To determine if an ion has a noble gas configuration, we examine its electron configuration and compare it with that of a nearby noble gas:
- Sr²⁺ (\(Z = 38\)) loses two electrons, resulting in the electron configuration \([Kr]\), which matches the noble gas krypton.
- Cs⁺ (\(Z = 55\)) loses one electron, resulting in the electron configuration \([Xe]\), matching xenon.
- La³⁺ (\(Z = 57\)) loses three electrons, resulting in the electron configuration \([Xe]\), also matching xenon.
- Yb²⁺ (\(Z = 70\)) loses two electrons, resulting in the electron configuration \([Xe]\), matching xenon.
On the other hand:
- Pb²⁺ does not match any noble gas configuration due to its partially filled \(d\)-orbitals.
- Fe²⁺ does not match a noble gas configuration either, as it retains electrons in the \(d\)-orbital.
Thus, only *Sr²⁺, Cs⁺, La³⁺, and Yb²⁺ have noble gas configurations, totaling four ions.
The Correct answer is: 2
Let \( y^2 = 12x \) be the parabola and \( S \) its focus. Let \( PQ \) be a focal chord of the parabola such that \( (SP)(SQ) = \frac{147}{4} \). Let \( C \) be the circle described by taking \( PQ \) as a diameter. If the equation of the circle \( C \) is: \[ 64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta, \] then \( \beta - \alpha \) is equal to:
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to:
Let $ A \in \mathbb{R} $ be a matrix of order 3x3 such that $$ \det(A) = -4 \quad \text{and} \quad A + I = \left[ \begin{array}{ccc} 1 & 1 & 1 \\2 & 0 & 1 \\4 & 1 & 2 \end{array} \right] $$ where $ I $ is the identity matrix of order 3. If $ \det( (A + I) \cdot \text{adj}(A + I)) $ is $ 2^m $, then $ m $ is equal to: