Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
The following table shows the ages of the patients admitted in a hospital during a year. Find the mode and the median of these data.\[\begin{array}{|c|c|c|c|c|c|c|} \hline Age (in years) & 5-15 & 15-25 & 25-35 & 35-45 & 45-55 & 55-65 \\ \hline \text{Number of patients} & \text{6} & \text{11} & \text{21} & \text{23} & \text{14} & \text{5} \\ \hline \end{array}\]
In the following figure \(\triangle\) ABC, B-D-C and BD = 7, BC = 20, then find \(\frac{A(\triangle ABD)}{A(\triangle ABC)}\).
In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \( P \), \( Q \), and \( R \)?
The radius of a circle with centre 'P' is 10 cm. If chord AB of the circle subtends a right angle at P, find area of minor sector by using the following activity. (\(\pi = 3.14\))
Activity : r = 10 cm, \(\theta\) = 90\(^\circ\), \(\pi\) = 3.14. A(P-AXB) = \(\frac{\theta}{360} \times \boxed{\phantom{\pi r^2}}\) = \(\frac{\boxed{\phantom{90}}}{360} \times 3.14 \times 10^2\) = \(\frac{1}{4} \times \boxed{\phantom{314}}\) <br>A(P-AXB) = \(\boxed{\phantom{78.5}}\) sq. cm.
Find the Derivative \( \frac{dy}{dx} \)Given:\[ y = \cos(x^2) + \cos(2x) + \cos^2(x^2) + \cos(x^x) \]
