Pure: Proof: Fluency and Exam Drill - Worksheets, Questions and Revision

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A-Level · Pure Mathematics

P1D Pure: Proof: Fluency and Exam Drill

EDEXCEL 9MA0 · Calculator allowed · about 50 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Write the integer 23 in the form 2k + 1, stating the value of k.

(Total for Question 1 is 1 mark)

State whether 78 is odd or even, and write it in the form 2n for an integer n.

(Total for Question 2 is 1 mark)

Expand and fully simplify (n + 1)² - n².

(Total for Question 3 is 1 mark)

Fully factorise n² - n.

(Total for Question 4 is 1 mark)

Expand and fully simplify (2k + 1)(2k - 1).

(Total for Question 5 is 1 mark)

Give one value of n that disproves the statement: "n² + 1 is even for every positive integer n."

(Total for Question 6 is 1 mark)

State the general algebraic form used to represent an even integer, and hence write 132 in that form.

(Total for Question 7 is 1 mark)

Simplify (n + 3) + (n + 4), and state whether the result is odd or even for every integer n.

(Total for Question 8 is 1 mark)

State a value of x, with 0 < x < 1, that could be used as a counter-example to disprove "x² > x for every non-zero real number x."

(Total for Question 9 is 1 mark)

Two odd integers are written as 2a + 1 and 2b + 1, for integers a and b. State, without proof, whether their sum is always odd or always even.

(Total for Question 10 is 1 mark)

Prove that the sum of any two even integers is always even.

(Total for Question 11 is 2 marks)

Show that (n + 2)² - n² = 4n + 4, and state whether this expression is always a multiple of 4.

(Total for Question 12 is 2 marks)

Prove that the product of any two odd integers is always odd.

(Total for Question 13 is 2 marks)

Fully factorise n³ - n, and hence explain why n³ - n is the product of three consecutive integers.

(Total for Question 14 is 2 marks)

Prove that n² - n + 3 is odd for every integer n.

(Total for Question 15 is 3 marks)

Priya claims: "n² - n + 41 is prime for every positive integer n." Use n = 41 to disprove Priya's claim.

(Total for Question 16 is 3 marks)

Prove by contradiction that if n² is a multiple of 3, where n is an integer, then n is a multiple of 3.

(Total for Question 17 is 4 marks)

Priya claims: "For every positive integer n, the value of n² + n + 2 is even."

(a)Verify Priya's claim for n = 1, n = 2 and n = 3.(2)

(b)Prove that Priya's claim is true for every positive integer n.(2)

(Total for Question 18 is 4 marks)

Prove that x² + 4y² ≥ 4xy for all real numbers x and y.

(Total for Question 19 is 3 marks)

A student claims: "n² + n + 5 is prime for every positive integer n."

(a)Verify the student's claim for n = 1, n = 2 and n = 3.(2)

(b)By finding a suitable value of n, prove that the student's claim is false.(3)

(Total for Question 20 is 5 marks)

Mark scheme · P1D Pure: Proof: Fluency and Exam Drill

Question 1

B1 k = 11, so 23 = 2(11) + 1
Answer: k = 11

Question 2

B1 even, 78 = 2(39), n = 39
Answer: Even; n = 39

Question 3

B1 2n + 1 cao
Answer: 2n + 1

Question 4

B1 n(n - 1) cao
Answer: n(n - 1)

Question 5

B1 4k² - 1 cao
Answer: 4k² - 1

Question 6

B1 n = 2 (gives 5, which is odd), or any other even n, oe
Answer: n = 2, since 2² + 1 = 5 is odd

Question 7

B1 2n, with n = 66
Answer: 2n, n = 66

Question 8

B1 2n + 7, which is odd for every integer n
Answer: 2n + 7, odd

Question 9

B1 any value with 0 < x < 1, e.g. x = 0.5
Answer: x = 0.5 (any value with 0 < x < 1)

Question 10

B1 always even
Answer: Always even

Question 11

M1 writes the two even integers as 2a and 2b, for integers a and b, and forms the sum 2a + 2b
A1 writes the sum as 2(a + b), a multiple of 2, hence even, cso
Answer: Proof: 2a + 2b = 2(a + b), which is even

Question 12

M1 expands (n + 2)² to n² + 4n + 4 and subtracts n²
A1 cso: obtains 4n + 4 = 4(n + 1), which is always a multiple of 4
Answer: 4n + 4 = 4(n + 1), always a multiple of 4

Question 13

M1 writes the two odd integers as 2p + 1 and 2q + 1, and expands (2p + 1)(2q + 1) = 4pq + 2p + 2q + 1
A1 writes this as 2(2pq + p + q) + 1, a multiple of 2 plus 1, hence odd, cso
Answer: Proof: (2p+1)(2q+1) = 2(2pq+p+q)+1, which is odd

Question 14

M1 takes out a factor of n, n³ - n = n(n² - 1), then factorises the difference of squares to n(n-1)(n+1)
A1 identifies (n-1), n and (n+1) as three consecutive integers, cao
Answer: n³ - n = (n-1)n(n+1), the product of three consecutive integers

Question 15

M1 factorises n² - n = n(n - 1)
A1 states that n and n-1 are consecutive integers, so exactly one of them is even, hence n(n-1) is always even
A1 cso: since n(n-1) is even, n(n-1)+3 = even + odd = odd for every integer n
Answer: Proof: n² - n + 3 = n(n-1) + 3 = even + 3, which is odd

Question 16

B1 substitutes n = 41
M1 computes 41² - 41 + 41 = 1681
A1 cao: states 1681 = 41 x 41, which is not prime, so Priya's claim is false
Answer: n = 41 gives 1681 = 41 x 41, not prime, so the claim is false

Question 17

M1 assumes, for contradiction, that n is not a multiple of 3, so n = 3k+1 or n = 3k+2 for an integer k
A1 case n = 3k+1: n² = 9k²+6k+1 = 3(3k²+2k)+1, not a multiple of 3
A1 case n = 3k+2: n² = 9k²+12k+4 = 3(3k²+4k+1)+1, not a multiple of 3
A1 cso: in both cases n² is not a multiple of 3, contradicting the assumption that n² is a multiple of 3, so n must be a multiple of 3
Answer: Proof by contradiction: if n² is a multiple of 3 then n is a multiple of 3

Question 18

(a) B1 correctly evaluates n=1: 4, n=2: 8, n=3: 14
(a) B1 states that all three values are even
(a) Answer: n=1: 4, n=2: 8, n=3: 14, all even
(b) M1 recognises n² + n = n(n+1), the product of two consecutive integers, so n(n+1) is always even
(b) A1 cso: n²+n+2 = n(n+1) + 2 = even + even = even for every positive integer n
(b) Answer: Proof: n²+n+2 = n(n+1)+2, and n(n+1) is always even, so the sum is always even

Question 19

B1 states (x - 2y)² ≥ 0 for all real x and y, since a square cannot be negative
M1 expands (x - 2y)² to x² - 4xy + 4y²
A1 rearranges x² - 4xy + 4y² ≥ 0 to x² + 4y² ≥ 4xy, cso
Answer: Proof: since (x-2y)² ≥ 0, x² + 4y² ≥ 4xy

Question 20

(a) B1 correctly evaluates n=1: 7, n=2: 11, n=3: 17
(a) B1 states that 7, 11 and 17 are all prime
(a) Answer: n=1: 7, n=2: 11, n=3: 17, all prime
(b) M1 evaluates n = 4: 16 + 4 + 5 = 25
(b) A1 states 25 = 5 x 5, which is not prime
(b) A1 cso: concludes n = 4 is a counter-example, so the student's claim is false
(b) Answer: n = 4 gives 25 = 5 x 5, not prime, so the claim is false

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Pure: Proof: Fluency and Exam Drill - Worksheets, Questions and Revision

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