Past Exam Questions

(a) Find the coordinates of the stationary points on the curve \(y=(2x+1)(x-3)^2\). Give your answers in exact form.

(b) On the axes below, sketch the graph of \(y=\left|(2x+1)(x-3)^2\right|\), stating the coordinates of the points where the curve meets the axes.

(c) Hence write down the value of the constant \(k\) such that \(\left|(2x+1)(x-3)^2\right|=k\) has exactly 3 distinct solutions.

The diagram shows the graph of the cubic function \(y=\mathrm f(x)\). The intercepts of the curve with the axes are all integers.

(a) Find the set of values of \(x\) for which \(\mathrm f(x)\lt 0\).

(b) Find an expression for \(\mathrm f(x)\).

On the axes below, sketch the graph of

\(y=-\frac14(2x+1)(x-3)(x+4),\)

stating the intercepts with the coordinate axes.

Sketch the graph of

\(y=\left|(x-2)(x+1)(x+2)\right|.\)

Show the coordinates of the points where the graph meets the axes.

(a) Sketch the graph of

\(y=-(x+2)(x-1)(x-6),\)

showing the coordinates of the points where the graph meets the coordinate axes.

(b) Hence solve

\(-(x+2)(x-1)(x-6)\leq0.\)

(a) On the axes, sketch the graph of \(y=(1-x)(x-5)(2x-5)\), stating the intercepts with the axes.

(b) Hence solve the inequality \((1-x)(x-5)(2x-5)\leqslant0\).

The diagram shows the graph of \(y=(x+1)(x-1)(x-2)\). Use the graph to solve the inequality \((x+1)(x-1)(x-2)\lt 1\).

(a) On the axes, sketch the graph of \(y=-\frac{1}{5}(x+2)(2 x-1)(x+5)\), stating the intercepts with the axes.

(b) Hence solve the inequality \(-\frac{1}{5}(x+2)(2 x-1)(x+5) \geqslant 0\).

The diagram shows the graph of the cubic polynomial \(y=f(x)\).

(a) Find an expression for \(f(x)\) in factorised form. Write each linear factor with its coefficients as integers.

(b) Write down the values of \(x\) such that \(f(x)\lt 0\).

The diagram shows the graph of \(y=f(x)\), where \(f(x)=a(x+b)^2(x+c)\) and \(a\), \(b\) and \(c\) are integers.

(a) Find the value of each of \(a\), \(b\) and \(c\).

(b) Hence solve the inequality \(f(x)\leqslant -1\).

(a) On the axes, sketch the graph of

\(y=5(x+1)(3x-2)(x-2),\)

stating the intercepts with the coordinate axes.

(b) Hence find the values of \(x\) for which

\(5(x+1)(3x-2)(x-2)\gt0.\)

(a) Sketch the graph of

\(y=-3(x-2)(x-4)(x+1),\)

showing the coordinates of the points where the curve intersects the coordinate axes.

(b) Hence find the values of \(x\) for which

\(-3(x-2)(x-4)(x+1)\gt 0.\)

The diagram shows the graph of a cubic curve \(y=f(x)\).

(a) Find an expression for \(f(x)\).

(b) Solve \(f(x)\leq0\).

(a) On the axes below, sketch the graph of

\(y=(x-2)(x+1)(3-x),\)

stating the intercepts on the coordinate axes.

(b) Hence write down the values of \(x\) such that

\((x-2)(x+1)(3-x)\gt 0.\)

Solve the equation \(\left|2x^2+x-10\right|=5\).

Solve the equation \(\left|x^2-5x\right|=6\).

Solve the equation \(x^{\frac13}+1=\frac{6}{x^{\frac13}}\).

Solve the equation \(6x^{\frac35}+1=\frac{12}{x^{\frac35}}\), giving your answers correct to 2 decimal places.

(a) Determine whether the equation \(\frac{(4 x+1)(3 x+2)}{5 x-3}=x+1\) has two distinct real roots, two equal roots or no real roots.

(b) Solve the equation \(\frac{12}{\sqrt[3]{x}}-\sqrt[3]{x}=4\).

Do not use a calculator in this question.

Solve the equation

\((2-\sqrt{10})x^2+x+(2+\sqrt{10})=0,\)

giving your answers in the form \(a+b\sqrt{10}\), where \(a\) and \(b\) are rational.