The set of all values of $$\mathrm{t\in \mathbb{R}}$$, for which the matrix
$$\left[ {\matrix{
{{e^t}} & {{e^{ - t}}(\sin t - 2\cos t)} & {{e^{ - t}}( - 2\sin t - \cos t)} \cr
{{e^t}} & {{e^{ - t}}(2\sin t + \cos t)} & {{e^{ - t}}(\sin t - 2\cos t)} \cr
{{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr
} } \right]$$ is invertible, is :
The value of the integral $$\int\limits_{1/2}^2 {{{{{\tan }^{ - 1}}x} \over x}dx} $$ is equal to :
The shortest distance between the lines $${{x - 1} \over 2} = {{y + 8} \over -7} = {{z - 4} \over 5}$$ and $${{x - 1} \over 2} = {{y - 2} \over 1} = {{z - 6} \over { - 3}}$$ is :
Let $$f$$ and $$g$$ be the twice differentiable functions on $$\mathbb{R}$$ such that
$$f''(x)=g''(x)+6x$$
$$f'(1)=4g'(1)-3=9$$
$$f(2)=3g(2)=12$$.
Then which of the following is NOT true?