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Maths Test 007

Maths Test 007

Note
Test 007 covers chapter 6 of the Edexcel Maths AS course.
There is no time limit - the average person should complete the test in an hour and a half.
The test will remain available until midnight on 30 July 2020.
The test total is 180 marks.

1. Calculate the missing point.                                                     (10)
   1. Find the midpoint of the line segment joining the points \( ( -2, -7) \) and \( ( -5, 1) \).
   2. Find the midpoint of the line segment joining the points \( ( \sqrt{2} - \sqrt{3}, 3\sqrt{2} + 4\sqrt{3}) \) and \( ( 3\sqrt{2} + \sqrt{3}, -\sqrt{2} + 2\sqrt{3}) \).
   3. The line segment \( AB \) is the diameter of a circle. Given that \( A \) and \( B \) are \( ( - 3, 2) \) and \( ( 1, 2) \) respectively, find the centre of the circle.
   4. Find the midpoint of the line segment joining the points \( ( -2a, -7b) \) and \( ( -5a, 1b) \).
   5. The line segment \( MN \) is the diameter of a circle with centre P. Given that P and M are \( ( 2, -5) \) and \( ( 3, -2) \) respectively, find the coordinates of point \( N \).


2. The line segment \( PQ \) is the diameter of a circle, where \( P \) and \( Q \) are \( ( -3, -4) \) and \( ( 6, 10) \) respectively.                     (10)
   1. Find the centre of the circle.
   2. Show that the centre of the circle lies on the line \( y = 2x \).
   3. Write the equation of the circle.
   4. Find point \( R \) when \( y = 4 \).
   5. Find the equation of the line joining \( R \) and \( P \).


3. A triangle has vertices at \( A(3,5) \), \( B(7,11) \), and \( C(p,q) \). The midpoint of side \( BC \) is \( M(8,5) \).           (20)
   1. Find the values of \( p \) and \( q \).
   2. Find the equation of the straight line joining th midpoint of \( AB \) to the point \( M \).
   3. Show that the line in part \( b \) is parallel to the line \( AC \).


4. The line \( FG \) is the diameter of a circle with centre \( C \), where \( F \) and \( G \) are the points \( (-2, 5) \) and \( (2, 9) \) respectively. The line \( l \) passes through point \( C \) and is perpendicular to the line segment \( FG \).   (20)
   1. Find the equation of \( FG \).
   2. Write the equation of the circle.
   3. Find the equation for line \( l \).
   4. Find the points where \( l \) intersects with the circle.


5. Points \( A \), \( B \), \( C \), and \( D \) have coordinates \( A(-4,-9) \), \( B(6,-3) \), \( C(11,5) \) and \( D(-1,9) \).           (20)
   1. Find the equation of the perpendicular bisector of line segment \( AB \).
   2. Find the equation of the perpendicular bisector of line segment \( CD \).
   3. Find the coordinates of the point of intersection \( E \) of the two perpendicular bisectors.
   4. Write the equation of the circle with centre \( E \) that passes through \( A \).


6. Find the radius and centre of each circle with the following equations.           (20)
   1. \( x^2 + y^2 - 2x + 8y = 8 \)
   2. \( x^2 + y^2 - 6y = 22x - 40 \)
   3. \( x^2 + 5x - y + y^2 + 4 = = 2y + 8 \)
   4. \( 5y - 6x + 2x^2 + 2y^2 = 2x - 3y - 3 \)


7. The line \( y + 2 - 2x = 0 \) meets the circle \( (x - 2)^2 +(y - 2)^2 - 20 = 0 \) at \( A \) and \( B \).           (20)
   1. Find the coordinates of the points \( A \) and \( B \).
   2. Show that \( AB \) is the diameter of the circle.
   3. Find the equation of the perpendicular bisector of \( AB \).
   4. Find the area of triangle \( OAB \).


8. The line with equation \( y = 4x - 1 \) does not intersect the circle with the equation \( x^2 + 2x + y^2 = k \). Find the range of possible values of \( k \).           (10)


9. The line with equation \( 2x + y - 5 = 0 \) is a tangent to the circle with equation \( (x - 3)^2 and (y - p)^2 = 5 \).           (10)
   1. Find the possible values of \( p \).
   2. Write down the centre of the circle.


10. The points \( R \) and \( S \) lie on circle with centre \( C(a,-2) \). The point \( R \) has coordinates \( (2, 3) \) and the point \( S \) has coordinates \( (10,1) \). \( M \) is the midpoint of the line segment \( RS \). The line \( l \) passes through \( M \) and \( C \) and meets the circle at \( A \) and \( B \) where \( A \) and \( M \) are on the same side of the centre.           (20)
    1. Find the equation for \( l \).
    2. Find the value of \( a \).
    3. Find the equation of the circle.
    4. Find the points of the intersection, \( A \) and \( B \), of the line \( l \) and the circle.


11. The circle \( C \) has a centre at \( (6,9) \) and a radius \( \sqrt{50} \). The line \( l1 \) with equation \( x + y - 21 = 0 \) intersects the cirlce at points \( P \) and \( Q \). \( l2 \) and \( l3 \) are tangents at \( P \) and \( Q \) respectively, and \( l4 \) is the perpendicular bisector of of the chord \( PQ \).           (20)
    1. Find the coordinates of the points \( P \) and \( Q \).
    2. Find the equations of the tangents \( l2 \) and \( l3 \).
    3. Find the equation of the perpendicular bisector \( l4 \).
    4. Show that the two tangents \( l2 \) and \( l3 \) and the perpendicular bisector \( l4 \) intersect at a common point \( R \) and find the coordinates of point \( R \).
    5. Name the quadrilateral \( APRQ \) and find the area of \( APRQ \).

End of Maths Test 007