Moderate -0.3 This is a straightforward application of the quotient rule to differentiate a trigonometric quotient, requiring knowledge of standard derivatives (cos x and sin x) and algebraic simplification. It's slightly easier than average because it's a single-step problem with no additional complications, though the simplification requires some care with trigonometric identities.
for reducing to a fraction with \(1 - \sin x\) or \(- \sin x + \sin^2 x + \cos^2 x\) in the numerator
A1
for correct final answer of \(\frac{1}{1-\sin x}\) or \((1 - \sin x)^{-1}\)
Question 6(i)(a)
If candidates use some long drawn-out method to find '\(a\)' instead of the direct route, allow
Answer
Marks
M1
as before, for producing the 3 equations
M1
for any satisfactory method which will/does produce '\(a\)', however involved
A2
for \(a = -2\)
Question 7(ii)
Marks for obtaining this Cartesian equation are not available in part (i).
If part (ii) is done first and then part (i) is attempted using the Cartesian equation, award marks as follow:
Method 1 where candidates differentiate implicitly
Answer
Marks
Guidance
M1
for attempt at implicit differentiation
A1
for \(\frac{dy}{dx} = \frac{2y - 2}{1 - 2x}\)
AEF
M1
for substituting parametric values of \(x\) and \(y\)
A2
for simplifying to \(\frac{2(t+1)}{(t+3)^2}\)
A1
for finish as in original method
Method 2 where candidates manipulate the Cartesian equation to find \(x = \) or \(y=\)
Answer
Marks
M1
for attempt to re-arrange so that either \(y = f(x)\) or \(x = g(y)\)
A1
for correct \(x = \frac{2-2x}{1-2y}\) AEF or \(x = \frac{2-2y}{2-2y}\) AEF
M1
for differentiating as a quotient
A2
for obtaining \(\frac{dy}{dx} = \frac{(1-2x)^2}{(2-2y)^2}\) or \(\frac{(2-2y)^2}{2}\)
A1
for finish as in original method
Question 8(ii)(b)
If definite integrals are used, then
Answer
Marks
Guidance
M2
for \(\int_7^y = \int_8^6\) or equivalent
or M1 for \(\int_y^7 = \int_6^8\) or equivalent
A2
for \(5, 5.0, 5.00 (5.002529)\) with caveat as in main scheme
dep M2
If $y = \frac{\cos x}{1 - \sin x}$ is changed into $y(1 - \sin x) = \cos x$, award
M1 | for clear use of the product rule (though possibly trig differentiation inaccurate)
A1 | for $- y \cos x + (1 - \sin x)\frac{dx}{dx} = -\sin x$ | AEF
B1 | for reducing to a fraction with $1 - \sin x$ or $- \sin x + \sin^2 x + \cos^2 x$ in the numerator
A1 | for correct final answer of $\frac{1}{1-\sin x}$ or $(1 - \sin x)^{-1}$
If $y = \frac{\cos x}{1 - \sin x}$ is changed into $y = \cos x(1 - \sin x)^{-1}$, award
M1 | for clear use of the product rule (though possibly trig differentiation inaccurate)
A1 | for $\left(\frac{dy}{dx}\right) = \cos^2 x(1 - \sin x)^{-2} + (1 - \sin x)^{-1} \cdot (-\sin x)$ | AEF
B1 | for reducing to a fraction with $1 - \sin x$ or $- \sin x + \sin^2 x + \cos^2 x$ in the numerator
A1 | for correct final answer of $\frac{1}{1-\sin x}$ or $(1 - \sin x)^{-1}$
## Question 6(i)(a)
If candidates use some long drawn-out method to find '$a$' instead of the direct route, allow
M1 | as before, for producing the 3 equations
M1 | for any satisfactory method which will/does produce '$a$', however involved
A2 | for $a = -2$
## Question 7(ii)
Marks for obtaining this Cartesian equation are not available in part (i).
If part (ii) is done first and then part (i) is attempted using the Cartesian equation, award marks as follow:
**Method 1 where candidates differentiate implicitly**
M1 | for attempt at implicit differentiation
A1 | for $\frac{dy}{dx} = \frac{2y - 2}{1 - 2x}$ | AEF
M1 | for substituting parametric values of $x$ and $y$
A2 | for simplifying to $\frac{2(t+1)}{(t+3)^2}$
A1 | for finish as in original method
**Method 2 where candidates manipulate the Cartesian equation to find $x = $ or $y=$**
M1 | for attempt to re-arrange so that either $y = f(x)$ or $x = g(y)$
A1 | for correct $x = \frac{2-2x}{1-2y}$ AEF or $x = \frac{2-2y}{2-2y}$ AEF
M1 | for differentiating as a quotient
A2 | for obtaining $\frac{dy}{dx} = \frac{(1-2x)^2}{(2-2y)^2}$ or $\frac{(2-2y)^2}{2}$
A1 | for finish as in original method
## Question 8(ii)(b)
If definite integrals are used, then
M2 | for $\int_7^y = \int_8^6$ or equivalent | or M1 for $\int_y^7 = \int_6^8$ or equivalent
A2 | for $5, 5.0, 5.00 (5.002529)$ with caveat as in main scheme | dep M2
2 Given that $y = \frac { \cos x } { 1 - \sin x }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, simplifying your answer.
\hfill \mbox{\textit{OCR C4 2010 Q2 [4]}}