Show dy/dx equals expression

4 The parametric equations of a curve are $$x = 1 + \ln ( t - 2 ) , \quad y = t + \frac { 9 } { t } , \quad \text { for } t > 2$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( t ^ { 2 } - 9 \right) ( t - 2 ) } { t ^ { 2 } }\).
2. Find the coordinates of the only point on the curve at which the gradient is equal to 0 .

7 The parametric equations of a curve are $$x = \mathrm { e } ^ { 3 t } , \quad y = t ^ { 2 } \mathrm { e } ^ { t } + 3$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { t ( t + 2 ) } { 3 \mathrm { e } ^ { 2 t } }\).
2. Show that the tangent to the curve at the point \(( 1,3 )\) is parallel to the \(x\)-axis.
3. Find the exact coordinates of the other point on the curve at which the tangent is parallel to the \(x\)-axis.

6 The parametric equations of a curve are $$x = 1 + 2 \sin ^ { 2 } \theta , \quad y = 4 \tan \theta$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sin \theta \cos ^ { 3 } \theta }\).
2. Find the equation of the tangent to the curve at the point where \(\theta = \frac { 1 } { 4 } \pi\), giving your answer in the form \(y = m x + c\).

4 The parametric equations of a curve are $$x = \ln ( 1 - 2 t ) , \quad y = \frac { 2 } { t } , \quad \text { for } t < 0$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 t } { t ^ { 2 } }\).
2. Find the exact coordinates of the only point on the curve at which the gradient is 3 .

5 The parametric equations of a curve are $$x = \cos 2 \theta - \cos \theta , \quad y = 4 \sin ^ { 2 } \theta$$ for \(0 \leqslant \theta \leqslant \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 \cos \theta } { 1 - 4 \cos \theta }\).
2. Find the coordinates of the point on the curve at which the gradient is - 4 .

7
\includegraphics[max width=\textwidth, alt={}, center]{250b4df9-2646-4246-bb6d-2be92bf29598-3_553_689_258_726} The parametric equations of a curve are $$x = 6 \sin ^ { 2 } t , \quad y = 2 \sin 2 t + 3 \cos 2 t$$ for \(0 \leqslant t < \pi\). The curve crosses the \(x\)-axis at points \(B\) and \(D\) and the stationary points are \(A\) and \(C\), as shown in the diagram.

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 3 } \cot 2 t - 1\).
2. Find the values of \(t\) at \(A\) and \(C\), giving each answer correct to 3 decimal places.
3. Find the value of the gradient of the curve at \(B\).

6
\includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-3_453_650_258_744} The diagram shows the curve with parametric equations $$x = 3 \cos t , \quad y = 2 \cos \left( t - \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant t < 2 \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 3 } ( \sqrt { } 3 - \cot t )\).
2. Find the equation of the tangent to the curve at the point where the curve crosses the positive \(y\)-axis. Give the answer in the form \(y = m x + c\).

7
\includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-3_533_698_735_717} The diagram shows the curve with parametric equations $$x = 4 \sin \theta , \quad y = 1 + 3 \cos \left( \theta + \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant \theta < 2 \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(k ( 1 + ( \sqrt { } 3 ) \tan \theta )\) where the exact value of \(k\) is to be determined.
2. Find the equation of the normal to the curve at the point where the curve crosses the positive \(y\)-axis. Give your answer in the form \(y = m x + c\), where the constants \(m\) and \(c\) are exact.

7
\includegraphics[max width=\textwidth, alt={}, center]{77672e56-a268-47b8-ab8b-cd84b4b3de4f-10_551_689_258_726} The parametric equations of a curve are $$x = 6 \sin ^ { 2 } t , \quad y = 2 \sin 2 t + 3 \cos 2 t$$ for \(0 \leqslant t < \pi\). The curve crosses the \(x\)-axis at points \(B\) and \(D\) and the stationary points are \(A\) and \(C\), as shown in the diagram.

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 3 } \cot 2 t - 1\).
2. Find the values of \(t\) at \(A\) and \(C\), giving each answer correct to 3 decimal places.
3. Find the value of the gradient of the curve at \(B\).

6 The parametric equations of a curve are \(x = \frac { 1 } { \cos t } , y = \ln \tan t\), where \(0 < t < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \cos t } { \sin ^ { 2 } t }\).
2. Find the equation of the tangent to the curve at the point where \(y = 0\).

4 The parametric equations of a curve are $$x = \frac { \cos \theta } { 2 - \sin \theta } , \quad y = \theta + 2 \cos \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 - \sin \theta ) ^ { 2 }\).

4 The parametric equations of a curve are $$x = 1 - \cos \theta , \quad y = \cos \theta - \frac { 1 } { 4 } \cos 2 \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).

5 The parametric equations of a curve are $$x = t \mathrm { e } ^ { 2 t } , \quad y = t ^ { 2 } + t + 3$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 t }\).
2. Hence show that the normal to the curve, where \(t = - 1\), passes through the point \(\left( 0,3 - \frac { 1 } { \mathrm { e } ^ { 4 } } \right)\).

3 The parametric equations of a curve are $$x = 3 - \cos 2 \theta , \quad y = 2 \theta + \sin 2 \theta$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).

4 The parametric equations of a curve are $$x = 2 t - \tan t , \quad y = \ln ( \sin 2 t )$$ for \(0 < t < \frac { 1 } { 2 } \pi\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot t\).

8 The parametric equations of a curve are $$x = \tan ^ { 2 } 2 t , \quad y = \cos 2 t$$ for \(0 < t < \frac { 1 } { 4 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { 2 } \cos ^ { 3 } 2 t\).
   \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-10_2716_38_109_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-11_2725_35_99_20}
2. Hence find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 8 } \pi\). Give your answer in the form \(y = m x + c\).

7 The parametric equations of a curve are $$x = 3 \sin 2 t , \quad y = \tan t + \cot t$$ for \(0 < t < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 \sin ^ { 2 } 2 t }\).
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-10_2716_40_109_2009}
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-11_2723_33_99_22}
2. Find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 4 } \pi\). Give your answer in the form \(p y + q x + r = 0\), where \(p , q\) and \(r\) are integers.

13. A curve \(C\) has parametric equations $$x = 6 \cos 2 t , \quad y = 2 \sin t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 }$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \operatorname { cosec } t\), giving the exact value of the constant \(\lambda\).
2. Find an equation of the normal to \(C\) at the point where \(t = \frac { \pi } { 3 }\) Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are simplified surds. The cartesian equation for the curve \(C\) can be written in the form $$x = f ( y ) , \quad - k < y < k$$ where \(\mathrm { f } ( y )\) is a polynomial in \(y\) and \(k\) is a constant.
3. Find \(\mathrm { f } ( y )\).
4. State the value of \(k\).

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = ( \sqrt { } 3 ) \sin 2 t , \quad y = 4 \cos ^ { 2 } t , \quad 0 \leqslant t \leqslant \pi$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k ( \sqrt { } 3 ) \tan 2 t\), where \(k\) is a constant to be determined.
2. Find an equation of the tangent to \(C\) at the point where \(t = \frac { \pi } { 3 }\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
3. Find a cartesian equation of \(C\).

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 6 t - 3 \sin 2 t \quad y = 2 \cos t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The curve meets the \(y\)-axis at 2 and the \(x\)-axis at \(k\), where \(k\) is a constant.

1. State the value of \(k\).
2. Use parametric differentiation to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \operatorname { cosec } t$$ where \(\lambda\) is a constant to be found. The point \(P\) with parameter \(\mathrm { t } = \frac { \pi } { 4 }\) lies on \(C\).
   The tangent to \(C\) at the point \(P\) cuts the \(y\)-axis at the point \(N\).
3. Find the exact \(y\) coordinate of \(N\), giving your answer in simplest form. The region bounded by the curve, the \(x\)-axis and the \(y\)-axis is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
4. 1. Show that the volume of this solid is given by $$\int _ { 0 } ^ { \alpha } \beta ( 1 - \cos 4 t ) d t$$ where \(\alpha\) and \(\beta\) are constants to be found.
   2. Hence, using algebraic integration, find the exact volume of this solid.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = \sqrt { 3 } \sin 2 t \quad y = 4 \cos ^ { 2 } t \quad 0 \leqslant t \leqslant \pi$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k \sqrt { 3 } \tan 2 t\), where \(k\) is a constant to be found.
2. Find an equation of the tangent to \(C\) at the point where \(t = \frac { \pi } { 3 }\) Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants.

4

1. A curve is defined by the parametric equations \(x = 3 \cos 2 \theta , y = 2 \cos \theta\).
   1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { k \cos \theta }\), where \(k\) is an integer.
   2. Find an equation of the normal to the curve at the point where \(\theta = \frac { \pi } { 3 }\).
2. Find the exact value of \(\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \sin ^ { 2 } x \mathrm {~d} x\).

11. A curve is defined parametrically by $$x = 2 \theta + \sin 2 \theta , \quad y = 1 + \cos 2 \theta$$

1. Show that the gradient of the curve at the point with parameter \(\theta\) is \(- \tan \theta\).
2. Find the equation of the tangent to the curve at the point where \(\theta = \frac { \pi } { 4 }\).