For any four vectors $\mathbf{a , b , c , d}$ the expression $(\mathrm{b} \times \mathrm{c}) \cdot(\mathrm{a} \times \mathrm{d})+(\mathrm{c} \times \mathrm{a}) \cdot(\mathrm{b} \times \mathrm{d})+(\mathrm{a} \times \mathrm{b}) \cdot(\mathrm{c} \times \mathrm{d})$ is always equal to

If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x)^{3 / 2}-\left(1+\frac{1}{2} x\right)^3}{(1-x)^{1 / 2}}$ may be approximate as

The focal chord of $y^2=16 x$ is a tangent to $(x-6)^2+y^2=2$, then the possible values of the slope of this chord are

Let $f(x)$ be a polynomial of degree 6 divisible by $x^3$ and having a point of extremum at $x=2$. If $f^{\prime}(x)$ is divisible by $1+x^2$, then find the value of $\frac{3 f(2)}{f(1)}$