Home
Class 12
MATHS
Let A(r) be the area of the region bound...

Let `A_(r)` be the area of the region bounded between the curves `y^(2)=(e^(-kr))x("where "k gt0,r in N)" and the line "y=mx ("where "m ne 0)`, k and m are some constants
`A_(1),A_(2),A_(3),…` are in G.P. with common ratio

A

`e^(-k)`

B

`e^(-2k)`

C

`e^(-4k)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
Solving the two equations, we get
`m^(2)x^(2)=(e^(-kr))x`
`x_(1)=0,x_(2)=(e^(-kr))/(m^(2)).`

`"So, "A_(r)=overset(x_(2))underset(0)int(e^((-kr)/(2))sqrt(x)-mx)dx`
`=(2)/(3)e^(-kr//2)x_(2)^(3//2)-m(x_(2)^(2))/(2)`
`=(2)/(3)e^(-kr//2)(e^(-3kr//2))/(m^(3))-(m)/(2)(e^(-2kr))/(m^(4))=(e^(-2kr))/(6m^(3)).`
`"Now, "(A_(r+1))/(A_(r))=(e^(-2k(r+1)))/(e^(-2kr))=e^(-2k)="constant".`
So, the sequence `A_(1),A_(2),A_(3),...` is in G.P.
`"Sum of n terms "=(e^(-2k))/(6m^(3))(e^(-2nk)-1)/(e^(-2k)-1)=(1)/(6m^(3))(e^(-2nk)-1)/(1-e^(2k))`
`"Sum to infinte terms "=A_(1)(1)/(1-e^(-2k))`
`=(e^(-2k))/(6m^(3))xx(e^(2k))/(e^(2k)-1)=(1)/(6m^(3)(e^(2k-1))).`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • AREA

    CENGAGE|Exercise Exercise (Matrix)|4 Videos

  • AREA

    CENGAGE|Exercise Exercise (Numerical)|16 Videos

  • AREA

    CENGAGE|Exercise Exercise (Multiple)|10 Videos

  • APPLICATIONS OF DERIVATIVES

    CENGAGE|Exercise Question Bank|29 Videos

  • AREA UNDER CURVES

    CENGAGE|Exercise Question Bank|20 Videos

Similar Questions

Explore conceptually related problems

If A_(m) represents the area bounded by the curve y=ln x^(m), the x -axis and the lines x=1 and x= e then A_(m)+mA_(m-1) is

If A_(1) denotes area of the region bounded by the curves C_(1):y=(x-1)e^(x) tangent to C_(1) at (1,0)&y -axis and A_(2) denotes the area of the region bounded by C_(1) and co-ordinate axes in the fourth quadrant then (A) A_(1)>A_(2)(B)A_(1)

Knowledge Check

  • Let A_(r) be the area of the region bounded between the curves y^(2)=(e^(-kr))x("where "k gt0,r in N)" and the line "y=mx ("where "m ne 0) , k and m are some constants lim_(n to oo)Sigma_(i=1)^(n)A_(i)=(1)/(48(e^(2k)-1)) then the value of m is

    A
    3
    B
    1
    C
    2
    D
    4

  • Let a_(1), a_(2), a_(3) …. a_(n) be in G.P. If the area bounded by the curve y^(2) = 4a_(n) X and y^(2) = 4a_(n) (a_(n) - x) be A_(n) , then the sequence A_(1), A_(2), A_(3) ……. A_(n) are in

    A
    A.P
    B
    G.P
    C
    H.P
    D
    none of these

  • Let A_(1) (r in N) be the area of the bounded region whose boundary is defined by (6y^(2) r - x) (6 pi^(2) y - x) = 0 then the value of underset(n to oo)(lim) (sqrt(A_(1) A_(2) A_(3)) + sqrt(A_(2) A_(3) A_(4)) + ….. Terms ) is

    A
    `pi^(9)`
    B
    `(1)/(2) pi^(9)`
    C
    `(1)/(3) pi^(9)`
    D
    `(1)/(pi) pi^(9)`

    • Similar Questions

    Explore conceptually related problems

    Let A_(1),A_(2) denote the area bounded by the curve y=x|x| , axis the lines x=-1,x=1 and the area enlosed between the curves y^(2)=x,y=|x| . Find (A_(1))/(A_(2))

    The line x = (pi)/(4) divides the area of the region bounded by y = sin x, y = cos x and X-axis (0lexle(pi)/(2)) into two regions of areas A_(1) and A_(2) . Then, A_(1) : A_(2) equals

    We know that , if a_(1),a_(2),….a_(n) are in A.P and vice versa . If a_(1),a_(2),…a_(n) are in A.P and vice versa . If a_(1),a_(2)….a_(n) are in A.P with common difference d, then for nay (b gt 0) the numbers b^(a_(1)),b^(a_(2)),b^(a_(3)),....,b^(a_(n)) are in G.P with common ratio b^(d) If a_(1),a_(2),.....a_(n) are positive and in G.P with common ratio r , then for any base b(b gt 0), log_(b) a_(1) , log _(b) a_(2) , ..., log_(b) a_(n) are in A.P with common difference log_(b)r If a,b,c,d are in G.P and a^(x) = b^(y) = c^(z) = d^(v) , then x, y , z , v are in

    We know that , if a_(1),a_(2),….a_(n) are in A.P and vice versa . If a_(1),a_(2),…a_(n) are in A.P and vice versa . If a_(1),a_(2)….a_(n) are in A.P with common difference d, then for nay b ( gt 0) the numbers b^(a_(1)),b^(a_(2)),b^(a_(3)),....,b^(a_(n)) are in G.P with common ratio b^(d) If a_(1),a_(2),.....a_(n) are positive and in G.P with common ratio r , then for any base b(b gt 0), log_(b) a_(1) , log _(b) a_(2) , ..., log_(b) a_(n) are in A.P with common difference log_(b)r If p,q,r are in A.P , then the pth , the qth , the rth terms of any G.P are in

    We know that , if a_(1),a_(2),….a_(n) are in A.P and vice versa . If a_(1),a_(2),…a_(n) are in A.P and vice versa . If a_(1),a_(2)….a_(n) are in A.P with common difference d, then for nay b ( gt 0) the numbers b^(a_(1)),b^(a_(2)),b^(a_(3)),....,b^(a_(n)) are in G.P with common ratio b^(d) If a_(1),a_(2),.....a_(n) are positive and in G.P with common ratio r , then for any base b(b gt 0), log_(b) a_(1) , log _(b) a_(2) , ..., log_(b) a_(n) are in A.P with common difference log_(b)r If x, y , z are respectively the pth , qth and the rth terms of an A.P ..., A.P .., as well as of a G.P , then x^(y-z),y^(z-x).z^(x-y) is equal to

    Home
    Profile