51.

If $\mathrm{f}\left(\mathrm{x}\right) = \left|\begin{array}{ccc}1& \mathrm{x}& \mathrm{x} + 1\\ 2\mathrm{x}& \mathrm{x}(\mathrm{x} - 1)& \left(\mathrm{x} + 1\right)\mathrm{x}\\ 3\mathrm{x}\left(\mathrm{x} - 1\right)& \mathrm{x}(\mathrm{x} - 1)\left(\mathrm{x} - 2\right)& \left(\mathrm{x} + 1\right)\mathrm{x}\left(\mathrm{x} - 1\right)\end{array}\right|$

Then, f(100) is equal to

• 0

• 1

• 100

• 10

If ${\sin }^{- 1}\left(\mathrm{x} - \frac{{\mathrm{x}}^2}{2} + \frac{{\mathrm{x}}^3}{4} - \frac{{\mathrm{x}}^4}{8} + \right) = \frac{\mathrm{\pi }}{6}, \mathrm{where} \left|\mathrm{x}\right| < 2$ then the value of x is

• $\frac{2}{3}$

• $\frac{3}{2}$

• - $\frac{2}{3}$

• - $\frac{3}{2}$

The number of distinct real roots of

$\left|\begin{array}{ccc}\sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ \cos \left(\mathrm{x}\right)& \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ \cos \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \sin \left(\mathrm{x}\right)\end{array}\right| = 0 \mathrm{in} \mathrm{the} \mathrm{interval} - \frac{\mathrm{\pi }}{4} \le \hspace{0.17em}\mathrm{x} \le \frac{\mathrm{\pi }}{4}$ is

• 0

• 2

• 1

• > 2

Let [x] denotes the greatest integer less than or equal to x. Then, the value of $\mathrm{\alpha } \mathrm{f}$ · which the function

f(x) = $$\begin{gathered} \left\{\frac{\sin \left[- {\mathrm{x}}^2\right]}{\left[- {\mathrm{x}}^2\right]}, \mathrm{x} \ne 0 \\ \mathrm{\alpha }, \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, is

• $\mathrm{\alpha } = 0$

• $\mathrm{\alpha } = \sin \left(- 1\right)$

• $\mathrm{\alpha } = \sin \left(1\right)$

• $\mathrm{\alpha } = 1$

For all real values of a₀, a₁, a₂, a₃ satisfying ${\mathrm{a}}_0 + \frac{{\mathrm{a}}_1}{2} + \frac{{\mathrm{a}}_2}{3} + \frac{{\mathrm{a}}_3}{4} = 0$, the equation a₀ + a₁x + a₂x + a₃x³ = 0 has a real root in the interval

• [0, 1]

• [- 1, 0]

• [1, 2]

• [- 2, - 1]

$$\begin{gathered} \mathrm{Let} \mathrm{f}:\hspace{0.17em}\mathrm{R} \to \mathrm{R} \mathrm{be} \mathrm{defined} \mathrm{as} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{0, \mathrm{x} \mathrm{is} \mathrm{irrational} \\ \sin \left|\mathrm{x}\right|, \mathrm{x} \mathrm{is} \mathrm{rational}\right\} \\ \mathrm{Then}, \mathrm{which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{true}? \end{gathered}$$

• f is discontinuous for all x

• f is continuous for all x

• f is discontinuous at x = k$\mathrm{\pi }$, where k is an integer

• f is continuous at x = k$\mathrm{\pi }$, where k is an integer

If $\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{axe}}^{\mathrm{x}} - \mathrm{blog}\left(1 + \mathrm{x}\right)}{{\mathrm{x}}^2} = 3$, then the value of a and b are, respectively

• 2, 2

• 1, 2

• 2, 1

• 2, 0

The trigonometric equation ${\sin }^{-1}\left(\mathrm{x}\right) = 2{\sin }^{-1}\left(2\mathrm{a}\right)$ has a real solution, if

• $\left|\mathrm{a}\right| > \frac{1}{\sqrt{2}}$

• $\frac{1}{2\sqrt{2}} < \left|\mathrm{a}\right| < \frac{1}{\sqrt{2}}$

• $\left|\mathrm{a}\right| > \frac{1}{2\sqrt{2}}$

• $\left|\mathrm{a}\right| \le \frac{1}{2\sqrt{2}}$

$$\begin{gathered} \mathrm{If} \mathrm{f} : \left[0, \mathrm{\pi }/2\right] \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{as} \\ \mathrm{f}\left(\mathrm{\theta }\right) = \left|\begin{array}{ccc}1& \tan \left(\mathrm{\theta }\right)& 1\\ - \tan \left(\mathrm{\theta }\right)& 1& \tan \left(\mathrm{\theta }\right)\\ - 1& - \tan \left(\mathrm{\theta }\right)& 1\end{array}\right| \mathrm{Then}, \mathrm{the} \mathrm{range} \mathrm{of} \mathrm{f} \mathrm{is} \end{gathered}$$

• $\left(2, \infty \right)$

• $(-\infty , - 2]$

• $[2, \infty )$

• $(- \infty , 2]$

If A and B are two matrices such that AB = B and BA = A, then A² + B² equals

• 2AB

• 2BA

• A + B

• AB