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Given that [(x-y)^{1/4}(x−y) 1/4 ] = 2 and (3x−2y) 1/2] = 5. What is the minimum possible integral value of y? [x] is equal to greatest integer less than or equal to x.
- a)- 217
- b)- 218
- c)- 219
- d)- 216
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Given that[(x-y)^{1/4}(x−y) 1/4 ] = 2and(3x−2y) 1/2] = 5.W...
Given in the question :
The square bracket represent integer function.
Let us assume x - y = A. (1)
The second equation mentioned was :
Let 3x-2y = B (2)
In the question we were asked for the minimum possible value of y .
Multiplying (1) by and subtracting this from (2) we get :
3x-2y -(3x-3y) = y.
Hence y can be written as B - 3*A
In order to minimise y we must try to make the difference as low as possible and to do this we minimise the value of B to as low as possible and maximise A so the difference gets to its lowest value.
The minimum value of B = minimum value of square of 3x-2y
Since the integral part of the square root of B is 5. This must be in the range of
The minimum value it can take is 5 and hence 3x-2y minimum value is 25.
The maximum value of A = maximum of x-y.
Since the integral part of fourth root of A is 2. This must be in the range of
The maximum value this cannot take is 81. But this takes values greater than 80 and less than 81 also.
So 3A can have a maximum value of 242.
The minimum value of B - 3A is 25 - 242 = -217
The square bracket represent integer function.Let us assume x - y = A. (1)
The second equation mentioned was :
Let 3x-2y = B (2)
In the question we were asked for the minimum possible value of y .
Multiplying (1) by and subtracting this from (2) we get :
3x-2y -(3x-3y) = y.
Hence y can be written as B - 3*A
In order to minimise y we must try to make the difference as low as possible and to do this we minimise the value of B to as low as possible and maximise A so the difference gets to its lowest value.
The minimum value of B = minimum value of square of 3x-2y
Since the integral part of the square root of B is 5. This must be in the range of
The minimum value it can take is 5 and hence 3x-2y minimum value is 25.
The maximum value of A = maximum of x-y.
Since the integral part of fourth root of A is 2. This must be in the range of
The maximum value this cannot take is 81. But this takes values greater than 80 and less than 81 also.
So 3A can have a maximum value of 242.
The minimum value of B - 3A is 25 - 242 = -217
Most Upvoted Answer
Given that[(x-y)^{1/4}(x−y) 1/4 ] = 2and(3x−2y) 1/2] = 5.W...
-2y)(1/3)] = 4, we can solve for x and y using the following steps:
1. Simplify the first equation by multiplying both sides by 4 and dividing by (x-y):
(x-y)(1/4) = 2
4(x-y)(1/4) = 4(2)
x-y = 8
2. Simplify the second equation by multiplying both sides by 3 and dividing by (3x-2y):
(3x-2y)(1/3) = 4
3(3x-2y)(1/3) = 3(4)
3x-2y = 12
3. Solve for y by adding the two equations together:
x-y + 3x-2y = 8 + 12
4x-3y = 20
3y = 4x-20
y = (4/3)x - 20/3
4. Substitute the value of y into one of the original equations to solve for x:
x - [(4/3)x - 20/3] = 8
(3/3)x - (4/3)x + 20/3 = 8
(-1/3)x + 20/3 = 8
(-1/3)x = -4
x = 12
5. Substitute the value of x back into the equation for y to get the final solution:
y = (4/3)(12) - 20/3
y = 8
Therefore, the solution to the system of equations is x = 12 and y = 8.
1. Simplify the first equation by multiplying both sides by 4 and dividing by (x-y):
(x-y)(1/4) = 2
4(x-y)(1/4) = 4(2)
x-y = 8
2. Simplify the second equation by multiplying both sides by 3 and dividing by (3x-2y):
(3x-2y)(1/3) = 4
3(3x-2y)(1/3) = 3(4)
3x-2y = 12
3. Solve for y by adding the two equations together:
x-y + 3x-2y = 8 + 12
4x-3y = 20
3y = 4x-20
y = (4/3)x - 20/3
4. Substitute the value of y into one of the original equations to solve for x:
x - [(4/3)x - 20/3] = 8
(3/3)x - (4/3)x + 20/3 = 8
(-1/3)x + 20/3 = 8
(-1/3)x = -4
x = 12
5. Substitute the value of x back into the equation for y to get the final solution:
y = (4/3)(12) - 20/3
y = 8
Therefore, the solution to the system of equations is x = 12 and y = 8.
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Given that[(x-y)^{1/4}(x−y) 1/4 ] = 2and(3x−2y) 1/2] = 5.What is the minimum possible integral value of y? [x] is equal to greatest integer less than or equal to x.a)- 217b)- 218c)- 219d)- 216Correct answer is option 'A'. Can you explain this answer?
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