Foundation November 2021 Paper 2 Q30

AQACurrent spec3 marksAngles (Including Parallel Lines)

30 \(A\), \(B\) and \(C\) are three points on a circle.

The radii from \(A\), \(B\) and \(C\) are shown.

Circle with radii to A, B and C. The angle between the radii to A and B is x, the angle between the radii to B and C is 2(2x + 20 degrees), and the reflex angle between the radii to C and A is 5x + 40 degrees

Not drawn accurately

Is \(AC\) a diameter of the circle?

You must show your working. [3 marks]

Mark scheme

Answer	Mark	Comments
Alternative method 1 Shows algebraically that the angles are equal
\(4x + 40\)	M1	may be embedded or on the diagram
\(x + 2(2x + 20)\) or \(x + 4x + 40\)	M1
\(x + 4x + 40 = 5x + 40\) and Yes	A1
Alternative method 2 Derives and solves an equation for angles at a point and substitutes into \(5x + 40\) or \(x + 2(2x + 20)\)
\(4x + 40\)	M1	may be embedded or on the diagram or implied eg implied by \(10x + 80 = 360\)
\(x + 2(2x + 20) + 5x + 40 = 360\) or \(x + 4x + 40 + 5x + 40 = 360\) or \((x =)\) 28	M1	oe equation eg \(10x + 80 = 360\) \((x =)\) 28 may be on the diagram
\(140 + 40 = 180\) and Yes or \(28 + 152 = 180\) and Yes	A1	oe must obtain \((x =)\) 28 from one expression and substitute \((x =)\) 28 into a different expression
Alternative method 3 Assumes line is a diameter. Derives and solves an equation for angles on a line using \(5x + 40\) and substitutes into \(x + 2(2x + 20)\) or \(x + 2(2x + 20) + 5x + 40\)
\(5x + 40 = 180\)	M1
\((x =)\ (180 - 40) \div 5\) or \((x =)\) 28	M1dep	oe \((x =)\) 28 may be on the diagram
\(28 + 152 = 180\) and Yes or \(28 + 152 + 140 + 40 = 360\) and Yes	A1	oe must obtain \((x =)\) 28 from one expression and substitute \((x =)\) 28 into a different expression
Alternative method 4 Assumes line is a diameter. Derives and solves an equation for angles on a line using \(x + 2(2x + 20)\) and substitutes into \(5x + 40\) or \(x + 2(2x + 20) + 5x + 40\)
\(x + 2(2x + 20) = 180\) or \(x + 4x + 40 = 180\)	M1
\((x =)\ (180 - 40) \div 5\) or \((x =)\) 28	M1dep	oe \((x =)\) 28 may be on the diagram
\(140 + 40 = 180\) and Yes or \(28 + 152 + 140 + 40 = 360\) and Yes	A1	oe must obtain \((x =)\) 28 from one expression and substitute \((x =)\) 28 into a different expression
Alternative method 5 Assumes line is a diameter. Derives and solves two equations for angles on a line/angles at a point
\(5x + 40 = 180\) or \(x + 2(2x + 20) = 180\) or \(x + 4x + 40 = 180\) or \(x + 2(2x + 20) + 5x + 40 = 360\) or \(x + 4x + 40 + 5x + 40 = 360\)	M1
\((x =)\ (180 - 40) \div 5\) or \((x =)\) 28	M1dep	oe \((x =)\) 28 may be on the diagram
Obtains \((x =)\) 28 from two equations for angles on a line/angles at a point and Yes	A1

Additional guidance

Choose the scheme that favours the student
Up to M2 may be awarded for correct work, with no or incorrect answer, even if this is seen amongst multiple attempts
Correct response with other incorrect work	M1M1A0
Alt 1 \(2(2x + 20) = 4x + 20\) followed by \(x + 4x + 20\) Alt 1 \(x + 4x + 20\) with \(2(2x + 20) = 4x + 20\) not seen Apply marks in a similar way in alts 2, 4 and 5	M0M1 M0M0
\((x =)\) 28	M1M1
Allow \((x =)\) 28 to be embedded	M1M1
No method marks scored with a value of \(x\) \((\ne 28)\) substituted into \(5x + 40\) and \(x + 2(2x + 20)\) giving the same value	M0M0A0
Yes can be implied eg Alt 1 \(x + 4x + 40 = 5x + 40\) and It is a diameter	M1M1A1