21. The domain of $$f\left( x \right) = {\sin ^{ - 1}}\left( {\frac{{1 + {x^2}}}{{2x}}} \right) + \sqrt {1 - {x^2}} \,$$ is :
A
$$\left\{ 1 \right\}$$
B
$$\left( { - 1,\,1} \right)$$
C
$$\left\{ {1,\, - 1} \right\}$$
D
none of these
Answer :
$$\left\{ {1,\, - 1} \right\}$$
22. Let $$f$$ be a function from $$R$$ to $$R$$ given by $$f\left( x \right) = \frac{{{x^2} - 4}}{{{x^2} + 1}}.$$ Then $$f\left( x \right)$$ is :
A
one-one and into
B
one-one and onto
C
many-one and into
D
many-one and onto
Answer :
many-one and into
23.
Let
\[\begin{array}{l}
{f_1}\left( x \right) = \left\{ \begin{array}{l}
x,\,\,\,0 \le x \le 1\\
1,\,\,\,x > 1\\
0,\,\,\,{\rm{otherwise}}
\end{array} \right.\\
{f_2}\left( x \right) = {f_1}\left( { - x} \right){\rm{ for\,\,all\,\, }}x\\
{f_3}\left( x \right) = - {f_2}\left( x \right){\rm{ for\,\,all\,\, }}x\\
{f_4}\left( x \right) = {f_3}\left( { - x} \right){\rm{ for\,\, all\,\, }}x
\end{array}\]
Which of the following is necessarily true ?
A
$${f_4}\left( x \right) = {f_1}\left( x \right){\text{ for all }}x$$
B
$${f_1}\left( x \right) = - {f_3}\left( { - x} \right){\text{ for all }}x$$
C
$${f_2}\left( { - x} \right) = {f_4}\left( x \right){\text{ for all }}x$$
D
$${f_1}\left( x \right) + {f_3}\left( x \right) = 0{\text{ for all }}x$$
Answer :
$${f_1}\left( x \right) = - {f_3}\left( { - x} \right){\text{ for all }}x$$
24. The entire graphs of the equation $$y = {x^2} + kx - x + 9$$ is strictly above the $$x$$-axis if and only if
A
$$k < 7$$
B
$$ - 5 < k < 7$$
C
$$k > - 5$$
D
None of these.
Answer :
$$ - 5 < k < 7$$
25. If $$F\left( x \right) = {\left( {f\left( {\frac{x}{2}} \right)} \right)^2} + {\left( {g\left( {\frac{x}{2}} \right)} \right)^2}$$ where $$f''\left( x \right) = - f\left( x \right)$$ and $$g\left( x \right) = f'\left( x \right)$$ and given that $$F\left( 5 \right) = 5,$$ then $$F\left( {10} \right)$$ is equal to
A
5
B
10
C
0
D
15
Answer :
5
26. Let $$f\left( x \right)$$ be a function whose domain is $$\left[ { - 5,\,7} \right].$$ Let $$g\left( x \right) = \left| {2x + 5} \right|.$$ Then the domain of $$\left( {f\,o\,g} \right)\left( x \right)$$ is :
A
$$\left[ { - 5,\,1} \right]$$
B
$$\left[ { - 4,\,0} \right]$$
C
$$\left[ { - 6,\,1} \right]$$
D
none of these
Answer :
$$\left[ { - 6,\,1} \right]$$
27. If $$f\left( {x + y,\,x - y} \right) = xy$$ then the arithmetic mean of $$f\left( {x,\,y} \right)$$ and $$f\left( {y,\,x} \right)$$ is :
A
$$x$$
B
$$y$$
C
0
D
none of these
Answer :
0
28. If the real-valued function $$f\left( x \right) = px + \sin \,x$$ is a bijective function then the set of possible values of $$p\, \in \,R$$ is :
A
$$R - \left\{ 0 \right\}$$
B
$$R$$
C
$$\left( {0,\, + \infty } \right)$$
D
none of these
Answer :
$$R - \left\{ 0 \right\}$$
29.
A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
| Interval | Function | |
|---|---|---|
| (a) | $$\left( { - \infty ,\,\infty } \right)$$ | $${x^3} - 3{x^2} + 3x + 3$$ |
| (b) | $$\left[ {2,\,\infty } \right)$$ | $$2{x^3} - 3{x^2} - 12x + 6$$ |
| (c) | $$\left( { - \infty ,\,\frac{1}{3}} \right]$$ | $$3{x^2} - 2x + 1$$ |
| (d) | $$\left( { - \infty ,\, - 4} \right)$$ | $${x^3} + 6{x^2} + 6$$ |
A
a
B
b
C
c
D
d
Answer :
c
30. Let $$f\left( x \right) = {\log _{{x^2}}}25$$ and $$g\left( x \right) = {\log _x}5$$ then $$f\left( x \right) = g\left( x \right)$$ holds for $$x$$ belonging to :
A
$$R$$
B
$$\left( {0,\,1} \right) \cup \left( {1,\, + \infty } \right)$$
C
$$\phi $$
D
none of these
Answer :
$$\left( {0,\,1} \right) \cup \left( {1,\, + \infty } \right)$$