A curve C has polar equation r² = 8 cosec 2θ for 0 < θ < ½π. Find a cartesian equation of C. Sketch C. Determine the exact area of the sector bounded by the arc of C between θ = (π/6) and θ = (π/3), the half-line θ = (π/6) and the half-line θ = (π/3). [It is given that ∫ cosec x dx = ln | tan (x/2)| + c.]

The curve C has polar equation r = 2 + 2 cos θ, for 0 ≤ θ ≤ π. Sketch the graph of C. Find the area of the region R e...

Find the area of the region enclosed by the curve with polar equation r = 2(1 + cos θ), for 0 ≤ θ < 2π.

The curve C has polar equation r = a(1 + sin θ), where a is a positive constant and 0 ≤ θ < 2π. Draw a sketch of C. F...

It is given that \( f(n) = 3^{3n} + 6^{n-1} \).

The line \( l_1 \) passes through the point with position vector \( 8\mathbf{i} + 8\mathbf{j} - 7\mathbf{k} \) and is...

The variables x and y are related by the differential equation \( y^2 \frac{d^2y}{dx^2} + 2y \left(\frac{dy}{dx}\righ...