Indefinite Integration 3 Question 3
3. The intercepts on $X$-axis made by tangents to the curve, $y=\int _0^{x}|t| d t, x \in R$, which are parallel to the line $y=2 x$, are equal to
(2013 Main)
(a) \pm 1
(b) \pm 2
(c) \pm 3
(d) \pm 4
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Answer:
Correct Answer: 3. (a)
Solution:
- Given, $y=\int _0^{x}|t| d t$
$\therefore \frac{d y}{d x}=|x| \cdot 1-0=|x| \quad$ [by Leibnitz’s rule]
$\because$ Tangent to the curve $y=\int _0^{x}|t| d t, x \in R$ are parallel to the line $y=2 x$
$\therefore$ Slope of both are equal $\Rightarrow x= \pm 2$
Points, $\quad y=\int _0^{ \pm 2}|t| d t= \pm 2$
Equation of tangent is
$$ y-2=2(x-2) \text { and } y+2=2(x+2) $$
For $x$ intercept put $y=0$, we get
$$ 0-2=2(x-2) \text { and } 0+2=2(x+2) $$
$$ \Rightarrow \quad x= \pm 1 $$
