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:

  1. 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

$$ 0-2=2(x-2) \text { and } 0+2=2(x+2) $$

$$ \Rightarrow \quad x= \pm 1 $$



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