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Given \(\log_x y\) and \(\log_y x\) are reciprocals, what conclusion can we draw if \(k + 1/k = 2\)?
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The only solution is \(k = 1\), leading to \(x = y\).
In the speed and distance problem, what is the distance covered if Geeta saves 1 hour by increasing her speed from 12 km/hr to 20 km/hr?
The distance is 30 km.
What is the solution range when solving the inequality \(\log_3 x\)?
The solution range is \([2, 81)\).
How is the final count of circular arrangements derived when considering seating restrictions?
Apply the subtraction principle; the final count is 960.
In modulus and geometry, how is the minimum value calculated using symmetry?
The calculated minimum value is 26.
What is the sum of the coefficients in the expansion of \((5x - 9)^4\)?
256
How do you find the next term in a GP formed from consecutive terms of an AP?
Identify the term forming both sequences and use the relations to find \(n = 344\).
In a triangle with a longest side of 40, what must be true about the possible shortest side length?
The shortest side must be greater than 19.
What is the maximum possible sum of integers if the difference constraints apply to the highest integer being 100?
The optimal sum is 3150.
How many triangles can be formed using 7 points on one line and 5 points on another line in combinatorial geometry?
175 triangles.
How would you find the 2022nd term in the sequence that omits perfect squares?
Determine the nearest square beyond 2022, recognizing 2025, resulting in 2067.
In the profit and loss problem, what is the ratio of apples to mangoes if there's no net profit or loss?
5:2
What is the simplified form of the equation \(\log_y x^2 + \log_x y^2 = 1\)?
\(\log_x y + \log_y x = 2\)
In geometry, how is the value of \(k\) found if lines are concurrent?
The value of \(k = 0\).
What are the roots of the quadratic equation \(x^2 - x - 30 = 0\) and which root is feasible if \(x\) cannot be negative?
The roots are \(x = 6, -5\), and the feasible root is \(x = 6\).
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