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Difficulty: HardBinary Operations

A binary operation ⋆\star defined on the set of real numbers R\mathbb{R} is given by a⋆b=2a−3b+aba \star b = 2a - 3b + ab. If (2⋆x)⋆3=16(2 \star x) \star 3 = 16, what is the value of xx?

  1. A
    114\frac{11}{4}
  2. −1-1Answer
  3. C
    −15-\frac{1}{5}
  4. D
    −35-\frac{3}{5}

Answer

−1-1
Applying the binary operation rule a⋆b=2a−3b+aba \star b = 2a - 3b + ab step-by-step gives 2⋆x=4−x2 \star x = 4 - x. Operating on this with 33 yields (4−x)⋆3=2(4−x)−3(3)+3(4−x)=11−5x(4 - x) \star 3 = 2(4 - x) - 3(3) + 3(4 - x) = 11 - 5x. Setting 11−5x=1611 - 5x = 16 gives x=−1x = -1.

Step-by-Step Solution

1
Evaluate the inner expression 2⋆x2 \star x using the given definition a⋆b=2a−3b+aba \star b = 2a - 3b + ab.
2⋆x=2(2)−3(x)+(2)(x)=4−3x+2x=4−x2 \star x = 2(2) - 3(x) + (2)(x) = 4 - 3x + 2x = 4 - x
The expression inside the parentheses must be simplified first.
2
Substitute 4−x4 - x into the outer operation (4−x)⋆3(4 - x) \star 3.
(4−x)⋆3=2(4−x)−3(3)+(4−x)(3)=8−2x−9+12−3x=11−5x(4 - x) \star 3 = 2(4 - x) - 3(3) + (4 - x)(3) = 8 - 2x - 9 + 12 - 3x = 11 - 5x
Apply the operation rule with first element a=4−xa = 4 - x and second element b=3b = 3.
3
Equate the resulting expression to 16 and solve for xx.
11−5x=16  ⟹  −5x=5  ⟹  x=−111 - 5x = 16 \implies -5x = 5 \implies x = -1
Solve the linear equation to determine the value of xx.

Key Concept

Non-commutative nested binary operation evaluation
Estimated Time:2m 0s
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Topics

Binary Operations

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