JEE Advance - Mathematics (2014 - Paper 1 Offline - No. 18)
Let f : [0, 4$$\pi$$] $$\to$$ [0, $$\pi$$] be defined by f(x) = cos$$-$$1 (cos x). The number of points x $$\in$$ [0, 4$$\pi$$] satisfying the equation $$f(x) = {{10 - x} \over {10}}$$ is
Answer
3
Explanation
Concept :
The number of solutions of equations involving trigonometric functions and algebraic functions are found using graphs of the curves.
We know, $${\cos ^{ - 1}}(\cos x) = \left\{ \matrix{ x,\,if\,x \in [0,\pi ] \hfill \cr 2\pi - x,\,if\,x \in [\pi ,2\pi ] \hfill \cr - 2\pi + x,\,if\,x \in [2\pi ,3\pi ] \hfill \cr 4\pi - x,\,if\,x \in [3\pi ,4\pi ] \hfill \cr} \right.$$
$$y = {{10 - x} \over {10}} = 1 - {x \over {10}}$$
From above figure, it is clear that $$y = {{10 - x} \over {10}}$$ and $$y = {\cos ^{ - 1}}(\cos x)$$ intersect at three distinct points, so number of solutions is 3.
The number of solutions of equations involving trigonometric functions and algebraic functions are found using graphs of the curves.
We know, $${\cos ^{ - 1}}(\cos x) = \left\{ \matrix{ x,\,if\,x \in [0,\pi ] \hfill \cr 2\pi - x,\,if\,x \in [\pi ,2\pi ] \hfill \cr - 2\pi + x,\,if\,x \in [2\pi ,3\pi ] \hfill \cr 4\pi - x,\,if\,x \in [3\pi ,4\pi ] \hfill \cr} \right.$$
$$y = {{10 - x} \over {10}} = 1 - {x \over {10}}$$
From above figure, it is clear that $$y = {{10 - x} \over {10}}$$ and $$y = {\cos ^{ - 1}}(\cos x)$$ intersect at three distinct points, so number of solutions is 3.
Comments (0)
