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Difficulty: Very hardBinary Operations

A binary operation ⊕\oplus defined on the set of real numbers is given by a⊕b=a+b+kaba \oplus b = a + b + kab, where kk is a non-zero real constant. If the inverse of 33 under this operation is −12-\frac{1}{2}, what is the inverse of 44 under the same operation?

  1. −1223-\frac{12}{23}Answer
  2. B
    −35-\frac{3}{5}
  3. C
    1217\frac{12}{17}
  4. D
    −512-\frac{5}{12}

Answer

−1223-\frac{12}{23}
To find the inverse of 44, first determine the identity element ee from a⊕e=aa \oplus e = a, which gives e=0e = 0. Next, set x⊕x−1=0x \oplus x^{-1} = 0 to get x−1=−x1+kxx^{-1} = -\frac{x}{1 + kx}. Using 3−1=−123^{-1} = -\frac{1}{2}, solve −31+3k=−12-\frac{3}{1 + 3k} = -\frac{1}{2} to find k=53k = \frac{5}{3}. Finally, substituting x=4x = 4 yields 4−1=−41+4(53)=−12234^{-1} = -\frac{4}{1 + 4(\frac{5}{3})} = -\frac{12}{23}.

Step-by-Step Solution

1
Determine the identity element ee of the operation.
e=0e = 0
By definition of identity element, a⊕e=a  ⟹  a+e+kae=a  ⟹  e(1+ka)=0  ⟹  e=0a \oplus e = a \implies a + e + kae = a \implies e(1 + ka) = 0 \implies e = 0 for all valid aa.
2
Derive the general formula for the inverse x−1x^{-1} of an element xx.
x−1=−x1+kxx^{-1} = -\frac{x}{1 + kx}
An element and its inverse yield the identity element under the operation: x⊕x−1=0  ⟹  x+x−1+kxx−1=0  ⟹  x−1(1+kx)=−xx \oplus x^{-1} = 0 \implies x + x^{-1} + kxx^{-1} = 0 \implies x^{-1}(1 + kx) = -x.
3
Use the given inverse of 33 to solve for the constant kk.
k=53k = \frac{5}{3}
Given 3−1=−123^{-1} = -\frac{1}{2}, substitute x=3x = 3 into the inverse formula: −31+3k=−12  ⟹  1+3k=6  ⟹  3k=5  ⟹  k=53-\frac{3}{1 + 3k} = -\frac{1}{2} \implies 1 + 3k = 6 \implies 3k = 5 \implies k = \frac{5}{3}.
4
Calculate the inverse of 44 using k=53k = \frac{5}{3}.
4−1=−12234^{-1} = -\frac{12}{23}
Substitute x=4x = 4 and k=53k = \frac{5}{3} into the inverse formula: 4−1=−41+4(53)=−41+203=−4233=−12234^{-1} = -\frac{4}{1 + 4\left(\frac{5}{3}\right)} = -\frac{4}{1 + \frac{20}{3}} = -\frac{4}{\frac{23}{3}} = -\frac{12}{23}.

Key Concept

Binary Operations: Identity and Inverse Elements with Unknown Parameters
Estimated Time:2m 0s
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Topics

Binary Operations

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