Moderate -0.3 This question requires understanding that roots give factors, expanding (x+3)(x-2.5)(x-4) with leading coefficient 2, then evaluating f(0) to find the y-intercept. While it involves multiple steps (factorization, expansion, evaluation), each step is straightforward application of standard techniques with no conceptual difficulty or novel insight required. Slightly easier than average due to being a routine procedural question.
The cubic polynomial \(f(x)\) is given by \(f(x) = 2x^3 + ax^2 + bx + c\), where \(a\), \(b\), \(c\) are constants. The graph of \(f(x)\) intersects the \(x\)-axis at the points with coordinates \((-3, 0)\), \((2.5, 0)\) and \((4, 0)\). Find the coordinates of the point where the graph of \(f(x)\) intersects the \(y\)-axis. [5]
Correct use of the Factor Theorem to find at least one factor of \(f(x)\)
M1
AO3
At least two factors of \(f(x)\)
A1
AO3 (accept \((x – 2·5)\) as a factor)
\(f(x) = (x + 3)(x – 4)(2x – 5)\)
A1
AO3 (c.a.o.)
Use of the fact that \(f(x)\) intersects the y-axis when \(x = 0\)
M1
AO3
\(f(x)\) intersects the y-axis at \((0, 60)\)
A1
AO3 (f.t. candidate's expression for \(f(x)\))
Total: [5]
Correct use of the Factor Theorem to find at least one factor of $f(x)$ | M1 | AO3
At least two factors of $f(x)$ | A1 | AO3 (accept $(x – 2·5)$ as a factor)
$f(x) = (x + 3)(x – 4)(2x – 5)$ | A1 | AO3 (c.a.o.)
Use of the fact that $f(x)$ intersects the y-axis when $x = 0$ | M1 | AO3
$f(x)$ intersects the y-axis at $(0, 60)$ | A1 | AO3 (f.t. candidate's expression for $f(x)$)
**Total: [5]**
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The cubic polynomial $f(x)$ is given by $f(x) = 2x^3 + ax^2 + bx + c$, where $a$, $b$, $c$ are constants. The graph of $f(x)$ intersects the $x$-axis at the points with coordinates $(-3, 0)$, $(2.5, 0)$ and $(4, 0)$. Find the coordinates of the point where the graph of $f(x)$ intersects the $y$-axis. [5]
\hfill \mbox{\textit{WJEC Unit 1 Q4 [5]}}