If y = (tan^-1(x)), then (x² + 1)²y₂ + 2x(x² + 1)y₁ is equal to

• 4

• 0

• 2

• 1

If f(x) = x³ and g(x) = x³ - 4x in - 2 $\le \mathrm{x} \le$, then consider the statements

(i) f(x) and g(x) satisfy mean value theorem.

(ii) f(x) and g(x) both satisfy Rolle's theorem.

(iii)  Only g(x) satisfies Rolle's theorem.

Of these statements.

• (i) and (ii) are correct

• only (i) is correct

• None is correct

• (i) and (iii) are correct

3.

The tangent to the curve y = x³ + 1 at (1, 2) makes an angle $\mathrm{\theta }$ with y-axis, then the value of $\tan \left(\mathrm{\theta }\right)$ is

• $- \frac{1}{3}$

• 3

• - 3

• $\frac{1}{3}$

If the function f(x) defined by $\mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{x}}^{100}}{100} + \frac{{\mathrm{x}}^{99}}{99} + ... + \frac{{\mathrm{x}}^2}{2} +\hspace{0.17em}\mathrm{x}\hspace{0.17em}+\hspace{0.17em}1$, then f'(0) is equal to

• 100f'(0)

• 100

• 1

• - 1

Let S be the set of all real numbers. A relation R has been defined on S by aRb $\iff \left|\mathrm{a} - \mathrm{b}\right| \le 1$, then R is

• symmetric and transitive but not reflexive

• reflexive and transitive but not symmetrIc

• reflexive and symmetric but not transitive

• an equivalence relation

For any two real numbers, an operation * defined by a * b  = 1 + ab is

• neither commutative nor associative

• commutative but not associative

• both commutative and associative

• associative but not commutative

Let f : N $\to$ N defined by f(n) = $$\begin{gathered} \left\{\frac{\mathrm{n} +1}{2}, \mathrm{if} \mathrm{n} \mathrm{is} \mathrm{odd} \\ \frac{\mathrm{n}}{2}, \mathrm{if} \mathrm{n} \mathrm{is} \mathrm{even}\right\} \end{gathered}$$, then f is

• onto but not one-one

• one-one and onto

• neither one-one nor onto

• one-one but not onto

Suppose f(x) = (x + 1)² for x $\ge$ - 1. If g(x) is a function whose graph is the reflection of the graph of f(x) in the line y = x, then g(x) is equal to

• $\frac{1}{{\left(\mathrm{x} + 1\right)}^2}\mathrm{x} > - 1$

• $- \sqrt{\mathrm{x}} - 1$

• $\sqrt{\mathrm{x}} + 1$

• $\sqrt{\mathrm{x}} - 1$

Given $0 \le \mathrm{x} \le \frac{1}{2}$, then the value of $\tan \left[{\sin }^{-1}\left\{\frac{\mathrm{x}}{\sqrt{2}} + \frac{\sqrt{1 - {\mathrm{x}}^2}}{\sqrt{2}}\right\} - {\sin }^{-1}\left(\mathrm{x}\right)\right]$ is

• 1

• $\sqrt{3}$

• - 1

• $\frac{1}{\sqrt{3}}$

The value of $\sin \left(2{\sin }^{-1}\left(0.8\right)\right)$ is equal to

• 0.48

• $\sin \left(1.2°\right)$

• $\sin \left(1.6°\right)$

• 0.96