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.
