1.02q Use intersection points: of graphs to solve equations

10 The equation of a curve is \(y = 3 + 4 x - x ^ { 2 }\).

1. Show that the equation of the normal to the curve at the point \(( 3,6 )\) is \(2 y = x + 9\).
2. Given that the normal meets the coordinate axes at points \(A\) and \(B\), find the coordinates of the mid-point of \(A B\).
3. Find the coordinates of the point at which the normal meets the curve again.

11 The equation of a curve is \(y = \frac { 9 } { 2 - x }\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine, with a reason, whether the curve has any stationary points.
2. Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
3. Find the set of values of \(k\) for which the line \(y = x + k\) intersects the curve at two distinct points.

4

1. Sketch the curve \(y = 2 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).
2. By adding a suitable straight line to your sketch, determine the number of real roots of the equation $$2 \pi \sin x = \pi - x$$ State the equation of the straight line.

11 \includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-5_710_931_255_607} The diagram shows parts of the curves \(y = 9 - x ^ { 3 }\) and \(y = \frac { 8 } { x ^ { 3 } }\) and their points of intersection \(P\) and \(Q\). The \(x\)-coordinates of \(P\) and \(Q\) are \(a\) and \(b\) respectively.

1. Show that \(x = a\) and \(x = b\) are roots of the equation \(x ^ { 6 } - 9 x ^ { 3 } + 8 = 0\). Solve this equation and hence state the value of \(a\) and the value of \(b\).
2. Find the area of the shaded region between the two curves.
3. The tangents to the two curves at \(x = c\) (where \(a < c < b\) ) are parallel to each other. Find the value of \(c\).

9 A line has equation \(y = k x + 6\) and a curve has equation \(y = x ^ { 2 } + 3 x + 2 k\), where \(k\) is a constant.

1. For the case where \(k = 2\), the line and the curve intersect at points \(A\) and \(B\). Find the distance \(A B\) and the coordinates of the mid-point of \(A B\).
2. Find the two values of \(k\) for which the line is a tangent to the curve.

4 The equation of a curve is \(y ^ { 2 } + 2 x = 13\) and the equation of a line is \(2 y + x = k\), where \(k\) is a constant.

1. In the case where \(k = 8\), find the coordinates of the points of intersection of the line and the curve.
2. Find the value of \(k\) for which the line is a tangent to the curve.

5 Find the set of values of \(k\) for which the line \(y = 2 x - k\) meets the curve \(y = x ^ { 2 } + k x - 2\) at two distinct points.

6 A curve has equation \(y = x ^ { 2 } - x + 3\) and a line has equation \(y = 3 x + a\), where \(a\) is a constant.

1. Show that the \(x\)-coordinates of the points of intersection of the line and the curve are given by the equation \(x ^ { 2 } - 4 x + ( 3 - a ) = 0\).
2. For the case where the line intersects the curve at two points, it is given that the \(x\)-coordinate of one of the points of intersection is - 1 . Find the \(x\)-coordinate of the other point of intersection.
3. For the case where the line is a tangent to the curve at a point \(P\), find the value of \(a\) and the coordinates of \(P\).

1 A line has equation \(y = 2 x - 7\) and a curve has equation \(y = x ^ { 2 } - 4 x + c\), where \(c\) is a constant. Find the set of possible values of \(c\) for which the line does not intersect the curve.

7 \includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-3_704_558_258_790} The diagram shows parts of the curves \(y = ( 2 x - 1 ) ^ { 2 }\) and \(y ^ { 2 } = 1 - 2 x\), intersecting at points \(A\) and \(B\).

1. State the coordinates of \(A\).
2. Find, showing all necessary working, the area of the shaded region.

2 Find the set of values of \(a\) for which the curve \(y = - \frac { 2 } { x }\) and the straight line \(y = a x + 3 a\) meet at two distinct points.

8 \includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-12_485_570_262_790} The diagram shows parts of the graphs of \(y = 3 - 2 x\) and \(y = 4 - 3 \sqrt { } x\) intersecting at points \(A\) and \(B\).

1. Find by calculation the \(x\)-coordinates of \(A\) and \(B\).
2. Find, showing all necessary working, the area of the shaded region.

7 \includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-10_503_853_260_641} The diagram shows part of the curve with equation \(y = k \left( x ^ { 3 } - 7 x ^ { 2 } + 12 x \right)\) for some constant \(k\). The curve intersects the line \(y = x\) at the origin \(O\) and at the point \(A ( 2,2 )\).

1. Find the value of \(k\).
2. Verify that the curve meets the line \(y = x\) again when \(x = 5\).
3. Find, showing all necessary working, the area of the shaded region.

6 A line has equation \(y = 3 k x - 2 k\) and a curve has equation \(y = x ^ { 2 } - k x + 2\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the line and curve meet at two distinct points.
2. For each of two particular values of \(k\), the line is a tangent to the curve. Show that these two tangents meet on the \(x\)-axis.

6 A curve has equation \(y = x ^ { 2 } - x + 3\) and a line has equation \(y = 3 x + a\), where \(a\) is a constant.

1. Show that the \(x\)-coordinates of the points of intersection of the line and the curve are given by the equation \(x ^ { 2 } - 4 x + ( 3 - a ) = 0\).
2. For the case where the line intersects the curve at two points, it is given that the \(x\)-coordinate of one of the points of intersection is - 1 . Find the \(x\)-coordinate of the other point of intersection.
3. For the case where the line is a tangent to the curve at a point \(P\), find the value of \(a\) and the coordinates of \(P\).

4

1. \includegraphics[max width=\textwidth, alt={}, center]{a1ea242a-c7f4-46b0-b4b8-bd13b3880557-05_753_944_278_630} The diagram shows the graph of \(y = 3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x }\).
   On the diagram, sketch the graph of \(y = | 5 x - 4 |\), and show that the equation \(3 - e ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) has exactly two real roots. It is given that the two roots of \(3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
2. Show by calculation that \(\alpha\) lies between 0.36 and 0.37 .
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 7 - \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to find \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5

1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 2 - x ^ { 2 }$$ has exactly one root.
2. Verify by calculation that the root lies between 1.0 and 1.4 .
3. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$ to determine the root correct to 2 decimal places, showing the result of each iteration.

6

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) that is a root of the equation \(x = 9 \mathrm { e } ^ { - 2 x }\).
2. Verify, by calculation, that this root lies between 1 and 2 .
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( \ln 9 - \ln x _ { n } \right)$$ converges, then it converges to the root of the equation given in part (i).
4. Use the iterative formula, with \(x _ { 1 } = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5

1. By sketching a suitable pair of graphs, show that the equation $$\sec x = 3 - x$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 1.0 and 1.2.
3. Show that this root also satisfies the equation $$x = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } } \right)$$ with initial value \(x _ { 1 } = 1.1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7

1. By sketching a suitable pair of graphs, show that the equation $$\mathrm { e } ^ { 2 x } = 2 - x$$ has only one root.
2. Verify by calculation that this root lies between \(x = 0\) and \(x = 0.5\).
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 2 - x _ { n } \right)$$ converges, then it converges to the root of the equation in part (i).
4. Use this iterative formula, with initial value \(x _ { 1 } = 0.25\), to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
5. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
6. By expressing \(\cot ^ { 2 } x\) in terms of \(\operatorname { cosec } ^ { 2 } x\) and using the result of part (i), show that $$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { 2 } x \mathrm {~d} x = 1 - \frac { 1 } { 4 } \pi$$
7. Express \(\cos 2 x\) in terms of \(\sin ^ { 2 } x\) and hence show that \(\frac { 1 } { 1 - \cos 2 x }\) can be expressed as \(\frac { 1 } { 2 } \operatorname { cosec } ^ { 2 } x\). Hence, using the result of part (i), find $$\int \frac { 1 } { 1 - \cos 2 x } \mathrm {~d} x$$

7

1. By sketching a suitable pair of graphs, show that the equation $$\mathrm { e } ^ { 2 x } = 14 - x ^ { 2 }$$ has exactly two real roots.
2. Show by calculation that the positive root lies between 1.2 and 1.3.
3. Show that this root also satisfies the equation $$x = \frac { 1 } { 2 } \ln \left( 14 - x ^ { 2 } \right) .$$
4. Use an iteration process based on the equation in part (iii), with a suitable starting value, to find the root correct to 2 decimal places. Give the result of each step of the process to 4 decimal places.
5. Express \(4 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
6. Solve the equation \(4 \sin \theta - 6 \cos \theta = 3\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
7. Find the greatest and least possible values of \(( 4 \sin \theta - 6 \cos \theta ) ^ { 2 } + 8\) as \(\theta\) varies.

6

1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 4 x - 2$$ where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.9\).
3. Show that this root also satisfies the equation $$x = \frac { 1 + 2 \tan x } { 4 \tan x }$$
4. Use the iterative formula \(x _ { n + 1 } = \frac { 1 + 2 \tan x _ { n } } { 4 \tan x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. By sketching a suitable pair of graphs, show that the equation $$3 \mathrm { e } ^ { x } = 8 - 2 x$$ has only one root.
2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.8\).
3. Show that this root also satisfies the equation $$x = \ln \left( \frac { 8 - 2 x } { 3 } \right)$$
4. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 8 - 2 x _ { n } } { 3 } \right)\) to determine this root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

4

1. By sketching a suitable pair of graphs, show that the equation $$3 \ln x = 15 - x ^ { 3 }$$ has exactly one real root.
2. Show by calculation that the root lies between 2.0 and 2.5.
3. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 15 - 3 \ln x _ { n } \right)\) to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

3

1. By sketching a suitable pair of graphs, show that the equation $$x ^ { 3 } = 11 - 2 x$$ has exactly one real root.
2. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 11 - 2 x _ { n } \right)$$ to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.