TS EAPCET (DEMO) MATHS INVERSE TRIGONOMETRIC FUNCTIONS PAPER

Domain of the function $$f(x)=\log _{\mathrm{e}} \cos ^{-1}\{\sqrt{x}\}$$ is, where {.} represents fractional part function –

The range of the function $$f(x)=\sin ^{-1}\left(\log _2\left(-x^2+2 x+3\right)\right)$$ is –

Range of $$f(x)=\cot ^{-1}\left(\log _e\left(1-x^2\right)\right)$$ is –

The function $$f(x)=\cot ^{-1} \sqrt{(x+3) x}+\cos ^{-1} \sqrt{x^2+3 x+1}$$ is defined on the set $$S$$, where $$S$$ is equal to:

The value of $$\cot ^{-1}\left\{\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\right\}$$, where $$\frac{\pi}{2}<x<\pi$$, is:

The inequality $$\sin ^{-1}(\sin 5)>x^2-4 x$$ holds for

Show that the roots r , s and t of the cubic $$x(x-2)(3 x-7)=2$$, are real and positive. Also compute the value of $$\tan ^{-1}(r)+\tan ^{-1}(s)+\tan ^{-1}(t)$$.

The complete solution set of the inequality $$\left[\cot ^{-1} x\right]^2-6\left[\cot ^{-1} x\right]+9 \leq 0$$, where [.] denotes greatest integer function, is

If $$\cos ^{-1}\left(2 x^2-1\right)=2 \pi-2 \cos ^{-1} x$$, then –

If $$\cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha$$, then $$4 x^2-4 x y \cos \alpha+y^2$$ is equal to-

$$\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x\right)+\tan \left(\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} x\right), x \neq 0$$ is equal to

$$
\text { If } \alpha=2
$$ arc tan
$$\left(\frac{1+x}{1-x}\right) \& \beta=$$ arc sin
$$
\left(\frac{1-x^2}{1+x^2}\right) \text {, what is the value of } \alpha+\beta \text { if } x>1 \text {. }
$$

The numerical value of $$\tan \left(2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right)$$ is

$$\sum_{r=0}^{\infty} \tan ^{-1}\left(\frac{r((r+1)!)}{(r+1)+((r+1)!)^2}\right)$$ is equal to –

$$\lim _{n \rightarrow \infty} \sum_{r=1}^n \tan ^{-1} \frac{2 r+1}{r^4+2 r^3+r^2+1}$$ is equal to –