第 1 題
1 分Solve the exponential equation \( 9^x = 27^{x-1} \) for \( x \).
IB Diploma Programme (DP) - SL & HL · Mathematics - Analysis and Approaches
科學記數法與指數律：練習題
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第 2 題
1 分The first three terms of an arithmetic sequence are \( 2k+1, 3k, 5k-2 \). Find the value of \( k \).
第 3 題
1 分The polynomial \( P(x) = x^3 - ax^2 + bx - 6 \) has roots \( \alpha, \beta, \gamma \). If \( \alpha + \beta + \gamma = 6 \) and \( \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{3}{2} \), find the value of \( a-b \).
第 4 題
1 分Which of the following expressions is equivalent to \( 2\log_a(x) - \log_a(y) \) for \( x, y, a > 0 \)?
第 5 題
1 分If \( 2^x = 3^y = 12^z \), express \( z \) in terms of \( x \) and \( y \).
第 6 題
2 分Find the 20th term of an arithmetic sequence where the first term is 5 and the common difference is -3.
先自己寫一次答案，再對照解題步驟。
第 7 題
3 分Determine the coefficient of the \( x^3y^2 \) term in the expansion of \( (2x + y)^5 \).
先自己寫一次答案，再對照解題步驟。
第 8 題
6 分Find the modulus and argument of the complex number \( z = \frac{(1-i\sqrt{3})^3}{(2+2i)^2} \).
先自己寫一次答案，再對照解題步驟。
第 9 題
3 分Find the coefficient of \( x^4 \) in the expansion of \( (2x + 1)^7 \).
先自己寫一次答案，再對照解題步驟。
第 10 題
5 分Consider the expansion of the expression \( \left( 3x^2 - \frac{k}{x} \right)^6 \), where \( k \) is a positive constant.
a) Given that the constant term in this expansion is 1215, show that \( k = \sqrt{3} \).
b) Using the value of \( k \) found in part (a), determine the coefficient of the term in \( x^3 \).
先自己寫一次答案，再對照解題步驟。
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