Questions covered in Advanced

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. The eccentricity of the ellipse $9x^2 + 25y^2 = 225$ is
2. The distance between the parallel lines $3x + 4y = 9$ and $6x + 8y = 15$ is
3. A line passes through $(2, 2)$ and is perpendicular to the line $3x + y = 3$. Its y-intercept is
4. The focus of the parabola $y^2 - 8x - 8y - 3 = 0$ is
5. The centroid of a triangle is $(2, 3)$ and two of its vertices are $(5, 6)$ and $(-1, 4)$. The third vertex is
6. Find the equation of the circle with center $(2, 2)$ and passing through the point $(4, 5)$.
7. Find the coordinates of the point which divides the line segment joining the points $(1, -2, 3)$ and $(3, 4, -5)$ in the ratio 2:3 internally.
8. Find the equation of the line passing through $(3, 4)$ such that the sum of its intercepts on the axes is 14.
9. Find the equation of the line passing through the intersection of the lines $x + y = 4$ and $2x - 3y = 1$ and perpendicular to the line $x - 2y + 5 = 0$.
10. Find the equation of the hyperbola with foci $(\pm 5, 0)$ and the length of the transverse axis is 8.
11. If $p$ and $q$ are the lengths of perpendiculars from the origin to the lines $x \cos \theta - y \sin \theta = k \cos 2\theta$ and $x \sec \theta + y \csc \theta = k$, respectively, prove that $p^2 + 4q^2 = k^2$.
12. Case Study: An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base.
    (i) Find the equation of the parabola taking the vertex as origin.
    (ii) How wide is it 2 m from the vertex of the parabola?
    (iii) Find the height of the arch at a point 1 m from the center (axis).