$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right) = \frac{2\mathrm{x} - {\sin }^{-1}\left(\mathrm{x}\right)}{2\mathrm{x} + {\tan }^{-1}\left(\mathrm{x}\right)} \mathrm{continuous} \mathrm{at} \mathrm{each} \mathrm{point} \mathrm{in} \\ \mathrm{its} \mathrm{domain},\mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{f}\left(0\right) ? \end{gathered}$$

• $- \frac{1}{3}$

• $\frac{1}{3}$

• $\frac{2}{3}$

• 2

A function f : A $\to$ R is defined by the equation f(x) = x² - 4x + 5 where A = (1, 4). What is the range of the function ?

• (2, 5)

• (1, 5)

• [1, 5)

• [1, 5]

23.

What is ${\int }_{\mathrm{a}}^{\mathrm{b}}\left[\mathrm{x}\right]d\mathrm{x} + {\int }_{\mathrm{a}}^{\mathrm{b}}\left[ - \mathrm{x}\right]d\mathrm{x}$ = ?

where [.] is the greatest integer function ?

• b - a

• a - b

• 0

• 2(b - a)

In which one of the following cases would you except to get a negative correlation ?

• The ages of husbands and wives

• Shoe size and intelligence

• Insurance companies profits and the number of claims they have to pay

• Amount of rainfall and yield of crop

Let S be the set of all persons living in Delhi. We say that x, y in S are related if they were born in Delhi on the same day. Which one of the following is correct ?

• The relation is an equivalent relation

• The relation is not reflexive but it is symmetric and transitive

• The relation is not symmetric but it is reflexive and transitive

• The relation is not transitive but it is reflexive and symmetric

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{x}}{\mathrm{x} - 1}, \mathrm{then} \mathrm{what} \mathrm{is} \frac{\mathrm{f}\left(\mathrm{a}\right)}{\mathrm{f}\left(\mathrm{a} + 1\right)} = ?$

• $\mathrm{f}\left( - \frac{\mathrm{a}}{\mathrm{a} +\hspace{0.17em}1}\right)$

• $\mathrm{f}{\left(\mathrm{a}\right)}^2$

• $\mathrm{f}\left(\frac{1}{\mathrm{a}}\right)$

• f( - a)

The function f: X $\to$ Y defined by f(0) = cos(x), where x $\in$ X, is one-one and onto if X and Y are respectively equal to

• $\left[0, \mathrm{\pi }\right] \mathrm{and} \left[ - 1, 1\right]$

• $\left[ - \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right] \mathrm{and} \left[ - 1, 1\right]$

• $\left[0, \mathrm{\pi }\right] \mathrm{and} \left( - 1, 1\right)$

• $\left[0, \mathrm{\pi }\right] \mathrm{and} \left[0, 1\right]$

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) : \left\{\mathrm{x}, \mathrm{x} \mathrm{is} \mathrm{rational} \\ 0, \mathrm{x} \mathrm{is} \mathrm{irrational}\right\} \\ \mathrm{and} \mathrm{g}\left(\mathrm{x}\right) :\hspace{0.17em}\left\{0, \mathrm{x} \mathrm{is} \mathrm{rational} \\ \mathrm{x}, \mathrm{x} \mathrm{is} \mathrm{irrational}\right\} \\ \mathrm{If} \mathrm{f} : \mathrm{R} \to \mathrm{R} \mathrm{and} \mathrm{g} : \mathrm{R}\to \mathrm{R}, \mathrm{then} \left(\mathrm{f} - \mathrm{g}\right) \mathrm{is} \end{gathered}$$

• one-one and into

• neither one-one nor onto

• many-one and onto 

• one-one and onto

Let f(x) be an indefinite integral of sin(2x).

Consider the following statements :

Statement 1 : The function f(x) satisfies f(x + $\mathrm{\pi }$) = f(x) for all real x.

Statement 2 : sin²(x + $\mathrm{\pi }$) = sin²(x) for all real x.

Which one of the following is correct in respect of the above statements ?

• Both the statements are true and Statement 2 is the correct explanation of Statement 1

• Both the statements are true but Statement 2 is· not the correct explanation of Statement 1

• Statement 1 is true but Statement 2 is false

• Statement I is false but Statement 2 is true

What is the maximum value of the function f(x) = 4sin²(x) + 1 ?

• 5

• 3

• 2

• 1