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.

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