1.07m Tangents and normals: gradient and equations

873 questions

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AQA C3 2016 June Q4
10 marks Moderate -0.3
4
  1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x - 5 }\).
  2. The normal to the curve \(y = \mathrm { e } ^ { 2 x - 5 }\) at the point \(P \left( 2 , \mathrm { e } ^ { - 1 } \right)\) intersects the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\). Show that the area of the triangle \(O A B\) is \(\frac { \left( \mathrm { e } ^ { 2 } + 1 \right) ^ { m } } { \mathrm { e } ^ { n } }\), where \(m\) and \(n\) are integers.
    [0pt] [6 marks]
Edexcel C3 Q4
10 marks Standard +0.3
  1. Use the derivatives of \(\sin x\) and \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$ The tangent to the curve \(y = 2 x \tan x\) at the point where \(x = \frac { \pi } { 4 }\) meets the \(y\)-axis at the point \(P\).
  2. Find the \(y\)-coordinate of \(P\) in the form \(k \pi ^ { 2 }\) where \(k\) is a rational constant.
Edexcel C3 Q2
9 marks Moderate -0.3
2. A curve has the equation \(y = \sqrt { 3 x + 11 }\). The point \(P\) on the curve has \(x\)-coordinate 3 .
  1. Show that the tangent to the curve at \(P\) has the equation $$3 x - 4 \sqrt { 5 } y + 31 = 0$$ The normal to the curve at \(P\) crosses the \(y\)-axis at \(Q\).
  2. Find the \(y\)-coordinate of \(Q\) in the form \(k \sqrt { 5 }\).
Edexcel C3 Q4
10 marks Standard +0.3
4. The curve with equation \(y = x ^ { \frac { 5 } { 2 } } \ln \frac { x } { 4 } , x > 0\) crosses the \(x\)-axis at the point \(P\).
  1. Write down the coordinates of \(P\). The normal to the curve at \(P\) crosses the \(y\)-axis at the point \(Q\).
  2. Find the area of triangle \(O P Q\) where \(O\) is the origin. The curve has a stationary point at \(R\).
  3. Find the \(x\)-coordinate of \(R\) in exact form.
Edexcel C3 Q6
13 marks Standard +0.3
6. $$\mathrm { f } ( x ) = \mathrm { e } ^ { 3 x + 1 } - 2 , \quad x \in \mathbb { R } .$$
  1. State the range of f . The curve \(y = \mathrm { f } ( x )\) meets the \(y\)-axis at the point \(P\) and the \(x\)-axis at the point \(Q\).
  2. Find the exact coordinates of \(P\) and \(Q\).
  3. Show that the tangent to the curve at \(P\) has the equation $$y = 3 \mathrm { e } x + \mathrm { e } - 2 .$$
  4. Find to 3 significant figures the \(x\)-coordinate of the point where the tangent to the curve at \(P\) meets the tangent to the curve at \(Q\).
Edexcel C3 Q8
14 marks Standard +0.8
8. The curve \(C\) has the equation \(y = \sqrt { x } + \mathrm { e } ^ { 1 - 4 x } , x \geq 0\).
  1. Find an equation for the normal to the curve at the point \(\left( \frac { 1 } { 4 } , \frac { 3 } { 2 } \right)\). The curve \(C\) has a stationary point with \(x\)-coordinate \(\alpha\) where \(0.5 < \alpha < 1\).
  2. Show that \(\alpha\) is a solution of the equation $$x = \frac { 1 } { 4 } [ 1 + \ln ( 8 \sqrt { x } ) ]$$
  3. Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 4 } \left[ 1 + \ln \left( 8 \sqrt { x _ { n } } \right) \right]$$ with \(x _ { 0 } = 1\) to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving the value of \(x _ { 4 }\) to 3 decimal places.
  4. Show that your value for \(x _ { 4 }\) is the value of \(\alpha\) correct to 3 decimal places.
  5. Another attempt to find \(\alpha\) is made using the iteration formula $$x _ { n + 1 } = \frac { 1 } { 64 } \mathrm { e } ^ { 8 x _ { n } - 2 }$$ with \(x _ { 0 } = 1\). Describe the outcome of this attempt.
Edexcel C3 Q5
10 marks Standard +0.3
5. $$f ( x ) = 3 - \frac { x - 1 } { x - 3 } + \frac { x + 11 } { 2 x ^ { 2 } - 5 x - 3 } , \quad x \in \mathbb { R } , \quad x < - 1$$
  1. Show that $$f ( x ) = \frac { 4 x - 1 } { 2 x + 1 }$$
  2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = - 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C3 Q8
14 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d17a1b86-d758-4470-834a-b32a41f90c89-4_478_937_251_450} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = 2 x - 3 \ln ( 2 x + 5 )\) and the normal to the curve at the point \(P ( - 2 , - 4 )\).
  1. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) intersects the curve again at the point \(Q\) with \(x\)-coordinate \(q\).
  2. Show that \(1 < q < 2\).
  3. Show that \(q\) is a solution of the equation $$x = \frac { 12 } { 7 } \ln ( 2 x + 5 ) - 2 .$$
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 12 } { 7 } \ln \left( 2 x _ { n } + 5 \right) - 2 ,$$ with \(x _ { 0 } = 1.5\), to find the value of \(q\) to 3 significant figures and justify the accuracy of your answer.
Edexcel C3 Q3
11 marks Standard +0.8
3. $$f ( x ) = x ^ { 2 } + 5 x - 2 \sec x , \quad x \in \mathbb { R } , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } .$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [1,1.5]. A more accurate estimate of this root is to be found using iterations of the form $$x _ { n + 1 } = \arccos \mathrm { g } \left( x _ { n } \right) .$$
  2. Find a suitable form for \(\mathrm { g } ( x )\) and use this formula with \(x _ { 0 } = 1.25\) to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 3 decimal places. The curve \(y = \mathrm { f } ( x )\) has a stationary point at \(P\).
  3. Show that the \(x\)-coordinate of \(P\) is 1.0535 correct to 5 significant figures.
Edexcel C3 Q4
8 marks Standard +0.3
4. A curve has the equation \(x = y \sqrt { 1 - 2 y }\).
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - 2 y } } { 1 - 3 y } .$$ The point \(A\) on the curve has \(y\)-coordinate - 1 .
  2. Show that the equation of tangent to the curve at \(A\) can be written in the form $$\sqrt { 3 } x + p y + q = 0$$ where \(p\) and \(q\) are integers to be found.
Edexcel C3 Q8
13 marks Standard +0.3
8. A curve has the equation \(y = x ^ { 2 } - \sqrt { 4 + \ln x }\).
  1. Show that the tangent to the curve at the point where \(x = 1\) has the equation $$7 x - 4 y = 11$$ The curve has a stationary point with \(x\)-coordinate \(\alpha\).
  2. Show that \(0.3 < \alpha < 0.4\)
  3. Show that \(\alpha\) is a solution of the equation $$x = \frac { 1 } { 2 } ( 4 + \ln x ) ^ { - \frac { 1 } { 4 } }$$
  4. Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( 4 + \ln x _ { n } \right) ^ { - \frac { 1 } { 4 } }$$ with \(x _ { 0 } = 0.35\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 5 decimal places. END
Edexcel C3 Q1
7 marks Standard +0.3
A curve has the equation \(y = ( 3 x - 5 ) ^ { 3 }\).
  1. Find an equation for the tangent to the curve at the point \(P ( 2,1 )\). The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
  2. Find the coordinates of \(Q\).
Edexcel C3 Q6
11 marks Standard +0.3
6.
  1. Use the derivative of \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$ The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
  2. Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
  3. Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).
Edexcel C3 Q7
12 marks Standard +0.3
7. \(\quad f ( x ) = x ^ { 2 } - 2 x + 5 , x \in \mathbb { R } , x \geq 1\).
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. State the range of f.
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  4. Describe fully two transformations that would map the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) onto the graph of \(y = \sqrt { x } , x \geq 0\).
  5. Find an equation for the normal to the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point where \(x = 8\).
Edexcel C3 Q8
13 marks Standard +0.3
8. A curve has the equation \(y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , \quad x \neq 0\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    [0pt]
  2. Show that the curve has a stationary point in the interval [1.3,1.4]. The point \(A\) on the curve has \(x\)-coordinate 2 .
  3. Show that the tangent to the curve at \(A\) passes through the origin. The tangent to the curve at \(A\) intersects the curve again at the point \(B\).
    The \(x\)-coordinate of \(B\) is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with \(x _ { 0 } = - 1\).
  4. Find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 7 significant figures and hence state the \(x\)-coordinate of \(B\) to 5 significant figures.
Edexcel C3 Q4
8 marks Standard +0.2
4. The curve \(C\) has the equation \(y = x ^ { 2 } - 5 x + 2 \ln \frac { x } { 3 } , x > 0\).
  1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3 x + 5 y + 21 = 0$$
  2. Find the \(x\)-coordinates of the stationary points of \(C\).
Edexcel C3 Q8
14 marks Standard +0.8
8. $$f ( x ) = 2 x + \sin x - 3 \cos x$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [0.7, 0.8].
  2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where it crosses the \(y\)-axis.
  3. Find the values of the constants \(a , b\) and \(c\), where \(b > 0\) and \(0 < c < \frac { \pi } { 2 }\), such that $$f ^ { \prime } ( x ) = a + b \cos ( x - c )$$
  4. Hence find the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\), giving your answers to 2 decimal places.
Edexcel C3 Q3
10 marks Standard +0.3
\(f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 } , x \in \mathbb { R }\).
  1. Find the values of the constants \(A\), \(B\), \(C\) and \(D\) such that $$f ( x ) = x ^ { 2 } + A x + B + \frac { C x + D } { x ^ { 2 } - 3 x + 3 }$$ The point \(P\) on the curve \(y = \mathrm { f } ( x )\) has \(x\)-coordinate 1.
  2. Show that the normal to the curve \(y = \mathrm { f } ( x )\) at \(P\) has the equation $$x + 5 y + 9 = 0$$
Edexcel C3 Q4
11 marks Standard +0.3
  1. Given that $$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } } .$$
  2. Find an equation for the tangent to the curve \(y = \sqrt { 3 + 2 \cos x }\) at the point where \(x = \frac { \pi } { 3 }\).
Edexcel C3 Q5
11 marks Moderate -0.3
5. $$\mathrm { f } ( x ) = 5 + \mathrm { e } ^ { 2 x - 3 } , \quad x \in \mathbb { R } .$$
  1. State the range of f .
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  3. Solve the equation \(\mathrm { f } ( x ) = 7\).
  4. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(y = 7\).
AQA C4 2011 January Q4
6 marks Standard +0.3
4 A curve is defined by the parametric equations $$x = 3 \mathrm { e } ^ { t } , \quad y = \mathrm { e } ^ { 2 t } - \mathrm { e } ^ { - 2 t }$$
    1. Find the gradient of the curve at the point where \(t = 0\).
    2. Find an equation of the tangent to the curve at the point where \(t = 0\).
  2. Show that the cartesian equation of the curve can be written in the form $$y = \frac { x ^ { 2 } } { k } - \frac { k } { x ^ { 2 } }$$ where \(k\) is an integer.
AQA C4 2012 January Q5
11 marks Moderate -0.3
5 A curve is defined by the parametric equations $$x = 8 t ^ { 2 } - t , \quad y = \frac { 3 } { t }$$
  1. Show that the cartesian equation of the curve can be written as \(x y ^ { 2 } + 3 y = k\), stating the value of the integer \(k\).
    (2 marks)
    1. Find an equation of the tangent to the curve at the point \(P\), where \(t = \frac { 1 } { 4 }\).
    2. Verify that the tangent at \(P\) intersects the curve when \(x = \frac { 3 } { 2 }\).
AQA C4 2013 June Q4
12 marks Standard +0.3
4 A curve is defined by the parametric equations \(x = 8 \mathrm { e } ^ { - 2 t } - 4 , y = 2 \mathrm { e } ^ { 2 t } + 4\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. The point \(P\), where \(t = \ln 2\), lies on the curve.
    1. Find the gradient of the curve at \(P\).
    2. Find the coordinates of \(P\).
    3. The normal at \(P\) crosses the \(x\)-axis at the point \(Q\). Find the coordinates of \(Q\).
  3. Find the Cartesian equation of the curve in the form \(x y + 4 y - 4 x = k\), where \(k\) is an integer.
    (3 marks)
AQA C4 2015 June Q5
11 marks Standard +0.8
5 A curve is defined by the parametric equations \(x = \cos 2 t , y = \sin t\).
The point \(P\) on the curve is where \(t = \frac { \pi } { 6 }\).
  1. Find the gradient at \(P\).
  2. Find the equation of the normal to the curve at \(P\) in the form \(y = m x + c\).
  3. The normal at \(P\) intersects the curve again at the point \(Q ( \cos 2 q , \sin q )\). Use the equation of the normal to form a quadratic equation in \(\sin q\) and hence find the \(x\)-coordinate of \(Q\).
    [0pt] [5 marks]
Edexcel C4 Q7
12 marks Standard +0.8
7.
  1. Prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( a ^ { x } \right) = a ^ { x } \ln a .$$ A curve has the equation \(y = 4 ^ { x } - 2 ^ { x - 1 } + 1\).
  2. Show that the tangent to the curve at the point where it crosses the \(y\)-axis has the equation $$3 x \ln 2 - 2 y + 3 = 0 .$$
  3. Find the exact coordinates of the stationary point of the curve.
    7. continued