If the equation of the parabola with vertex \( \left( \frac{3}{2}, 3 \right) \) and the directrix \( x + 2y = 0 \) is \[ ax^2 + b y^2 - cxy - 30x - 60y + 225 = 0, \text{ then } \alpha + \beta + \gamma \text{ is equal to:} \]
The equation of the parabola is given in general form. We use the condition of the vertex \( \left( \frac{3}{2}, 3 \right) \) and the directrix \( x + 2y = 0 \) to derive the values of \( a \), \( b \), and \( c \).
Then, we calculate \( \alpha + \beta + \gamma \).
Final Answer: \( \alpha + \beta + \gamma = 6 \).
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:
Match List - I with List - II:
List - I:
(A) Amylase
(B) Cellulose
(C) Glycogen
(D) Amylopectin
List - II:
(I) β-C1-C4 plant
(II) α-C1-C4 animal
(III) α-C1-C4 α-C1-C6 plant
(IV) α-C1-C4 plant
Match List - I with List - II:
Choose the correct answer from the given below options
1.24 g of \(AX_2\) (molar mass 124 g mol\(^{-1}\)) is dissolved in 1 kg of water to form a solution with boiling point of 100.105°C, while 2.54 g of AY_2 (molar mass 250 g mol\(^{-1}\)) in 2 kg of water constitutes a solution with a boiling point of 100.026°C. \(Kb(H)_2\)\(\text(O)\) = 0.52 K kg mol\(^{-1}\). Which of the following is correct?