Let $$\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}, \vec{b}=3 \hat{i}+4 \hat{j}-5 \hat{k}$$ and a vector $$\vec{c}$$ be such that $$\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times \vec{c}=\hat{i}+8 \hat{j}+13 \hat{k}$$. If $$\vec{a} \cdot \vec{c}=13$$, then $$(24-\vec{b} \cdot \vec{c})$$ is equal to _______.

Let $$L_1, L_2$$ be the lines passing through the point $$P(0,1)$$ and touching the parabola $$9 x^2+12 x+18 y-14=0$$. Let $$Q$$ and $$R$$ be the points on the lines $$L_1$$ and $$L_2$$ such that the $$\triangle P Q R$$ is an isosceles triangle with base $$Q R$$. If the slopes of the lines $$Q R$$ are $$m_1$$ and $$m_2$$, then $$16\left(m_1^2+m_2^2\right)$$ is equal to __________.

A bullet of mass $$50 \mathrm{~g}$$ is fired with a speed $$100 \mathrm{~m} / \mathrm{s}$$ on a plywood and emerges with $$40 \mathrm{~m} / \mathrm{s}$$. The percentage loss of kinetic energy is :

The specific heat at constant pressure of a real gas obeying $$P V^2=R T$$ equation is: