CUET UG 2025 — General Test Quantitative Reasoning

Converting to base 2:

$$64 = 2^6$$64=26

$$128 = 2^7$$128=27

$$2 = 2^1$$2=21

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Rewriting the first equation:

$$2^{(x+y)} = 64$$2(x+y)=64

$$2^{(x+y)} = 2^6$$2(x+y)=26

Since the bases are equal, the exponents must be equal:

$x + y = 6$x+y=6 ... (Equation 1)

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Rewriting the second equation:

$$128^{(x-y)} = 2$$128(x−y)=2

$$(2^7)^{(x-y)} = 2^1$$(27)(x−y)=21

$$2^{7(x-y)} = 2^1$$27(x−y)=21

Since the bases are equal, the exponents must be equal:

$$7(x - y) = 1$$7(x−y)=1

$x - y = \dfrac{1}{7}$x−y=71​ ... (Equation 2)

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Solving the system of equations by adding Equation 1 and Equation 2:

$$x + y + x - y = 6 + \dfrac{1}{7}$$x+y+x−y=6+71​

$$2x = 6 + \dfrac{1}{7}$$2x=6+71​

$$2x = \dfrac{42}{7} + \dfrac{1}{7}$$2x=742​+71​

$$2x = \dfrac{43}{7}$$2x=743​

$$x = \dfrac{43}{14}$$x=1443​

Therefore, the value of $x = \dfrac{43}{14}$x=1443​.