Three fair coins numbered 1 and 0 are tossed simultaneously. Then variance Var (X) of the probability distribution of random variable $$\mathrm{X}$$, where $$\mathrm{X}$$ is the sum of numbers on the uppermost faces, is

$$\int \frac{1}{\sin (x-a) \sin x} d x=$$

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