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

Given that can be written in the form where and are constants,

find the value of , the value of and the value of .

(i) Hence write down the maximum value of ,

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

The curve has equation

The line with equation intersects at two points.

Find the coordinates of these two points.

The finite region bounded by the curve and the line is rotated about the -axis.

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

Given that can be written in the form , where and are integers,

find the value of , the value of and the value of

Given that can be written in the form , where and are integers,

(i) Hence, or otherwise, find the value of for which is a maximum

(ii) find the maximum value of

Show that

Hence, or otherwise, evaluate

Given that

find the value of

The curve has equation where and are integers and

One intersection of with the coordinate axes is at the point with coordinates

The asymptote parallel to the -axis has equation

Find the value of and the value of

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

The straight line with equation has no points of intersection with

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

where and are integers to be found.

The quadratic equation has roots and

Without solving the equation,

form a quadratic equation with integer coefficients, that has roots and

Figure 3 shows part of the curve with equation and part of the curve with equation

The curve and the curve intersect at the point

Find the coordinates of point

The finite region , shown shaded in Figure 3, bounded by the curve , the curve and the straight line is rotated through 360º about the -axis.

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

Given that can be expressed in the form where , and are positive constants

find the value of , the value of and the value of

Given that can be expressed in the form where , and are positive constants

Hence write down the maximum value of

The equation has roots and

Without solving the equation , form a quadratic equation, with integer coefficients, that has roots and

Show that

where and are constants

The equation has roots and where and are the roots of the equation

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

Given that can be written in the form where , and are constants,

find the value of , the value of and the value of

Given that can be written in the form where , and are constants,

Hence, or otherwise, find

(i) the value of for which has its greatest value

(ii) the greatest value of

The curve has equation

The curve with equation intersects curve at two points.

Find the coordinate of each of these two points.

Given that can be written in the form where , and are constants,

The curve has equation

The curve with equation intersects curve at two points.

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

The roots of a quadratic equation are and where

Given that and

show that

(i)

(ii)

Without solving

(i) find the value of

(ii) find the exact value of

Given that where and are positive integers,

find the value of and the value of

Express in the form where , and are rational numbers to be found.

Find

(i) the minimum value of

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

Hence, or otherwise, write down

(i) the minimum value of

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

The equation where is a constant, has real unequal roots.

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

Figure 1 shows part of the curve with equation
where , and are constants.

The points , and with coordinates (– 2, 0), (6, 0) and (4, – 6) respectively lie on

Show that an equation of is

Figure 1 shows part of the curve with equation
where , and are constants.

The line is the normal to at the point

Show that an equation of is

Figure 1 shows part of the curve with equation
where , and are constants.

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

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

Two numbers and are such that

Show that

Given that can vary,

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

Find the minimum value of

The roots of a quadratic equation are and where

Show that

The roots of a quadratic equation are and where

Form a quadratic equation with integer coefficients, that has roots

Given that is a non‑zero constant

curve has equation

straight line has equation

The point is the only point that lies on both and .

Find the value of

Hence, find the coordinates of .

Given that can be written in the form where , and are constants,

find the value of , the value of and the value of .

Hence, or otherwise, find,

(i) the value of for which has a minimum,

(ii) the minimum value of .

The curve has equation .

Find the coordinate of each of the points where crosses the -axis.

The straight line has equation

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

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

sketch the curve and the straight line .

The quadratic equation

where is a positive constant, has roots and

Given that and that where is an integer,

find the value of