1.07m Tangents and normals: gradient and equations

873 questions

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Edexcel C1 Q7
9 marks Standard +0.3
  1. Describe fully a single transformation that maps the graph of \(y = \frac{1}{x}\) onto the graph of \(y = \frac{3}{x}\). [2]
  2. Sketch the graph of \(y = \frac{3}{x}\) and write down the equations of any asymptotes. [3]
  3. Find the values of the constant \(c\) for which the straight line \(y = c - 3x\) is a tangent to the curve \(y = \frac{3}{x}\). [4]
Edexcel C1 Q10
11 marks Standard +0.3
A curve has the equation \(y = x + \frac{3}{x}\), \(x \neq 0\). The point \(P\) on the curve has \(x\)-coordinate \(1\).
  1. Show that the gradient of the curve at \(P\) is \(-2\). [3]
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = mx + c\). [4]
  3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again. [4]
Edexcel C1 Q9
11 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the curve \(C\) with the equation \(y = x^3 + 3x^2 - 4x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). [3]
The line \(l\) is the tangent to \(C\) at \(O\).
  1. Find an equation for \(l\). [4]
  2. Find the coordinates of the point where \(l\) intersects \(C\) again. [4]
Edexcel C1 Q10
12 marks Moderate -0.3
The curve \(C\) with equation \(y = \text{f}(x)\) is such that $$\frac{\text{d}y}{\text{d}x} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that \(C\) passes through the points \((0, -2)\) and \((2, 18)\),
  1. show that \(k = 2\) and find an equation for \(C\), [7]
  2. show that the line with equation \(y = x - 2\) is a tangent to \(C\) and find the coordinates of the point of contact. [5]
Edexcel C1 Q7
10 marks Moderate -0.8
A curve has the equation \(y = \frac{x}{2} + 3 - \frac{1}{x}\), \(x \neq 0\). The point \(A\) on the curve has \(x\)-coordinate 2.
  1. Find the gradient of the curve at \(A\). [4]
  2. Show that the tangent to the curve at \(A\) has equation $$3x - 4y + 8 = 0.$$ [3]
The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).
  1. Find the coordinates of \(B\). [3]
Edexcel C1 Q10
13 marks Moderate -0.3
The curve \(C\) has the equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 8x - \frac{2}{x^3}, \quad x \neq 0,$$ and that the point \(P(1, 1)\) lies on \(C\),
  1. find an equation for the tangent to \(C\) at \(P\) in the form \(y = mx + c\), [3]
  2. find an equation for \(C\), [5]
  3. find the \(x\)-coordinates of the points where \(C\) meets the \(x\)-axis, giving your answers in the form \(k\sqrt{2}\). [5]
Edexcel C1 Q6
8 marks Moderate -0.3
The curve with equation \(y = x^2 + 2x\) passes through the origin, \(O\).
  1. Find an equation for the normal to the curve at \(O\). [5]
  2. Find the coordinates of the point where the normal to the curve at \(O\) intersects the curve again. [3]
Edexcel C1 Q10
13 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = \text{f}(x)\). The curve meets the \(x\)-axis at the origin and at the point \(A\). Given that $$\text{f}'(x) = 3x^{\frac{1}{2}} - 4x^{-\frac{1}{2}},$$
  1. find f\((x)\). [5]
  2. Find the coordinates of \(A\). [2]
The point \(B\) on the curve has \(x\)-coordinate 2.
  1. Find an equation for the tangent to the curve at \(B\) in the form \(y = mx + c\). [6]
Edexcel C1 Q9
13 marks Standard +0.3
A curve has the equation \(y = (\sqrt{x} - 3)^2\), \(x \geq 0\).
  1. Show that \(\frac{dy}{dx} = 1 - \frac{3}{\sqrt{x}}\). [4]
The point \(P\) on the curve has \(x\)-coordinate 4.
  1. Find an equation for the normal to the curve at \(P\) in the form \(y = mx + c\). [5]
  2. Show that the normal to the curve at \(P\) does not intersect the curve again. [4]
Edexcel C1 Q10
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 2 + 3x - x^2\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.
  1. Find an equation for \(l\). [5]
The line \(m\) is the normal to the curve at the point \(B\). Given that \(l\) and \(m\) are parallel,
  1. find the coordinates of \(B\). [6]
OCR C1 Q8
11 marks Moderate -0.3
\includegraphics{figure_8} The diagram shows the curve \(C\) with the equation \(y = x^3 + 3x^2 - 4x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). [3]
The line \(l\) is the tangent to \(C\) at \(O\).
  1. Find an equation for \(l\). [4]
  2. Find the coordinates of the point where \(l\) intersects \(C\) again. [4]
OCR C1 Q9
12 marks Moderate -0.3
The curve with equation \(y = 2x^3 - 8x^{\frac{1}{3}}\) has a minimum at the point \(A\).
  1. Find \(\frac{dy}{dx}\). [3]
  2. Find the \(x\)-coordinate of \(A\). [3]
The point \(B\) on the curve has \(x\)-coordinate 2.
  1. Find an equation for the tangent to the curve at \(B\) in the form \(y = mx + c\). [6]
OCR C1 Q9
10 marks Standard +0.3
A curve has the equation \(y = x + \frac{3}{x}\), \(x \neq 0\). The point \(P\) on the curve has \(x\)-coordinate \(1\).
  1. Show that the gradient of the curve at \(P\) is \(-2\). [3]
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = mx + c\). [3]
  3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again. [4]
OCR C1 Q5
8 marks Moderate -0.3
  1. Find in exact form the coordinates of the points where the curve \(y = x^2 - 4x + 2\) crosses the \(x\)-axis. [4]
  2. Find the value of the constant \(k\) for which the straight line \(y = 2x + k\) is a tangent to the curve \(y = x^2 - 4x + 2\). [4]
OCR C1 Q8
11 marks Standard +0.3
\includegraphics{figure_8} The diagram shows the curve with equation \(y = 2 + 3x - x^2\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.
  1. Find an equation for \(l\). [5]
The line \(m\) is the normal to the curve at the point \(B\). Given that \(l\) and \(m\) are parallel,
  1. find the coordinates of \(B\). [6]
OCR C1 Q7
11 marks Moderate -0.8
The point \(A\) has coordinates \((4, 6)\). Given that \(OA\), where \(O\) is the origin, is a diameter of circle \(C\),
  1. find an equation for \(C\). [4]
Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\). \begin{enumerate}[label=(\roman*)] \setcounter{enumi}{1} \item Find the coordinates of \(B\). [2] \item Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [5]
OCR C1 Q9
13 marks Moderate -0.3
A curve has the equation \(y = x^3 - 5x^2 + 7x\).
  1. Show that the curve only crosses the \(x\)-axis at one point. [4]
The point \(P\) on the curve has coordinates \((3, 3)\).
  1. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [6]
The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
  1. Show that triangle \(OQR\), where \(O\) is the origin, has area \(28\frac{1}{8}\). [3]
AQA C2 2009 June Q5
13 marks Standard +0.3
The diagram shows part of a curve with a maximum point \(M\). \includegraphics{figure_5} The equation of the curve is $$y = 15x^{\frac{3}{2}} - x^{\frac{5}{2}}$$
  1. Find \(\frac{dy}{dx}\). [3]
  2. Hence find the coordinates of the maximum point \(M\). [4]
  3. The point \(P(1, 14)\) lies on the curve. Show that the equation of the tangent to the curve at \(P\) is \(y = 20x - 6\). [3]
  4. The tangents to the curve at the points \(P\) and \(M\) intersect at the point \(R\). Find the length of \(RM\). [3]
Edexcel C2 Q9
12 marks Moderate -0.3
\includegraphics{figure_2} The curve \(C\), shown in Fig. 2, represents the graph of $$y = \frac{x^2}{25}, x \geq 0.$$ The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates \(5\) and \(10\) respectively.
  1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
  2. Find an equation of the tangent to \(C\) at \(A\). [4]
The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.
  1. For points \((x, y)\) on \(C\), express \(x\) in terms of \(y\). [2]
  2. Use integration to find the area of \(R\). [5]
Edexcel C2 Q5
10 marks Standard +0.3
The curve \(C\) with equation \(y = p + qe^x\), where \(p\) and \(q\) are constants, passes through the point \((0, 2)\). At the point \(P\) (ln 2, \(p + 2q\)) on \(C\), the gradient is 5.
  1. Find the value of \(p\) and the value of \(q\). [5]
The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).
  1. Show that the area of \(\triangle OLM\), where \(O\) is the origin, is approximately 53.8 [5]
Edexcel C2 Q8
12 marks Moderate -0.3
\includegraphics{figure_1} The curve \(C\), shown in Fig. 1, represents the graph of \(y = \frac{x^2}{25}\), \(x \geq 0\). The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates \(5\) and \(10\) respectively.
  1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
  2. Find an equation of the tangent to \(C\) at \(A\). [4]
The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.
  1. For points \((x, y)\) on \(C\), express \(x\) in terms of \(y\). [2]
  2. Use integration to find the area of \(R\). [5]
Edexcel C2 Q9
15 marks Moderate -0.3
For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),
  1. find \(\frac{dy}{dx}\), [2]
  2. find the coordinates of each of the stationary points, [5]
  3. determine the nature of each stationary point. [3]
The point \(A\), on the curve \(C\), has \(x\)-coordinate \(1\).
  1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
Edexcel C2 Q7
11 marks Moderate -0.3
\includegraphics{figure_1} Fig. 1 shows part of the curve \(C\) with equation \(y = \frac{1}{3}x^2 - \frac{1}{4}x^3\). The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).
  1. Show that \(p = 6\). [1]
  2. Find an equation of the tangent to \(C\) at \(A\). [4]
The curve \(C\) has a maximum at the point \(P\).
  1. Find the \(x\)-coordinate of \(P\). [2]
The shaded region \(R\), in Fig. 1, is bounded by \(C\) and the \(x\)-axis.
  1. Find the area of \(R\). [4]
OCR MEI C2 2010 January Q8
5 marks Moderate -0.3
Find the equation of the tangent to the curve \(y = 6\sqrt{x}\) at the point where \(x = 16\). [5]
OCR MEI C2 2013 January Q10
11 marks Standard +0.3
Fig. 10 shows a sketch of the curve \(y = x^2 - 4x + 3\). The point A on the curve has \(x\)-coordinate 4. At point B the curve crosses the \(x\)-axis. \includegraphics{figure_10}
  1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at C\((16, 0)\). [6]
  2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis. [5]