Past Exam Questions

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 \(y=600\).

When \(y^3\) is plotted against \(\ln x\), a straight line graph is obtained, passing through the points \((1,5)\) and \((6,15)\). Find \(y\) in terms of \(x\).

Two variables, \(x\) and \(y\), are related by an equation of the form \(y=A x^{b}\), where \(A\) and \(b\) are constants. The following pairs of values of \(x\) and \(y\) are given.

(b) Use your graph to find the values of \(A\) and \(b\).

Variables \(x\) and \(y\) are such that when \(\mathrm{e}^y\) is plotted against \(x^3\), a straight-line graph is obtained. This line passes through the points \((1,13.5)\) and \((7.5,0.5)\).

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

(b) Find the values of \(x\) for which your equation is valid.

Variables \(x\) and \(y\) are such that when \(\ln y\) is plotted against \(x\), a straight-line graph is obtained. The line passes through the points \((1,\ln15)\) and \((2,\ln75)\).

Show that \(y=Ab^x\), where \(A\) and \(b\) are integers to be found.

When \(\ln y\) is plotted against \(x^3\), a straight line passing through the points \((2,5)\) and \((-8,25)\) is obtained.

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

(b) Find the value of \(x\) when \(y=\mathrm{e}^{25}\).

Variables \(x\) and \(y\) are such that when \(\sqrt{y}\) is plotted against \(x^{3}\) a straight line graph passing through the points \((2,5)\) and \((10,21)\) is obtained.

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

When \(\mathrm e^y\) is plotted against \(x^2\), a straight-line graph with gradient \(-3\) is obtained.

The line passes through the point \((4.30,5.85)\).

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

(b) Find the values of \(x\) for which \(y\) exists.

When \(\mathrm{e}^{5 y}\) is plotted against \(x^{3}\), a straight line passing through the points \((-2.56,4.38)\) and \((6.54,9.84)\) is obtained. (a) Find \(y\) in terms of \(x\).

(b) Find the values of \(x\) for which \(y\) can exist.

The table shows the variables \(x\) and \(y\) which are related by the equation \(y=A b^{x^{2}}\), where \(A\) and \(b\) are constants.

Variables \(y\) and \(x\) are known to be connected by the relationship \(y=A b^{x}\) where \(A\) and \(b\) are constants. The table shows values of \(y\) for certain values of \(x\).

(b) Use your graph to find values of \(A\) and \(b\), giving each to 1 significant figure.

(c) Find an estimate of \(x\) when \(y=1500\).

When \(\mathrm{e}^{2 y}\) is plotted against \(x^{3}\), a straight line graph that passes through the points \((2,5)\) and \((6.4,7.2)\) is obtained. (a) Find \(y\) in terms of \(x\).

(b) Find the values of \(x\) for which \(y\) exists.

An experiment was carried out and values of \(y\) for certain values of \(x\) were recorded. The table shows the values recorded.

The relationship between \(y\) and \(x\) is modelled by \(y=A \mathrm{e}^{k x}, \quad\) where \(A\) and \(k\) are constants. (a) Draw a straight line graph for \(\ln y\) against \(x\).

(b) Find the equation of the line in part (a) and hence find the values of \(A\) and \(k\). Give each value correct to 1 significant figure.

(c) Find the value of \(x\) for which \(y=17\).

The table shows values of the variables \(x\) and \(y\), which are related by an equation of the form

\(y=Ab^{x^2},\)

where \(A\) and \(b\) are constants.

(a) Use the data to draw a straight line graph of \(\ln y\) against \(x^2\).

(b) Use your graph to estimate the values of \(A\) and \(b\). Give your answers correct to 1 significant figure.

(c) Estimate the value of \(y\) when \(x=1.75\).

(d) Estimate the positive value of \(x\) when \(y=20\).

The table shows values of the variables \(x\) and \(y\) which are related by an equation of the form \(y=Ax^b\), where \(A\) and \(b\) are constants.

(a) Use the data to draw a straight line graph of \(\ln y\) against \(\ln x\).

(b) Use your graph to estimate the values of \(A\) and \(b\).

(c) Estimate the value of \(x\) when \(y=100\).

Variables \(x\) and \(y\) are such that when \(\lg y\) is plotted against \(\sqrt{x}\), a straight line passing through the points \((1,5)\) and \((2.5,8)\) is obtained. Show that \(y=A\times b^{\sqrt{x}}\), where \(A\) and \(b\) are constants to be found.

Variables \(P\) and \(T\) are known to be connected by the relationship

\(P=Ab^T,\)

where \(A\) and \(b\) are constants. Values of \(P\) are found for certain values of time, \(T\).

(a) Show that a graph of \(\lg P\) against \(T\) will be a straight line.

(b) The diagram shows the graph of \(\lg P\) against \(T\). The graph passes through \((0,6)\) and \((14,12)\). Find the values of \(A\) and \(b\).

(c) Using the graph or otherwise, find the length of time for which \(P\) is between \(100\) million and \(1000\) million.