Variables \(x\) and \(y\) are such that, when \(\lg(2y+1)\) is plotted against \(x^2\), a straight line graph passing through the points \((1,1)\) and \((2,5)\) is obtained.

(a) Find \(y\) in terms of \(x\).

(b) Find the value of \(y\) when \(x=\frac{\sqrt3}{2}\).

(c) Find the value of \(x\) when \(y=2\).

Variables \(x\) and \(y\) are such that when \(\mathrm e^{4y}\) is plotted against \(x\), a straight line of gradient \(\frac25\), passing through \((10,2)\), is obtained.

(a) Find \(y\) in terms of \(x\).

(b) Find the value of \(y\) when \(x=45\), giving your answer in the form \(\ln p\).

(c) Find the values of \(x\) for which \(y\) can be defined.

When \(\sqrt[3]{y}\) is plotted against \(x^2\), the graph is a straight line passing through the points \((9,8)\) and \((16,1)\). Find \(y\) as a function of \(x\).

It is known that

\(y=A\times10^{bx^2},\)

where \(A\) and \(b\) are constants. When \(\lg y\) is plotted against \(x^2\), a straight line passing through the points \((3.63,5.25)\) and \((4.83,6.88)\) is obtained.

(a) Find the value of \(A\) and of \(b\).

Using your values of \(A\) and \(b\), find

(b) the value of \(y\) when \(x=2\),

(c) the positive value of \(x\) when \(y=4\).

Variables \(x\) and \(y\) are such that when \(y^2\) is plotted against \(e^{2x}\), a straight line is obtained which passes through the points \((1.5,5.5)\) and \((3.7,12.1)\).

Find

(i) \(y\) in terms of \(e^{2x}\),

(ii) the value of \(y\) when \(x=3\),

(iii) the value of \(x\) when \(y=50\).

It is given that \(y=A(10^{bx})\), where \(A\) and \(b\) are constants. The straight line graph obtained when \(\lg y\) is plotted against \(x\) passes through the points \((0.5,2.2)\) and \((1.0,3.7)\).

(i) Find \(A\) and \(b\).

Using your values of \(A\) and \(b\), find

(ii) the value of \(y\) when \(x=0.6\),

(iii) the value of \(x\) when \(y=600\).