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

$f(x)=2x^{3}+px^{2}+qx+12p,q\inℤ$

Given that $(x+3)$ is a factor of $f(x)$ and that when $fʹ(x)$ is divided by $(x+3)$ the remainder is $37$

show that $p=1$and find the value of $q$

$f(x)=2x^{3}+px^{2}+qx+12p,q\inℤ$

Given that $(x+3)$ is a factor of $f(x)$ and that when $fʹ(x)$ is divided by $(x+3)$ the remainder is $37$

hence factorise $f(x)$completely

show that the equation $f(x)=0$ has only one real root.

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

Given that $( 3 x - 2 )$ is a factor of $f(x)$

show that $a=21$

Hence show algebraically that the curve $y=f(x)$ has only one intersection with the $x$-axis.

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

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

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

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

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

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

The equation $\mathrm{f} ( x ) = 0$ has roots $α$ and $β$

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

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

$g ( x ) = 3 x^{2} + q x + r$ where $q$and $r$ are constants

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

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 $( 4 x - 1 )$ is a factor of

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

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

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

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

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

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 $S_{n}>191.9$

$f'(x)=18x^{2}-2x+13$

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

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

$g'(x)=mx^{2}-10x-37$ where $m$ is an integer

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

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

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

$g'(x)=mx^{2}-10x-37$ where $m$ is an integer

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

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

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

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 $\tan \text{angle} A B C$. Give your answer in its simplest form.

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

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

(i) show that $p=-2$

(ii) find the value of $q$

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

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.