New chat PYQ Library Doubts Rank report

Sign in to see Recents

Sign in to save progress →

Application of Integration question

2015 · Shift 1 · Q31

Guest · try one question or challengeGuest · try one question or challenge

PYQ Library
JEE Advanced
Mathematics
Application of Integration
2015 · Shift 1 · Q31

Application of Integration question

2015 · Shift 1 · Q31

JEE AdvancedMathematicsApplication of IntegrationNumerical+4 / −1

Let

F\left( x \right) = \int\limits_x^{{x^2} + {\pi \over 6}} {2{{\cos }^2}t\left( {dt} \right)}

for all

x \in R

and

f:\left[ {0,{1 \over 2}} \right] \to \left[ {0,\infty } \right]

be a continuous function. For

a \in \left[ {0,{1 \over 2}} \right],\,

F'(a)+2

is the area of the region bounded by

x=0, y=0, y=f(x)

and

x=a,

then

f(0)

is

Numerical answer

More from Application of Integration

Let $f : R \to R$ be a continuous odd function, which vanishes exactly at one point and $f (1) = \frac{1}{2.}$ Suppose that $F (x) = - 1 \int x f (t) d t$ for all $x \in [- 1, 2]$ …2015 · Numerical
Let $F : R \to R$ be a thrice differentiable function. Suppose that $F (1) = 0, F (3) = - 4$ and $F (x) < 0$ for all $x \in (\frac{1}{2}, 3) .$ Let $f (x) = x F (x)$ …2015 · Multiple correct
The area enclosed by the curves $y = sin x + cosx o l imi t s$ and $y = ∣ cos x - sin x ∣$ over the interval $[0, \frac{π}{2}]$ is2013 · MCQ
Let $S$ be the area of the region enclosed by $y = e^{- x^{2}}$ , $y = 0$ , $x = 0$ , and $x = 1$ ; then2012 · Multiple correct
Let the straight line $x = b$ divide the area enclosed by $y = (1 - x)^{2}, y = 0,$ and $x = 0$ into two parts $R_{1} (0 \leq x \leq b)$ and $R_{2} (b \leq x \leq 1)$ such that $R_{1} - R_{2} = \frac{1}{4} .$ …2011 · MCQ
Let f $:$ $[- 1, 2] \to [0, \infty]$ be a continuous function such that $f (x) = f (1 - x)$ for all $x \in [- 1, 2]$ Let $R_{1} = - 1 \int 2 x f (x) d x,$ …2011 · MCQ
Let $f$ be a real-valued function defined on the interval $(0, \infty)$ by $f (x) = ln x + 0 \int x 1 + sin t d t .$ then which of the following statement(s) is (are) true?2010 · Multiple correct
Consider the polynomial $f (x) = 1 + 2 x + 3 x^{2} + 4 x^{3} .$ Let $s$ be the sum of all distinct real roots of $f (x)$ and let $t = ∣ s ∣ .$ The area bounded by the curve $y = f (x)$ and the lines $x = 0, y = 0$ and $x = t,$ …2010 · MCQ