Quadratic Functions (Edexcel IGCSE Further Pure Maths): Exam Questions

$\mathrm{f}\left(x\right)=7+4x–2x^2$

Given that $\mathrm{f}\left(x\right)$ can be written in the form $P{(x+Q)}^2+R$ where $P,Q$ and $R$ are constants,

find the value of $P$, the value of $Q$and the value of $R$.

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

(ii) Write down the value of $x$ for which this maximum occurs.

The curve $C$ has equation $y=7+4x–2x^2$

The line $l$ with equation $y=4–x$ intersects $C$ at two points.

Find the $x$ coordinates of these two points.

The finite region bounded by the curve $C$ and the line $l$ is rotated $360°$ about the $x$-axis.

Use algebraic integration to find, to 3 significant figures, the volume of the solid generated.

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

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

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

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

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

(i) Hence, or otherwise, find the value of $x$ for which $\frac{1}{\mathrm{f}\left(x\right)}$is a maximum

(ii) find the maximum value of $\frac{1}{\mathrm{f}\left(x\right)}$

Show that $\sum _{r=1}^n\left(5r-3\right)=\frac{n}{2}\left(5n-1\right)$

Hence, or otherwise, evaluate $\sum _{r=31}^{60}\left(5r-3\right)$

Given that $\sum _{r=1}^n\left(5r-3\right)=3783$

find the value of $n$

The curve $C$ has equation $y=\frac{ax-5}{b-x}$ where $a$ and $b$ are integers and $x\ne b$

One intersection of $C$ with the coordinate axes is at the point with coordinates $\left(\frac{5}{4},0\right)$

The asymptote parallel to the $y$-axis has equation $x=3$

Find the value of $a$ and the value of $b$

Sketch $C$, showing clearly the asymptotes with their equations and the coordinates of the points of intersection with the coordinate axes.

The straight line $l$ with equation $4y-7x=k$ has no points of intersection with $C$

Show, using algebra, that the range of possible values of $k$ can be written as

$m<k<n$

where $m$ and $n$ are integers to be found.

The quadratic equation $3x^2-5x+1=0$ has roots $\alpha$ and $\beta$

Without solving the equation,

form a quadratic equation with integer coefficients, that has roots $\frac{\alpha }{2\beta }$ and $\frac{\beta }{2\alpha }$

Figure 3 shows part of the curve $C$ with equation $y=\frac{1}{4x},x>0$ and part of the curve $S$ with equation $y=2x^2,x⩾0$

The curve $C$ and the curve $S$ intersect at the point $A$

Find the coordinates of point $A$

The finite region $R$ , shown shaded in Figure 3, bounded by the curve $C$, the curve $S$ and the straight line $y=4$ is rotated through 360º about the $y$-axis.

Find, using algebraic integration, the exact volume of the solid formed.

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

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 $r$

$\mathrm{f}\left(x\right)=10+6x-x^2$

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

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

$\mathrm{f}\left(x\right)=10+6x-x^2$

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

Hence, or otherwise, find

(i) the value of $x$ for which $\mathrm{f}\left(x\right)$ has its greatest value

(ii) the greatest value of $\mathrm{f}\left(x\right)$

The curve $C$ has equation $y=\mathrm{f}\left(x\right)$

The curve $S$ with equation $y=x^2-x+13$ intersects curve $C$ at two points.

Find the $x$ coordinate of each of these two points.

$\mathrm{f}\left(x\right)=10+6x-x^2$

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

The curve $C$ has equation $y=\mathrm{f}\left(x\right)$

The curve $S$ with equation $y=x^2-x+13$ intersects curve $C$ at two points.

Use algebraic integration to find the exact area of the finite region bounded by the curve $C$ and the curve $S$

The roots of a quadratic equation $E$ are $\alpha$and $\beta$ where $\alpha >\beta >0$

Given that $\alpha -\beta =2\sqrt{6}$ and ${\alpha }^2+{\beta }^2=30$

show that

(i) $\alpha \beta =3$

(ii) $\alpha +\beta =6$

Without solving $E$

(i) find the value of ${\alpha }^4+{\beta }^4$

(ii) find the exact value of ${\alpha }^4–{\beta }^4$

Given that $a^{4=}P+Q\sqrt{6}$ where $P$ and $Q$ are positive integers,

find the value of $P$ and the value of $Q$

$\mathrm{g}\left(x\right)=2x^2+\frac{1}{2}x-3$

Express $\mathrm{g}\left(x\right)$ in the form $p{\left(x+q\right)}^2$ where $p$, $q$ and $r$ are rational numbers to be found.

Find

(i) the minimum value of $\mathrm{g}\left(x\right)$

(ii) the value of $x$at which this minimum occurs.

$\mathrm{h}\left(x\right)=2x^6+\frac{1}{2}{x}^3-3$

Hence, or otherwise, write down

(i) the minimum value of $\mathrm{h}\left(x\right)$

(ii) the value of $x$ at which this minimum occurs.

The equation $k{x}^2+8x+3k=0$ where $k$ is a constant, has real unequal roots.

Find the set of values of $k$ giving your answer in an exact simplified form.

Figure 1 shows part of the curve $S$ with equation $y=p{x}^2+qx+r$
where $p$, $q$ and $r$ are constants.

The points $A$, $B$ and $P$ with coordinates (– 2, 0), (6, 0) and (4, – 6) respectively lie on $S$

Show that an equation of $S$ is $y=\frac{x^2}{2}-2x-6$

Figure 1 shows part of the curve $S$ with equation $y=p{x}^2+qx+r$
where $p$, $q$ and $r$ are constants.

The line $l$ is the normal to $S$ at the point $P$

Show that an equation of $l$ is $2y+x+8=0$

Figure 1 shows part of the curve $S$ with equation $y=p{x}^2+qx+r$
where $p$, $q$ and $r$ are constants.

The finite region shown shaded in Figure 1 is bounded by $S$ and $l$

Use algebraic integration to find the exact area of the shaded region.

Two numbers $x$ and $y$ are such that $3x-y=4$

$S=5x^3+y^2$

Show that $S=5x^3+9x^2-24x+16$

Given that $x$ can vary,

use calculus to find the value of $x$for which $S$ is a minimum, justifying that this value of $x$ gives a minimum value of $S$

Find the minimum value of $S$

The roots of a quadratic equation are $\alpha$ and $\beta$ where

$\alpha +\beta =-\frac{5}{2}\text{and}{\alpha }^3+{\beta }^3=\frac{115}{8}$

Show that $\alpha \beta =4$

The roots of a quadratic equation are $\alpha$ and $\beta$ where

$\alpha +\beta =-\frac{5}{2}\text{and}{\alpha }^3+{\beta }^3=\frac{115}{8}$

Form a quadratic equation with integer coefficients, that has roots

$\frac{{\alpha }^2+1}{\beta }\text{and}\frac{{\beta }^2+1}{\alpha }$

Given that $k$ is a non‑zero constant

curve $C$ has equation $k{x}^2-xy+\left(k+1\right)x=1$

straight line $l$ has equation $y=\frac{k}{2}x+1$

The point $A$is the only point that lies on both $C$ and $l$.

Find the value of $k$

Hence, find the coordinates of $A$.

$\mathrm{f}\left(\mathrm{x}\right)=8x^2+10x-3$

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

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

Hence, or otherwise, find,

(i) the value of $x$ for which $\mathrm{f}\left(x\right)$ has a minimum,

(ii) the minimum value of $\mathrm{f}\left(x\right)$.

The curve $C$ has equation $y=\mathrm{f}\left(x\right)$.

Find the $x$ coordinate of each of the points where $C$ crosses the $x$-axis.

The straight line $l$ has equation $y=2x+13$

Use algebra to find the coordinates of the two points of intersection of $C$ and $l$ .

Using the same axes and the results of parts (b), (c) and (d),

sketch the curve $C$ and the straight line $l$ .

The quadratic equation

$x^2-4k\sqrt{2} x+2k^4-1=0$

where $k$ is a positive constant, has roots $\alpha$ and $\beta$

Given that ${\alpha }^2+{\beta }^2=66$ and that ${\alpha }^3+{\beta }^3=p\sqrt{2}$ where $p$ is an integer,

find the value of $p$