Let the parametric equations of the line be:
\[ x = 1 + 2t, \quad y = -1 + 3t, \quad z = 4t. \]
Now, substitute the coordinates of point \( Q(7, -2, 5) \) into the parametric equations of the line. Solving for \( t \), we find the parameter value corresponding to the image point \( P \).
Next, we find the coordinates of point \( R(5, p, q) \) on the line. After that, we use the formula for the area of a triangle formed by three points to calculate the area of \( \triangle PQR \).
The square of the area of \( \triangle PQR \) is \( 25 \).
Final Answer: The square of the area of \( \triangle PQR \) is \( 25 \).
If \[ \int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left( \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| \right) + C, \] where \( C \) is the constant of integration, then \( \alpha + 2\beta \) is equal to ________________
If the system of equations \[ x + 2y - 3z = 2, \quad 2x + \lambda y + 5z = 5, \quad 14x + 3y + \mu z = 33 \]
has infinitely many solutions, then \( \lambda + \mu \) is equal to:
If the equation of the parabola with vertex \( \left( \frac{3}{2}, 3 \right) \) and the directrix \( x + 2y = 0 \) is \[ ax^2 + b y^2 - cxy - 30x - 60y + 225 = 0, \text{ then } \alpha + \beta + \gamma \text{ is equal to:} \]
For the reaction:
The correct order of set of reagents for the above conversion is :
Match List - I with List - II.