question_answer

The area (in sq. units) of the region described by \[A=\{(x,y)|y\ge {{x}^{2}}-5x+4,x+y\ge 1,y\le 0\}\]is :   JEE Main Online Paper (Held On 09 April 2016)

A) \[\frac{7}{2}\]                                   

B) \[\frac{13}{6}\]

C) \[\frac{17}{6}\]                                

D) \[\frac{19}{6}\]

Correct Answer: D

Solution :

                \[A=\{(x,y)|y\ge {{x}^{2}}-5x+4,x+y\ge 1,y\le 0\}\] Here \[y\ge {{x}^{2}}-5x+4,x+y\ge 1,y\le 0\] Required area \[=\frac{1}{2}.22+\int\limits_{3}^{4}{(5x-{{x}^{2}}-4)dx}\] \[=2+\left[ \frac{5{{x}^{2}}}{2}-\frac{{{x}^{3}}}{3}-4x \right]_{3}^{4}\] \[=2+\frac{5}{2}(16-9)-\frac{1}{3}(64-27)-4(4-3)\] \[-2+\frac{35}{2}-\frac{37}{3}-4=\frac{19}{6}\]