1.02n Sketch curves: simple equations including polynomials

7

1. Given that \(z = \frac { \sin x } { \cos x }\), use the quotient rule to show that \(\frac { \mathrm { d } z } { \mathrm {~d} x } = \sec ^ { 2 } x\).
2. Sketch the curve with equation \(y = \sec x\) for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
3. The region \(R\) is bounded by the curve \(y = \sec x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid formed when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer to three significant figures.

7. (a) Sketch on the same diagram the graphs of \(y = 4 a ^ { 2 } - x ^ { 2 }\) and \(y = | 2 x - a |\), where \(a\) is a positive constant. Show, in terms of \(a\), the coordinates of any points where each graph meets the coordinate axes.
(b) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$

6. The function f is defined by $$\mathrm { f } ( x ) \equiv 3 - x ^ { 2 } , \quad x \in \mathbb { R } , \quad x \geq 0 .$$

1. State the range of f.
2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$\mathrm { g } ( x ) \equiv \frac { 8 } { 3 - x } , \quad x \in \mathbb { R } , \quad x \neq 3 .$$
4. Evaluate \(\mathrm { fg } ( - 3 )\).
5. Solve the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ( x ) .$$

5. (a) Sketch the graph of \(y = 2 + \sec \left( x - \frac { \pi } { 6 } \right)\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\). Show on your sketch the coordinates of any turning points and the equations of any asymptotes.
(b) Find, in terms of \(\pi\), the \(x\)-coordinates of the points where the graph crosses the \(x\)-axis.

4. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\). The graph has a minimum at \(\left( \frac { \pi } { 2 } , - 1 \right)\), a maximum at \(\left( \frac { 3 \pi } { 2 } , - 5 \right)\) and an asymptote with equation \(x = \pi\).

1. Showing the coordinates of any stationary points, sketch the graph of \(y = | \mathrm { f } ( x ) |\). Given that $$f : x \rightarrow a + b \operatorname { cosec } x , \quad x \in \mathbb { R } , \quad 0 < x < 2 \pi , \quad x \neq \pi ,$$
2. find the values of the constants \(a\) and \(b\),
3. find, to 2 decimal places, the \(x\)-coordinates of the points where the graph of \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.

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. The curve \(C\) has equation \(y = \frac { x } { 4 + x ^ { 2 } }\).

1. Use calculus to find the coordinates of the turning points of \(C\). Using the result \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 x \left( x ^ { 2 } - 12 \right) } { \left( 4 + x ^ { 2 } \right) ^ { 3 } }\), or otherwise,
2. determine the nature of each of the turning points.
3. Sketch the curve \(C\).

7 A projectile P travels in a vertical plane over level ground. Its position vector \(\mathbf { r }\) at time \(t\) seconds after projection is modelled by $$\mathbf { r } = \binom { x } { y } = \binom { 0 } { 5 } + \binom { 30 } { 40 } t - \binom { 0 } { 5 } t ^ { 2 }$$ where distances are in metres and the origin is a point on the level ground.

1. Write down
   (A) the height from which P is projected,
   (B) the value of \(g\) in this model.
2. Find the displacement of P from \(t = 3\) to \(t = 5\).
3. Show that the equation of the trajectory is $$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 }$$

1 A girl throws a small stone with initial speed \(14 \mathrm {~ms} { } ^ { 1 }\) at an angle of \(60 ^ { \circ }\) to the horizontal from a point 1 m above the ground. She throws the stone directly towards a vertical wall of height 6 m standing on horizontal ground. The point O is on the ground directly below the point of projection, as shown in Fig. 8. Air resistance is negligible. \begin{figure}[h] \end{figure}

1. Write down an expression in terms of \(t\) for the horizontal displacement of the stone from O , \(t\) seconds after projection. Find also an expression for the height of the stone above O at this time. The stone is at the top of its trajectory when it passes over the wall.
2. (A) Find the time it takes for the stone to reach its highest point.
   (B) Calculate the distance of O from the base of the wall.
   (C) Show that the stone passes over the wall with 2.5 m clearance.
3. Find the cartesian equation of the trajectory of the stone referred to the horizontal and vertical axes, \(\mathrm { O } x\) and \(\mathrm { O } y\). There is no need to simplify your answer. The girl now moves away a further distance \(d \mathrm {~m}\) from the wall. She throws a stone as before and it just passes over the wall.
4. Calculate \(d\).

4 A curve has equation $$y = \frac { 6 x } { x - 1 }$$

1. Write down the equations of the two asymptotes to the curve.
2. Sketch the curve and the two asymptotes.
3. Solve the inequality $$\frac { 6 x } { x - 1 } < 3$$

8 A curve has equation \(y ^ { 2 } = 12 x\).

1. Sketch the curve.
2. 1. The curve is translated by 2 units in the positive \(y\) direction. Write down the equation of the curve after this translation.
   2. The original curve is reflected in the line \(y = x\). Write down the equation of the curve after this reflection.
3. 1. Show that if the straight line \(y = x + c\), where \(c\) is a constant, intersects the curve \(y ^ { 2 } = 12 x\), then the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + ( 2 c - 12 ) x + c ^ { 2 } = 0$$
   2. Hence find the value of \(c\) for which the straight line is a tangent to the curve.
   3. Using this value of \(c\), find the coordinates of the point where the line touches the curve.
   4. In the case where \(c = 4\), determine whether the line intersects the curve or not.

5 A curve has equation $$y = \frac { x } { x ^ { 2 } - 1 }$$

1. Write down the equations of the three asymptotes to the curve.
2. Sketch the curve.
   (You are given that the curve has no stationary points.)
3. Solve the inequality $$\frac { x } { x ^ { 2 } - 1 } > 0$$

8 A curve \(C\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$$

1. Find the \(y\)-coordinates of the points on \(C\) for which \(x = 10\), giving each answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
2. Sketch the curve \(C\), indicating the coordinates of any points where the curve intersects the coordinate axes.
3. Write down the equation of the tangent to \(C\) at the point where \(C\) intersects the positive \(x\)-axis.
4. 1. Show that, if the line \(y = x - 4\) intersects \(C\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$16 x ^ { 2 } - 200 x + 625 = 0$$
   2. Solve this equation and hence state the relationship between the line \(y = x - 4\) and the curve \(C\).

6 A curve has equation $$y = \frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) }$$

1. 1. Write down the equations of the three asymptotes of this curve.
   2. State the coordinates of the points at which the curve intersects the \(x\)-axis.
   3. Sketch the curve.
      (You are given that the curve has no stationary points.)
2. Hence, or otherwise, solve the inequality $$\frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) } < 0$$

9 A hyperbola \(H\) has equation $$x ^ { 2 } - \frac { y ^ { 2 } } { 2 } = 1$$

1. Find the equations of the two asymptotes of \(H\), giving each answer in the form \(y = m x\).
2. Draw a sketch of the two asymptotes of \(H\), using roughly equal scales on the two coordinate axes. Using the same axes, sketch the hyperbola \(H\).
3. 1. Show that, if the line \(y = x + c\) intersects \(H\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$x ^ { 2 } - 2 c x - \left( c ^ { 2 } + 2 \right) = 0$$
   2. Hence show that the line \(y = x + c\) intersects \(H\) in two distinct points, whatever the value of \(c\).
   3. Find, in terms of \(c\), the \(y\)-coordinates of these two points.

7 A graph has equation $$y = \frac { x - 4 } { x ^ { 2 } + 9 }$$

1. Explain why the graph has no vertical asymptote and give the equation of the horizontal asymptote.
2. Show that, if the line \(y = k\) intersects the graph, the \(x\)-coordinates of the points of intersection of the line with the graph must satisfy the equation $$k x ^ { 2 } - x + ( 9 k + 4 ) = 0$$
3. Show that this equation has real roots if \(- \frac { 1 } { 2 } \leqslant k \leqslant \frac { 1 } { 18 }\).
4. Hence find the coordinates of the two stationary points on the curve.
   (No credit will be given for methods involving differentiation.)

7 A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 9 } - y ^ { 2 } = 1$$

1. Find the equations of the asymptotes of \(H\).
2. The asymptotes of \(H\) are shown in the diagram opposite. On the same diagram, sketch the hyperbola \(H\). Indicate on your sketch the coordinates of the points of intersection of \(H\) with the coordinate axes.
3. The hyperbola \(H\) is now translated by the vector \(\left[ \begin{array} { r } - 3 \\ 0 \end{array} \right]\).
   1. Write down the equation of the translated curve.
   2. Calculate the coordinates of the two points of intersection of the translated curve with the line \(y = x\).
4. From your answers to part (c)(ii), deduce the coordinates of the points of intersection of the original hyperbola \(H\) with the line \(y = x - 3\). \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-4_675_1157_1932_495}

9 A curve has equation $$y = \frac { x } { x - 1 }$$

1. Find the equations of the asymptotes of this curve.
2. Given that the line \(y = - 4 x + c\) intersects the curve, show that the \(x\)-coordinates of the points of intersection must satisfy the equation $$4 x ^ { 2 } - ( c + 3 ) x + c = 0$$
3. It is given that the line \(y = - 4 x + c\) is a tangent to the curve.
   1. Find the two possible values of \(c\).
      (No credit will be given for methods involving differentiation.)
   2. For each of the two values found in part (c)(i), find the coordinates of the point where the line touches the curve.

7 A curve has equation $$y = \frac { 3 x - 1 } { x + 2 }$$

1. Write down the equations of the two asymptotes to the curve.
2. Sketch the curve, indicating the coordinates of the points where the curve intersects the coordinate axes.
3. Hence, or otherwise, solve the inequality $$0 < \frac { 3 x - 1 } { x + 2 } < 3$$

7 A curve \(C\) has equation $$y = 7 + \frac { 1 } { x + 1 }$$

1. Define the translation which transforms the curve with equation \(y = \frac { 1 } { x }\) onto the curve \(C\).
2. 1. Write down the equations of the two asymptotes of \(C\).
   2. Find the coordinates of the points where the curve \(C\) intersects the coordinate axes.
3. Sketch the curve \(C\) and its two asymptotes.

6 An ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Sketch the ellipse \(E\), showing the coordinates of the points of intersection of the ellipse with the coordinate axes.
2. The ellipse \(E\) is stretched with scale factor 2 parallel to the \(y\)-axis. Find and simplify the equation of the curve after the stretch.
3. The original ellipse, \(E\), is translated by the vector \(\left[ \begin{array} { l } a \\ b \end{array} \right]\). The equation of the translated ellipse is $$4 x ^ { 2 } + 3 y ^ { 2 } - 8 x + 6 y = 5$$ Find the values of \(a\) and \(b\).

8 A curve has equation $$y = \frac { x ^ { 2 } } { ( x - 1 ) ( x - 5 ) }$$

1. Write down the equations of the three asymptotes to the curve.
2. Show that the curve has no point of intersection with the line \(y = - 1\).
3. 1. Show that, if the curve intersects the line \(y = k\), then the \(x\)-coordinates of the points of intersection must satisfy the equation $$( k - 1 ) x ^ { 2 } - 6 k x + 5 k = 0$$
   2. Show that, if this equation has equal roots, then $$k ( 4 k + 5 ) = 0$$
4. Hence find the coordinates of the two stationary points on the curve.

9 A curve has equation $$y = \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 2 } - 2 x - 3 }$$

1. Find the equations of the three asymptotes of the curve.
2. 1. Show that if the line \(y = k\) intersects the curve then $$( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$$
   2. Given that the equation \(( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0\) has real roots, show that $$k ^ { 2 } - k \geqslant 0$$
   3. Hence show that the curve has only one stationary point and find its coordinates.
      (No credit will be given for solutions based on differentiation.)
3. Sketch the curve and its asymptotes.

1 A family of functions is defined as $$f ( x ) = a x + \frac { x ^ { 2 } } { 1 + x } , \quad x \neq - 1$$ where the parameter \(a\) is a real number. You may find it helpful to use a slider (for \(a\) ) to investigate the family of curves \(y = f ( x )\). \begin{enumerate}[label=(\alph*)] \item \begin{enumerate}[label=(\roman*)] \item On the axes in the Printed Answer Booklet, sketch the curve \(y = f ( x )\) in each of the following cases.

• \(a = - 2\)
• \(a = - 1\)
• \(a = 0\)
• State a feature which is common to the curve in all three cases, \(a = - 2\), \(a = - 1\) and \(a = 0\).
• State a feature of the curve for the cases \(a = - 2 , a = - 1\) that is not a feature of the curve in the case \(a = 0\).
• 1. Determine the equation of the oblique asymptote to the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) in terms of \(a\).
  2. For \(b \neq - 1,0,1\) let \(A\) be the point with coordinates ( \(- b , \mathrm { f } ( - b )\) ) and let \(B\) be the point with coordinates ( \(b , \mathrm { f } ( b )\) ).

Show that the \(y\)-coordinate of the point at which the chord to the curve \(y = f ( x )\) between \(A\) and \(B\) meets the \(y\)-axis is independent of \(a\).

With \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), determine the range of values of \(a\) for which

Find its coordinates and fully justify that it is a cusp.

1 A family of curves is given by the equation $$y = \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 }$$ where the parameter \(a\) is a real number.

1. 1. On the axes in the Printed Answer Booklet, sketch the curve in each of these cases.
      • \(a = - 0.5\)
      • \(a = - 0.1\)
      • \(a = 0.5\)
      • State one feature of the curve for the cases \(a = - 0.5\) and \(a = - 0.1\) that is not a feature of the curve in the case \(a = 0.5\).
      • By using a slider for \(a\), or otherwise, write down the non-zero value of \(a\) for which the points on the curve (\textit{) all lie on a straight line.
      • Write down the equation of the vertical asymptote of the curve (}).
      The equation of the curve (*) can be written in the form \(y = x + A + \frac { a ^ { 2 } - a } { x - 1 }\), where \(A\) is a constant.
   2. Show that \(A = 0\).
   3. Hence, or otherwise, find the value of $$\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 } - x \right) .$$
   4. Explain the significance of the result in part (a)(vi) in terms of a feature of the curve (*).
   5. In this part of the question the value of the parameter \(a\) satisfies \(0 < a < 1\). For values of \(a\) in this range the curve intersects the \(x\)-axis at points X and Y . The point Z has coordinates \(( 0 , - 1 )\). These three points form a triangle XYZ.
      1. Determine, in terms of \(a\), the area of the triangle XYZ.
      2. Find the maximum area of the triangle XYZ.