Standard +0.3 This is a straightforward multi-step calculus problem requiring students to use the condition that dy/dx = 0 at stationary points to find a and b, then integrate to find the original function and evaluate k. While it involves several steps (forming equations, solving simultaneously, integrating with a constant, applying boundary conditions), each step uses standard AS-level techniques with no novel insight required. Slightly above average difficulty due to the number of steps and algebraic manipulation needed.
A curve is such that \(\frac{\text{d}y}{\text{d}x} = 3x^2 + ax + b\). The curve has stationary points at \((-1, 2)\) and \((3, k)\). Find the values of the constants \(a\), \(b\) and \(k\). [8]
FT correct integration for their a,b (numerical a, b)
2=−1−3+9+c
M1
Sub x = ‒1, y = 2 into their integrated f(x). c must be present
c = ‒3
A1
FT from their f(x)
f ( 3 )=k → k =27−27−27−3
M1
Sub x = 3, y = k into their integrated f(x) (Allow c omitted)
k =−30
A1
8
Answer
Marks
Guidance
Question
Answer
Marks
Question 8:
8 | f′ ( −1 )=0 → 3−a+b=0 f′ ( 3 )=0 → 27+3a+b=0 | M1 | Stationary points at x = ‒1 & x = 3 gives sim. equations in a & b
a = ‒6 | A1 | Solve simultaneous equation
b = ‒9 | A1
( )=3x2 ( )= (+c )
Hence f′ x −6x−9 → f x x3 −3x2 −9x | B1 | FT correct integration for their a,b (numerical a, b)
2=−1−3+9+c | M1 | Sub x = ‒1, y = 2 into their integrated f(x). c must be present
c = ‒3 | A1 | FT from their f(x)
f ( 3 )=k → k =27−27−27−3 | M1 | Sub x = 3, y = k into their integrated f(x) (Allow c omitted)
k =−30 | A1
8
Question | Answer | Marks | Guidance
A curve is such that $\frac{\text{d}y}{\text{d}x} = 3x^2 + ax + b$. The curve has stationary points at $(-1, 2)$ and $(3, k)$. Find the values of the constants $a$, $b$ and $k$. [8]
\hfill \mbox{\textit{CAIE P1 2019 Q8 [8]}}