Home
Class 12
MATHS
Let f:RtoR be a differentiable function ...

Let `f:RtoR` be a differentiable function such that `f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt`.
`y=f(x)` is

A

The curve y=f(x) passes through the point (1,2)

B

The curve y=f(x) passes through the point (2,-1)

C

The area of the region `{(x,y) in [0,1]xxR:f(x)leylesqrt(1-x^(2))}" is "(pi-2)/(4)`

D

The area of the region `{(x,y)in [0,1]xxR:f(x)leylesqrt(1-x^(2))}" is "(pi-1)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
B, C
`f(x)=1-2x+overset(x)underset(0)inte^(x-1)f(t)dt`
`rArr" "f(x)=1-2x+e^(x)overset(x)underset(0)inte^(-t)f(t)dt" ...(i)"`
Differentiating w.r.t. x, we get
`f'(x)=-2+e^(x)e^(-x)f(x)+e^(x)overset(x)underset(0)inte^(-t)f(t)dt" ...(ii)"`
Subtracting (i) from (ii), we get
`f'(x)-f(x)=-2+f(x)-1+2x`
`rArr" "f'(x)-2f(x)=2x-3`
This is linear differential equation.
`I.F. =e^(int-2dx)=e^(-2x)`
Therefore, solution is
`ycdote^(-2x)=int(2x-3)e^(-2x)dx+c`
`rArr" "ycdote^(-2x)=(2x-3)cdot(e^(-2x))/(-2)-int2xx(e^(-2x))/(-2)dx+c`
`rArr" "ycdote^(-2x)=-((2x-3)e^(-2x))/(2)-(e^(-2x))/(2)+c`
`rArr" "ycdote^(-2x)=(1-x)e^(-2x)+c`
`rArr" "y=(1-x)+c.e^(2x)" ...(iii)"`
Putting x=0 in (i), we get f(0)=1.
So, from above equation, we have
`1=1+c rArr c=0`
Now, (iii) reduces to y=1-x, which passes through point (2,-1).
`"Now,"f(x)leylesqrt(1-x^(2))`
`rArr" "1-xleylesqrt(1-x^(2))`
`1-xley` represents the region above the line `1-x=y and yle sqrt(1-x^(2))` represents the region inside the semicircle (lying above x-axis) `x^(2)+y^(2)=1.`
So, common region is as the shaded one in the following figure :

Area of the shaded region
=Area of quarter of the circle - Area of triangle OAB
`=(1)/(4)xxpi(1)^(2)-(1)/(2)xx1xx1=(pi-2)/(4)`
Promotional Banner

Topper's Solved these Questions

  • AREA

    CENGAGE ENGLISH|Exercise Linkded Comprehension Type|21 Videos

  • AREA

    CENGAGE ENGLISH|Exercise Matrix Match Type|5 Videos

  • AREA

    CENGAGE ENGLISH|Exercise Exercises - Single Correct Answer Type|40 Videos

  • APPLICATIONS OF DERIVATIVES

    CENGAGE ENGLISH|Exercise Comprehension Type|5 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Matrix|4 Videos

Similar Questions

Explore conceptually related problems

Let f:R to R be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt . f(x) increases for

Let f:RtoR be a differntiable function satisfying f(x)=x^(2)+3int_(0)^(x)e^(-t^(3)).f(x-t^(3))dt . Then find f(x) .

Let f(x) be a differentiable function such that f(x)=x^2 +int_0^x e^-t f(x-t) dt then int_0^1 f(x) dx=

Let f be a differentiable function such that f'(x) = f(x) + int_(0)^(2) f(x) dx and f(0) = (4-e^(2))/(3) . Find f(x) .

Let f(x) be a differentiable function satisfying f(x)=int_(0)^(x)e^((2tx-t^(2)))cos(x-t)dt , then find the value of f''(0) .

If f:RrarrR,f(x) is a differentiable function such that (f(x))^(2)=e^(2)+int_(0)^(x)(f(t)^(2)+(f'(t))^(2))dtAAx inR . The values f(1) can take is/are

Let f(x) and g(x) be differentiable functions such that f(x)+ int_(0)^(x) g(t)dt= sin x(cos x- sin x) and (f'(x))^(2)+(g(x))^(2) = 1,"then" f(x) and g (x) respectively , can be

Let f : (0, oo) rarr R be a continuous function such that f(x) = int_(0)^(x) t f(t) dt . If f(x^(2)) = x^(4) + x^(5) , then sum_(r = 1)^(12) f(r^(2)) , is equal to

Let f be a differentiable function satisfying int_(0)^(f(x))f^(-1)(t)dt-int_(0)^(x)(cost-f(t)dt=0 and f((pi)/2)=2/(pi) The value of lim_(x to 0)(cosx)/(f(x)) is equal to where [.] denotes greatest integer function

Let f be a differentiable function satisfying int_(0)^(f(x))f^(-1)(t)dt-int_(0)^(x)(cost-f(t)dt=0 and f((pi)/2)=2/(pi) The value of int_(0)^(pi//2) f(x)dx lies in the interval