1.07m Tangents and normals: gradient and equations

6 The equation of a curve is \(y = ( 2 k - 3 ) x ^ { 2 } - k x - ( k - 2 )\), where \(k\) is a constant. The line \(y = 3 x - 4\) is a tangent to the curve. Find the value of \(k\).

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

1. Find the equation of the normal to the curve at the point (4,4), giving your answer in the form \(y = m x + c\).
2. Find the coordinates of the stationary point.
3. Determine the nature of the stationary point.
4. Find the exact area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 4\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The point \(A\) has coordinates \(( 1,5 )\) and the line \(l\) has gradient \(- \frac { 2 } { 3 }\) and passes through \(A\). A circle has centre \(( 5,11 )\) and radius \(\sqrt { 52 }\).

1. Show that \(l\) is the tangent to the circle at \(A\).
2. Find the equation of the other circle of radius \(\sqrt { 52 }\) for which \(l\) is also the tangent at \(A\).

3 A line with equation \(y = m x - 6\) is a tangent to the curve with equation \(y = x ^ { 2 } - 4 x + 3\).
Find the possible values of the constant \(m\), and the corresponding coordinates of the points at which the line touches the curve.

10 Points \(A ( - 2,3 ) , B ( 3,0 )\) and \(C ( 6,5 )\) lie on the circumference of a circle with centre \(D\).

1. Show that angle \(A B C = 90 ^ { \circ }\).
2. Hence state the coordinates of \(D\).
3. Find an equation of the circle.
   The point \(E\) lies on the circumference of the circle such that \(B E\) is a diameter.
4. Find an equation of the tangent to the circle at \(E\).

11 \includegraphics[max width=\textwidth, alt={}, center]{aaba3158-b5be-464e-bea3-1a4c460f9637-16_622_1091_260_525} The diagram shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } + k ^ { 2 } x ^ { - \frac { 1 } { 2 } }\), where \(k\) is a positive constant.

1. Find the coordinates of the minimum point of the curve, giving your answer in terms of \(k\).
   The tangent at the point on the curve where \(x = 4 k ^ { 2 }\) intersects the \(y\)-axis at \(P\).
2. Find the \(y\)-coordinate of \(P\) in terms of \(k\).
   The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = \frac { 9 } { 4 } k ^ { 2 }\) and \(x = 4 k ^ { 2 }\).
3. Find the area of the shaded region in terms of \(k\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The equation of a curve is \(y = 4 x ^ { 2 } - k x + \frac { 1 } { 2 } k ^ { 2 }\) and the equation of a line is \(y = x - a\), where \(k\) and \(a\) are constants.

1. Given that the curve and the line intersect at the points with \(x\)-coordinates 0 and \(\frac { 3 } { 4 }\), find the values of \(k\) and \(a\).
2. Given instead that \(a = - \frac { 7 } { 2 }\), find the values of \(k\) for which the line is a tangent to the curve. [5]

8 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + a x + b y - 12 = 0\). The points \(A ( 1,1 )\) and \(B ( 2 , - 6 )\) lie on the circle.

1. Find the values of \(a\) and \(b\) and hence find the coordinates of the centre of the circle.
2. Find the equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y = k\), where \(p , q\) and \(k\) are integers.

10 Functions \(f\) and \(g\) are defined as follows: $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 2 x + 1 } { 2 x - 1 } & \text { for } x \neq \frac { 1 } { 2 } \\ \mathrm {~g} ( x ) = x ^ { 2 } + 4 & \text { for } x \in \mathbb { R } \end{array}$$

1. \includegraphics[max width=\textwidth, alt={}, center]{bb7595c9-93ae-49e8-9cc5-9ecc802e6060-16_773_1182_555_511} The diagram shows part of the graph of \(y = \mathrm { f } ( x )\).
   State the domain of \(\mathrm { f } ^ { - 1 }\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. Find \(\mathrm { gf } ^ { - 1 } ( 3 )\).
4. Explain why \(\mathrm { g } ^ { - 1 } ( x )\) cannot be found.
5. Show that \(1 + \frac { 2 } { 2 x - 1 }\) can be expressed as \(\frac { 2 x + 1 } { 2 x - 1 }\). Hence find the area of the triangle enclosed by the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = 1\) and the \(x\) - and \(y\)-axes.

8 \includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-12_577_1088_260_523} The diagram shows the curve with equation \(y = x ^ { \frac { 1 } { 2 } } + 4 x ^ { - \frac { 1 } { 2 } }\). The line \(y = 5\) intersects the curve at the points \(A ( 1,5 )\) and \(B ( 16,5 )\).

1. Find the equation of the tangent to the curve at the point \(A\).
2. Calculate the area of the shaded region.

11 The point \(P\) lies on the line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. A curve has equation \(y = - \frac { m } { x }\). There is a single point \(P\) on the curve such that the straight line is a tangent to the curve at \(P\).

1. Find the coordinates of \(P\), giving the \(y\)-coordinate in terms of \(m\).
   The normal to the curve at \(P\) intersects the curve again at the point \(Q\).
2. Find the coordinates of \(Q\) in terms of \(m\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The line with equation \(y = k x - k\), where \(k\) is a positive constant, is a tangent to the curve with equation \(y = - \frac { 1 } { 2 x }\). Find, in either order, the value of \(k\) and the coordinates of the point where the tangent meets the curve. [5]

10 \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-14_631_689_274_721} The diagram shows part of the curve with equation \(y = \frac { 4 } { ( 2 x - 1 ) ^ { 2 } }\) and parts of the lines \(x = 1\) and \(y = 1\). The curve passes through the points \(A ( 1,4 )\) and \(B , \left( \frac { 3 } { 2 } , 1 \right)\).

1. Find the exact volume generated when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. A triangle is formed from the tangent to the curve at \(B\), the normal to the curve at \(B\) and the \(x\)-axis. Find the area of this triangle.

9 A curve which passes through \(( 0,3 )\) has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 1 - \frac { 2 } { ( x - 1 ) ^ { 3 } }\).

1. Find the equation of the curve.
   The tangent to the curve at \(( 0,3 )\) intersects the curve again at one other point, \(P\).
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(( 2 x + 1 ) ( x - 1 ) ^ { 2 } - 1 = 0\).
3. Verify that \(x = \frac { 3 } { 2 }\) satisfies this equation and hence find the \(y\)-coordinate of \(P\).

10 \includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-14_832_830_276_653} The diagram shows the points \(A \left( 1 \frac { 1 } { 2 } , 5 \frac { 1 } { 2 } \right)\) and \(B \left( 7 \frac { 1 } { 2 } , 3 \frac { 1 } { 2 } \right)\) lying on the curve with equation \(y = 9 x - ( 2 x + 1 ) ^ { \frac { 3 } { 2 } }\).

1. Find the coordinates of the maximum point of the curve.
2. Verify that the line \(A B\) is the normal to the curve at \(A\).
3. Find the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-12_764_967_292_555} The diagram shows the curve with equation \(y = \sqrt { 2 x ^ { 3 } + 10 }\).

1. Find the equation of the tangent to the curve at the point where \(x = 3\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-12_2716_35_141_2013}
2. The region shaded in the diagram is enclosed by the curve and the straight lines \(x = 1 , x = 3\) and \(y = 0\). Find the volume of the solid obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

11 \includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-18_497_1049_264_548} The diagram shows the curve with equation \(y = 9 \left( x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } } \right)\). The curve crosses the \(x\)-axis at the point \(A\).

1. Find the \(x\)-coordinate of \(A\).
2. Find the equation of the tangent to the curve at \(A\).
3. Find the \(x\)-coordinate of the maximum point of the curve.
4. Find the area of the region bounded by the curve, the \(x\)-axis and the line \(x = 9\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 At the point \(( 4 , - 1 )\) on a curve, the gradient of the curve is \(- \frac { 3 } { 2 }\). It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { - \frac { 1 } { 2 } } + k\), where \(k\) is a constant.

1. Show that \(k = - 2\).
2. Find the equation of the curve.
3. Find the coordinates of the stationary point.
4. Determine the nature of the stationary point.

5 A curve has the equation \(\mathrm { y } = \frac { 3 } { 2 \mathrm { x } ^ { 2 } - 5 }\).
Find the equation of the normal to the curve at the point \(( 2,1 )\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.

10

1. Find \(\int _ { 1 } ^ { \infty } \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\). \includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-16_499_689_1322_726} The diagram shows the curve with equation \(y = \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } }\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). The shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Find the volume of revolution.
   The normal to the curve at the point \(( 1,1 )\) crosses the \(y\)-axis at the point \(A\).
3. Find the \(y\)-coordinate of \(A\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 \includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-16_505_1166_258_486} The diagram shows the line \(x = \frac { 5 } { 2 }\), part of the curve \(y = \frac { 1 } { 2 } x + \frac { 7 } { 10 } - \frac { 1 } { ( x - 2 ) ^ { \frac { 1 } { 3 } } }\) and the normal to the curve at the point \(A \left( 3 , \frac { 6 } { 5 } \right)\).

1. Find the \(x\)-coordinate of the point where the normal to the curve meets the \(x\)-axis.
2. Find the area of the shaded region, giving your answer correct to 2 decimal places.

8 \includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-12_570_961_260_591} The diagram shows the curves with equations \(y = x ^ { - \frac { 1 } { 2 } }\) and \(y = \frac { 5 } { 2 } - x ^ { \frac { 1 } { 2 } }\). The curves intersect at the points \(A \left( \frac { 1 } { 4 } , 2 \right)\) and \(B \left( 4 , \frac { 1 } { 2 } \right)\).

1. Find the area of the region between the two curves.
2. The normal to the curve \(y = x ^ { - \frac { 1 } { 2 } }\) at the point \(( 1,1 )\) intersects the \(y\)-axis at the point \(( 0 , p )\). Find the value of \(p\).

2 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 \left( \frac { 1 } { 2 } x - 1 \right) ^ { - 4 }\). It is given that the curve passes through the point \(P ( 6,4 )\).

1. Find the equation of the tangent to the curve at \(P\).
2. Find the equation of the curve.

10 \includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-16_942_933_262_605} Curves with equations \(y = 2 x ^ { \frac { 1 } { 2 } } + 1\) and \(y = \frac { 1 } { 2 } x ^ { 2 } - x + 1\) intersect at \(A ( 0,1 )\) and \(B ( 4,5 )\), as shown in the diagram.

1. Find the area of the region between the two curves.
   The acute angle between the two tangents at \(B\) is denoted by \(\alpha ^ { \circ }\), and the scales on the axes are the same.
2. Find \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-18_951_725_267_703} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\). Tangents touching the circle at points \(B\) and \(C\) pass through the point \(A ( 0,10 )\).
   1. By letting the equation of a tangent be \(y = m x + 10\), find the two possible values of \(m\).
   2. Find the coordinates of \(B\) and \(C\).
      The point \(D\) is where the circle crosses the positive \(x\)-axis.
   3. Find angle \(B D C\) in degrees.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 A curve has a stationary point at \(( 2 , - 10 )\) and is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the curve.
3. Find the coordinates of the other stationary point and determine its nature.
4. Find the equation of the tangent to the curve at the point where the curve crosses the \(y\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-18_689_828_276_646} The diagram shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 40\). Parallel tangents, each with gradient 1 , touch the circle at points \(A\) and \(B\).
   1. Find the equation of the line \(A B\), giving the answer in the form \(y = m x + c\).
   2. Find the coordinates of \(A\), giving each coordinate in surd form.
   3. Find the equation of the tangent at \(A\), giving the answer in the form \(y = m x + c\), where \(c\) is in surd form.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.