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Exponential equation: substitute t = a^x and solve the resulting quadratic

This type of question, asked 4 times: CDS 1 2017, CDS 2 2019, CDS 2 2023, CDS 2 2025. All 4 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 1 2017

    29
    The values of xx which satisfy the equation 51+x+51−x=265^{1+x} + 5^{1-x} = 26 are

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  2. Question 2

    CDS 2 2019

    12
    Which of the following pair of numbers is the solution of the equation 3x+2+3−x=103^{x + 2} + 3^{-x} = 10 ?

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  3. Question 3

    CDS 2 2023

    15
    If 3x−1+33−x=63^{x - 1} + 3^{3 - x} = 6, then what is 2x−1+23−x2^{x - 1} + 2^{3 - x} equal to ?

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  4. Question 4

    CDS 2 2025

    6
    If (2+3)x+(2−3)x=2(2+\sqrt{3})^x+(2-\sqrt{3})^x=2, then what is (2+3)x−(2−3)x(2+\sqrt{3})^x-(2-\sqrt{3})^x equal to?

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