A-LevelMathematicsDifferentiation9709/32 · Feb/Mar 2024 · Q67 Marks
The equation of a curve is 2y² + 3xy + x = x².
- (a)Show that dy/dx = (2x - 3y - 1) / (4y + 3x).[4]
- (b)Hence show that the curve does not have a tangent that is parallel to the x-axis.[3]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.
- [B1] State or imply 4y dy/dx as the derivative of 2y²
- [B1] State or imply 3y + 3x dy/dx as the derivative of 3xy
- [M1] Complete the differentiation, all 4 terms, isolate dy/dx terms on LHS or bracket dy/dx terms and solve for dy/dx
- [A1] Obtain dy/dx = (2x - 3y - 1) / (4y + 3x)
- [M1*] Equate numerator to zero, obtaining 2x = 3y + 1 or 3y = 2x -1 and form equation in x only or y only from 2y² +3xy + x = x²
- [DM1] Obtain (2x−1)² = -x² or a 3 term quadratic in one unknown and try to solve. If errors in quadratic formulation allow solution, applying usual rules for solution of quadratic equation, and allow M1
- [A1] Conclude that the equation has no [real] roots
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