[Maximum number: 3]

Solve the equation |5 x-2|=|4 x+9|.

[Maximum number: 3]

Find the gradient of the curve $y=\ln (5 x+1)$ at the point where x=4.

[Maximum number: 3]

Given that $y=\frac{\ln x}{x^{2}}$ , find the exact value of $\frac{\mathrm{d} y}{\mathrm{~d} x}$ when x=e.

A curve is defined for $0<\theta<\frac{1}{2} \pi$ by the parametric equations

x=\tan \theta, \quad y=2 \cos ^{2} \theta \sin \theta

Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=6 \cos ^{5} \theta-4 \cos ^{3} \theta$ .

[Maximum number: 4]

The equation of a curve is $y=\frac{\sin x}{1+\cos x}$ , for $-\pi<x<\pi$ . Show that the gradient of the curve is positive for all x in the given interval.

[Maximum number: 5]

The curve with equation $y=\frac{\mathrm{e}^{-2 x}}{1-x^{2}}$ has a stationary point in the interval -1<x<1. Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ and hence find the x-coordinate of this stationary point, giving the answer correct to 3 decimal places.

The curve with equation $y=\frac{\sin 2 x}{\mathrm{e}^{2 x}}$ has one stationary point in the interval $0 \leqslant x \leqslant \frac{1}{2} \pi$ . Find the exact x-coordinate of this point.

[Maximum number: 6]

Find the gradient of each of the following curves at the point for which x=0.

(a)

$y=3 \sin x+\tan 2 x$

[ 3 ]

(b)

$y=\frac{6}{1+\mathrm{e}^{2 x}}$

[ 3 ]

[Maximum number: 4]

A curve has equation $y=3 \ln (2 x+9)-2 \ln x$ .

(a)

Find the x-coordinate of the stationary point.

[ 4 ]

[Maximum number: 5]

A curve has equation $y=7+4 \ln (2 x+5)$ .
Find the equation of the tangent to the curve at the point ( -2,7 ), giving your answer in the form y=m x+c.