A ray of light is incident on an interface separating two media at an angle of incidence equal to 45°, for which the angle of refraction is 30°. The refractive index of the second medium with respect to the first is equal to:
\(\frac{\sqrt{3}}{2}\)
\(\frac{1}{\sqrt{2}}\)
\(\frac{1}{2}\)
\(\sqrt{2}\)
This problem can be solved using Snell's Law, which relates the angles of incidence and refraction to the refractive indices of the two media. Snell's Law states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the refractive index of the second medium to the refractive index of the first medium.
Identify the given values:
Recall Snell's Law:
Snell's Law is given by:
\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]where \(n_1\) is the refractive index of the first medium and \(n_2\) is the refractive index of the second medium.
Determine the refractive index of the second medium with respect to the first:
The refractive index of the second medium with respect to the first is defined as the ratio \(\frac{n_2}{n_1}\). From Snell's Law, we can write:
\[ \frac{n_2}{n_1} = \frac{\sin(\theta_1)}{\sin(\theta_2)} \]Substitute the given values into the formula:
\[ \frac{n_2}{n_1} = \frac{\sin(45^\circ)}{\sin(30^\circ)} \]Calculate the sine values:
Perform the division:
\[ \frac{n_2}{n_1} = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}} = \frac{1}{\sqrt{2}} \times 2 = \frac{2}{\sqrt{2}} \]Simplify the expression:
\[ \frac{2}{\sqrt{2}} = \frac{\sqrt{2} \times \sqrt{2}}{\sqrt{2}} = \sqrt{2} \]Thus, the refractive index of the second medium with respect to the first is \(\sqrt{2}\).
D) The refractive index of the second medium with respect to the first is \(\sqrt{2}\), which matches our calculation based on Snell's Law.
If the mean of the following data is 4, find the missing frequency.
| x | 3 | 5 | 9 | 2 |
| f | 3 | 4 | - | 8 |
To find the missing frequency when the mean of a grouped data is given, we use the formula for the mean of grouped data. We set up an equation where the given mean is equal to the sum of (x * f) divided by the sum of frequencies, and then solve for the unknown frequency.
Understand the data: We are given values of 'x' (data points) and 'f' (frequencies). One frequency is missing.
| x | f | f * x |
|---|---|---|
| 3 | 3 | 9 |
| 5 | 4 | 20 |
| 9 | y (let's assume 'y' is the missing frequency) | 9y |
| 2 | 8 | 16 |
Calculate the sum of frequencies (Σf):
Let the missing frequency be \(y\).
\[ \sum f = 3 + 4 + y + 8 = 15 + y \]
Calculate the sum of (f * x) (Σfx):
\[ \sum fx = (3 \times 3) + (5 \times 4) + (9 \times y) + (2 \times 8) \]
\[ \sum fx = 9 + 20 + 9y + 16 \]
\[ \sum fx = 45 + 9y \]
Apply the mean formula:
The mean (\(\bar{x}\)) of grouped data is given by the formula:
\[ \bar{x} = \frac{\sum fx}{\sum f} \]
We are given that the mean is 4. So, we can set up the equation:
\[ 4 = \frac{45 + 9y}{15 + y} \]
Solve for the missing frequency (y):
Multiply both sides by \((15 + y)\):
\[ 4(15 + y) = 45 + 9y \]
\[ 60 + 4y = 45 + 9y \]
Subtract \(4y\) from both sides:
\[ 60 = 45 + 9y - 4y \]
\[ 60 = 45 + 5y \]
Subtract 45 from both sides:
\[ 60 - 45 = 5y \]
\[ 15 = 5y \]
Divide by 5:
\[ y = \frac{15}{5} \]
\[ y = 3 \]
D) 3 — The calculated missing frequency is 3.
To solve this problem, we need to perform addition and subtraction of decimal numbers. It's crucial to align the decimal points correctly before performing the operations.
Thus, the final value is \(355.085\).
Correct Option: C) 355.085
This problem requires us to deduce the relative scores of six friends based on a series of comparative statements. We will use inequalities to represent the given information and then combine them to determine who scored the most.
C) Arun — Based on the combined inequalities, Arun scored more than Suresh, Priyanka, and Rahul. Since Suresh scored more than Priyanka, and Dilan scored less than Rahul, and Ritu scored the least, Arun emerges as the highest scorer among all six friends.
The question asks how making the manufacturing of refrigerators that do not release chlorofluorocarbons (CFCs) mandatory helps prevent ozone depletion. This requires understanding the role of CFCs in ozone depletion and the mechanism by which they affect the ozone layer.
A) It will reduce the release of CFCs that react with ozone molecules. — This is correct because the primary mechanism of ozone depletion involves CFCs releasing chlorine atoms, which then catalytically destroy ozone molecules. Reducing CFC emissions directly addresses this cause.
The question asks for the headquarters of the Archaeological Survey of India (ASI). This is a factual question about a prominent government organization in India. Knowing the headquarters of key national institutions is important for competitive exams.
Correct Option: D) New Delhi
The Archaeological Survey of India (ASI) is an attached office of the Ministry of Culture, Government of India. Its headquarters are situated in New Delhi, the capital city of India. It is responsible for archaeological research and the protection of cultural heritage.
The question asks about the number of tiers of administration in Panchayati Raj Institutions (PRIs) in states with a population exceeding 2 million. This is a direct factual question related to the structure of local self-governance in India, as mandated by the 73rd Constitutional Amendment Act.
Correct Option: A) The 73rd Constitutional Amendment Act mandates a three-tier system of Panchayati Raj Institutions for states with a population exceeding 2 million: Gram Panchayat (village level), Panchayat Samiti (intermediate/block level), and Zila Parishad (district level).
The problem presents two letter-cluster pairs (STM : RUL and PJR : OKQ) that follow a specific logical rule. We need to identify this rule and then apply it to the given options to find the pair that adheres to the same logic.
C) TCS : SDR
This pair follows the \(-1, +1, -1\) logic.
The question asks about the location within a cell where glucose is converted into pyruvate. This process is known as glycolysis, a fundamental metabolic pathway.
Correct Option: A) Cytoplasm is the correct answer because glycolysis, the process of converting glucose to pyruvate, exclusively takes place in the cytoplasm of the cell.
The question asks about the direction of magnetic field lines inside a bar magnet. It's crucial to distinguish between the direction of field lines inside and outside the magnet.
A) south pole to north pole — This is the correct direction for magnetic field lines inside a bar magnet, completing the continuous closed loops that start from the North pole and end at the South pole outside the magnet.