to identify the mathematical operation or pattern connecting the first number to the second number in each given pair.
Correct Option: B) 324 is the correct answer because \(18^2 = 324\).
A man walking at a speed of 10 km/h crosses a bridge in 7\( \frac{1}{2} \) minutes. The length of the bridge is ____ metres.
To find the length of the bridge, we need to use the fundamental formula relating speed, distance, and time. The man's speed is given in km/h, and the time is given in minutes. The required answer is in metres. Therefore, the first step is to convert all units to a consistent system, preferably metres per second (m/s) for speed and seconds for time, to directly obtain the distance in metres.
The man's speed is given as 10 km/h.
\[ \text{Speed} = 10 \text{ km/h} \]\[ \text{Speed} = 10 \times \frac{1000 \text{ m}}{3600 \text{ s}} \]\[ \text{Speed} = 10 \times \frac{5}{18} \text{ m/s} \]\[ \text{Speed} = \frac{50}{18} \text{ m/s} \]\[ \text{Speed} = \frac{25}{9} \text{ m/s} \]The time taken to cross the bridge is \( 7\frac{1}{2} \) minutes.
\[ \text{Time} = 7\frac{1}{2} \text{ minutes} = 7.5 \text{ minutes} \]\[ \text{Time} = 7.5 \times 60 \text{ seconds} \]\[ \text{Time} = 450 \text{ seconds} \]The formula relating speed, distance, and time is:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]\[ \text{Distance} = \frac{25}{9} \text{ m/s} \times 450 \text{ s} \]\[ \text{Distance} = 25 \times \frac{450}{9} \text{ m} \]\[ \text{Distance} = 25 \times 50 \text{ m} \]\[ \text{Distance} = 1250 \text{ m} \]C) 1250 metres is the correct length of the bridge.
The question asks to identify the class of compounds whose functional group does not contain an oxygen atom. We need to examine the functional groups of each given option and determine if oxygen is present.
Based on this analysis, only halo compounds have a functional group without an oxygen atom.
A) halo compounds — The functional group of halo compounds is a halogen atom (\(X\)) attached to a carbon chain (\(R-X\)). Halogens (Fluorine, Chlorine, Bromine, Iodine) do not contain oxygen atoms. Therefore, this class of compounds fits the description.
The question asks about the number of main divisions or 'folds' of the Yajurveda. This is a factual question related to the classification of Vedic texts in ancient Indian literature.
Therefore, the Yajurveda is two-fold.
B) 2 — The Yajurveda is traditionally divided into two main parts: the Shukla Yajurveda and the Krishna Yajurveda.
To solve this problem, we need to simplify the expression step-by-step following the order of operations (PEMDAS/BODMAS). We will first evaluate the terms inside the parentheses, then deal with the negative exponents, and finally perform the addition/subtraction and squaring operations.
The expression contains \((1 - 2)\). Let's calculate its value:
\[ 1 - 2 = -1 \]The original expression is \(\{1 + (1 - 2)^{-1} + 2 - (1 - 2)^{-2}\}^2\).
Substituting \((1 - 2) = -1\), we get:
\[ \{1 + (-1)^{-1} + 2 - (-1)^{-2}\}^2 \]Recall that \(a^{-n} = \frac{1}{a^n}\).
\[ (-1)^{-1} = \frac{1}{(-1)^1} = \frac{1}{-1} = -1 \]
\[ (-1)^{-2} = \frac{1}{(-1)^2} = \frac{1}{1} = 1 \]
\[ \{1 + (-1) + 2 - (1)\}^2 \]\[ \{1 - 1 + 2 - 1\}^2 \]
\[ \{0 + 2 - 1\}^2 \]\[ \{2 - 1\}^2 \]\[ \{1\}^2 \]
\[ 1^2 = 1 \]
C) The value of the expression is \(1\).
This question tests the understanding of how a parabolic dish antenna works, specifically the placement of its receiver. The key principle is the reflective property of a parabola.
B) in front of the dish at F
Placing the receiver at the focal point (F) of the parabolic dish ensures that all incoming parallel radio waves are concentrated onto it, leading to the strongest and sharpest signal.
All 30 students are standing in a straight line facing the north. Amit is at 12th position from the extreme left end. What is the position of Amit when considered from the extreme right end?
This problem involves determining a person's position in a line from the opposite end, given their position from one end and the total number of people. We can use a standard formula for this type of ranking problem.
Identify the total number of students in the line.
Total number of students \( = 30 \)
Identify Amit's position from the extreme left end.
Amit's position from the left \( = 12^{\text{th}} \)
To find Amit's position from the extreme right end, we use the formula:
\[ \text{Position from Right} = (\text{Total number of students} - \text{Position from Left}) + 1 \]Substitute the given values into the formula:
\[ \text{Position from Right} = (30 - 12) + 1 \]Perform the subtraction:
\[ \text{Position from Right} = 18 + 1 \]Perform the addition:
\[ \text{Position from Right} = 19 \]Therefore, Amit is at the 19th position from the extreme right end.
D) 19 th — As calculated, Amit's position from the right end is 19th.
A) 17 th — This would be obtained if you subtracted 12 from 30 and then subtracted 1 again (i.e., \(30 - 12 - 1 = 17\)), which is incorrect.
B) 18 th — This would be obtained if you simply subtracted Amit's position from the total number of students (i.e., \(30 - 12 = 18\)). This is incorrect because it doesn't account for Amit himself being counted in both directions.
C) 20 th — This would be obtained if you added 1 to Amit's position from the left and then subtracted from total (i.e., \(30 - (12+1) = 17\)) or some other incorrect calculation. It does not follow the correct formula.
The question asks to identify a mechanical barrier method of contraception from the given options. We need to understand what a mechanical barrier method is and then evaluate each option.
D) Use of condom — A condom acts as a physical barrier, preventing sperm from reaching the egg, thus classifying it as a mechanical barrier method of contraception.