1.07m Tangents and normals: gradient and equations

3 In this question you must show detailed reasoning. Find the equation of the normal to the curve \(y = 4 \sqrt { x } - 3 x + 1\) at the point on the curve where \(x = 4\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

15 In this question you must show detailed reasoning. The equation of a curve is $$\ln y + x ^ { 3 } y = 8$$ Find the equation of the normal to the curve at the point where \(y = 1\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are constants to be found.

12 In this question you must show detailed reasoning. Fig. 12 shows part of the graph of \(y = x ^ { 2 } + \frac { 1 } { x ^ { 2 } }\). \begin{figure}[h] \end{figure} The tangent to the curve \(\mathrm { y } = \mathrm { x } ^ { 2 } + \frac { 1 } { \mathrm { x } ^ { 2 } }\) at the point \(\left( 2 , \frac { 17 } { 4 } \right)\) meets the \(x\)-axis at A and meets the \(y\)-axis at B . O is the origin.

1. Find the exact area of the triangle OAB .
2. Use calculus to prove that the complete curve has two minimum points and no maximum point. \section*{END OF QUESTION PAPER}

10 In this question you must show detailed reasoning. The equation of a curve is \(y = \frac { x ^ { 2 } } { 4 } + \frac { 2 } { x } + 1\). A tangent and a normal to the curve are drawn at the point where \(x = 2\). Calculate the area bounded by the tangent, the normal and the \(x\)-axis. \section*{END OF QUESTION PAPER}

14 In this question you must show detailed reasoning. The equation of a curve is \(y = 16 \sqrt { x } + \frac { 8 } { x }\).
Determine the equation of the tangent to the curve at the point where \(x = 4\).

10 In this question you must show detailed reasoning. Fig. 10 shows the curve given parametrically by the equations \(\mathrm { x } = \frac { 1 } { \mathrm { t } ^ { 2 } } , \mathrm { y } = \frac { 1 } { \mathrm { t } ^ { 3 } } - \frac { 1 } { \mathrm { t } }\), for \(t > 0\). \begin{figure}[h] \end{figure}

1. Show that \(\frac { d y } { d x } = \frac { 3 - t ^ { 2 } } { 2 t }\).
2. Find the coordinates of the point on the curve at which the tangent to the curve is parallel to the line \(4 \mathrm { y } + \mathrm { x } = 1\).
3. Find the cartesian equation of the curve. Give your answer in factorised form.

6 In this question you must show detailed reasoning. A curve has equation \(y = \frac { 2 x } { 3 x - 1 } + \sqrt { 5 x + 1 }\). Show that the equation of the tangent to the curve at the point where \(x = 3\) is \(19 x - 32 y + 95 = 0\).

5 The curve \(C\) has equation $$3 x ^ { 2 } - 5 x y + \mathrm { e } ^ { 2 y - 4 } + 6 = 0$$ The point \(P\) with coordinates \(( 1,2 )\) lies on \(C\). The tangent to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\) and the normal to \(C\) at \(P\) meets the \(y\)-axis at the point \(B\). Find the exact area of triangle \(A B P\).

2 Points \(A\) and \(B\) have coordinates \(( 3,0 )\) and \(( 9,8 )\) respectively. The line \(A B\) is a diameter of a circle.

1. Find the coordinates of the centre of the circle.
2. Find the equation of the tangent to the circle at the point \(B\).

1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1\).

The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \sec t , 2 \tan t )\).

1. Use calculus to show that an equation of \(l _ { 1 }\) is $$2 y \sin t = x - 4 \cos t$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\).
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(Q\).
2. Show that, as \(t\) varies, an equation of the locus of \(Q\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$

16. \section*{Figure 3} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 3.

1. Find the coordinates of the centre of the circle.
2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.

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 \(P ( 2,6 )\),

1. find \(y\) in terms of \(x\).
2. Verify that \(C\) passes through the point ( 4,0 ).
3. Find an equation of the tangent to \(C\) at \(P\). The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
4. Calculate the exact \(x\)-coordinate of \(Q\).
   21. $$y = 7 + 10 x ^ { \frac { 3 } { 2 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find \(\int y \mathrm {~d} x\).
   

22.

1. Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\).
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 \(Q\).

1. Calculate the coordinates of the mid-point of \(P Q\).
2. 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 \(R\).
3. Calculate the exact coordinates of \(R\).
   24. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$

1. Use integration to find \(y\) in terms of \(x\).
2. 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

1. \(6 x - 7 < 2 x + 3\),
2. \(2 x ^ { 2 } - 11 x + 5 < 0\),
3. 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. 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,
2. 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 \(B\).

1. 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 .
2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
3. Find the exact coordinates of the mid-point of \(A C\).
   28. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
2. 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 $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$$

1. Write down the first two terms of the series.
2. Find the common difference of the series. Given that \(n = 50\),
3. find the sum of the series.
   

30.

1. Solve the equation \(4 x ^ { 2 } + 12 x = 0\). $$f ( x ) = 4 x ^ { 2 } + 12 x + c$$ where \(c\) is a constant.
2. 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.

1. Find the gradient of \(A B\). The point \(M\) is the mid-point of \(A B\).
2. Find an equation for the line through \(C\) and \(M\). Given that the \(x\)-coordinate of \(C\) is 4 ,
3. find the \(y\)-coordinate of \(C\),
4. 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.

1. Show that \(p = 4\).
2. Find the value of the 40th term of this series.
3. Prove that the sum of the first \(n\) terms of the series is a perfect square.
   35. $$\mathrm { f } ( x ) = x ^ { 2 } - k x + 9 , \text { where } k \text { is a constant. }$$

1. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has no real solutions. Given that \(k = 4\),
2. 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 $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0 .$$

1. 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 \(( 4,30 )\).
2. 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 \(Q\).

1. Write down the coordinates of \(A\).
2. Find, using algebra, the coordinates of \(P\) and \(Q\).
3. 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.

1. 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.
2. 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.

1. Find the coordinates of the mid-point of \(A B\).
2. 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

1. \(( 3 - \sqrt { } 8 ) ^ { 2 }\),
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 ,

1. form a linear inequality in \(x\). Given that the area of the pitch must be greater than \(4800 \mathrm {~m} ^ { 2 }\),
2. form a quadratic inequality in \(x\).
3. 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 \(y + 3 x = 0\).

1. In the space below, sketch \(C\) and \(l\) on the same axes.
2. Write down the coordinates of the points at which \(C\) meets the coordinate axes.
3. Using algebra, find the coordinates of the points at which \(l\) intersects \(C\).
   43. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$

1. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
2. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).

6 The curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-5_428_563_372_740} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and cuts the \(y\)-axis at the point \(B\).

1. 1. State the coordinates of the point \(B\) and hence find the area of the triangle \(A O B\), where \(O\) is the origin.
   2. Find \(\int \left( 3 x ^ { 5 } + 2 x + 5 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve and the line \(A B\).
2. 1. Find the gradient of the curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) at the point \(A ( - 1,0 )\).
   2. Hence find an equation of the tangent to the curve at the point \(A\).

6

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\).
   1. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\) as the product of three linear factors.
2. The curve with equation \(y = x ^ { 3 } - 7 x - 6\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{de4f827d-f237-488a-9177-3d85d0cb1771-4_403_762_651_641} The curve cuts the \(x\)-axis at the point \(A\) and the points \(B ( - 1,0 )\) and \(C ( 3,0 )\).
   1. State the coordinates of the point \(A\).
   2. Find \(\int _ { - 1 } ^ { 3 } \left( x ^ { 3 } - 7 x - 6 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 7 x - 6\) and the \(x\)-axis between \(B\) and \(C\).
   4. Find the gradient of the curve \(y = x ^ { 3 } - 7 x - 6\) at the point \(B\).
   5. Hence find an equation of the normal to the curve at the point \(B\).

5 A circle with centre \(C\) has equation \(( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 25\).

1. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. 1. Verify that the point \(N ( 0 , - 2 )\) lies on the circle.
   2. Sketch the circle.
   3. Find an equation of the normal to the circle at the point \(N\).
3. The point \(P\) has coordinates (2, 6).
   1. Find the distance \(P C\), leaving your answer in surd form.
   2. Find the length of a tangent drawn from \(P\) to the circle.

7 The circle \(S\) has centre \(C ( 8,13 )\) and touches the \(x\)-axis, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-4_444_755_356_641}

1. Write down an equation for \(S\), giving your answer in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. The point \(P\) with coordinates \(( 3,1 )\) lies on the circle.
   1. Find the gradient of the straight line passing through \(P\) and \(C\).
   2. Hence find an equation of the tangent to the circle \(S\) at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
   3. The point \(Q\) also lies on the circle \(S\), and the length of \(P Q\) is 10 . Calculate the shortest distance from \(C\) to the chord \(P Q\).

3 The curve with equation \(y = x ^ { 5 } + 20 x ^ { 2 } - 8\) passes through the point \(P\), where \(x = - 2\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Verify that the point \(P\) is a stationary point of the curve.
3. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(P\).
   2. Hence, or otherwise, determine whether \(P\) is a maximum point or a minimum point.
4. Find an equation of the tangent to the curve at the point where \(x = 1\).

5 A circle with centre \(C\) has equation $$( x - 5 ) ^ { 2 } + ( y + 12 ) ^ { 2 } = 169$$

1. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. 1. Verify that the circle passes through the origin \(O\).
   2. Given that the circle also passes through the points \(( 10,0 )\) and \(( 0 , p )\), sketch the circle and find the value of \(p\).
3. The point \(A ( - 7 , - 7 )\) lies on the circle.
   1. Find the gradient of \(A C\).
   2. Hence find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

6 A curve \(C\) is defined for \(x > 0\) by the equation \(y = x + 1 + \frac { 4 } { x ^ { 2 } }\) and is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-4_545_784_420_628}

1. 1. Given that \(y = x + 1 + \frac { 4 } { x ^ { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
   3. Find an equation of the normal to \(C\) at the point ( 1,6 ).
2. 1. Find \(\int \left( x + 1 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
   2. Hence find the area of the region bounded by the curve \(C\), the lines \(x = 1\) and \(x = 4\) and the \(x\)-axis.

5 A curve is defined for \(x > 0\) by the equation $$y = \left( 1 + \frac { 2 } { x } \right) ^ { 2 }$$ The point \(P\) lies on the curve where \(x = 2\).

1. Find the \(y\)-coordinate of \(P\).
2. Expand \(\left( 1 + \frac { 2 } { x } \right) ^ { 2 }\).
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
4. Hence show that the gradient of the curve at \(P\) is - 2 .
5. Find the equation of the normal to the curve at \(P\), giving your answer in the form \(x + b y + c = 0\), where \(b\) and \(c\) are integers.

9

1. Given that \(y = \frac { 4 x } { 4 x - 3 }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { ( 4 x - 3 ) ^ { 2 } }\), where \(k\) is an integer.
2. 1. Given that \(y = x \ln ( 4 x - 3 )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the tangent to the curve \(y = x \ln ( 4 x - 3 )\) at the point where \(x = 1\).
3. 1. Use the substitution \(u = 4 x - 3\) to find \(\int \frac { 4 x } { 4 x - 3 } \mathrm {~d} x\), giving your answer in terms of \(x\).
   2. By using integration by parts, or otherwise, find \(\int \ln ( 4 x - 3 ) \mathrm { d } x\).

1

1. Differentiate \(\ln x\) with respect to \(x\).
2. Given that \(y = ( x + 1 ) \ln x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Find an equation of the normal to the curve \(y = ( x + 1 ) \ln x\) at the point where \(x = 1\).

2 A curve is defined by the parametric equations $$x = 3 - 4 t \quad y = 1 + \frac { 2 } { t }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the equation of the tangent to the curve at the point where \(t = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Verify that the cartesian equation of the curve can be written as $$( x - 3 ) ( y - 1 ) + 8 = 0$$

1 A curve is defined by the parametric equations $$x = 1 + 2 t , \quad y = 1 - 4 t ^ { 2 }$$

1. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t }\).
      (2 marks)
   2. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find an equation of the normal to the curve at the point where \(t = 1\).
3. Find a cartesian equation of the curve.

5 A curve is defined by the parametric equations \(x = 2 t + \frac { 1 } { t ^ { 2 } } , \quad y = 2 t - \frac { 1 } { t ^ { 2 } }\).

1. At the point \(P\) on the curve, \(t = \frac { 1 } { 2 }\).
   1. Find the coordinates of \(P\).
   2. Find an equation of the tangent to the curve at \(P\).
2. Show that the cartesian equation of the curve can be written as $$( x - y ) ( x + y ) ^ { 2 } = k$$ where \(k\) is an integer.