Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa

Let \mathrm{\left [ t \right ]} denote the greatest integer less than or equal to t. Then, the value of the integral \mathrm{\int_{0}^{1}\left [ -8x^{2}+6x-1 \right ]dx} is equal to

Option: 1

\mathrm{-1}


Option: 2

\mathrm{\frac{-5}{4}}


Option: 3

\mathrm{\frac{\sqrt{17}-13}{8}}


Option: 4

\mathrm{\frac{\sqrt{17}-16}{8}}


Answers (1)

best_answer

\mathrm{Let \: f(x)=-8 x^{2}+6 x-1, \quad f(0)=-1, f(1)=-3},

\mathrm{f\left(\frac{3}{8}\right)=-8 \times \frac{9}{64}+6 \times \frac{3}{8}-1=\frac{1}{8}\: \Rightarrow \: f(x) \in[-3,1 / 8]in \, x\in[0,1]. }

So when
\mathrm{-8 x^{2}+6 x-1=-3 ; \: 8 x^{2}-6 x-2=0 \Rightarrow 4 x^{2}-3 x-1=0 \Rightarrow 4 x^{2}-4 x+x-1=0 }
\mathrm{\Rightarrow x=1 \:\: OR\: \: x=-1 / 4}.

\mathrm{When\: \: -8 x^{2}+6 x-1=-2 ; \: \Rightarrow 8 x^{2}-6 x-1=0 \Rightarrow x=\frac{6 \pm \sqrt{36+32}}{16}=\frac{3 \pm \sqrt{17}}{8}}
\mathrm{When\: -8 x^{2}+6 x-1=-1\: \Rightarrow\: -8 x^{2}+6 x=0 \: \Rightarrow\: x=0 \quad o R \quad x=3 / 4.}
\mathrm{When\: -8 x^{2}+6 x-1= 0\: \Rightarrow x=1 / 4\: or\: 1 / 2.}



\mathrm{So\: \: \int_{0}^{1}\left[-8 x^{2}+6 x-1\right] d x }
\mathrm{=\int_{0}^{1 / 4}(-1) d x+\int_{1 / 4}^{1 / 2} 0 d x+\int_{1 / 2}^{3 / 4}(-1) d x+\int_{3 / 4}^{\frac{3+\sqrt{7}}{2}}(-2) d x}
\mathrm{+\int_{\frac{3+\sqrt{17}}{2}}^{1}(-3) d x}

\mathrm{= \frac{-1}{4}+0-\frac{1}{4}-2\left ( \frac{3+\sqrt{17}}{8}-\frac{3}{4} \right )-3\left (1-\frac{3+\sqrt{17}}{8} \right )}
\mathrm{=\frac{-2-2-6-2\sqrt{17}+12-24+9+3\sqrt{17}}{8}= \frac{\sqrt{17}-13}{8}}

Option (C).

 

Posted by

Divya Prakash Singh

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2025: Most important chapters for physics, chemistry, mathematics to score good marks
anam-khan

JEE Main 2025 session 2 dates clash with ISC, PSEB 12th exams; no impact on major science subjects
anam-khan

Delhi police EOW books FIITJEE for fraud; 192 complaints filed

Latest Question