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Question

Let y=y(x) be a curve passing through the point (1,1) and satisfying dydx+(x21)(y21)xy=0. If the curve passes through the point (2,k), then the largest value of |[k]| is
(Here, [.] represents the greatest integer function)

A
2
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B
1
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C
5
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D
7
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Solution

The correct option is A 2
dydx+(x21)(y21)xy=0
yy21dy=x21xdx
yy21dy=x21xx21dx

Let y21=t22ydy=2tdt
dt=xx21dx+1xx21dx
t=x21+sec1x+C
y21=x21+sec1x+C

The curve passes through (1,1).
C=0
Hence, the curve is y21=x21+sec1x
Also, (2,k) lies on the curve.
k21=1+π4
k2=(π41)2+1
k2=m+1 for some m(0,1)
k=±m+1
[k]=1 or 2
The largest value of |[k]| is 2

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