1.07m Tangents and normals: gradient and equations

A curve has the equation \(y = e^{-2x}\) At point \(P\) on the curve the tangent is parallel to the line \(x + 8y = 5\) Find the coordinates of \(P\) stating your answer in the form \((\ln p, q)\), where \(p\) and \(q\) are rational. [7 marks]

Curve \(C\) has equation \(y = \frac{\sqrt{2}}{x^2}\)

1. Find an equation of the tangent to \(C\) at the point \(\left(2, \frac{\sqrt{2}}{4}\right)\) [4 marks]
2. Show that the tangent to \(C\) at the point \(\left(2, \frac{\sqrt{2}}{4}\right)\) is also a normal to the curve at a different point. \includegraphics{figure_10} [5 marks]

At a point \(P\) on a curve, the gradient of the tangent to the curve is 10 State the gradient of the normal to the curve at \(P\) Circle your answer. [1 mark] \(-10\) \quad \(-0.1\) \quad \(0.1\) \quad \(10\)

A curve has equation \(y = 6x\sqrt{x} + \frac{32}{x}\) for \(x > 0\)

1. Find \(\frac{dy}{dx}\) [4 marks]
2. The point \(A\) lies on the curve and has \(x\)-coordinate 4 Find the coordinates of the point where the tangent to the curve at \(A\) crosses the \(x\)-axis. [5 marks]

A curve has equation \(y = e^{2x}\) Find the coordinates of the point on the curve where the gradient of the curve is \(\frac{1}{2}\) Give your answer in an exact form. [5 marks]

A curve with equation \(y = f(x)\) passes through the point \((3, 7)\) Given that \(f'(3) = 0\) find the equation of the normal to the curve at \((3, 7)\) Circle your answer. [1 mark] \(y = \frac{7}{3}x\) \(y = 0\) \(x = 3\) \(x = 7\)

\includegraphics{figure_3} The curve \(C_1\), shown in Figure 3, has equation \(y = 4x^2 - 6x + 4\). The point \(P\left(\frac{1}{2}, 2\right)\) lies on \(C_1\) The curve \(C_2\), also shown in Figure 3, has equation \(y = \frac{1}{2}x + \ln(2x)\). The normal to \(C_1\) at the point \(P\) meets \(C_2\) at the point \(Q\). Find the exact coordinates of \(Q\). (Solutions based entirely on graphical or numerical methods are not acceptable.) [8]

A circle \(C\) with centre at \((-2, 6)\) passes through the point \((10, 11)\).

1. Show that the circle \(C\) also passes through the point \((10, 1)\). [3]

The tangent to the circle \(C\) at the point \((10, 11)\) meets the \(y\) axis at the point \(P\) and the tangent to the circle \(C\) at the point \((10, 1)\) meets the \(y\) axis at the point \(Q\).

1. Show that the distance \(PQ\) is \(58\) explaining your method clearly. [7]

A curve with centre \(C\) has equation $$x^2 + y^2 + 2x - 6y - 40 = 0$$

1. 1. State the coordinates of \(C\).
   2. Find the radius of the circle, giving your answer as \(r = n\sqrt{2}\). [3]
2. The line \(l\) is a tangent to the circle and has gradient \(-7\). Find two possible equations for \(l\), giving your answers in the form \(y = mx + c\). [8]

\includegraphics{figure_5} Figure 5 shows a sketch of part of the curve \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The point \(A(4, \frac{7}{2})\) lies on C. The line \(l\) is the tangent to C at the point A. The region \(R\), shown shaded in figure 5 is bounded by the line \(l\), the curve C, the line with equation \(x = 1\) and the \(x\)-axis. Find the exact area of \(R\). (Solutions based entirely on graphical or numerical methods are not acceptable.)

In this question you must show detailed reasoning. A circle has equation \(x^2 + y^2 - 6x - 4y + 12 = 0\). Two tangents to this circle pass through the point \((0, 1)\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same. Find the angle between these two tangents. [7]

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 diagram below shows a sketch of the curve \(y = f(x)\), where \(f(x) = 10x + 3x^2 - x^3\). The curve intersects the \(x\)-axis at the origin \(O\) and at the points \(A(-2, 0)\), \(B(5, 0)\). The tangent to the curve at the point \(C(2, 24)\) intersects the \(y\)-axis at the point \(D\). \includegraphics{figure_11}

1. Find the coordinates of \(D\). [5]
2. Find the area of the shaded region. [6]
3. Determine the range of values of \(x\) for which \(f(x)\) is an increasing function. [4]

The curve \(C\) has equation \(y = 2x^2 + 5x - 12\) and the line \(L\) has equation \(y = mx - 14\), where \(m\) is a real constant.

1. Given that \(L\) is a tangent to \(C\),
   1. show that \(m\) satisfies the equation $$m^2 - 10m + 9 = 0,$$ [5]
   2. find the coordinates of the two possible points of contact of \(C\) and \(L\). [6]
2. Given instead that \(L\) intersects \(C\) at two distinct points, find the range of values of \(m\). [2]

The circle \(C\) has centre \(A\) and equation $$x^2 + y^2 - 2x + 6y - 15 = 0.$$

1. Find the coordinates of \(A\) and the radius of \(C\). [3]
2. The point \(P\) has coordinates \((4, -7)\) and lies on \(C\). Find the equation of the tangent to \(C\) at \(P\). [4]

\includegraphics{figure_17} The diagram above shows a sketch of the curve \(y = 3x - x^2\). The curve intersects the \(x\)-axis at the origin and at the point \(A\). The tangent to the curve at the point \(B(2, 2)\) intersects the \(x\)-axis at the point \(C\).

1. Find the equation of the tangent to the curve at \(B\). [4]
2. Find the area of the shaded region. [8]

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 \(-\tan\theta\). [6]
2. Find the equation of the tangent to the curve at the point where \(\theta = \frac{\pi}{4}\). [4]

The diagram below shows part of a curve C with equation \(y = 1 + 3x - \frac{1}{2}x^2\). \includegraphics{figure_7}

1. The curve crosses the \(y\) axis at the point A. The straight line L is normal to the curve at A and meets the curve again at B. Find the equation of L and the \(x\) coordinate of the point B. [4]
2. The region R is bounded by the curve C and the line L. Find the exact area of R. [3]

A curve has equation \(y = \frac{1}{4}x^4 - x^3 - 2x^2\).

1. Find \(\frac{dy}{dx}\). [1]
2. Hence sketch the gradient function for the curve. [4]
3. Find the equation of the tangent to the curve \(y = \frac{1}{4}x^4 - x^3 - 2x^2\) at \(x = 4\). [2]

$$\text{f}(x) = x^2 - 2x - 3, \quad x \in \mathbb{R}, x \geq 1.$$

1. Write down the domain and range of \(\text{f}^{-1}\) [2]
2. Sketch the graph of \(\text{f}^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]
3. Find the gradient of \(f^{-1}(x)\) when \(f^{-1}(x) = \frac{5}{3}\) [3]

$$\text{f}(x) = x^2 - 2x - 3, \quad x \in \mathbb{R}, x \geq 1.$$

1. Write down the domain and range of \(\text{f}^{-1}\) [2]
2. Sketch the graph of \(\text{f}^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]
3. Find the gradient of \(f^{-1}(x)\) when \(f^{-1}(x) = -\frac{5}{3}\) [3]