If \( m = a\cos^{3}\beta \) and \( n = b\sin^{3}\beta \), then find the value of \( \left(\frac{m}{a}\right)^{\frac{2}{3}} + \left(\frac{n}{b}\right)^{\frac{2}{3}} \).
The problem provides two equations relating \(m\), \(n\), \(a\), \(b\), and \(\beta\). We need to find the value of an expression involving \[\left(\frac{m}{a}\right)^{\frac{2}{3}}\] and \[\left(\frac{n}{b}\right)^{\frac{2}{3}}\]. The key is to simplify the terms \(\frac{m}{a}\) and \(\frac{n}{b}\) first, then raise them to the power of \(\frac{2}{3}\), and finally use the fundamental trigonometric identity \(\cos^2\beta + \sin^2\beta = 1\).
Express \(\frac{m}{a}\) and \(\frac{n}{b}\) in terms of \(\cos\beta\) and \(\sin\beta\):
Given \(m = a\cos^{3}\beta\), we can write:
\[ \frac{m}{a} = \cos^{3}\beta \]Given \(n = b\sin^{3}\beta\), we can write:
\[ \frac{n}{b} = \sin^{3}\beta \]Calculate \[\left(\frac{m}{a}\right)^{\frac{2}{3}}\] and \[\left(\frac{n}{b}\right)^{\frac{2}{3}}\]:
Substitute the expressions from Step 1:
\[ \left(\frac{m}{a}\right)^{\frac{2}{3}} = \left(\cos^{3}\beta\right)^{\frac{2}{3}} \]Using the exponent rule \((x^p)^q = x^{pq}\):
\[ \left(\cos^{3}\beta\right)^{\frac{2}{3}} = \cos^{3 \cdot \frac{2}{3}}\beta = \cos^{2}\beta \]Similarly for the second term:
\[ \left(\frac{n}{b}\right)^{\frac{2}{3}} = \left(\sin^{3}\beta\right)^{\frac{2}{3}} \] \[ \left(\sin^{3}\beta\right)^{\frac{2}{3}} = \sin^{3 \cdot \frac{2}{3}}\beta = \sin^{2}\beta \]Substitute these values into the required expression:
The expression to find is \[\left(\frac{m}{a}\right)^{\frac{2}{3}} + \left(\frac{n}{b}\right)^{\frac{2}{3}}\].
Substitute the results from Step 2:
\[ \left(\frac{m}{a}\right)^{\frac{2}{3}} + \left(\frac{n}{b}\right)^{\frac{2}{3}} = \cos^{2}\beta + \sin^{2}\beta \]Apply the trigonometric identity:
We know the fundamental trigonometric identity \(\cos^{2}\beta + \sin^{2}\beta = 1\).
Therefore, the value of the expression is \(1\).
C) 1
To find the possible value of x, we need to use the property of LCM (Least Common Multiple). The LCM of a set of numbers is the smallest number that is a multiple of all the numbers. This means that each number in the set must be a factor of their LCM. We will find the prime factorization of the given numbers and the LCM, and then determine the prime factors that x must contain or can contain.
B) 200 is the correct option because its prime factorization \(2^3 \times 5^2\) is consistent with the prime factors of 40, 78, and the LCM 7800. Specifically, it contains \(5^2\), which is necessary for the LCM to be 7800, and its other prime factors (2) do not exceed the powers in the LCM.
The question asks to identify the most complex and extensive traditional method of Vedic chanting among the given options. This requires knowledge of the different "pathas" or recitation styles used in Vedic tradition.
Correct Option: D) Ghanapatha is the most complex and extensive traditional method of Vedic chanting. It involves a highly intricate pattern of word recitation (forward, backward, and then forward again) to ensure perfect preservation of the Vedic texts.
The question asks for the keyboard shortcut to start editing text within a selected shape or text box in MS PowerPoint 365. We need to identify the correct function key or key combination for this action.
F2 key after selecting a shape or text box immediately places the cursor inside it, allowing you to start typing or editing the text content.D) F2 is the correct shortcut. When a shape or text box is selected in PowerPoint, pressing F2 activates text editing mode for that object.
Enter is used for a new line, Alt + Enter is not a standard shortcut for starting text editing in a selected shape or text box in PowerPoint. In some contexts, Alt + Enter might be used to open the properties dialog box for a selected object, but not for direct text editing.The problem presents a series of letter-number clusters. To find the missing cluster, we need to identify the pattern in the letters and the numbers separately.
The first letters are D, E, F, G. This is a simple alphabetical sequence where each letter is the next in the English alphabet.
Following this pattern, the next first letter should be G \(\xrightarrow{+1}\) H.
The second letters are J, K, L, M. This is also a simple alphabetical sequence.
Following this pattern, the next second letter should be M \(\xrightarrow{+1}\) N.
The numbers are 26, 37, 48, 59. Let's find the difference between consecutive numbers:
The pattern for the numbers is an arithmetic progression with a common difference of 11. Therefore, the next number in the series will be \(59 + 11 = \textbf{70}\).
Combining the next first letter (H), the next second letter (N), and the next number (70), the complete cluster is HN70.
B) HN70 is the correct cluster as it follows all the identified patterns for letters and numbers.
The question asks for the primary objective of the Sagarmala Project. To answer this, one needs to recall the core focus and stated goals of this initiative by the Ministry of Shipping (now Ministry of Ports, Shipping and Waterways).
Correct Option: C) Port-led development is the primary objective of the Sagarmala Project. This encompasses all aspects of port modernization, enhanced connectivity, and industrialization around ports to foster economic growth.
To find the missing number in the series, we need to identify the pattern of differences between consecutive terms.
The differences are 14, 13, 12, 11. This shows a decreasing sequence of differences, where each subsequent difference is 1 less than the previous one.
Following the pattern, the next difference should be \(11 - 1 = 10\).
Subtract this predicted difference from the last given term:
\[ 137 - 10 = 127 \]Thus, the missing number in the series is 127.
Correct Option: A) 127
This is a linear arrangement puzzle. We need to deduce the seating order of seven people (A, B, C, D, S, T, U) based on the given clues, all facing north. The key is to start with the most restrictive clues and build the arrangement step-by-step.
Clue 1: No one sits to the right of D.
Since everyone is facing north, "right" means to the east. This implies D is at the extreme right end of the row.
Arrangement so far: _ _ _ _ _ _ D
Clue 2: Only three people sit between D and S.
Counting three positions to the left of D, we place S.
Arrangement so far: S _ _ _ D
Combining with the full row: S _ _ _ _ _ D (This means S is at the first position)
So the arrangement is: S _ _ D _ _ D (Incorrect, as D is at the end)
Let's re-evaluate: S _ _ _ D. If D is at the extreme right, and there are 7 people, then S must be at position 3 from the left.
Positions: 1 2 3 4 5 6 7
D is at position 7.
Three people between D and S means S is at position \(7 - 3 - 1 = 3\).
Arrangement: _ _ S _ _ _ D
Clue 3: Only two people sit between S and T.
S is at position 3. T can be to the left or right of S.
Arrangement: _ _ S _ _ T D
Clue 4: C sits third to the left of U.
This means C _ _ U. There are three remaining empty spots: positions 1, 2, 4, 5.
Possible placements for C and U:
Clue 5: A sits to the immediate right of U.
This means U A. Let's combine this with the previous clue (C _ _ U A).
So, U is at position 4, A is at position 5.
From C _ _ U, if U is at 4, then C is at position 1 (\(4 - 2 - 1 = 1\)).
Arrangement: C _ S U A T D
Final Placement:
We have C, S, U, A, T, D placed. The remaining person is B. The only empty spot is position 2.
Final arrangement: C B S U A T D
Answer the question: How many people sit between B and T?
From the final arrangement: C B S U A T D
The people between B and T are S, U, A. There are three people.
B) Three
The problem asks us to identify a sequence of numbers from the given options that follows the same pattern of mathematical operations as the two example sequences. We need to find the operations between consecutive numbers in the given sets and then apply these operations to the options.
So, the pattern for the first set is: Multiply by 2, Add 10, Multiply by 2.
So, the pattern for the second set is also: Multiply by 2, Add 10, Multiply by 2.
C) 5 - 10 - 20 - 40
This option perfectly follows the derived pattern.
The question asks to identify a scenario where a three-dimensional (3D) reference in MS Excel is most effectively used. A 3D reference refers to the same cell or range of cells across multiple worksheets. It is particularly useful when performing calculations on data that is structured identically across several sheets.
=SUM(Sheet1:Sheet12!A1) would sum the value in cell A1 from Sheet1 through Sheet12.=Sheet2!A1>10.=SUM(Jan:Dec!A1) would efficiently sum all those values. This directly matches the definition and utility of a 3D reference.C) Summing values from a specific cell across a sequence of identically structured worksheets representing monthly data for a year. This scenario perfectly describes the utility of a 3D reference, allowing a single formula like =SUM(Sheet1:Sheet12!A1) to aggregate data from the same cell across a range of sheets.