Moderate -0.3 This question requires finding the equation of a parallel line (using gradient from the given line), then finding intercepts and calculating a triangle area. While it involves multiple steps, each is routine: rearranging to find gradient, using point-slope form, finding intercepts, and applying the triangle area formula. It's slightly easier than average due to being a standard multi-step coordinate geometry exercise with no conceptual challenges.
A line \(L\) is parallel to the line \(x + 2y = 6\) and passes through the point \((10, 1)\). Find the area of the region bounded by the line \(L\) and the axes. [5]
or B2; must be simplified; or evidence of correct 'stepping' using \((10, 1)\) eg may be on diagram
\((12, 0)\) or ft
M1
or 'when \(y = 0, x = 12'\) etc or using 12 or ft as a limit of integration; intersections must ft from their line or 'stepping' diagram using their gradient; NB the equation of the line is not required; correct intercepts obtained will imply this A1
NB for intersections with axes, if both Ms are not gained, it must be clear which coord is being found eg M0 for intn with \(x\) axis 6 from correct eqn; if the intersections are not explicit, they may be implied by the area calculation eg use of ht = 6 or the correct ft area found
\((0, 6)\) or ft
M1
or integrating to give \(-\frac{1}{4}x^2 + 6x\) or ft their line; allow ft from the given line as well as others for both these intersection Ms
\(36\) [sq units] cao
A1
or B3 www
$x + 2y = k$ $(k \neq 6)$ or $y = -\frac{1}{2}x + c$ $(c \neq 3)$ | M1 | for attempt to use gradients of parallel lines the same; M0 if just given line used
$x + 2y = 12$ or $\left[y = \right] -\frac{1}{2}x + 6$ oe | A1 | or B2; must be simplified; or evidence of correct 'stepping' using $(10, 1)$ eg may be on diagram
$(12, 0)$ or ft | M1 | or 'when $y = 0, x = 12'$ etc or using 12 or ft as a limit of integration; intersections must ft from their line or 'stepping' diagram using their gradient; NB the equation of the line is not required; correct intercepts obtained will imply this A1
| NB for intersections with axes, if both Ms are not gained, it must be clear which coord is being found eg M0 for intn with $x$ axis 6 from correct eqn; if the intersections are not explicit, they may be implied by the area calculation eg use of ht = 6 or the correct ft area found |
$(0, 6)$ or ft | M1 | or integrating to give $-\frac{1}{4}x^2 + 6x$ or ft their line; allow ft from the given line as well as others for both these intersection Ms
$36$ [sq units] cao | A1 | or B3 www | NB A0 if 36 is incorrectly obtained eg after intersection $x = -12$ seen (which earns M0 from correct line)
A line $L$ is parallel to the line $x + 2y = 6$ and passes through the point $(10, 1)$. Find the area of the region bounded by the line $L$ and the axes. [5]
\hfill \mbox{\textit{OCR MEI C1 2011 Q9 [5]}}