- The curve C is defined by the equations \(y = x - 4 \sqrt { x }\), \(x \geq 0\)
a) Find \(\frac { d y } { d x }\)
b) Find the coordinates of the turning point of \(C\)
c) Find the coordinates of the two \(x\)-intercepts of \(C\). - The cubic polynomial \(f ( \mathrm { x } )\) is defined by \(f ( x ) = x ^ { 3 } + k x ^ { 2 } + 9 x - 20\)
a) Given that \(( x - 5 )\) is a factor of \(f ( x )\), find the value of \(k\).
b) Show clearly that there is only one real solution to the equations \(f ( x ) = 0\)
c) Given also that the function \(g ( x )\) is defined as \(g ( x ) = \log _ { 2 } x\), \(x > 0\)
Solve \(f g ( x ) = 0\)
3)
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The diagram above shows part of the curve with equation \(y = k \sin \left( x + \frac { \pi } { 3 } \right)\)
The curve meets the y -axis at \(( 0 , \sqrt { 3 } )\) and the x -axis at \(( p , 0 )\) and \(( q , 0 )\)
a) Find the value of the constant \(k\)
b) Find the value of \(p\) and the value of \(q\).
4) On the axes provided, sketch the curve \(y = \tan \left( \frac { x } { 2 } \right) , - 2 \pi \leq x \leq 2 \pi\)
Mark clearly the coordinates of any points the curves crosses the coordinate axes and the equations of any asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{a202ddae-5ecd-4803-9a05-33c37d1880cd-05_661_979_2131_568}
5)
\includegraphics[max width=\textwidth, alt={}, center]{a202ddae-5ecd-4803-9a05-33c37d1880cd-06_643_661_132_667}
The diagram above shows the cross-section of a small shed.
The straight line \(A B\) is vertical and has length 2.12 m . The straight line \(A D\) is horizontal and has length 1.86 m .
The curve \(B C\) is an arc of a circle with centre \(A\), and \(C D\) is a straight line.
Given that the size of angle BAC is 0.65 radians:
a) Find the size of angle CAD giving your answer in radians to 2 dp .
b) Find the area of the cross-section \(A B C D\)
c) Find the perimeter of the cross-section ABCD
6) Prove, from first principles, that if \(f ( x ) = 3 x ^ { 2 }\) then the derivative \(f ^ { \prime } ( x )\) is given by \(f ^ { \prime } ( x ) = 6 x\)
7) Given \(f ( x ) = x ^ { 2 } + 1 , \quad x < - 1\)
Find \(f ^ { - 1 } ( x )\) stating its domain and range
8) The curve \(C\) has equation \(y = x ^ { 2 } ( x - 6 ) + \frac { 4 } { x } , \quad x > 0\)
The points P and Q lie on C and have x -coordinates 1 and 2 respectively.
a) Show that the length of PQ is V 170 .
b) Show that the tangents to C at P and Q are parallel.
c) Find an equation for the normal to C at P , giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
a) Solve, for \(0 \leq x < 360 ^ { \circ }\), giving your answers to 1 decimal place.
$$5 \sin 2 x = 2 \cos 2 x$$
b) Solve for \(0 \leq x \leq 4 \pi\) giving your answers in radians to 3 significant figures.
$$4 \sin ^ { 2 } x = 6 - 9 \cos x$$