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Algebra

Function Graphs

Function Graphs: Level 4 Challenges

         

Given the equation \(4x^2+2\sqrt{3}xy+2y^2=1\).
Through what angles \(\theta \in [0,\pi]\) should the axes be rotated so that the term \(xy\) is removed from the transformed equation?

Give the answer as the sum of all values of \(\theta\)(in degrees).

The graph above shows the figure created by the function \(x^2+(y+x)^2=1\).

Find the area of this tilted ellipse correct to 2 decimal places.

For each integer \(n>1\), let \(F(n)\) be the number of solutions to the equation \(\sin{x} = \sin{(nx)}\) on the interval \([0,\pi]\). What is \(\displaystyle\sum_{n=2}^{2007}\)\(F(n)\)?

If \(f(x)=\dfrac{x^2}{1-\cos x}\), where \(0<x<1\). Which of the following describes the function of \(f(x) \)?

If \(f(x) \) and \( g(x) \) have the same fundamental period, then so does \( f(x) + g(x) \).

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