Tangent parallel to given line - Tangents, normals and gradients

Edexcel C1 2006 January Q9

12 marks Easy -1.2

9. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{815e288c-0140-4c12-9e89-b0bb4fb1a8c1-12_812_1088_317_427}

\end{figure} Figure 2 shows part of the curve $C$ with equation $$y = ( x - 1 ) \left( x ^ { 2 } - 4 \right) .$$ The curve cuts the $x$-axis at the points $P , ( 1,0 )$ and $Q$, as shown in Figure 2.

Write down the $x$-coordinate of $P$, and the $x$-coordinate of $Q$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 2 x - 4$.
Show that $y = x + 7$ is an equation of the tangent to $C$ at the point ( $- 1,6$ ). The tangent to $C$ at the point $R$ is parallel to the tangent at the point ( $- 1,6$ ).
Find the exact coordinates of $R$.

Edexcel C1 2013 January Q11

12 marks Moderate -0.8

11. The curve $C$ has equation $$y = 2 x - 8 \sqrt { } x + 5 , \quad x \geqslant 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving each term in its simplest form. The point $P$ on $C$ has $x$-coordinate equal to $\frac { 1 } { 4 }$
Find the equation of the tangent to $C$ at the point $P$, giving your answer in the form $y = a x + b$, where $a$ and $b$ are constants. The tangent to $C$ at the point $Q$ is parallel to the line with equation $2 x - 3 y + 18 = 0$
Find the coordinates of $Q$.

Edexcel C1 2014 January Q10

10 marks Moderate -0.3

10. The curve $C$ has equation $y = x ^ { 3 } - 2 x ^ { 2 } - x + 3$ The point $P$, which lies on $C$, has coordinates $( 2,1 )$.

Show that an equation of the tangent to $C$ at the point $P$ is $y = 3 x - 5$ The point $Q$ also lies on $C$.
Given that the tangent to $C$ at $Q$ is parallel to the tangent to $C$ at $P$,
find the coordinates of the point $Q$.

Edexcel C1 2005 June Q10

11 marks Moderate -0.3

10. The curve $C$ has equation $y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3$. The point $P$ has coordinates $( 3,0 )$.

Show that $P$ lies on $C$.
Find the equation of the tangent to $C$ at $P$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants. Another point $Q$ also lies on $C$. The tangent to $C$ at $Q$ is parallel to the tangent to $C$ at $P$.
Find the coordinates of $Q$.

Edexcel C1 Q6

9 marks Easy -1.2

6. $$f ( x ) = \frac { ( 2 x + 1 ) ( x + 4 ) } { \sqrt { x } } , \quad x > 0$$

Show that $\mathrm { f } ( x )$ can be written in the form $P x ^ { \frac { 3 } { 2 } } + Q x ^ { \frac { 1 } { 2 } } + R x ^ { - \frac { 1 } { 2 } }$, stating the values of the constants $P , Q$ and $R$.
Find f ${ } ^ { \prime } ( x )$.
Show that the tangent to the curve with equation $y = \mathrm { f } ( x )$ at the point where $x = 1$ is parallel to the line with equation $2 y = 11 x + 3$.
(3)

6. continued Leave blank

Edexcel C1 2008 June Q9

8 marks Moderate -0.3

The curve $C$ has equation $y = k x ^ { 3 } - x ^ { 2 } + x - 5$, where $k$ is a constant.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$. The point $A$ with $x$-coordinate $- \frac { 1 } { 2 }$ lies on $C$. The tangent to $C$ at $A$ is parallel to the line with equation $2 y - 7 x + 1 = 0$. Find
the value of $k$,
the value of the $y$-coordinate of $A$.

OCR C1 Q9

10 marks Moderate -0.3

9. 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 .

Find the gradient of the curve at $A$.
Show that the tangent to the curve at $A$ has equation $$3 x - 4 y + 8 = 0$$ The tangent to the curve at the point $B$ is parallel to the tangent at $A$.
Find the coordinates of $B$.

OCR C3 Q3

7 marks Standard +0.3

3. A curve has the equation $y = ( 3 x - 5 ) ^ { 3 }$.

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 $P$.
Find the coordinates of $Q$.

OCR PURE Q9

6 marks Standard +0.3

9 In this question you must show detailed reasoning.
Find the equation of the straight line with positive gradient that passes through $( 0,2 )$ and is a tangent to the curve $y = x ^ { 2 } - x + 6$.

Edexcel C1 Q9

14 marks Standard +0.3

9. The curve $C$ has equation $y = \mathrm { f } ( x )$. Given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29$ and that $C$ passes through the point $P ( 2,6 )$,

find $y$ in terms of $x$.
Verify that $C$ passes through the point $( 4,0 )$.
Find an equation of the tangent to $C$ at $P$. The tangent to $C$ at the point $Q$ is parallel to the tangent at $P$.
Calculate the exact $x$-coordinate of $Q$.

AQA C2 2010 January Q5

10 marks Standard +0.3

5 A curve has equation $y = \frac { 1 } { x ^ { 3 } } + 48 x$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Hence find the equation of each of the two tangents to the curve that are parallel to the $x$-axis.
Find an equation of the normal to the curve at the point $( 1,49 )$.

AQA C2 2015 June Q8

5 marks Moderate -0.3

8 The point $A$ lies on the curve with equation $y = x ^ { \frac { 1 } { 2 } }$. The tangent to this curve at $A$ is parallel to the line $3 y - 2 x = 1$. Find an equation of this tangent at $A$.
[0pt] [5 marks]

OCR MEI AS Paper 2 Specimen Q11

6 marks Standard +0.8

11 In this question you must show detailed reasoning. Fig. 11 shows the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x )$ is a cubic function. Fig. 11 also shows the coordinates of the turning points and the points of intersection with the axes. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-11_805_620_543_317} \captionsetup{labelformat=empty} \caption{Fig. 11}

\end{figure} Show that the tangent to $y = \mathrm { f } ( x )$ at $x = t$ is parallel to the tangent to $y = \mathrm { f } ( x )$ at $x = - t$ for all values of $t$.

Edexcel M2 Q20

14 marks Moderate -0.5

20. The curve $C$ has equation $y = \mathrm { f } ( x )$. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29$$ and that $C$ passes through the point $P ( 2,6 )$,

find $y$ in terms of $x$.
Verify that $C$ passes through the point ( 4,0 ).
Find an equation of the tangent to $C$ at $P$. The tangent to $C$ at the point $Q$ is parallel to the tangent at $P$.
Calculate the exact $x$-coordinate of $Q$.
21. $$y = 7 + 10 x ^ { \frac { 3 } { 2 } }$$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Find $\int y \mathrm {~d} x$.
22. (a) Given that $3 ^ { x } = 9 ^ { y - 1 }$, show that $x = 2 y - 2$.
Solve the simultaneous equations $$\begin{gathered} x = 2 y - 2 \\ x ^ { 2 } = y ^ { 2 } + 7 \end{gathered}$$
1. The straight line $l _ { 1 }$ with equation $y = \frac { 3 } { 2 } x - 2$ crosses the $y$-axis at the point $P$. The point $Q$ has coordinates $( 5 , - 3 )$.
The straight line $l _ { 2 }$ is perpendicular to $l _ { 1 }$ and passes through $Q$.
Calculate the coordinates of the mid-point of $P Q$.
Find an equation for $l _ { 2 }$ in the form $a x + b y = c$, where $a$, b and $c$ are integer constants. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $R$.
Calculate the exact coordinates of $R$.
24. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$
Use integration to find $y$ in terms of $x$.
Given that $y = 7$ when $x = 1$, find the value of $y$ at $x = 2$.
25. Find the set of values for $x$ for which
$6 x - 7 < 2 x + 3$,
$2 x ^ { 2 } - 11 x + 5 < 0$,
both $6 x - 7 < 2 x + 3$ and $2 x ^ { 2 } - 11 x + 5 < 0$.
[0pt] [P1 June 2003 Question 2]
26. In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by $x$ phones per month for the next 35 months, so that $( 280 + x )$ phones will be sold in the second month, $( 280 + 2 x )$ in the third month, and so on. Using this model with $x = 5$, calculate
1. the number of phones sold in the 36th month,
2. the total number of phones sold over the 36 months. The shop sets a sales target of 17000 phones to be sold over the 36 months.
  Using the same model,
find the least value of $x$ required to achieve this target.
[0pt] [P1 June 2003 Question 3]
27. The points $A$ and $B$ have coordinates $( 4,6 )$ and $( 12,2 )$ respectively. The straight line $l _ { 1 }$ passes through $A$ and $B$.
Find an equation for $l _ { 1 }$ in the form $a x + b y = c$, where $a$, b and $c$ are integers. The straight line $l _ { 2 }$ passes through the origin and has gradient - 4 .
Write down an equation for $l _ { 2 }$. The lines $l _ { 1 }$ and $l _ { 2 }$ intercept at the point $C$.
Find the exact coordinates of the mid-point of $A C$.
28. For the curve $C$ with equation $y = x ^ { 4 } - 8 x ^ { 2 } + 3$,
find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, The point $A$, on the curve $C$, has $x$-coordinate 1 .
Find an equation for the normal to $C$ at $A$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
[0pt] [P1 June 2003 Question 8*]
29. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$$
Write down the first two terms of the series.
Find the common difference of the series. Given that $n = 50$,
find the sum of the series.
30. (a) Solve the equation $4 x ^ { 2 } + 12 x = 0$. $$f ( x ) = 4 x ^ { 2 } + 12 x + c$$ where $c$ is a constant.
Given that $\mathrm { f } ( x ) = 0$ has equal roots, find the value of $c$ and hence solve $\mathrm { f } ( x ) = 0$.
31. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 1 = 0 \\ & x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{aligned}$$
1. A container made from thin metal is in the shape of a right circular cylinder with height $h \mathrm {~cm}$ and base radius $r \mathrm {~cm}$. The container has no lid. When full of water, the container holds $500 \mathrm {~cm} ^ { 3 }$ of water.
Show that the exterior surface area, $A \mathrm {~cm} ^ { 2 }$, of the container is given by $$A = \pi r ^ { 2 } + \frac { 1000 } { r } .$$ 33. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-15_668_748_358_699}
The points $A$ and $B$ have coordinates $( 2 , - 3 )$ and $( 8,5 )$ respectively, and $A B$ is a chord of a circle with centre $C$, as shown in Fig. 1.
Find the gradient of $A B$. The point $M$ is the mid-point of $A B$.
Find an equation for the line through $C$ and $M$. Given that the $x$-coordinate of $C$ is 4 ,
find the $y$-coordinate of $C$,
show that the radius of the circle is $\frac { 5 \sqrt { } 17 } { 4 }$.
34. The first three terms of an arithmetic series are $p , 5 p - 8$, and $3 p + 8$ respectively.
Show that $p = 4$.
Find the value of the 40th term of this series.
Prove that the sum of the first $n$ terms of the series is a perfect square.
35. $$\mathrm { f } ( x ) = x ^ { 2 } - k x + 9 , \text { where } k \text { is a constant. }$$
Find the set of values of $k$ for which the equation $\mathrm { f } ( x ) = 0$ has no real solutions. Given that $k = 4$,
express $\mathrm { f } ( x )$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are constants to be found,
36. The curve $C$ with equation $y = \mathrm { f } ( x )$ is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0 .$$
Show that, when $x = 8$, the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ is $9 \sqrt { } 2$. The curve $C$ passes through the point $( 4,30 )$.
Using integration, find $\mathrm { f } ( x )$.
37. \section*{Figure 2}
\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-17_687_1074_351_539}
Figure 2 shows the curve with equation $y ^ { 2 } = 4 ( x - 2 )$ and the line with equation $2 x - 3 y = 12$.
The curve crosses the $x$-axis at the point $A$, and the line intersects the curve at the points $P$ and $Q$.
Write down the coordinates of $A$.
Find, using algebra, the coordinates of $P$ and $Q$.
Show that $\angle P A Q$ is a right angle.
38. A sequence is defined by the recurrence relation $$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ where $a$ is a constant.
Given that $a = 20$ and $u _ { 1 } = 3$, find the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$, giving your answers to 2 decimal places.
Given instead that $u _ { 1 } = u _ { 2 } = 3$,
1. calculate the value of $a$,
2. write down the value of $u _ { 5 }$.
  [0pt] [P2 January 2004 Question 2]
  39. The points $A$ and $B$ have coordinates $( 1,2 )$ and $( 5,8 )$ respectively.
Find the coordinates of the mid-point of $A B$.
Find, in the form $y = m x + c$, an equation for the straight line through $A$ and $B$.
40. Giving your answers in the form $a + b \sqrt { 2 }$, where $a$ and $b$ are rational numbers, find
$( 3 - \sqrt { } 8 ) ^ { 2 }$,
$\frac { 1 } { 4 - \sqrt { 8 } }$.
41. The width of a rectangular sports pitch is $x$ metres, $x > 0$. The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,
form a linear inequality in $x$. Given that the area of the pitch must be greater than $4800 \mathrm {~m} ^ { 2 }$,
form a quadratic inequality in $x$.
by solving your inequalities, find the set of possible values of $x$.
42. The curve $C$ has equation $y = x ^ { 2 } - 4$ and the straight line $l$ has equation $y + 3 x = 0$.
In the space below, sketch $C$ and $l$ on the same axes.
Write down the coordinates of the points at which $C$ meets the coordinate axes.
Using algebra, find the coordinates of the points at which $l$ intersects $C$.
43. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
Show that $\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }$.
Hence, or otherwise, differentiate $\mathrm { f } ( x )$ with respect to $x$.

CAIE P1 2019 March Q10

12 marks Standard +0.3

\includegraphics{figure_10} The diagram shows the curve with equation $y = 4x^{\frac{1}{3}}$.

The straight line with equation $y = x + 3$ intersects the curve at points $A$ and $B$. Find the length of $AB$. [6]
The tangent to the curve at a point $T$ is parallel to $AB$. Find the coordinates of $T$. [3]
Find the coordinates of the point of intersection of the normal to the curve at $T$ with the line $AB$. [3]

CAIE P3 2018 June Q8

8 marks Moderate -0.3

The tangent to the curve $y = x^3 - 9x^2 + 24x - 12$ at a point $A$ is parallel to the line $y = 2 - 3x$. Find the equation of the tangent at $A$. [6]
The function f is defined by $\mathrm{f}(x) = x^3 - 9x^2 + 24x - 12$ for $x > k$, where $k$ is a constant. Find the smallest value of $k$ for f to be an increasing function. [2]

Edexcel C1 Q9

12 marks Moderate -0.8

\includegraphics{figure_2} Figure 2 shows part of the curve $C$ with equation $$y = (x - 1)(x^2 - 4).$$ The curve cuts the $x$-axis at the points $P$, $(1, 0)$ and $Q$, as shown in Figure 2.

Write down the $x$-coordinate of $P$ and the $x$-coordinate of $Q$. [2]
Show that $\frac{dy}{dx} = 3x^2 - 2x - 4$. [3]
Show that $y = x + 7$ is an equation of the tangent to $C$ at the point $(-1, 6)$. [2]

The tangent to $C$ at the point $R$ is parallel to the tangent at the point $(-1, 6)$.

Find the exact coordinates of $R$. [5]

Edexcel C1 Specimen Q9

11 marks Easy -1.2

The curve $C$ has equation $y = \text{f}(x)$ and the point $P(3, 5)$ lies on $C$. Given that $$\text{f}(x) = 3x^2 - 8x + 6,$$

find $\text{f}'(x)$. [4]
Verify that the point $(2, 0)$ lies on $C$. [2]

The point $Q$ also lies on $C$, and the tangent to $C$ at $Q$ is parallel to the tangent to $C$ at $P$.

Find the $x$-coordinate of $Q$. [5]

Edexcel M2 2014 January Q10

10 marks Moderate -0.3

The curve $C$ has equation $y = x^3 - 2x^2 - x + 3$ The point $P$, which lies on $C$, has coordinates $(2, 1)$.

Show that an equation of the tangent to $C$ at the point $P$ is $y = 3x - 5$ [5]

The point $Q$ also lies on $C$. Given that the tangent to $C$ at $Q$ is parallel to the tangent to $C$ at $P$,

find the coordinates of the point $Q$. [5]

OCR C1 2013 January Q10

10 marks Standard +0.3

Find the coordinates of the points on the curve $y = \frac{1}{3}x^3 + \frac{9}{x}$ at which the tangent is parallel to the line $y = 8x + 3$. [10]

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.

Find the gradient of the curve at $A$. [4]
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 $A$.

Find the coordinates of $B$. [3]

AQA AS Paper 1 2021 June Q6

7 marks Standard +0.3

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]