Solutions to this question by accurate drawing will not be accepted.

Find the \(x\)-coordinates of the points where the curve \(y=(2x-9)(x^2+5)+42\) cuts the \(x\)-axis.

DO NOT USE A CALCULATOR IN THIS QUESTION. The polynomial p is defined by \(\mathrm{p}(x)=a x^{3}-3 x^{2}-3 x+b, \quad\) where \(a\) and \(b\) are constants. (a) Given that \(x=2\) and \(x=-1\) are roots of the equation \(\mathrm{p}(x)=0\), find \(a\) and \(b\).

(b) Solve the equation \(\mathrm{p}(x)=0\).

The polynomial p is such that \(\mathrm{p}(x)=6 x^{3}+x^{2}-12 x+5\). (a) Find the remainder when \(\mathrm{p}(x)\) is divided by \(x-2\).

(b) (i) Show that \(2 x-1\) is a factor of \(\mathrm{p}(x)\).

(ii) Hence write \(\mathrm{p}(x)\) as a product of linear factors.

(iii) Hence solve the equation \(6 \sin ^{3} \theta+\sin ^{2} \theta-12 \sin \theta+5=0\) for \(0^{\circ} \leqslant \theta \leqslant 90^{\circ}\).

Do not use a calculator in this question.

(a) Show that \(x+3\) is a factor of

\(-12+23x+3x^2-2x^3.\)

(b) The curve

\(y=-5+33x+3x^2-2x^3\)

and the line

\(y=10x+7\)

intersect at three points, \(A\), \(B\) and \(C\). These points are such that the \(x\)-coordinate of \(A\) has the least value and the \(x\)-coordinate of \(C\) has the greatest value. Show that \(B\) is the midpoint of \(AC\).

(a) The diagram shows the graph of \(y=|f(x)|\), where \(f(x)\) is a quadratic function. Write down the two possible expressions for \(f(x)\).

(b) The three roots of \(p(x)=0\), where

\(p(x)=5x^3+ax^2+bx-2,\)

are \(x=\frac15\), \(x=n\) and \(x=n+1\), where \(a\) and \(b\) are positive integers and \(n\) is a negative integer. Find \(p(x)\), simplifying your coefficients.

(a) Show that \(x-1\) is a factor of

\(x^3-2x^2-19x+20.\)

(b) Hence write

\(x^3-2x^2-19x+20\)

as a product of linear factors.

(c) Hence find the exact solutions of

\(\mathrm e^{3y}-2\mathrm e^{2y}-19\mathrm e^y+20=0.\)

The polynomial \(\mathrm p\) is given by

\(\mathrm p(x)=36x^3-15x^2-2x+1.\)

(a) Show that \(x=-0.25\) is a root of the equation \(\mathrm p(x)=0\).

(b) Show that the equation \(\mathrm p(x)=0\) has a repeated root.

The three roots of \(p(x)=0\), where

\(p(x)=2x^3+ax^2+bx+c,\)

are \(x=\frac12\), \(x=n\) and \(x=-n\), where \(a\), \(b\), \(c\) and \(n\) are integers. The \(y\)-intercept of the graph of \(y=p(x)\) is \(4\).

Find \(p(x)\), simplifying the coefficients.

The diagram shows the graph of \(y=p(x)\), where \(p(x)\) is a cubic function. Find the two possible expressions for \(p(x)\).

The diagram shows the graph of \(y=\mathrm{f}(x)\), where \(\mathrm{f}(x)\) is a cubic function.

(a) Find an expression for \(\mathrm{f}(x)\).

(b) Hence solve the inequality \(\mathrm{f}(x)\lt 0\).

(a) The cubic equation \(x^3+ax^2+bx-40=0\) has three positive integer roots. Two of the roots are \(2\) and \(4\). Find the other root and the value of each of the integers \(a\) and \(b\).

(b) Do not use a calculator in this question. Solve

\(x^3-5x^2-46x-40=0,\)

given that it has three integer roots, only one of which is positive.

The roots of the equation

\(x^3+ax^2+bx+24=0\)

are \(2\), \(3\) and \(p\), where \(p\) is an integer.

(i) Find the value of \(p\).

(ii) Show that \(a=-1\) and find the value of \(b\).

Given that a curve has equation \(y=x^3-x^2+bx+24\), find, using your value of \(b\),

(iii) \(\displaystyle \frac{dy}{dx}\),

(iv) the integer value of \(x\) for which the gradient of the curve is \(2\), and the corresponding value of \(y\).

The coordinates of point \(P\) on the curve are given by the values of \(x\) and \(y\) found in part (iv).

(v) Find the equation of the tangent to the curve at \(P\).

The cubic equation

\(x^3+ax^2+bx-36=0\)

has a repeated positive integer root.

(i) If the repeated root is \(x=3\), find the other positive root and the value of \(a\) and of \(b\).

(ii) There are other possible values of \(a\) and \(b\) for which the cubic equation has a repeated positive integer root. In each case state all three integer roots of the equation.