Question

Assertion :$f:R\to [0,\frac{\pi }{2})$ defined by $f\left(x\right)={\tan }^{-1}(x^2+x+a)$ is onto for all $a\in (-\infty ,\frac{1}{4})$ Reason: For onto function codomain of $f=$ Range of $f$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

For Onto function

codomain $=[0,\frac{\pi }{2})=$ Range

Which is possiable where $x^2+x+a\ge 0$

Now using fact $f\left(x\right)\ge 0\Rightarrow D\le 0$

$(A=1=b)andC=a$

$\Rightarrow 1-4a\le 0$

$\Rightarrow a\ge \frac{1}{4}$

$\therefore$ Assertion is false, Reason is true