Extra Questions

The following topics are arranged in no particular order. They represent problems perhaps more difficult than encountered in the previous units. Solutions to all of these problems are available upon request.

STEP 1998, Math I, #3: Which of the following statements are true and which are false? justify your answers.

(i)

for all .

(ii) for all real .

(iii) there exists a polynomial

such that for all real .

(iv) for all .

STEP 2013, Math I, #1: (i) Use the substitution

(where ) to find the real root of the equation

(ii) Find all real roots of the following equations:

(a) ;

(b)

STEP 2006, Math II, #5: The notation denotes the greatest integer less than or equal to the real number .

Thus, for example,

, and .

(i) Two curves are given by

and . Sketch the curves, for , on the same axes.

Find the area between the two curves for

, where is a positive integer.

(ii) Two curves are given by

and . Sketch the curves, for , on the same axes.

Show that the area between the two curves for

, where is a positive integer, is

STEP 2009, Math II, 4: The polynomial

is of degree 9 and is exactly divisible by .

(i) Find the value of .

(ii) Show that

is exactly divisible by .

(iii) Given also that

is exactly divisible by , find .

STEP 2010, Math III, #5: The vertices A , B , C and D of a square have coordinates , , and , respectively. The points P and Q have coordinates

and respectively, where . The line CP produced meets DA produced at R and the line CQ produced meets BA produced at S. The line PQ produced meets line RS produced at T. Show that TA is perpendicular to AC.

Explain how, given a square of area , a square of area may eb constructed using only a straight-edge.

[Note: a straight-edge is a ruler with no markings on it; no measurements (an no use of compasses) are allowed in the construction.]

STEP 2006, Math II, #3: (i) Show that

is an integer.

Show also that

Hence determine, with clear reasoning, the value of correct to four decimal places.

(ii) If

is an integer greater than 1, show that , where is a positive integer, differs from the integer nearest

to it by less than

STEP 2002, Math II, #4: Give a sketch to show that, if for , then

(i) By considering show that, if and , then .

(ii) By considering show that, if and , then .

(iii) Show that, if

, and , then

STEP 1998, Math III, #2: Let

(i) Show that .

(ii) Show that .

9iii) Show that when and are positive and hence calculate when and are positive integers.

STEP 2005, Math III, #4: The sequence

satisfies the recurrence relation

where

is a constant.

and , where and are non-zero and , prove by induction that

for

, where is a constant to be found in terms of , and . Hence express and in terms of , , and .

Find conditions on

, and in the three cases:

(i) the sequence

is geometric;

(ii) the sequence

has period 2;

(iii) the sequence

has period 4 .

STEP 2012, Math III, #8: The sequence is defined by , and, for ,

(i) Show that .

(ii) Find the values of in the two cases that arise.

(iii) Prove that, for ,

and hence evaluate the following sum (which you may assume converges):

STEP 2013, Math I, #2: In this question,

denotes the greatest integer that is less than or equal to , so that

and .

The function

is defined, for , by .

(i) Sketch the graph of for (with ).

(ii) By considering the line on your graph, or otherwise, solve the equation .

Solve also the equations and .

(iii) Find the largest root of the equation .

Give necessary and sufficient conditions, in the form of inequalities, for the equation to have exactly roots, where .

STEP 1999, Math III, #7: Let

be a non-zero real number and define a binary operation on the set of real numbers by

Show that the operation

is associative.

Show that is a group, where is the set of all real numbers except for one number which you should identify.

Find a subgroup of which has exactly 2 elements.

STEP 2009, Math III, #8: Let

be a positive integer and let be a non-negative integer.

(i) Use the result

to show that

By writing

as show that

(ii) Let

Show that

and hence evaluate .

(iii) Show that

STEP 2011, Math I, #8: (i) The numbers

and satisfy

. (*)

(a) Show that

. Show also that if and only if . Deduce that unless .

(b) Hence show that the only solutions of (*) for which both

and are integers are and .

(ii) Find all integer solutions of the equation

STEP 2002, Math III, #4: Show that if

and are positive and then .

Show further that if

, then .

Prove that there does not exist a pair of positive integers such that the difference of their cubes is equal to the sum of their squares.

Find all the pairs of integers such that the difference of their cubes is equal to the sum of their squares.

STEP 2004, Math III, #3: Given that when , explain with the aid of a sketch why

By choosing suitable

, and , show that

where

is an integer greater than 1.

Deduce that

Show that

and hence show that

STEP 2007, Math II, #3: By writing

, show that, for ,

(i) Let

(a) Evaluate

(b) Use the substitution

to show that

(ii) Evaluate

STEP 2003, Math III, #7: In the xy-plane, the point A has coordinates and the point B has coordinates , where and are positive. The point P, which is distinct from A and B has coordinates

. X and Y are the feet of the perpendiculars from P to the x-axis and y-axis respectively, and N is the foot of the perpendicular from P to the line AB. Show that the coordinates

of N are given by

, .

Show that, if

then N lies on the line XY.

Give a geometrical interpretation of this result.

STEP 2008, Math III, #2: Let

where

is a positive integer, so that

and

(i) By considering

show that

. (*)

Obtain simplified expressions for and .

(ii) Explain, using (*), why is a polynomial of degree in . Show that in this polynomial the constant term is zero and the sum of the coefficients is 1.

STEP 2024, Math III, #7: (i) Let be a solution of the differential equation

with

and at , and let

Show by differentiation that E is constant and deduce that for all .

(ii) Let be a solution of the differential equation

with

and at , and let

Show that for and deduce that for .

(iii) Let be a solution of the differential equation

with and at . Show that for .

STEP 1999, Math II, #4: By considering the expansion in powers of

of both sides of the identity

show that

where

By considering similar identities, or otherwise, show also that:

(i) If

is an even integer, then

(ii)

STEP 2010, Math II, #3: The first four terms of a sequence are given by , , and . The general term is given by

, (*)

where

, , and are independent of , and is positive.

(i) Show that , and find the values of , and .

(ii) Use (*) to evaluate .

(iii) Evaluate

STEP 2001, Math I, #6: A spherical loaf of bread is cut into parallel slices of equal thickness. Show that, after any number of the slices have been eaten, the area of crust remaining is proportional to the number of slices remaining.

A European ruling decrees that a parallel-sliced loaf can only be referred to as 'crusty' if the ratio of volume

(in cubic metres) of bread remaining to area

(in square metres) of crust remaining after any number of slices have been eaten satisfies

. Show that the radius of a crusty parallel-sliced spherical loaf must be less than metres.

[The area

and volume formed by rotating a curve in the xy-plane round the x-axis from to are given by

, .]

STEP 2003, Math I, #7: Let

be an integer satisfying . Show that and that is divisible by 3.

For each 3-digit number

, where , let be the sum of the hundreds digit, the square of the tens digit, and the cube of the units digit. Find the numbers

such that .

[Hint: write

where , and are the digits of .]

STEP 2001, Math III, #1: Given that , show that

Prove by induction that, for

where

and .

Using this result in the case

, or otherwise, show that the Maclaurin series for begins

and find the next non-zero term.

STEP 1999, Math II, #8: Prove that

(*)

(i) Deduce that, when

is large,

(ii) By differentiating (*) with resepct to

or otherwise, show that, when is large,

[The approximations, valid for small

and may be assumed.]

STEP 2009, Math II, #8: The non-collinear points A , B and C have position vectors

, and , respectively. The points P and Q have position vectors and

, respectively, given by

and

where

and . Draw a diagram showing A , B , C , P and Q .

Given that , find in terms of , and show that, for all values of , the line passes through the fixed point D , with position vector given by

What can be said about the quadrilateral ABDC ?

STEP 2003, Math III, #2: Show that

for

(i) Give the first four terms of the binomial series for .

By choosing a suitable value for

in this series, or otherwise, show that

(ii) Show that

[Note:

is an alternative notation for for , and .]

STEP 2011, Math II, #2: Write down the cubes of the integers .

The positive integers

, and , where , satisfy

where

is a given positive integer.

(i) In the case , show that

Use your sketch to show that only one real value of

STEP 2011, Math II, #1: (i) Sketch the curve

, find two solutions of (*) when

(ii) By considering the case

Use these results to find a solution of (*) when

is a perfect square and that

Deduce that

satisfies

(ii) Determine graphically the number of real values of

that satisfy

Solve the equation.

STEP 2002, Math III, #2: Prove that

and

when

Prove by induction that, for ,

and hence find

Hence prove that

STEP 2004, Math I, #5: The positive integers can be split into five distinct arithmetic progressions, as shown:

A : 1, 6, 11, 16, ...

B : 2, 7, 12, 17, ...

C : 3, 8, 13, 18, ...

D : 4, 9, 14, 19, ...

E : 5, 10, 15, 20, ...

Write down an expression for the value of the general term in each of the five progressions.

Hence prove that the sum of any term in B and any term in C is a term in E.

Prove also that the square of any term in B is a term in D. State and prove a similar claim about the square of every term in C.

(i) Prove that there are no positive integers

and such that

(ii) Prove also that there are no positive integers

and such that

. Explain by means of a sketch, or otherwise, why

when

STEP 1999, Math I, #8: The function satisfies

, or otherwise, show that, if

By considering

and deduce that

deduce that as

, show also that if

Noting that

is an operation that takes polynomials in

STEP 2006, Math III, #8:

. These rules are as follows:

(i)

using the rules that define

which is obtained from

, there is a polynomial called

to polynomials in ; that is, given any polynomial

;

(ii)

for any constant

;

(iii)

for any polynomials and

and any polynomial .

(iv)

;

for any polynomials and

Prove that

and

. Calculate

Using these rules show that, if is a polynomial of degree zero (that is, a constant), then

, then

. You should make it clear whenever you use one of the above rules in your proof.

STEP 2005, Math I, #8: Show that,

for any polynomial

in this result, find the solution of the differential equation

(i) By setting

. Describe geometrically this solution.

(ii) Find the solution

when

for which

STEP 2008, Math I, #1: What does it mean to say that a number

is irrational ?

Prove by contradiction statements A and B below, where

when

are real numbers.

A: If

and

is irrational, then at least one of

is irrational.

B: If

and

is irrational.

Disprove by means of a counterexample statement C below, where

are real numbers.

C: If

and

are irrational, then

and

satisfies

is rational.

STEP 2012, Math I, #7: A sequence of numbers

are irrational, prove that at most one of the numbers

and

is irrational.

If the numbers

where

for all values of

are real. Throughout this question,

, , and are non-zero real numbers.

(i) Show that, if

and

for all values of

and

can be any (non-zero_ real number.

(ii) Show that if

, then

, unless

. Deduce that either

, then

for all values of

and

take certain values that you should identify.

(iii) Show that, if

and

, then

STEP 2010, Math I, #8: (i) Suppose that

, and are integers that satisfy the equation

. Hence show that either

Deduce that either

Explain why

(ii) Suppose that

must be divisible by 3, and show further that both and must also be divisible by 3. Hence show that the only integer solution is

are integers that satisfy the equation

and

By considering the possible final digit of each term, or otherwise, show that

, show that the curve described by B is given parametrically by

and B initially at

is attached to a point on the circumference of a fixed circle of unit radius and centre O. Initially the string is straight and tangent to the circle. The string is then wrapped round the circle until the end b comes into contact with the circle. The string remains taut during the motion, so that a section of the string is in contact with the circumference and the remaining section is straight.

Taking O to be the origin of cartesian coordinates with A at

STEP 2011, Math II, #8: The end A of an inextensible string AB of length

are divisible by 5. Hence show that the only integer solution is

and

, by

are defined for

Find the area swept out by the string (that is, the area between the curve described by B and the semicircle shown in the diagram).

STEP 2009, Math III, #7: (i) The functions

to find the area between the curve and the x-axis for

for which

takes its maximum value on the curve, and sketch the curve.

Use the area integral

, of

is the angle shown in the diagram.

Find the value,

where

and

Prove, for

, that

, by

are defined for

(ii) The functions

Prove, for

and

Find expressions for

, that

is a polynomial of degree

and that

defined by

STEP 2009, Math III, #4: For any given (suitable) function , the Laplace transform of is the function

, where

(ii) Show that the Laplace transform of

, is

, where

(i) Show that the Laplace transform of

, show that

(iv) In the case

(iii) Show that the Laplace transform of

, is

is defined by

, the sequence

STEP 2004, Math III, #6: Given a sequence

and

, where

Using only these four results, find the Laplace transform of

and satisfies

and

has

(i) The sequence

for

Show that

for

Prove that

and satisfies

has

(ii) A sequence

(b) Show that the sequence, with period 2, defined by

, either

and prove that, for each

. (*)

(a) Find

with period 4 which has

satisfies (*).

, and satisfies (*).

STEP 2013, Math III, #8: Evaluate

and the angle

in terms of

be the distance from P to O. Write down an expression for

be the distance from P to D and let

from a fixed line D. For any point P, let

The fixed point O is a distance

is a fixed angle and

where

, where is as shown in the diagram below.

holds, where

The curve E shown in the diagram is such that, for any point P on E, the relation

Each of the

is a fixed number with

. The line

passes through O and the angle between adjacent lines is

lines

. Show that for

intersects E in two two points forming a chord of length

STEP 2007, Math III, #7: The functions

and

, and the real number

, are defined by

to show that

(i) Use the substitution

For this question, do not evaluate any of the above integrals explicitly in terms of inverse trigonometric functions or the number

in terms of

Hence evaluate

and deduce that

(ii) Let

in terms of

Express

, and show that

, show that

By making a substitution in the integral for

Deduce that

(iii) Let

Show that

and hence that

, prove that

, for

STEP 2010, Math III, #7: Given that

Obtain the first three non-zero terms of the Maclaurin series for

and

Conjecture and prove a relation between

and

, and a similar equation relating

and

Obtain a similar equation relating

beginning

may be written as a polynomial in

. Show that, if is an even integer,

State the degree of the polynomial.

STEP 2000, Math III, #5: Given two non-zero vectors

and

Let A , B and C be points with position vectors

we define

(ii) Show that

such that P and Q are the images of A and B under the transformation represented by

matrix

, respectively, none of which are parallel.

(i) Show that there exists a

, and , respectively, no two of which are parallel. Let P , Q and R be points with position vectors , and

is that

matrix

Hence, or otherwise, prove that a necessary and sufficient condition for the points P , Q and R to be the images of points A , B and C under the transformation represented by some

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