KCET · Maths · Mathematical Reasoning
On the set of all non-zero reals, an operation * is defined as \(a^{*} b=\frac{3 a b}{2}\). In this group, a solution of \(\left(2^{*} x\right) * 3^{-1}=4^{-1}\) is
- A 6
- B 1
- C \(1 / 6\)
- D \(3 / 2\)
Answer & Solution
Correct Answer
(C) \(1 / 6\)
Step-by-step Solution
Detailed explanation
Given binary operation is
\[
a^{*} b=\frac{3 a b}{2}
\]
Now, \(\quad 2 * x=\frac{3}{2} \cdot 2 x=3 x[\) from Eq. (i) \(] \ldots\) (ii) and \((3 x)^{*} \frac{1}{3}=\frac{3}{2} \cdot 3 x \cdot \frac{1}{3}=\frac{3 x}{2}\)
Then, \(\left(2^{*} x\right)^{*} 3^{-1}=4^{-1}\)
\(\begin{aligned} \Rightarrow &(3 x)^{\frac{*}{n}} \frac{1}{3}=\frac{1}{4} & \text { [from Eq. (ii)] } \\ \Rightarrow & \frac{3 x}{2} &=\frac{1}{4} & \text { [from Eq. (iii)] } \\ \Rightarrow & x &=\frac{1}{6} & \end{aligned}\)
\[
a^{*} b=\frac{3 a b}{2}
\]
Now, \(\quad 2 * x=\frac{3}{2} \cdot 2 x=3 x[\) from Eq. (i) \(] \ldots\) (ii) and \((3 x)^{*} \frac{1}{3}=\frac{3}{2} \cdot 3 x \cdot \frac{1}{3}=\frac{3 x}{2}\)
Then, \(\left(2^{*} x\right)^{*} 3^{-1}=4^{-1}\)
\(\begin{aligned} \Rightarrow &(3 x)^{\frac{*}{n}} \frac{1}{3}=\frac{1}{4} & \text { [from Eq. (ii)] } \\ \Rightarrow & \frac{3 x}{2} &=\frac{1}{4} & \text { [from Eq. (iii)] } \\ \Rightarrow & x &=\frac{1}{6} & \end{aligned}\)
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