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Curves with equations \(y = 2 x ^ { \frac { 1 } { 2 } } + 1\) and \(y = \frac { 1 } { 2 } x ^ { 2 } - x + 1\) intersect at \(A ( 0,1 )\) and \(B ( 4,5 )\), as shown in the diagram.
- Find the area of the region between the two curves.
The acute angle between the two tangents at \(B\) is denoted by \(\alpha ^ { \circ }\), and the scales on the axes are the same. - Find \(\alpha\).
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The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\). Tangents touching the circle at points \(B\) and \(C\) pass through the point \(A ( 0,10 )\). - By letting the equation of a tangent be \(y = m x + 10\), find the two possible values of \(m\).
- Find the coordinates of \(B\) and \(C\).
The point \(D\) is where the circle crosses the positive \(x\)-axis. - Find angle \(B D C\) in degrees.
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