Proof, Perms and Combs Past
Paper Questions [204 marks]
1a. Show that
(2n − 1)2 + (2n + 1)2 = 8n2 + 2, where n ∈ Z.
[2 marks]
1b. Hence, or otherwise, prove that the sum of the squares of any two
consecutive odd integers is even.
[3 marks]
4k or 4k + 1 or
[2 marks]
2a. Explain why any integer can be written in the form
4k + 2 or 4k + 3, where k ∈ Z.
2b. Hence prove that the square of any integer can be written in the form 4t [6 marks]
or 4t + 1, where t ∈ Z+ .
3a. Write down and simplify the first three terms, in ascending powers of x
1
3
[3 marks]
, in the Extended Binomial expansion of (1 − x) .
3
3b. By substituting x = 1 find a rational approximation to √9
.
9
[3 marks]
This question will investigate power series, as an extension to the Binomial
Theorem for negative and fractional indices.
A power series in x is defined as a function of the form
f (x) = a0 + a1 x + a2 x2 + a3 x3 +. . . where the ai ∈ R.
It can be considered as an infinite polynomial.
4a. Expand
(1 + x)5 using the Binomial Theorem.
[2 marks]
This is an example of a power series, but is only a finite power series, since only a
finite number of the a i are non-zero.
4b. Consider the power series 1 − x + x2
− x3 + x4 −. . .
[4 marks]
By considering the ratio of consecutive terms, explain why this series is equal to
(1 + x)−1 and state the values of x for which this equality is true.
4c. Differentiate the equation obtained part (b) and hence, find the first four [2 marks]
−2
terms in a power series for (1 + x) .
4d. Repeat this process to find the first four terms in a power series for
(1 + x)−3 .
[2 marks]
4e. Hence, by recognising the pattern, deduce the first four terms in a
−n
power series for (1 + x) , n ∈ Z+ .
[3 marks]
We will now attempt to generalise further.
(1 + x)q , q ∈ Q can be written as the power series
a0 + a1 x + a2 x2 + a3 x3 +. . ..
Suppose
4f. By substituting x = 0, find the value of a 0 .
[1 mark]
4g. By differentiating both sides of the expression and then substituting
x = 0, find the value of a1 .
[2 marks]
4h. Repeat this procedure to find a 2 and a 3 .
[4 marks]
4i. Hence, write down the first four terms in what is called the Extended
q
Binomial Theorem for (1 + x) , q ∈ Q .
4j. Write down the power series for
1
.
1+x2
[1 mark]
[2 marks]
4k. Hence, using integration, find the power series for arctan x, giving the
first four non-zero terms.
dn
[4 marks]
5. Use mathematical induction to prove that dn (xepx )= pn−1 (px + n)epx [7 marks]
d xn
+
for n ∈ Z , p ∈ Q .
6. Consider the function f (x)
= x e2x , where x ∈ R. The nth derivative of [7 marks]
f (x) is denoted by f (n) (x).
Prove, by mathematical induction, that
7a. Solve the inequality x2
f (n) (x) = (2n x + n2n−1 ) e2x , n ∈ Z+ .
[2 marks]
> 2x + 1.
7b. Use mathematical induction to prove that 2n+1
> n2 for n ∈ Z, n ⩾ 3.
[7 marks]
A team of four is to be chosen from a group of four boys and four girls.
[3 marks]
8a. Find the number of different possible teams that could be chosen.
8b. Find the number of different possible teams that could be chosen, given [2 marks]
that the team must include at least one girl and at least one boy.
[6 marks]
n
9.
∑
Use mathematical induction to prove that r=1 r (r!)
n ∈ Z+ .
= (n + 1) ! − 1, for
[7 marks]
10. Use the principle of mathematical induction to prove that
2
3
n−1
1 + 2 ( 12 ) + 3( 12 ) + 4( 12 ) + … + n( 12 )
11a.
11b.
Express the binomial coefficient
(
= 4−
n+2
, where
2n−1
n ∈ Z+ .
3n + 1
) as a polynomial in n.
3n − 2
Hence find the least value of n for which
(
3n + 1
) > 106 .
3n − 2
(1 − )n > 1 −
[3 marks]
[3 marks]
12. Use mathematical induction to prove that
{n : n ∈ Z+ , n ⩾ 2} where 0 < a < 1.
(1 − a)n > 1 − na for
[7 marks]
Chloe and Selena play a game where each have four cards showing capital letters
A, B, C and D.
Chloe lays her cards face up on the table in order A, B, C, D as shown in the
following diagram.
Selena shuffles her cards and lays them face down on the table. She then turns
them over one by one to see if her card matches with Chloe’s card directly above.
Chloe wins if no matches occur; otherwise Selena wins.
13a. Show that the probability that Chloe wins the game is 3 .
8
[6 marks]
Chloe and Selena repeat their game so that they play a total of 50 times.
Suppose the discrete random variable X represents the number of times Chloe
wins.
13b. Determine the mean of X.
[3 marks]
13c. Determine the variance of X.
[2 marks]
Consider the function fn (x)
= (cos 2x)(cos 4x) … (cos 2n x), n ∈ Z+ .
14a. Determine whether fn is an odd or even function, justifying your
answer.
[2 marks]
14b. By using mathematical induction, prove that
[8 marks]
fn (x) =
sin 2n+1x
,
2n sin 2x
x≠
mπ where
2
m ∈ Z.
14c. Hence or otherwise, find an expression for the derivative of fn (x) with
respect to x.
>1
[3 marks]
14d. Show that, for n > 1, the equation of the tangent to the curve
y = fn (x) at x = π4 is 4x − 2y − π = 0.
[8 marks]
15. Three girls and four boys are seated randomly on a straight bench. Find [5 marks]
the probability that the girls sit together and the boys sit together.
16. Use the method of mathematical induction to prove that 4n + 15n − 1 is [6 marks]
divisible by 9 for n ∈ Z+ .
17. Prove by mathematical induction that
2
3
4
n−1
n
( )+( )+( )+…+(
) = ( ), where n ∈ Z, n ⩾ 3
2
2
2
2
3
[9 marks]
.
18. In a trial examination session a candidate at a school has to take 18
[6 marks]
examination papers including the physics paper, the chemistry paper
and the biology paper. No two of these three papers may be taken consecutively.
There is no restriction on the order in which the other examination papers may be
taken.
Find the number of different orders in which these 18 examination papers may be
taken.
19. Use mathematical induction to prove that n(n2
n ∈ Z+ .
Let
+ 5) is divisible by 6 for [8 marks]
y(x) = xe3x , x ∈ R.
20a. Find dy .
d
[2 marks]
x
20b. Prove by induction that dny
d xn
= n3n−1 e3x + x3n e3x for n ∈ Z+ .
[7 marks]
20c. Find the coordinates of any local maximum and minimum points on the [5 marks]
graph of y(x).
Justify whether any such point is a maximum or a minimum.
20d. Find the coordinates of any points of inflexion on the graph of y(x).
Justify whether any such point is a point of inflexion.
[5 marks]
21a. Show that
1
+√
n
n+1
√
= √n + 1 − √n where n ≥ 0, n ∈ Z.
21b. Hence show that √2 − 1
<
[2 marks]
1
.
√2
[2 marks]
[9 marks]
r=n
21c.
∑
Prove, by mathematical induction, that r=1 1
√r
> √n for n ≥ 2, n ∈ Z.
From a group of five males and six females, four people are chosen.
22a. Determine how many possible groups can be chosen.
[2 marks]
22b. Determine how many groups can be formed consisting of two males
and two females.
[2 marks]
22c. Determine how many groups can be formed consisting of at least one
female.
[2 marks]
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