The value of $\sin \left[\frac{\mathrm{\pi }}{2} - {\sin }^{-1}\left(- \frac{\sqrt{3}}{2}\right)\right]$ is

• $\frac{1}{2}$

• $- \frac{1}{2}$

• 1

• - 1

2.

If $\mathrm{a} \le {\sin }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{x}\right) \le \mathrm{b}$, then

• $\mathrm{a} = 0, \mathrm{b} = \mathrm{\pi }$

• $\mathrm{a} = 0, \mathrm{b} = \frac{\mathrm{\pi }}{2}$

• $\mathrm{a} = \frac{\mathrm{\pi }}{2}, \mathrm{b} = \mathrm{\pi }$

• None of these

If A is invertible matrix and B is any matrix, then

• Rank (AB) = Rank(A)

• Rank (AB) = Rank (B)

• Rank (AB) > Rank (A)

• Rank (AB) > Rank (B)

Rank of the matrix $\mathrm{A} = \left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$ is

• 0

• 1

• 2

• 3

The equation of tangent of the curve y = be^-x/a at the point, where the curve meet y-axis is

• bx + ay - ab = 0

• ax + by - ab = 0

• bx - ay - ab = 0

• ax + by - ab = 0

Inverse of function f(x) = $\frac{{10}^{\mathrm{x}} - {10}^{- \mathrm{x}}}{{10}^{\mathrm{x}} + {10}^{- \mathrm{x}}}$ is

• log₁₀(2 - x)

• $\frac{1}{2}{\log }_{10}\left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right)$

• $\frac{1}{2}{\log }_{10}\left(2\mathrm{x} - 1\right)$

• $\frac{1}{4}{\log }_{10}\left(\frac{2\mathrm{x}}{2 - \mathrm{x}}\right)$

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(\mathrm{y}\right) = \frac{2\mathrm{\pi }}{3}$ and ${\cos }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{y}\right) = \frac{\mathrm{\pi }}{3}$. Then, (x, y) is equal to

• (0, 1)

• (1/2, 1)

• (1, 1/2)

• $\left(\sqrt{3}/2, 1\right)$

If y² = P(x) be a cubic polynomial, then $2\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{y}}^3\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)$ is equal to

• P'''(x) + P'(x)

• P''(x)P'''(x)

• P(x)P'''(x)

• constant

Let f : R - {x} $\to$ R be a function defined by f(x) =  $\frac{\mathrm{x} - \mathrm{m}}{\mathrm{x} - \mathrm{n}}$, where m $\ne$ n. Then

• f is one-one onto

• f is one-one into

• f is many one onto

• f is many one into

For the equations x + 2y + 3z = 1, 2x + y + 3z = 2 and 5x + 5y + 9z = 4

• there is only one solution

• there exists infinitely many solution

• there is no solution

• None of the above