3 - Set Theory and Relations Questions Answers

number of real  values of x satisfying 2log2(x+7x+1)+log2(x+1x-1)=1  is

Asked By: SHUBHAM VED
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Joshi sir comment

given eq. can be written as log2 [(x+7)/(x+1)]2 + log2 [(x+1)/(x-1)] = 1

or log2 [(x+7)2(x+1)/(x+1)2(x-1)] = 1

or log2 [(x+7)2/(x2-1)] = 1

or (x+7)2/(x2-1) = 2

or x2 + 14x +49 = 2x- 2

or x2-14x -51 = 0

or x = -3, 17

since at x = -3 first part will become negative inside root so it will not be a root.

so only 1 root

The number of solutions of :  4x²-4x+2 = sin² y   &     x²+y² ≤ 3    are ?    (please solve it graphically i am getting one solution by using graph but the answer is 2)

Asked By: AMIT DAS
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Joshi sir comment

Given eq. can be written as (2x-1)2 + 1 = sin2y

and max. of siny is 1 so only possible value of x = 1/2

and possible values of y = ±nπ/2  where n is odd integer

but by drawing the circle of 2nd eq. we find that only y = ±π/2 are possible values so answer will be 2

What is the range of the function  y = (e^-x)/ (1+[x]) is ?

Asked By: AMIT DAS
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Joshi sir comment

Domain of the given function is (-∞, -1) υ [0, ∞)

Now at x=0, y=1

and at x=∞, y=0

similarly left limiting value of y = -e

and for x=-∞, y=-∞

so range is (-∞, -e] υ (0, 1]

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