Tangent with specified gradient

6 The equation of a curve is \(y = 2 + \sqrt { 25 - x ^ { 2 } }\).
Find the coordinates of the point on the curve at which the gradient is \(\frac { 4 } { 3 }\).

8

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

7

1. A straight line passes through the point \(( 2,0 )\) and has gradient \(m\). Write down the equation of the line.
2. Find the two values of \(m\) for which the line is a tangent to the curve \(y = x ^ { 2 } - 4 x + 5\). For each value of \(m\), find the coordinates of the point where the line touches the curve.
3. Express \(x ^ { 2 } - 4 x + 5\) in the form \(( x + a ) ^ { 2 } + b\) and hence, or otherwise, write down the coordinates of the minimum point on the curve.

4 The line \(y = \frac { x } { k } + k\), where \(k\) is a constant, is a tangent to the curve \(4 y = x ^ { 2 }\) at the point \(P\). Find

1. the value of \(k\),
2. the coordinates of \(P\).

11
\includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-18_428_857_260_644} The diagram shows the curve \(y = ( x - 1 ) ^ { \frac { 1 } { 2 } }\) and points \(A ( 1,0 )\) and \(B ( 5,2 )\) lying on the curve.

1. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
2. Find, showing all necessary working, the equation of the tangent to the curve which is parallel to \(A B\).
3. Find the perpendicular distance between the line \(A B\) and the tangent parallel to \(A B\). Give your answer correct to 2 decimal places.

3 The line \(y = a x + b\) is a tangent to the curve \(y = 2 x ^ { 3 } - 5 x ^ { 2 } - 3 x + c\) at the point \(( 2,6 )\). Find the values of the constants \(a , b\) and \(c\).

6 A straight line has gradient \(m\) and passes through the point ( \(0 , - 2\) ). Find the two values of \(m\) for which the line is a tangent to the curve \(y = x ^ { 2 } - 2 x + 7\) and, for each value of \(m\), find the coordinates of the point where the line touches the curve.

3
\includegraphics[max width=\textwidth, alt={}, center]{55794ceb-2d52-459c-8724-6a6a29ab159a-2_705_737_591_703} The diagram shows the part of the curve \(y = \frac { 1 } { 2 } \tan 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinates of the points on this part of the curve at which the gradient is 4 .

13 In this question you must show detailed reasoning.
The equation of a curve is \(y = 3 x + \frac { 7 } { x } - \frac { 3 } { x ^ { 2 } }\).
Determine the coordinates of the points on the curve where the curve is parallel to the line \(y = 2 x\).
[0pt] [9] END OF QUESTION PAPER

11. A curve \(C\) has equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 5 x + 2\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2 . The \(x\)-coordinate of \(P\) is 3 .
2. Find the \(x\)-coordinate of \(Q\).
3. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. This tangent intersects the coordinate axes at the points \(R\) and \(S\).
4. Find the length of \(R S\), giving your answer as a surd.

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