The derivative of $$\mathrm{f}(\sec x)$$ with respect to $$g(\tan x)$$ at $$x=\frac{\pi}{4}$$, where $$f^{\prime}(\sqrt{2})=4$$ and $$g^{\prime}(1)=2$$, is

The scalar product of the vector $$\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$$ with a unit vector along the sum of the vectors $$2 \hat{i}+4 \hat{j}-5 \hat{k}$$ and $$\lambda \hat{i}+2 \hat{j}+3 \hat{k}$$ is equal to 1 , then value of $$\lambda$$ is

The number of discontinuities of the greatest integer function $$\mathrm{f}(x)=[x], x \in\left(-\frac{7}{2}, 100\right)$$

If $$[(\bar{a}+2 \bar{b}+3 \bar{c}) \times(\bar{b}+2 \bar{c}+3 \bar{a})] \cdot(\bar{c}+2 \bar{a}+3 \bar{b})=54$$ then the value of $$\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$$ is