The sum of sigma (σ) and pi (π) bonds in Hex-1,3-dien-5-yne is ________.

Let $ L_1: \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{2} $ and $ L_2: \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z}{1} $ be two lines. Let $ L_3 $ be a line passing through the point $(\alpha, \beta, \gamma)$ and be perpendicular to both $ L_1 $ and $ L_2 $. If $ L_3 $ intersects $ L_1 $, then $ 5 \alpha = 11 \beta - 8 \gamma $ equals :

Let $ P $ be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in $ P $ are formed by using the digits 1, 2 and 3 only, then the number of elements in the set $ P $ is :

Let $ A = \begin{bmatrix} a_{ij} \end{bmatrix} = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} $. If $ A_{ij} $ is the cofactor of $ a_{ij} $, $ C_{ij} = \sum\limits_{k=1}^{2} a_k A_{jk} , 1 \leq i, j \leq 2 $, and $ C=[C_{ij}] $, then $ 8|C| $ is equal to :