### Sequences And Series Question 14

#### Question 14 - 31 January - Shift 1

Let $y=f(x)$ represent a parabola with focus $(-\frac{1}{2}, 0)$ and directrix $y=-\frac{1}{2}$.

Then

$S={x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)}+\sin ^{-1}(\sqrt{f(x)+1}))=\frac{\pi}{2}}:$

(1) contains exactly two elements

(2) contains exactly one element

(3) is an infinite set

(4) is an empty set

## Show Answer

#### Answer: (1)

#### Solution:

#### Formula: Important terms and other forms of a standard parabola (iii) and (iv)

$(x+\frac{1}{2})^{2}=(y+\frac{1}{4})$

$y=(x^{2}+x)$

$\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\pi / 2$

$0 \leq x^{2}+x+1 \leq 1$

$x^{2}+x \leq 0$

Also $x^{2}+x \geq 0$

$\therefore x^{2}+x=0 \Rightarrow x=0,-1$

$S$ contains 2 element.