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If the function f: R-> R defined as $f(x) =|x|(x -sin x)$, then which of the following statement is true

(a) f is bijective

(b) f is one-one but not Onto

(c) f is onto but not one-one

(d) f is neither one-one nor onto

Answers

If the function f:$[1,\infty)$ -> [2,\infy)$ defined as $f(x) =x + \frac {1}{x}$, then $f^{-1}$ x is

(a) $\frac {x}{1+x^2}$

(b) $x + \frac {1}{x}$

(c) $\frac {x + \sqrt {x^2 -4}}{2}$

(d) $\frac {x - \sqrt {x^2 -4}}{2}$

Answers

The function $f:[0,\infty)$ -> R given by $f(x)= \frac {x}{x+1}$ is

(a) one -one and onto

(b) one-one but not onto

(c) onto but not one-one

(d) Neither one-one nor onto

Answers

if $g(f(x))=sin^2 x$ and $f(g(x)) =(sin \sqrt x)^2$ then

(a) $f(x) =x^2$ and $g(x) = sin \sqrt x$

(b) $f(x) =sinx $ and $g(x) = |x|$

(c) $f(x) =sin^2 x $ and $g(x) = \sqrt x$

(d) f and g cannot be determined

Answers

The function $f(x)=[x] sin (\frac {\pi}{[x+1]})$ where [.] denotes the greatest integer function. The domain of the function f is

(a) $(-\infty ,\infty) - {-1}$

(b) $[0 ,\infty)$

(c) $(-\infty ,-1)$

(d) $(-\infty ,-1) \cup [0, \infty)$

Answers

The domain of the function $f(x)=\frac {log_2 (x+3)}{x^2 + 3x +2}$?

(a) $R - {-1,-2,-3}$

(b) $R - {-1,-2}$

(c) $(-2,\infty)$

(d) $(-3, \infty) - {-1,-2}$

Answers

Consider the below two relations A= {(x,y)|x,y are real numbers and x=wy for some rational number}

B= {(m/n,p/q)|m,n, p and q are integers such that n and q are not zero and qm=pn$ (a) A is a equivalence relation but B is not

(b) A and B both are equivalence relation

(c) A is not a equivalence relation but B is a equivalence relation

(d) A and B both are not equivalence relations

Answers

let $f(x) = \frac {1}{\sqrt {|x| - x}}$, the domain of f(x) is

(a) $(-\infty ,\infty)$

(b) $(-\infty ,0)

(c) $(-\infty ,\infty) - {0}$

(d) $(0 ,\infty)

Answers

Let f: R -> [-1/2,1/2] and $f(x) =\frac {x}{x^2 +1}$ then f(x) is

(a) one-one but not onto

(b) onto but not one -one

(c) one-one and onto

(d) neither one-one nor onto

Answers

If $a \in R$ and the equation $-3(x-[x])^2+2(x−[x])+a^2=0$ (where [x] denotes the greatest integer ) has a no integral solution , then all possible values of a lie in the interval

(a)(1,2)

(b)(-2,-1)

(c)$(-1,0) \cup (0,1)$

(d) $(-2,0) \cup (0,2)$

Answers

(i) Let f :[0,1] -> R be defined by

$f(x) = \frac {4^x}{4^x +2}$

Then the value of

$f(\frac {1}{40}) + f(\frac {2}{40}) + f(\frac {3}{40}) + .............. + f(\frac {39}{40})$ is ______

(ii) if $f(x)= \frac {1}{2} (sin^2 x + sin^2(x + \pi/3) + cosx cos(x + \pi/3))$ and g(5/8) =1, then g(f(x)) is _____

(iii) The domain of the function $cos^{-1} (log_e (\frac {x}{x-1}))$ is _______

Answers

Let $f(x)=\frac {x^2 -6x +5}{x^2 -5x + 6}$

Then match the column A and B

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Let $f(x)=\frac {x^2 -3x -10}{x^2 -2x -3}$

Then match the column A and B

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(i) The function f(x)=sin(log(x + \sqrt {x^2 +1})) is odd function

(ii)If the function $f:[1, \infty) -> [1, \infty)$ is defined by $f(x)=2^{x(x−1)}, then $f^{−1}(x) is given as $\frac {1}{2}[1 + \sqrt {1+4log_2 x}]$

(iii) The relation R on the set A = {1, 2, 3} defined as R = {{1, 1), (1, 2), (2, 1), (3, 3)} is reflexive, symmetric and transitive.

Answers

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