Convert to Cartesian (sin/cos identities)

3. The curve \(C\) has parametric equations $$x = 3 + 2 \sin t \quad y = \frac { 6 } { 7 + \cos 2 t } \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$

1. Show that \(C\) has Cartesian equation $$y = \frac { 12 } { ( 7 - x ) ( 1 + x ) } \quad p \leqslant x \leqslant q$$ where \(p\) and \(q\) are constants to be found.
2. Hence, find a Cartesian equation for \(C\) in the form $$y = \frac { a } { x + b } + \frac { c } { x + d } \quad p \leqslant x \leqslant q$$ where \(a , b , c\) and \(d\) are constants.

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

1. Show that $$x + y = 2 \sqrt { 3 } \cos t$$
2. Show that a cartesian equation of \(C\) is $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where \(a\) and \(b\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-22_2673_1948_107_118}

5 A curve \(C\) has parametric equations $$x = \cos t , \quad y = 3 + 2 \cos 2 t , \quad \text { where } 0 \leqslant t \leqslant \pi$$

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and hence show that the gradient at any point on \(C\) cannot exceed 8 .
2. Show that all points on \(C\) satisfy the cartesian equation \(y = 4 x ^ { 2 } + 1\).
3. Sketch the curve \(y = 4 x ^ { 2 } + 1\) and indicate on your sketch the part which represents \(C\).

3 The parametric equations of a curve are $$x = \cos 2 \theta , \quad y = \sin \theta \cos \theta \quad \text { for } 0 \leqslant \theta < \pi$$ Show that the cartesian equation of the curve is \(x ^ { 2 } + 4 y ^ { 2 } = 1\).
Sketch the curve.

4 Find the first three terms in the binomial expansion of \(\sqrt { 4 + x }\) in ascending powers of \(x\).
State the set of values of \(x\) for which the expansion is valid.
show that \(\frac { y - 2 } { y + 1 } = A \mathrm { e } ^ { x ^ { 3 } }\), where \(A\) is a constant.
(ii) Hence, given that \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } ( y - 2 ) ( y + 1 ) ,$$ 6 Solve the equation \(\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\). Section B (36 marks)
7 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, \(\mathrm { O } y\) due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.

9 Two astronomers wish to model the path of motion of a particle in two dimensions.
Experimental results show that the position of the particle can be found using the parametric equations $$x = 2 \cos \theta - \sin \theta + 2 \quad y = \cos \theta + 2 \sin \theta - 1 \quad \left( 0 \leq \theta \leq 360 ^ { \circ } \right)$$ One astronomer uses trigonometry.

1. Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants to be determined. Show also that, for the same values of \(R\) and \(\alpha\), $$\cos \theta + 2 \sin \theta = R \sin ( \theta + \alpha )$$
2. Hence, or otherwise, show that the path of particle may be written in the form $$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 5$$ Describe the path of the particle. The second astronomer sets up a first order differential equation with the condition that \(x = 4\) when \(y = 0\).
3. Verify that the point with parameter \(\theta = 0\) has coordinates \(( 4,0 )\).
4. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Deduce that \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x - 2 } { y + 1 }$$
5. Solve this differential equation, using the condition that \(y = 0\) when \(x = 4\). Hence show that the two solutions give the same cartesian equation for the path of particle.

4 The parametric equations of a curve are $$x = \cos 2 \theta , \quad y = \sin \theta \cos \theta \quad \text { for } 0 \leqslant \theta < \pi$$ Show that the cartesian equation of the curve is \(x ^ { 2 } + 4 y ^ { 2 } = 1\).
Sketch the curve.

8 A curve is defined by the parametric equations $$x = \sin ^ { 2 } \theta , \quad y = 4 \sin \theta - \sin ^ { 3 } \theta ,$$ where \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 - 3 \sin ^ { 2 } \theta } { 2 \sin \theta }\).
2. Find the coordinates of the point on the curve at which the gradient is 2 .
3. Show that the curve has no stationary points.
4. Find a cartesian equation of the curve, giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

3 The parametric equations of a curve are $$x = \sin \theta , \quad y = \sin 2 \theta , \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi .$$

1. Find the exact value of the gradient of the curve at the point where \(\theta = \frac { 1 } { 6 } \pi\).
2. Show that the cartesian equation of the curve is \(y ^ { 2 } = 4 x ^ { 2 } - 4 x ^ { 4 }\).

14. A curve \(C\) has parametric equations $$x = 1 - \cos t , \quad y = 2 \cos 2 t , \quad 0 \leq t < \pi$$ a. Show that the cartesian equation of the curve can be written as \(y = k ( 1 - x ) ^ { 2 } - 2\) where \(k\) is an integer.
b. i. Sketch the curve C .
ii. Explain briefly why C does not include all points of \(y = k ( 1 - x ) ^ { 2 } - 2 , x \in \mathbb { R }\). The line with equation \(y = k - x\), where \(k\) is a constant, intersects C at two distinct points.
(c) State the range of values of \(k\), writing your answer in set notation.

16. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations \(x = - 3 + 6 \sin \theta , \quad y = 9 \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 4 }\) where \(\theta\) is a parameter.
a. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\) The line \(l\) is normal to \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 6 }\) b. Show that an equation for \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 9 } { 2 }$$ c. The cartesian equation for the curve \(C\) can be written in the form $$y = a - \frac { 1 } { 2 } ( x + b ) ^ { 2 }$$ where \(a\) and \(b\) are integers to be found. The straight line with equation $$y = \frac { 1 } { 3 } x + k$$ where \(k\) is a constant intersects \(C\) at two distinct points.
d. Find the range of possible values for \(k\).

1. A curve \(C\) has parametric equations

$$x = 3 + 2 \sin t , \quad y = 4 + 2 \cos 2 t , \quad 0 \leqslant t < 2 \pi$$

1. Show that all points on \(C\) satisfy \(y = 6 - ( x - 3 ) ^ { 2 }\)
2. 1. Sketch the curve \(C\).
   2. Explain briefly why \(C\) does not include all points of \(y = 6 - ( x - 3 ) ^ { 2 } , \quad x \in \mathbb { R }\) The line with equation \(x + y = k\), where \(k\) is a constant, intersects \(C\) at two distinct points.
3. State the range of values of \(k\), writing your answer in set notation.

14. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \cos \left( t + \frac { \pi } { 6 } \right) , y = 2 \sin t , \quad 0 < t \leqslant 2 \pi$$ Show that a Cartesian equation of \(C\) can be written in the form $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where \(a\) and \(b\) are integers to be found.

6 The parametric equations of a circle are \(x = 2 \cos \theta - 3\) and \(y = 2 \sin \theta + 1\).
Determine the cartesian equation of the circle in the form \(( \mathrm { x } - \mathrm { a } ) ^ { 2 } + ( \mathrm { y } - \mathrm { b } ) ^ { 2 } = \mathrm { k }\), where \(a , b\) and \(k\) are integers.

5 A curve is defined by the parametric equations $$x = 2 \cos \theta , \quad y = 3 \sin 2 \theta$$

1. 1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = a \sin \theta + b \operatorname { cosec } \theta$$ where \(a\) and \(b\) are integers.
   2. Find the gradient of the normal to the curve at the point where \(\theta = \frac { \pi } { 6 }\).
2. Show that the cartesian equation of the curve can be expressed as $$y ^ { 2 } = p x ^ { 2 } \left( 4 - x ^ { 2 } \right)$$ where \(p\) is a rational number.

2. The curve \(C\) is described by the parametric equations $$x = 3 \cos t , \quad y = \cos 2 t , \quad 0 \leq t \leq \pi .$$

1. Find a cartesian equation of the curve \(C\).
2. Draw a sketch of the curve \(C\).

7. \begin{figure}[h] \end{figure} Figure 2 shows the curve with parametric equations $$x = - 1 + 4 \cos \theta , \quad y = 2 \sqrt { 2 } \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The point \(P\) on the curve has coordinates \(( 1 , \sqrt { 6 } )\).

1. Find the value of \(\theta\) at \(P\).
2. Show that the normal to the curve at \(P\) passes through the origin.
3. Find a cartesian equation for the curve.
   7. continued

6 A curve is given by the parametric equations $$x = \cos \theta \quad y = \sin 2 \theta$$

1. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} \theta }\).
      (2 marks)
   2. Find the gradient of the curve at the point where \(\theta = \frac { \pi } { 6 }\).
2. Show that the cartesian equation of the curve can be written as $$y ^ { 2 } = k x ^ { 2 } \left( 1 - x ^ { 2 } \right)$$ where \(k\) is an integer.

1 A curve is defined by the parametric equations $$x = \cos \theta \text { and } y = \sin \theta \quad \text { where } 0 \leq \theta \leq 2 \pi$$ Which of the options shown below is a Cartesian equation for this curve?
Circle your answer. $$\frac { y } { x } = \tan \theta \quad x ^ { 2 } + y ^ { 2 } = 1 \quad x ^ { 2 } - y ^ { 2 } = 1 \quad x ^ { 2 } y ^ { 2 } = 1$$

\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 4\cos\left(t + \frac{\pi}{6}\right), \quad y = 2\sin t, \quad 0 \leq t < 2\pi$$

1. [(a)] Show that $$x + y = 2\sqrt{3}\cos t$$ \hfill [3]
2. [(b)] Show that a cartesian equation of \(C\) is $$(x + y)^2 + ay^2 = b$$ where \(a\) and \(b\) are integers to be determined. \hfill [2]

The parametric equations of a curve are \(x = 2 + 5\cos\theta\) and \(y = 1 + 5\sin\theta\), where \(0 \leq \theta < 2\pi\).

1. Determine the cartesian equation of the curve. [3]
2. Hence or otherwise, find the equation of the tangent to the curve at the point \((5, -3)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers to be determined. [4]

The equation of a curve \(C\) is given by the parametric equations $$x = \cos 2\theta, \quad y = \cos\theta.$$

1. Find the Cartesian equation of \(C\). [2]
2. Show that the line \(x - y + 1 = 0\) meets \(C\) at the point \(P\), where \(\theta = \frac{\pi}{3}\), and at the point \(Q\), where \(\theta = \frac{\pi}{2}\). Write down the coordinates of \(P\) and \(Q\). [5]
3. Determine the equations of the tangents to \(C\) at \(P\) and \(Q\). Write down the coordinates of the point of intersection of the two tangents. [7]