Review practice · Section 1 of 2

Find the cartesian equation of the curve with parametric equations: $x=-4t-\dfrac1t$, $y=3t+\dfrac1{2t}$.4 marks

The curve $C$ is defined by $x=t^2$, $y=\ln t$, $t>0$.

(i) Sketch $C$ for $2\le x\le5$, indicating the coordinates of the endpoints.2 marks

(ii) Find the cartesian equation of $C$.2 marks

(a) Find the cartesian equation for $x=\cos\theta$, $y=2\sin^2\theta$.2 marks

(b) A curve has $x=t^3+t$, $y=2t-1$, $-1\le t\le1$.

(i) Sketch the curve, indicating its axial intercepts.3 marks

(ii) Without using a calculator, find its intersection with $y=\tfrac12$.2 marks

A curve has $x=t/(1+t)$, $y=\ln(\cos t)$, where the printed exclusions are $t\ne-1,\pi/2$. Find $dy/dx$ in terms of $t$.3 marks

A curve has $x=3t-1/t$, $y=t+4/(3t)$. Find the exact value(s) of $t$ at which the gradient is $1/6$.4 marks

The curve $C$ has $x=t^2(t+6)$, $y=t^2+t-6$, $t<0$.

(i) Find the equation(s) of the tangent to $C$ parallel to the y-axis.3 marks

(ii) Sketch $C$, showing its axial intercepts.2 marks