GCSE Maths (Year 10) 3 Questions about proofs
Text 2
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)(2a-1)=2a2-2a-2a+1=2a2-4a+1
so (2a – 1)2 – (2b – 1) 2= 2a2-4a-2b2-4b
(2b-1)(2b-1)= 2b2-2b-2b+1=2b2-4b+1
4(a-b)= 4a-4b
(4a-4b)(a+b-1)= 4a2+4ab-4a-4ab-4bb-4b= 2a2-4a-2b2-4b
(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:
(2x3)-1=5
32=9 52=25 difference =16 (8x2)
(2x7)-1=13
72=49 132=169 difference=120 (8x15)
(3)
(Total 6 marks)
GCSE Maths 3 Questions about Proofs Text 2
1
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 and SR are parallel and so are equal, both triangles share the line SQ so the side is also
equal. SP and RQ are also parallel and the parallel lines must be equal to make a
parallelogram.
(3)
Question 19 b) In quadrilateral PQRS, angle SPQ is obtuse.
Explain why PQRS cannot be a cyclic quadrilateral.
Student answer to 19b:
Because SPQ and SRQ are equal and so both obtuse. Cyclic quadrilaterals have opposite
angles that add up to 180 and so PQRS is not a cyclic quadrilateral.
(2)
(Total 5 marks)
GCSE Maths 3 Questions about Proofs Text 2
2
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:
Both the triangles share the angle at P and also have an angle of 90 which is subtended from
where the tangents meet the circumference of the circle. Two tangents drawn from the same
point are always equal so PSQ and PTR are equal making PQT and PRS 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:
Because these 2 triangles are congruent as all three lengths are equal- SR is equal to QT,
QS is equal to RT and the triangles both share the side ST. As they are congruent the area
must be equal as well.
(2)
(Total 5 marks)
GCSE Maths 3 Questions about Proofs Text 2
3