A wire of a given material has length ‘l’ and resistance ‘R’. Another wire of the same material having three times the length and twice the area of cross section will have a resistance equal to:
\( \frac{2}{3} \) R
\( \frac{3}{2} \) R
3 R
\( \frac{1}{2} \) R
The resistance of a wire depends on its material, length, and cross-sectional area. We will use the formula for resistance to find the relationship between the resistance of the first wire and the second wire, given their respective dimensions.
Recall the formula for resistance: The resistance \(R\) of a wire is given by the formula:
\[ R = \rho \frac{l}{A} \]
where:
Analyze the first wire: For the first wire, we are given:
So, for the first wire, the resistance is:
\[ R = \rho \frac{l}{A} \quad \text{(Equation 1)} \]
Analyze the second wire: For the second wire, we are given:
Let the resistance of the second wire be \( R_2 \). Using the formula for resistance:
\[ R_2 = \rho \frac{l_2}{A_2} \]
Substitute the given values for \( l_2 \) and \( A_2 \):
\[ R_2 = \rho \frac{3l}{2A} \]
Compare the resistances: We can rewrite the expression for \( R_2 \) as:
\[ R_2 = \frac{3}{2} \left( \rho \frac{l}{A} \right) \]
From Equation 1, we know that \( R = \rho \frac{l}{A} \). Substitute this into the expression for \( R_2 \):
\[ R_2 = \frac{3}{2} R \]
Correct Option: B) \( \frac{3}{2} \) R
The question asks to identify the Indian long jumper who secured a silver medal at the U-20 World Athletics Championships in 2021. This requires knowledge of recent achievements in Indian athletics, specifically in the long jump category at junior international events.
B) Shaili Singh won a silver medal in the women's long jump at the U-20 World Athletics Championships in 2021, held in Nairobi, Kenya. Her performance was a significant achievement for India in junior athletics.
The question asks about the process by which plants produce oxygen as a waste product and when this process occurs. This points directly to understanding the fundamental processes of photosynthesis and respiration in plants.
C) By photosynthesis during the day — Photosynthesis is the process that releases oxygen as a byproduct, and it requires sunlight, hence it occurs during the day.
The question asks to identify the name of India's first intelligent messenger launched by Paytm Money in January 2022. This is a factual recall question related to current affairs and financial technology.
In January 2022, Paytm Money indeed launched an intelligent messenger for its users. This service was designed to provide market information, news, and updates in a conversational format.
D) Pops
Paytm Money launched "Pops" in January 2022. It is India's first intelligent messenger that allows users to get specific information about their investments, market movements, and other financial news in a chat-like interface. It aims to simplify investment-related queries and provide real-time updates.
This problem involves setting up and solving a system of linear equations based on the given relationships between the costs of a television and a printer. We will assign variables to the costs and then use the information provided to form two equations. Finally, we will solve for the cost of the television.
D) Rs 35,000 is the correct cost of the television, as derived from the equations.
The question asks about the initial composition of the Supreme Court of India as envisaged in the original Constitution of 1950. This is a factual question related to the Indian polity and constitutional history.
Correct Option: D) 7
A group of class IX students conducted a survey for the number of family members in a household on 40 households in a locality, which resulted in the given frequency table. What is the mode of the data?
| Family size | Number of families |
| 2-4 | 10 |
| 4-6 | 8 |
| 6-8 | 12 |
| 8-10 | 6 |
| 10-12 | 4 |
To find the mode of grouped data, we first identify the modal class, which is the class interval with the highest frequency. Then, we use the formula for the mode of grouped data, which involves the lower limit of the modal class, the class size, and the frequencies of the modal class, the class preceding it, and the class succeeding it.
| Family size | Number of families (Frequency) |
|---|---|
| 2-4 | 10 |
| 4-6 | 8 |
| 6-8 | 12 |
| 8-10 | 6 |
| 10-12 | 4 |
The highest frequency is 12, which corresponds to the class interval 6-8. Therefore, the modal class is 6-8.
The formula for the mode is:
\[ \text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \]Thus, the mode of the given data is 6.8.
D) 6.8 — This is the correct mode calculated using the formula for grouped data.
The question asks to identify the organization responsible for releasing the 'World Economic Outlook Update – January 2022' report, which projected India's GDP growth. This requires knowledge of major international economic reports and their publishers.
B) International Monetary Fund (IMF) — The International Monetary Fund (IMF) is well-known for its semi-annual 'World Economic Outlook' report, along with periodic updates. The January 2022 update, which included the mentioned GDP projections for India, was indeed released by the IMF.
To divide decimal numbers, it is often easiest to convert them into fractions or to shift the decimal point in both the numerator and the denominator until the denominator becomes a whole number. This simplifies the division process.
The expression \(0.0486 \div 0.002\) can be written as:
\[ \frac{0.0486}{0.002} \]To remove the decimal points, we need to multiply both the numerator and the denominator by a power of 10. The denominator \(0.002\) has three decimal places, and the numerator \(0.0486\) has four decimal places. To make both numbers integers, we should multiply by \(10000\) (since \(0.0486\) has the most decimal places, i.e., four).
\[ \frac{0.0486 \times 10000}{0.002 \times 10000} = \frac{486}{20} \]Alternatively, we can make the denominator a whole number by multiplying both by \(1000\):
\[ \frac{0.0486 \times 1000}{0.002 \times 1000} = \frac{48.6}{2} \]Now, divide \(48.6\) by \(2\):
\[ \frac{48.6}{2} = 24.3 \]Correct Option: B) 24.3
To solve this problem, we need to use the divisibility rules for 90. A number is divisible by 90 if it is divisible by both 9 and 10. We will apply these rules to the given number 115xy to find the possible values of x and y, and then calculate x + y.
Divisibility by 10: A number is divisible by 10 if its last digit is 0. In the number 115xy, the last digit is y. Therefore, for 115xy to be divisible by 10, y must be 0.
\[ y = 0 \]Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. The digits of the number 115xy are 1, 1, 5, x, and y. Since we found \(y=0\), the sum of the digits is:
\[ 1 + 1 + 5 + x + y = 1 + 1 + 5 + x + 0 = 7 + x \]For the number to be divisible by 9, the sum of its digits \((7+x)\) must be a multiple of 9. Since x is a digit, its value can range from 0 to 9. We need to find a value of x in this range such that \(7+x\) is a multiple of 9.
The only value of x (between 0 and 9) for which \(7+x\) is a multiple of 9 is \(x=2\).
Calculate x + y: We found \(x=2\) and \(y=0\). Now, we calculate their sum:
\[ x + y = 2 + 0 = 2 \]Correct Option: B) 2