If (3a/2b)2x-4 = (2b/3a)2x-4 , for some a and b , then the value of x is

Option d: 2. Solution: Let us define y = 3a/2b . Assuming y ≠ 0 and y ≠ ± 1 , the given equation: (3a/2b)2x-4 = (2b/3a)2x-4 can be written as: y2x-4 = (1/y)2x-4 Using the laws of exponents ( 1/y = y-1 ): y2x-4 = (y-1)2x-4 y2x-4 = y-(2x-4) Since the bases are the same, we can equate the exponents: $ 2x-4 = -(2x-4) 重 2x-4 = -2x+4 4x = 8 x = 2 Thus, the mathematically correct value of x is 2 . This corresponds to Option D. Note: The textbook answer key designates Option B ( 6 ) as the correct answer. This is a typographical error. If one of the exponents was written differently, such as the RHS exponent being (x-14) instead of (2x-4) , we would have 2x-4 = -(x-14) 3x = 18 x = 6 (Option B). However, for the given equation, the correct mathematical solution is x=2 (Option D). Hence, Option D is the correct answer.

If $\left(\frac{3a}{2b}\right)^{2x-4} = \left(\frac{2b}{3a}\right)^{2x-4}$, for some $a$ and $b$, then the value of $x$ is

The correct answer is Option d: 2. Let us define $y = \frac{3a}{2b}$. Assuming $y \neq 0$ and $y \neq \pm 1$, the given equation: $$\left(\frac{3a}{2b}\right)^{2x-4} = \left(\frac{2b}{3a}\right)^{2x-4}$$ can be written as: $$y^{2x-4} = \left(\frac{1}{y}\right)^{2x-4}$$ Using the laws of exponents ($\frac{1}{y} = y^{-1}$): $$y^{2x-4} = \left(y^{-1}\right)^{2x-4}$$ $$y^{2x-4} = y^{-(2x-4)}$$ Since the bases are the same, we can equate the exponents: $$2x-4 = -(2x-4)$重 $$2x-4 = -2x+4$$ $$4x = 8 \implies x = 2$$ Thus, the mathematically correct value of $x$ is $2$. This corresponds to **Option D**. Note: The textbook answer key designates **Option B** ($6$) as the correct answer. This is a typographical error. If one of the exponents was written differently, such as the RHS exponent being $(x-14)$ instead of $(2x-4)$, we would have $2x-4 = -(x-14) \implies 3x = 18 \implies x = 6$ (Option B). However, for the given equation, the correct mathematical solution is $x=2$ (Option D). Hence, **Option D** is the correct answer.

Ratio, Proportion, Indices, LogarithmMCQPYQ Dec. 21Question 884 of 305

All Questions

If $\displaystyle \left(\frac{3a}{2b}\right)^{2x-4} = \left(\frac{2b}{3a}\right)^{2x-4}$ , for some $\displaystyle a$ and $\displaystyle b$ , then the value of $\displaystyle x$ is

Options

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option d — 2

All Options:

Detailed Solution & Explanation

Let us define

\displaystyle y = \frac{3a}{2b}

. Assuming

\displaystyle y \neq 0

and

\displaystyle y \neq \pm 1

, the given equation:

\left(\frac{3a}{2b}\right)^{2x-4} = \left(\frac{2b}{3a}\right)^{2x-4}

can be written as:

y^{2x-4} = \left(\frac{1}{y}\right)^{2x-4}

Using the laws of exponents (

\displaystyle \frac{1}{y} = y^{-1}

y^{2x-4} = \left(y^{-1}\right)^{2x-4}

y^{2x-4} = y^{-(2x-4)}

Since the bases are the same, we can equate the exponents:

\displaystyle &#x27; in math mode at position 15: 2x-4 = -(2x-4)

̲重<br>" style="color:#cc0000">2x-4 = -(2x-4)

\displaystyle 重&lt;br&gt;</span>2x-4 = -2x+4<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>&lt;</mo><mi>b</mi><mi>r</mi><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">&lt;br&gt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span></span></span></span></span>4x = 8 \implies x = 2

\displaystyle <br>Thus, the mathematically correct value of

\displaystyle is

\displaystyle . This corresponds to **Option D**.<br>Note: The textbook answer key designates **Option B** (

\displaystyle ) as the correct answer. This is a typographical error. If one of the exponents was written differently, such as the RHS exponent being

(x-14)

\displaystyle instead of

(2x-4)

\displaystyle , we would have

2x-4 = -(x-14) \implies 3x = 18 \implies x = 6

\displaystyle (Option B). However, for the given equation, the correct mathematical solution is

x=2$ (Option D).
Hence, **Option D** is the correct answer.

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

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If (3a2b)2x−4=(2b3a)2x−4\displaystyle \left(\frac{3a}{2b}\right)^{2x-4} = \left(\frac{2b}{3a}\right)^{2x-4}(2b3a​)2x−4=(3a2b​)2x−4, for some a\displaystyle aa and b\displaystyle bb, then the value of x\displaystyle xx is