KVPY SX SB Exam 2015 Question Paper

Question : 88 of 120

Marks: +1, -0

The figure shows a portion of the graph

y = 2x- 4x^3

. The line y = c is such that the areas of the regions marked I and II are equal. If a, b are the x-coordinates of A, B respectively, then a + b equals -

$2/√7$
$3/√7$
$4/√7$
$5/√7$

Validate

Solution:

∫↖{b}↙{a}(2x - 4x^3)dx = 2 (b- a)c

(x^2 -x^4)_a^b = 2 (b - a) c

(a + b) (1 - (a^2 + b^2 )) = 2c

(a + b) (1 - (a + b)^2 + 2ab) = 2c

…(1)
again

2x -4x^3 = c

\table a + b + α = 0,\clearly a + b = - α,…(2);ab + (a + b)α = 1/2,ab = α^ 2-1/2,…(3);abα = -c/4,c =-4α(α^2-1/2),…(4)

put value of (a + b), ab, c from eq. (2), (3), (4) in equation (1) and solve it

\{1-α^2+2(α^2-1/2)\}=-8α(α^2-1/2)

1 – α^ 2 + 2α^ 2- 1 = 8α^ 2- 4

α^2 = 8α^ 2- 4

7α^ 2 = 4

α=2/√7

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