Download GCSE Maths (Year 10) 3 Questions about proofs

GCSE Maths (Year 10) 3 Questions about proofs
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Question 1 (6a and 6b)
Question 6 a) Show that (2a – 1)2 – (2b – 1)2 = 4(a – b)(a + b – 1)
Student answer to 6a:
(2a – 1)2 – (2b – 1)2 = (4a2-4a+1)-(4b2-4b+1) = 4a2-4b2-4a+4b
4(a-b)(a+b-1)= (4a-4b)(a+b-1)
=4a2+4ab-4ab-4b2+4b=4a2-4b2-4a+4b
(3)
Question 6 b) Prove that the difference between the squares of any two odd
numbers is a multiple of 8. (You may assume that any odd number can be written in
the form 2r – 1, where r is an integer).
Student answer to 6b:
From above the difference between the squares of any two odd numbers = 4(a-b)(a+b-1)
If either (a-b) or (a+b-1) is even then 4(a-b)(a+b-1) must be a multiple of 8.
If a and b are both odd or both even then a-b is even
If a is odd and b is even or the other way round then (a+b-1) is even
So we have 4 times an even number times something else which must be a multiple of 8
(3)
(Total 6 marks)
GCSE Maths (Year 10)
3 Questions about proofs
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Question 2 (19a and 19b)
Question 19.
PQRS is a quadrilateral.
P
S
Q
R
PQ is parallel to SR.
SP is parallel to RQ.
(a)
Prove that triangle PQS is congruent to triangle RSQ.
Student answer to 19a:
PQ=SR because opposite sides of a parallelogram are equal
PS=QR because opposite sides of a parallelogram are equal
SQ is a shared side
2 of the angles are alternate angles in each triangle so are equal which means the third
angle must also be equal
(3)
Question 19 b) In quadrilateral PQRS, angle SPQ is obtuse.
Explain why PQRS cannot be a cyclic quadrilateral.
Student answer to 19b:
In cyclic quadrilaterals opposite angles add up to 180. These angles are obtuse
(2)
(Total 5 marks)
GCSE Maths (Year 10)
3 Questions about proofs
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Question 3 (45a and 45b)
Question 45.
P
S
T
O
Q
R
Diagram NOT accurately drawn
S and T are points on a circle, centre O.
PSQ and PTR are tangents to the circle.
SOR and TOQ are straight lines.
(a)
Prove that triangle PQT and triangle PRS are congruent.
Student answer to 45a:
Angle P is common to both
Angle PSR and PTQ are both 90 because tangents joined to the centre of a circle at 90
PS=PT because tangents from a point are equal so by ASA the triangles are congruent
(3)
Question 45 b)
Asif says that triangle STQ and triangle STR have equal areas.
(b) Explain why Asif is correct.
Student answer to 45b:
The triangles are congruent because they are the same as the congruent triangles above
with triangle PST removed. They must have equal areas.
(2)
(Total 5 marks)
GCSE Maths (Year 10)
3 Questions about proofs
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