To find the wavelength of a wave, we use the fundamental relationship between wave speed, frequency, and wavelength. This relationship is given by the formula: Speed = Frequency × Wavelength.
The relationship between speed, frequency, and wavelength (\(\lambda\)) is:
\[ v = f \times \lambda \]To find the wavelength, we can rearrange the formula as:
\[ \lambda = \frac{v}{f} \]Substitute the speed and frequency into the rearranged formula:
\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{71 \times 10^4 \text{ Hz}} \]Perform the division:
\[ \lambda = \frac{3 \times 10^8}{71 \times 10^4} \text{ m} \]\[ \lambda = \frac{3}{71} \times 10^{(8-4)} \text{ m} \]\[ \lambda = \frac{3}{71} \times 10^4 \text{ m} \]\[ \lambda \approx 0.0422535 \times 10^4 \text{ m} \]\[ \lambda \approx 422.535 \text{ m} \]The calculated wavelength is approximately \(422.5 \text{ m}\), which matches option C.
Correct Option: C) 422.5 m
The question asks to identify the essential reproductive organs in a flower. This requires knowledge of flower anatomy and the function of each part in the reproductive process.
Correct Option: B) the stamen and pistil
The stamen is the male reproductive part of the flower, producing pollen. The pistil (also known as carpel) is the female reproductive part, containing the ovules. Both are directly involved in the process of sexual reproduction in flowering plants.
To find out how much money Radha received, we need to understand how to divide a total amount according to a given ratio. The ratio 1 : 2 means that for every 1 part James receives, Radha receives 2 parts. The total number of parts is the sum of the ratio terms.
Identify the given ratio: The money is divided between James and Radha in the ratio 1 : 2.
Calculate the total number of parts: Add the terms of the ratio.
\[ \text{Total parts} = 1 + 2 = 3 \]Determine Radha's share in terms of parts: From the ratio 1 : 2, Radha's share corresponds to 2 parts.
Calculate the value of one part: Divide the total amount of money by the total number of parts.
\[ \text{Value of one part} = \frac{\text{Total money}}{\text{Total parts}} = \frac{\text{Rs. } 87}{3} = \text{Rs. } 29 \]Calculate Radha's share: Multiply the value of one part by Radha's number of parts.
\[ \text{Radha's share} = \text{Value of one part} \times \text{Radha's parts} = \text{Rs. } 29 \times 2 = \text{Rs. } 58 \]Correct Option: D) Rs. 58
The question asks to identify the Indian tabla player who won a Grammy Award in 2017 for his collaboration with 'The Silk Road Ensemble'. This requires knowledge of prominent Indian musicians and their international accolades, specifically Grammy Awards.
Correct Option: A) Sandeep Das
Sandeep Das, a celebrated tabla virtuoso, won the Grammy Award in 2017 as part of Yo-Yo Ma's 'The Silk Road Ensemble'. Their album "Sing Me Home" received the award for Best World Music Album.
To find the area of a triangle given its vertices, we can use the determinant formula. This formula is derived from the concept of vectors and cross products, or by using the shoelace formula. The formula is:
\[ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the three vertices of the triangle.
Given the vertices are \((2, 4)\), \((-3, -1)\), and \((5, 3)\).
Substitute the coordinates into the formula:
\[ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \] \[ \text{Area} = \frac{1}{2} |2(-1 - 3) + (-3)(3 - 4) + 5(4 - (-1))| \]D) The calculated area of the triangle is 10 sq. units, which matches option D.
This is a syllogism problem that can be solved using Venn diagrams or by analyzing the logical relationships between the given statements. We will draw Venn diagrams for the statements and then check each conclusion against these diagrams.
C) Only conclusion 2 follows
In what time will Rs. 4,400 become Rs. 4,576 at 8% per annum interest compounded half-yearly?
To find the time it takes for a principal amount to reach a certain value with interest compounded half-yearly, we will use the compound interest formula. We need to adjust the annual interest rate and the time period for half-yearly compounding.
Identify the given values:
Adjust the interest rate and time for half-yearly compounding:
Since the interest is compounded half-yearly, the annual rate \(R\) is divided by 2, and the number of compounding periods \(n\) will be twice the number of years \(T\).
Apply the compound interest formula:
The formula for compound interest is \(A = P \left(1 + \frac{r}{100}\right)^N\).
Substitute the known values:
\[ 4576 = 4400 \left(1 + \frac{4}{100}\right)^N \]Simplify the equation:
\[ 4576 = 4400 \left(1 + 0.04\right)^N \] \[ 4576 = 4400 \left(1.04\right)^N \]Isolate the term with \(N\):
\[ \frac{4576}{4400} = (1.04)^N \] \[ 1.04 = (1.04)^N \]Solve for \(N\):
From the equation \(1.04 = (1.04)^N\), it is clear that \(N = 1\).
This means there is 1 compounding period.
Convert \(N\) back to time in years/months:
Since \(N\) represents the number of half-years, \(N=1\) means 1 half-year.
1 half-year = 6 months.
A) 6 months is the correct time period for the amount to grow from Rs. 4,400 to Rs. 4,576 at 8% per annum compounded half-yearly.
We need to find the number of questions answered wrongly. We can set up a system of two linear equations based on the total number of questions attempted and the total marks obtained. Let 'c' be the number of correct answers and 'w' be the number of wrong answers. We are given the total questions attempted, the marks for a correct answer, and the deduction for a wrong answer, along with the total marks obtained.
Let \(c\) be the number of correct answers and \(w\) be the number of wrong answers.
The total number of questions attempted is 94. So, we have the equation:
\[ c + w = 94 \quad \text{(Equation 1)} \]For every correct answer, 4 marks are given. For every wrong answer, 1 mark is deducted. The total marks obtained are 141. So, we have the equation:
\[ 4c - 1w = 141 \quad \text{(Equation 2)} \]From Equation 1, we can express \(c\) in terms of \(w\):
\[ c = 94 - w \]Substitute this expression for \(c\) into Equation 2:
\[ 4(94 - w) - w = 141 \]Distribute the 4:
\[ 376 - 4w - w = 141 \]Combine the \(w\) terms:
\[ 376 - 5w = 141 \]Subtract 376 from both sides:
\[ -5w = 141 - 376 \]\[ -5w = -235 \]Divide by -5 to find \(w\):
\[ w = \frac{-235}{-5} \]\[ w = 47 \]So, the number of questions wrongly answered by X is 47.
A) 47 is the correct answer. This is derived directly from solving the system of equations representing the given conditions.
We need to determine if the given statements, individually or combined, are sufficient to identify the person in the middle of a line of five people (P, Q, R, S, T).
Analyze Statement 1: Q is to the left of T.
Analyze Statement 2: S is in between P and T.
Analyze Statements 1 and 2 combined.
D) Neither 1 nor 2 is sufficient to answer the given question.
\(\frac{\sin30^\circ}{1+\cos30^\circ}\) + \(\frac{1+\cos30^\circ}{\sin30^\circ}\) = ?
To solve this problem, we need to evaluate the given trigonometric expression. First, we will recall the standard values of \(\sin30^\circ\) and \(\cos30^\circ\). Then, we will substitute these values into the expression and simplify it by finding a common denominator and performing algebraic operations.
Recall the standard trigonometric values:
Substitute these values into the given expression:
\[ \frac{\sin30^\circ}{1+\cos30^\circ} + \frac{1+\cos30^\circ}{\sin30^\circ} = \frac{\frac{1}{2}}{1+\frac{\sqrt{3}}{2}} + \frac{1+\frac{\sqrt{3}}{2}}{\frac{1}{2}} \]Simplify the denominators and numerators:
\[ = \frac{\frac{1}{2}}{\frac{2+\sqrt{3}}{2}} + \frac{\frac{2+\sqrt{3}}{2}}{\frac{1}{2}} \]Perform the divisions:
\[ = \frac{1}{2} \cdot \frac{2}{2+\sqrt{3}} + \frac{2+\sqrt{3}}{2} \cdot \frac{2}{1} \]\[ = \frac{1}{2+\sqrt{3}} + (2+\sqrt{3}) \]To simplify \(\frac{1}{2+\sqrt{3}}\), rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is \(2-\sqrt{3}\):
\[ \frac{1}{2+\sqrt{3}} = \frac{1}{2+\sqrt{3}} \cdot \frac{2-\sqrt{3}}{2-\sqrt{3}} = \frac{2-\sqrt{3}}{2^2 - (\sqrt{3})^2} = \frac{2-\sqrt{3}}{4-3} = \frac{2-\sqrt{3}}{1} = 2-\sqrt{3} \]Now substitute this back into the expression:
\[ = (2-\sqrt{3}) + (2+\sqrt{3}) \]Combine the terms:
\[ = 2 - \sqrt{3} + 2 + \sqrt{3} \]\[ = 2 + 2 \]\[ = 4 \]D) The simplified value of the expression is 4.