A-LevelMathematicsIntegration9709/21 · May/June 2024 · Q79 Marks
The polynomial p(x) is defined by p(x) = 9x³ + 6x² +12x+k, where k is a constant.
- (a)Find the quotient when p(x) is divided by (3x+2) and show that the remainder is (k-8).[3]
- (b)It is given that ∫´° 6 p(x) 3x+2 dx = a+ln64, where a is an integer. Find the values of a and k.[6]
Mark scheme
Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.
- [M1] Carry out division at least as far as 3x² + n₁
- [A1] Obtain quotient 3x² + 4
- [A1] Confirm remainder is k-8
- [*M1] Integrate to obtain at least a term in x³ and term of form n₂ ln(3x + 2)
- [A1] Obtain x³ +4x+(k-8)ln(3x+2)
- [DM1] Apply limits correctly to expression with three terms
- [A1] Obtain a = 235
- [DM1] Equate logarithm term to ln 64 and apply appropriate logarithm properties
- [A1] Obtain k=17
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