Let $C$ be the circle $x^2 + (y - 1)^2 = 2$, $E_1$ and $E_2$ be two ellipses whose centres lie at the origin and major axes lie on the $x$-axis and $y$-axis respectively. Let the straight line $x + y = 3$ touch the curves $C$, $E_1$, and $E_2$ at $P(x_1, y_1)$, $Q(x_2, y_2)$, and $R(x_3, y_3)$ respectively. Given that $P$ is the mid-point of the line segment $QR$ and $PQ = \frac{2\sqrt{2}}{3}$, the value of $9(x_1 y_1 + x_2 y_2 + x_3 y_3)$ is equal to
If $$ \int \frac{\left( \sqrt{1 + x^2} + x \right)^{10}}{\left( \sqrt{1 + x^2} - x \right)^9} \, dx = \frac{1}{m} \left( \left( \sqrt{1 + x^2} + x \right)^n \left( n\sqrt{1 + x^2} - x \right) \right) + C, $$ $\text{where } m, n \in \mathbb{N} \text{ and }$ $C \text{ is the constant of integration, then } m + n$ $\text{ is equal to:}$
Let \( S = \left\{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \right\} \), where
Then \( n(S) \) is equal to ______.
If the area of the region $\{ (x, y) : |x - 5| \leq y \leq 4\sqrt{x} \}$ is $A$, then $3A$ is equal to
Convert Propanoic acid to Ethane
मुझसे हँसा नहीं जाता । (वाच्य पहचानकर भेद का नाम लिखिए)
Match List-I with List-II\[\begin{array}{|c|c|} \hline \textbf{Provision} & \textbf{Case Law} \\ \hline \text{(A) Strict Liability} & \text{(1) Ryland v. Fletcher} \\ \hline \text{(B) Absolute Liability} & \text{(II) M.C. Mehta v. Union of India} \\ \hline \text{(C) Negligence} & \text{(III) Nicholas v. Marsland} \\ \hline \text{(D) Act of God} & \text{(IV) MCD v. Subhagwanti} \\ \hline \end{array}\]