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Three unequal resistors in parallel are equivalent to resistance 1 Ω . If two of them are in the ratio 1 : 2 and if no resistance value is fractional, the largest of the three resistances in ohms is
- a)4
- b)6
- c)8
- d)12
Correct answer is option 'B'. Can you explain this answer?
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Three unequal resistors in parallel are equivalent to resistance 1 . ...
To solve this problem, let's assign variables to the resistances. Let the resistances be R1, R2, and R3.
According to the problem, R1, R2, and R3 are connected in parallel and their equivalent resistance is 1 ohm.
Using the formula for the equivalent resistance of resistors in parallel, we have:
1/R = 1/R1 + 1/R2 + 1/R3
Since the resistances are unequal, we can write R2 = 2R1 and R3 = kR1, where k is a constant.
Substituting these values into the equation above, we get:
1/R = 1/R1 + 1/(2R1) + 1/(kR1)
To simplify the equation, we can find a common denominator:
1/R = (2 + 1/k) / (2R1)
Now, we can equate the numerator of the right side of the equation to 1, since the equivalent resistance is given as 1 ohm:
2 + 1/k = 1
Simplifying the equation, we have:
2 + 1/k = 1
Subtracting 2 from both sides, we get:
1/k = -1
Taking the reciprocal of both sides, we have:
k = -1
Since resistance values cannot be negative, we can discard the negative solution. Therefore, k = -1 is not valid.
Now, let's substitute k = 1 into the equation:
1/R = (2 + 1/1) / (2R1)
Simplifying further, we have:
1/R = 3 / (2R1)
Multiplying both sides by 2R1, we get:
2R1 / R = 3
Simplifying the left side of the equation, we have:
2 = 3
This is not a valid solution.
Therefore, the only valid solution is k = 1, which means R3 = R1.
Since the resistances are in the ratio 1:2, R1:R2:R3 = 1:2:2.
Since we know that the sum of the resistances is 1 ohm, we can write:
R1 + R2 + R3 = 1
Substituting the values, we have:
R1 + 2R1 + 2R1 = 1
Simplifying the equation, we get:
5R1 = 1
Dividing both sides by 5, we find:
R1 = 1/5 = 0.2 ohms
Since the largest resistance is R3, we can conclude that R3 = 2R1 = 2(0.2) = 0.4 ohms.
Therefore, the largest of the three resistances is 0.4 ohms, which corresponds to option B.
According to the problem, R1, R2, and R3 are connected in parallel and their equivalent resistance is 1 ohm.
Using the formula for the equivalent resistance of resistors in parallel, we have:
1/R = 1/R1 + 1/R2 + 1/R3
Since the resistances are unequal, we can write R2 = 2R1 and R3 = kR1, where k is a constant.
Substituting these values into the equation above, we get:
1/R = 1/R1 + 1/(2R1) + 1/(kR1)
To simplify the equation, we can find a common denominator:
1/R = (2 + 1/k) / (2R1)
Now, we can equate the numerator of the right side of the equation to 1, since the equivalent resistance is given as 1 ohm:
2 + 1/k = 1
Simplifying the equation, we have:
2 + 1/k = 1
Subtracting 2 from both sides, we get:
1/k = -1
Taking the reciprocal of both sides, we have:
k = -1
Since resistance values cannot be negative, we can discard the negative solution. Therefore, k = -1 is not valid.
Now, let's substitute k = 1 into the equation:
1/R = (2 + 1/1) / (2R1)
Simplifying further, we have:
1/R = 3 / (2R1)
Multiplying both sides by 2R1, we get:
2R1 / R = 3
Simplifying the left side of the equation, we have:
2 = 3
This is not a valid solution.
Therefore, the only valid solution is k = 1, which means R3 = R1.
Since the resistances are in the ratio 1:2, R1:R2:R3 = 1:2:2.
Since we know that the sum of the resistances is 1 ohm, we can write:
R1 + R2 + R3 = 1
Substituting the values, we have:
R1 + 2R1 + 2R1 = 1
Simplifying the equation, we get:
5R1 = 1
Dividing both sides by 5, we find:
R1 = 1/5 = 0.2 ohms
Since the largest resistance is R3, we can conclude that R3 = 2R1 = 2(0.2) = 0.4 ohms.
Therefore, the largest of the three resistances is 0.4 ohms, which corresponds to option B.
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Three unequal resistors in parallel are equivalent to resistance 1 . If two of them are in the ratio 1 : 2 and if no resistance value is fractional, the largest of the three resistances in ohms isa)4b)6c)8d)12Correct answer is option 'B'. Can you explain this answer?
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