IB Math · Topic practice · IB Mathematics AA HL · Unit 2
Functions (AHL) A mixed exercise set, like a textbook: every problem type this unit covers, shuffled together and getting harder as you go. Work on paper, reveal to check — and hit
All levels Level 1 · Basics Level 2 · Standard Level 3 · Exam
1. Find the remainder when p ( x ) = x 3 + x 2 + 2 x + 1 p(x) = x^3 + x^2 + 2x + 1 p ( x ) = x 3 + x 2 + 2 x + 1 is divided by ( x − 1 ) (x - 1) ( x − 1 ) .
2. Find the sum of the roots of the equation 2 x 3 + 3 x 2 + x − 2 = 0 2x^3 + 3x^2 + x - 2 = 0 2 x 3 + 3 x 2 + x − 2 = 0 .
3. For which value of k k k does x 2 + 4 x + k = 0 x^2 + 4x + k = 0 x 2 + 4 x + k = 0 have a repeated root?
4. Given that ( x − 1 ) (x - 1) ( x − 1 ) is a factor of p ( x ) = x 3 + k x + 1 p(x) = x^3 + kx + 1 p ( x ) = x 3 + k x + 1 , find the value of k k k .
5. The equation x 2 + x + 1 = 0 x^2 + x + 1 = 0 x 2 + x + 1 = 0 has roots α \alpha α and β \beta β . Find α 2 + β 2 \alpha^2 + \beta^2 α 2 + β 2 .
6. f ( x ) = x − 6 x + 1 f(x) = \dfrac{x - 6}{x + 1} f ( x ) = x + 1 x − 6 . Find f − 1 ( 2 ) f^{-1}(2) f − 1 ( 2 ) .
7. Find the remainder when p ( x ) = x 3 + 3 x 2 + 4 x − 4 p(x) = x^3 + 3x^2 + 4x - 4 p ( x ) = x 3 + 3 x 2 + 4 x − 4 is divided by ( x − 1 ) (x - 1) ( x − 1 ) .
8. Find the sum of the roots of the equation 2 x 3 + 5 x 2 + 2 x + 7 = 0 2x^3 + 5x^2 + 2x + 7 = 0 2 x 3 + 5 x 2 + 2 x + 7 = 0 .
9. For which value of k k k does x 2 + 30 x + k = 0 x^2 + 30x + k = 0 x 2 + 30 x + k = 0 have a repeated root?
10. Given that ( x + 1 ) (x + 1) ( x + 1 ) is a factor of p ( x ) = x 3 + k x − 2 p(x) = x^3 + kx - 2 p ( x ) = x 3 + k x − 2 , find the value of k k k .
11. The equation x 2 + 2 x + 5 = 0 x^2 + 2x + 5 = 0 x 2 + 2 x + 5 = 0 has roots α \alpha α and β \beta β . Find α 2 + β 2 \alpha^2 + \beta^2 α 2 + β 2 .
12. f ( x ) = x − 6 x + 5 f(x) = \dfrac{x - 6}{x + 5} f ( x ) = x + 5 x − 6 . Find f − 1 ( 3 ) f^{-1}(3) f − 1 ( 3 ) .
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