Challenging +1.2 This is a multi-step integration problem requiring finding a tangent equation, setting up a composite area calculation with both integration and triangle geometry, and working with algebraic fractions. While it involves several techniques (differentiation, integration of x^(-2), coordinate geometry), each step follows standard AS-level procedures without requiring novel insight. The 'exact area' requirement adds minor algebraic complexity but this is typical for higher-mark AS questions.
\includegraphics{figure_5}
Figure 5 shows a sketch of part of the curve \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\).
The point \(A(4, \frac{7}{2})\) lies on C. The line \(l\) is the tangent to C at the point A.
The region \(R\), shown shaded in figure 5 is bounded by the line \(l\), the curve C, the line with equation \(x = 1\) and the \(x\)-axis.
Find the exact area of \(R\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Finding equation of line \(l\): \(y = \frac{7}{4}x - \frac{7}{2}\), intersection with x-axis: \(x = 2\)
M1, A1, M1, A1
Attempts to differentiate \(y = 2x + 8x^{-2} - 5\) with at least one index reduced by one; Correct derivative obtained; Correct method to find equation of tangent; Find correct intercept with x-axis
Area of region: \(\frac{5}{2}\)
M1, M1, A1, M1, A1, M1, A1
Complete strategy of finding the areas under the curve between \(x = 1\) and \(x = 4\) and area of triangle; Method for integration; Correct integration; Attempts to find a value for the area of shaded region; Correct value for the area of shaded region
Finding equation of line $l$: $y = \frac{7}{4}x - \frac{7}{2}$, intersection with x-axis: $x = 2$ | M1, A1, M1, A1 | Attempts to differentiate $y = 2x + 8x^{-2} - 5$ with at least one index reduced by one; Correct derivative obtained; Correct method to find equation of tangent; Find correct intercept with x-axis
Area of region: $\frac{5}{2}$ | M1, M1, A1, M1, A1, M1, A1 | Complete strategy of finding the areas under the curve between $x = 1$ and $x = 4$ and area of triangle; Method for integration; Correct integration; Attempts to find a value for the area of shaded region; Correct value for the area of shaded region
\includegraphics{figure_5}
Figure 5 shows a sketch of part of the curve $y = 2x + \frac{8}{x^2} - 5$, $x > 0$.
The point $A(4, \frac{7}{2})$ lies on C. The line $l$ is the tangent to C at the point A.
The region $R$, shown shaded in figure 5 is bounded by the line $l$, the curve C, the line with equation $x = 1$ and the $x$-axis.
Find the exact area of $R$.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
\hfill \mbox{\textit{Edexcel AS Paper 1 Q15}}