Equations & Identities (Edexcel IGCSE Further Pure Maths): Exam Questions

$\mathrm{f}\left(x\right)=6x^3-13x^2+ax-10$ where $a$ is a constant

Given that $\left(3x-2\right)$ is a factor of $\mathrm{f}\left(x\right)$

show that $a=21$

Hence show algebraically that the curve $y=\mathrm{f}\left(x\right)$ has only one intersection with the $x$-axis.

$\mathrm{f}\left(x\right)=3-4x-9x^2$

Given that $\mathrm{f}\left(x\right)$can be expressed in the form $A-B{\left(x+C\right)}^2$ where $A$, $B$ and $C$ are positive constants

find the value of $A$, the value of $B$and the value of $C$

$\mathrm{f}\left(x\right)=3-4x-9x^2$

Given that $\mathrm{f}\left(x\right)$can be expressed in the form $A-B{\left(x+C\right)}^2$ where $A$, $B$ and $C$ are positive constants

Hence write down the maximum value of $\mathrm{f}\left(x\right)$

The equation $\mathrm{f}\left(x\right)=0$ has roots $\alpha$ and $\beta$

Without solving the equation $\mathrm{f}\left(x\right)=0$, form a quadratic equation, with integer coefficients, that has roots $\frac{3\alpha }{\beta }$ and $\frac{3\beta }{\alpha }$

Show that $(x+y{)}^3=x^3+y^3+3xy(x+y)$

$g\left(x\right)=3x^2+qx+r$ where $q$and $r$ are constants

The equation $\mathrm{g}\left(x\right)=0$ has roots ${\alpha }^2-\beta$ and ${\beta }^2-\alpha$ where $\alpha$ and $\beta$are the roots of the equation $\mathrm{f}\left(x\right)=0$

Using your answer to part (d), find in simplified exact form, the value of $q$ and the value of $r$

Use the factor theorem to show that $\left(4x-1\right)$ is a factor of

$\mathrm{f}\left(x\right)=64x^3-64x^2+3$

Use the factor theorem to show that $\left(4x-1\right)$ is a factor of

$\mathrm{f}\left(x\right)=64x^3-64x^2+3$

Hence, or otherwise, find the exact roots of the equation

$\mathrm{f}\left(x\right)=0$

A geometric series $G$ has first term $a$ and common ratio $r$
The third term of $G$ is 9 and the sum to infinity of $G$ is 192

Show that $64r^3-64r^2+3=0$

A geometric series $G$ has first term $a$ and common ratio $r$
The third term of $G$ is 9 and the sum to infinity of $G$ is 192

Given that $r$ is a rational number

write down the value of $r$

A geometric series $G$ has first term $a$ and common ratio $r$
The third term of $G$ is 9 and the sum to infinity of $G$ is 192

show that $a$ =144

The sum to $n$ terms of $G$ is $S_n$

Using logarithms, find the least value of $n$ such that $S_n>191.9$

$\mathrm{f}\text{'}\left(x\right)=18x^2-2x+13$

Given that $\left(2x-1\right)$ is a factor of $\mathrm{f}\left(x\right)$

show that the curve with equation $y=\mathrm{f}\left(x\right)$has only one intersection with the $x$-axis.

$\mathrm{g}\text{'}\left(x\right)=m{x}^2-10x-37$ where $m$ is an integer

The curve $y=\mathrm{g}\left(x\right)$ passes through the point with coordinates $\left(1,20\right)$

Given that $\left(x-5\right)$is a factor of $\mathrm{g}\left(x\right)$

show that $\mathrm{g}\left(x\right)=2x^3-5x^2-37x+60$

$\mathrm{g}\text{'}\left(x\right)=m{x}^2-10x-37$ where $m$ is an integer

The curve $y=\mathrm{g}\left(x\right)$ passes through the point with coordinates $\left(1,20\right)$

Given that $\left(x-5\right)$is a factor of $\mathrm{g}\left(x\right)$

Hence, or otherwise, use algebra to solve the equation $\mathrm{g}\left(\mathrm{x}\right)=0$

The point $A$ has coordinates (–5, 3), the point $B$ has coordinates (4, 0) and the point $C$ has coordinates (–1, 5).

The line $l$ passes through $C$ and is perpendicular to $AB$.

Find an equation of $l$ .
Give your answer in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

The line $l$ intersects $AB$ at the point $D$.

Show that the coordinates of $D$ are (-2, 2).

The point $A$ has coordinates (–5, 3), the point $B$ has coordinates (4, 0) and the point $C$ has coordinates (–1, 5).

Show that $l$ is not the perpendicular bisector of $AB$.

Find the value of $\tan \angle ABC$.
Give your answer in its simplest form.

$\mathrm{f}\left(x\right)=x^3+p{x}^2+qx+6$ where $p$ and $q$ are constants.

Given that $\left(x-1\right)$ is a factor of $\mathrm{f}\left(x\right)$ and that when $\mathrm{f}\left(x\right)$ is divided by $\left(x+1\right)$ the remainder is 8

(i) show that $p=-2$

(ii) find the value of $q$

Hence, solve the equation $\mathrm{f}\left(x\right)=0$

The region $R$, shown shaded in Figure 2, is bounded by the curve with equation $y=x^2+1$and the curve with equation $x^2+y^2=11$

The two curves intersect at the point $A$ and at the point $B$.

Find the $x$coordinate of the point $A$and the $x$ coordinate of the point $B$.

The region $R$ is rotated through 360° about the $x$‑axis.

Use algebraic integration to find the volume, to 2 decimal places, of the solid generated.