# 3 - Set Theory and Relations Questions Answers

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number of real values of x satisfying 2log_{2}(x+7x+1)+log_{2}(x+1x-1)=1 is

**Asked By: SHUBHAM VED**

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**Joshi sir comment**

given eq. can be written as log_{2} [(x+7)/(x+1)]^{2} + log_{2} [(x+1)/(x-1)] = 1

or log_{2} [(x+7)^{2}(x+1)/(x+1)^{2}(x-1)] = 1

or log_{2} [(x+7)^{2}/(x^{2}-1)] = 1

or (x+7)^{2}/(x^{2}-1) = 2

or x^{2} + 14x +49 = 2x^{2 }- 2

or x^{2}-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 = sin^{2}y

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]