Core Mathematics 2 Coordinate Geometry

Size: px
Start display at page:

Download "Core Mathematics 2 Coordinate Geometry"

Transcription

1 Core Mathematics 2 Coordinate Geometry Edited by: K V Kumaran Core Mathematics 2 Coordinate Geometry 1

2 Coordinate geometry in the (x, y) plane Coordinate geometry of the circle using the equation of a circle in the form (x a) 2 + (y b) 2 = r 2 and including use of the following circle properties: (i) the angle in a semicircle is a right angle; (ii) the perpendicular from the centre to a chord bisects the chord; (iii) the perpendicularity of radius and tangent. Students should be able to find the radius and the coordinates of the centre of the Circle given the equation of the circle, and vice versa. Core Mathematics 2 Coordinate Geometry 2

3 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint of AB = (2,2) B (3,3) 2 The formula for the midpoint is:- A (1,1) 2 x + x 1 2, y + y Where (x 1, y 1 ) and (x 2, y 2 ) are 2 given points on the line Example 1. If A(3,7) and B(11, 3) Find the midpoint of AB Midpoint of AB = x + x 1 2, y + y = , = 14 2, 4 2 = (7,2) Diameter of circles are often used in this topic because the midpoint will always be the centre the circle. Core Mathematics 2 Coordinate Geometry 3

4 Example 2. If A(2,3) and B is(5,9) and the centre of the circle. If AC is the diameter of the circle find the coordinates of C Midpoint of AB = x + x 1 2, y + y (5, 9) = 2 + x, 3 + y = 2 + x 2 and 9 = 3 + y 2 10 = 2 + x 18 = 3 + y 8 = x 15 = y C = (5,15) if x = 10 4 then y = x 1 y = y = 3 5 The Centre of the circle is (2.5, 3.5) Core Mathematics 2 Coordinate Geometry 4

5 Chords and Perpendicular Lines A chord is a line that passes from one side of a circle to the other but which does not pass through the centre. A perpendicular line always cuts at 90. If it bisects a line then it cuts it exactly in half. It is often called a perpendicular bisector. When questions are talking about this then you need to use the equation of a normal and the midpoints. The perpendicular bisector of a chord always passes through the centre of a circle. The key to success is that you always need to draw a sketch so you know what is going on. Example 1. The Lines AB and CD are chords of a circle. The line y = 3x 11 is the perpendicular bisector of AB. The line y = x 1 is the perpendicular bisector of CD. Find the coordinates of the centre of the circle. We know the perpendicular bisector of a chord passes through the centre so the centre of the circle is where the lines meet! So solve simultaneously Core Mathematics 2 Coordinate Geometry 5

6 y = 3x 11 y = x 1 3x 11 = x 1 4x 11 = 1 4x = 10 x = 10 4 if x = 10 4 then y = x 1 y = y = 3 5 The Centre of the circle is (2.5, 3.5) Core Mathematics 2 Coordinate Geometry 6

7 Distance Between Two Points To work out the distance between two points we use Pythagoras Midpoint of AB = (2,2) B (6,8) 4 A (3,4) 3 The formula for the distance between two points is:- (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Where (x 1, y 1 ) and (x 2, y 2 ) are 2 given points on the line Example 1. PQ is the diameter of a circle where p( 1,3) and Q(6, 3). Find the radius of the circle First we need to remember that Radius = half the Diameter PQ = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 PQ = (6 1) 2 + ( 3 3) 2 PQ = (7 2 + ( 6) 2 ) PQ = 85 Radius = 85 2 Angles in a semicircle An angle in a semicircle is always 90 when one side of the triangle is the diameter and all 3 sides sit on the circumference of the circle Core Mathematics 2 Coordinate Geometry 7

8 Example 2. The points A(2,6), B(5,7) and C(8, 2) lie on a circle. Show that ABC is a right angled triangle and find the area of the triangle Length AB = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 AB = (5 2) 2 + (7 6) 2 AB = AB = 10 Length BC = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 BC = (8 5) 2 + ( 2 7) 2 BC = ( 9) 2 BC = 90 Length AC = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 AC = (8 2) 2 + ( 2 6) 2 AC = ( 8) 2 AC = 100 AC = 10 Using pythagoras to prove ABC is a right angled triangle AC 2 = AB 2 + BC = ( 10) 2 + ( 90) = = 100 This proves the triangle is a right angled triangle Core Mathematics 2 Coordinate Geometry 8

9 Area of a triangle = 1 2 base height A = 1 2 AB BC A = A = A = 15 units 2 Core Mathematics 2 Coordinate Geometry 9

10 Equation of a Circle An equation of a circle is always in the form (x a) 2 + (y b) 2 = r 2 where r is the radius and (a,b) is the centre of the circle. Example 1. If a circle has a radius of 7 and a centre at (2,6), what is the equation of the circle? (x a) 2 + (y b) 2 = r 2 where a = 2, b = 6, r = 7 (x 2) 2 + (y 6) 2 = 7 2 (x 2) 2 + (y 6) 2 = 49 The equation of the cirlce is (x 2) 2 + (y 6) 2 = 49 Example 2. Given the equation (x 2 3) 2 + (y + 7) 2 = 144, find the radius of the centre of the circle. (x a) 2 + (y b) 2 = r 2 where a = 2 3, b = 7, r 2 = 144 centre is (2 3, 7) r 2 = 144 r = 12 as radius cannot be negative we can ingnore the negative value r = 12 Example 3. equation Prove that (1,2) lies on the circumference of the circle which has the (x 2) 2 + (y + 3) 2 = 26 when x = 1 y = 2 (1 2) 2 + (2 + 3) 2 = 26 ( 1) 2 + (5) 2 = = = 26 (1,2) lies on the circumference of the circle Core Mathematics 2 Coordinate Geometry 10

11 The angle between the tangent and a radius is 90. A tangent only touches at one point. This circle theorem is often used in questions as it can relate closely to perpendicular bisectors. Tangents Example 1. The line 4x 3y 40 = 0 touches the circle (x 2) 2 + (y 6) 2 = 100 at P(10,0). Show that the radius at P is perpendicular to the line. This mean the centre A is (2,6) Gradient of AP = y 2 y 1 x 2 x 1 A = = 6 8 P (10,0) gradient of AP is 3 4 Gradient of tangent 4x 3y 40 = 0 4x 40 = 3y 4x 40 3 = y gradient of tangent is 4 3 Using m 1 m 2 = 1 where m 1 = 3 4 and m 2 = = 1 lines are perpendicualr Core Mathematics 2 Coordinate Geometry 11

12 Finding Points of Intersection If you need to find where a circle meets a line then solve the two equations simultaneously. Example 1. Find where the line y = x + 5 meets the circle x 2 + (y 2) 2 = 29 Substitute y = x + 5 into x 2 + (y 2) 2 = 29 x 2 + ((x + 5) 2) 2 = 29 x 2 + (x + 3) 2 = 29 x 2 + x 2 + 6x + 9 = 29 2x 2 + 6x 20 = 0 x 2 + 3x 10 = 0 (x + 5)(x 2) = 0 x = 5 or x = 2 if x = 5 y = x + 5 y = y = 0 ( 5,0) if x = 2 y = x + 5 y = y = 7 (2,7) So the line meets the circle at (-5,0) and (2,7). If you get no solutions when you try and solve two equations then it means the lines do not meet Core Mathematics 2 Coordinate Geometry 12

13 (C2-4.1) Name: Homework Questions 1 Midpoint of a Line 1. If A (9,15) and B (13,21). Find the midpoint of the line AB 2. If CD is the diameter of a circle, where C(2,6) and D (8,14) respectively. Find the coordinates of the centre of the circle. 3. If AB is the diameter of a circle and A(a, 4b) and B(3a, 3b+1) Find the value of a and b given that the centre of the circle is (4,11) 4. The line RS is a diameter of a circle centre (0,4). Given R is (3,10), find the coordinate of S 5. Look at the diagram below. Find the midpoint of AB B (2,15) C (2,7) A( 6,7) Core Mathematics 2 Coordinate Geometry 13

14 (C2-4.2) Name: Homework Questions 2 Perpendicular Bisectors from Chords 1. The Line PQ is a diameter of a circle centre C where P(1,2) and Q(3,4). The line T passes through C and is perpendicular to PQ. a) Find the midpoint of PQ b) Find the gradient of PQ c) Hence find the equation of T 2. The Line RS is a diameter of a circle centre P where R(-1,-2) and S(6,-5). The Line L passes through P and is perpendicular to RS, find the equations of L in the form ax+by+c=0, where a,b and c are integers. Core Mathematics 2 Coordinate Geometry 14

15 3. The Line AB is a chord on the circle with centre (3,-2). If A(6,-2) and B(3,1). The line L is perpendicular to AB and bisects it. Find the equation of L give your answer in the form y=mx+c 4. The points R(-2,5) S(2,1) T(-6,1) lie on the circumference of a circle. Find the equation of RS and RT and hence find the coordinate of the centre of the circle Core Mathematics 2 Coordinate Geometry 15

16 Core Mathematics 2 Coordinate Geometry 16

17 (C2-4.3) Name: Homework Questions 3 Distance Between 2 Points 1. If P(3,-2) and Q(-4,-3). Find the distance between PQ 2. If A (2,6) and B (8,14). Find the distance between AB 3. If CD is the diameter of a circle where C (3, -4), and D(2, 5). Find the radius of the circle. 4. The point (-4,9) lies on the circle centre (-4,3). Find the diameter of the circle 5. Points A(2,-1) and B(-2,-5) lie on the circumference of a circle centre C(2,-5). a) Prove if ABC is a right angled triangle or not b) Calculate the diameter of the circle Core Mathematics 2 Coordinate Geometry 17

18 C2-4.4) Name: Homework Questions 4 Equations of Circles 1. Plot the following graphs on the axis below a) x 2 + y 2 = 16 y 5 b) x 2 + y 2 = x Find the coordinate of the center of the following circle a) x 2 + y 2 = 36 b) (x 3) 2 + y 2 = 56 c) (x + 2) 2 + (y 5) 2 = 76 d) x 2 + (y + 7) 2 = Find the length of the radius of the following circles a) x 2 + y 2 = 100 Core Mathematics 2 Coordinate Geometry 18

19 b) (x + 2) 2 + (y 3) 2 = c) (x 5) 2 + (y + 8) 2 = Draw the graph and state the centre and radius of the circle (x 3) 2 + (y 1) 2 = 36 y x Draw the graph of (x + 2) 2 + y 2 = 16 y x Show that the following circle passess through the point (5,9) (x 3) 2 + (y 4) 2 = 29 Core Mathematics 2 Coordinate Geometry 19

20 7. The point P(2,-3) lies on the circle centre (7,5). a) Find the equation of the circle b) Find the equation of the tangent to the circle at P Core Mathematics 2 Coordinate Geometry 20

21 C2-4.5) Name: Homework Questions 5 Finding Points of Intersections for Circles 1. Find where the circles below meet the x-axis a) (x + 2) 2 + (y 5) 2 = 50 b) ((x 3) 2 + (y + 6) 2 = Find where the circles below meet the y-axis a) (x 3) 2 + (y 1) 2 = 10 b) (x + 4) 2 + (y 5) 2 = Find the points of intersection of the following line and circle x 2 + y 2 = 4 y = x Show that the line below is a tangent to the circle y = 5 (x 1) 2 + (y 2) 2 = 9 Core Mathematics 2 Coordinate Geometry 21

22 Past examination questions 1. The points A and B have coordinates (5, 1) and (13, 11) respectively. (a) Find the coordinates of the mid-point of AB. Given that AB is a diameter of the circle C, (b) find an equation for C. (4) (C2, Jan2005 Q2) 2. The circle C, with centre at the point A, has equation x 2 + y 2 10x + 9 = 0. Find (a) the coordinates of A, (b) the radius of C, (c) the coordinates of the points at which C crosses the x-axis. Given that the line l with gradient 2 7 is a tangent to C, and that l touches C at the point T, (d) find an equation of the line which passes through A and T. (3) (C2, June2005 Q8) Core Mathematics 2 Coordinate Geometry 22

23 3. Figure 1 y B C P O A x In Figure 1, A(4, 0) and B(3, 5) are the end points of a diameter of the circle C. Find (a) the exact length of AB, (b) the coordinates of the midpoint P of AB, (c) an equation for the circle C. (3) (C2, Jan2006 Q3) 4. The line joining points ( 1, 4) and (3, 6) is a diameter of the circle C. Find an equation for C. (6) (C2, Jan2007 Q3) Core Mathematics 2 Coordinate Geometry 23

24 5. Figure 1 y y = 3x 4 P(2, 2) C Q O x The line y = 3x 4 is a tangent to the circle C, touching C at the point P(2, 2), as shown in Figure 1. The point Q is the centre of C. (a) Find an equation of the straight line through P and Q. (3) Given that Q lies on the line y = 1, (b) show that the x-coordinate of Q is 5, (c) find an equation for C. (1) (4) (C2, May 2006 Q7) 6. y B Core Mathematics 2 Coordinate Geometry 24 M (3, 1)

25 Figure 3 The points A and B lie on a circle with centre P, as shown in Figure 3. The point A has coordinates (1, 2) and the mid-point M of AB has coordinates (3, 1). The line l passes through the points M and P. (a) Find an equation for l. (4) Given that the x-coordinate of P is 6, (b) use your answer to part (a) to show that the y-coordinate of P is 1, (c) find an equation for the circle. (1) (4) (C2, May 2007 Q7) Core Mathematics 2 Coordinate Geometry 25

26 7. A circle C has centre M(6, 4) and radius 3. (a) Write down the equation of the circle in the form (x a) 2 + (y b) 2 = r 2. y T Figure 3 C 3 M (6, 4) Q P (12, 6) x Figure 3 shows the circle C. The point T lies on the circle and the tangent at T passes through the point P (12, 6). The line MP cuts the circle at Q. (b) Show that the angle TMQ is radians to 4 decimal places. (4) The shaded region TPQ is bounded by the straight lines TP, QP and the arc TQ, as shown in Figure 3. (c) Find the area of the shaded region TPQ. Give your answer to 3 decimal places. (5) (C2, Jan 2008 Q8) 8. The circle C has centre (3, 1) and passes through the point P(8, 3). (a) Find an equation for C. (b) Find an equation for the tangent to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers. (5) Core Mathematics 2 Coordinate Geometry 26 (4) C2, June 2008 Q5)

27 9. Figure 2 The points P( 3, 2), Q(9, 10) and R(a, 4) lie on the circle C, as shown in Figure 2. Given that PR is a diameter of C, (a) show that a = 13, (b) find an equation for C. 10. The circle C has equation (3) (5) (C2, Jan 2009 Q5) x 2 + y 2 6x + 4y = 12 (a) Find the centre and the radius of C. (5) The point P( 1, 1) and the point Q(7, 5) both lie on C. (b) Show that PQ is a diameter of C. The point R lies on the positive y-axis and the angle PRQ = 90. (c) Find the coordinates of R. (4) (C2, June 2009 Q6) Core Mathematics 2 Coordinate Geometry 27

28 11. y C O N x A 12 B P Figure 3 Figure 3 shows a sketch of the circle C with centre N and equation (a) Write down the coordinates of N. (x 2) 2 + (y + 1) 2 = (b) Find the radius of C. (1) The chord AB of C is parallel to the x-axis, lies below the x-axis and is of length 12 units as shown in Figure 3. (c) Find the coordinates of A and the coordinates of B. (5) (d) Show that angle ANB = 134.8, to the nearest 0.1 of a degree. The tangents to C at the points A and B meet at the point P. (e) Find the length AP, giving your answer to 3 significant figures. (C2, Jan 2010 Q8) Core Mathematics 2 Coordinate Geometry 28

29 12. The circle C has centre A(2,1) and passes through the point B(10, 7). (a) Find an equation for C. (4) The line l 1 is the tangent to C at the point B. (b) Find an equation for l 1. (4) The line l 2 is parallel to l 1 and passes through the mid-point of AB. Given that l 2 intersects C at the points P and Q, (c) find the length of PQ, giving your answer in its simplest surd form. (3) (C2, June 2010 Q10) 13. The points A and B have coordinates ( 2, 11) and (8, 1) respectively. Given that AB is a diameter of the circle C, (a) show that the centre of C has coordinates (3, 6), (b) find an equation for C. (c) Verify that the point (10, 7) lies on C. (d) Find an equation of the tangent to C at the point (10, 7), giving your answer in the form y = mx + c, where m and c are constants. (4) (C2, Jan 2011 Q9) (1) (4) (1) 14. The circle C has equation x 2 + y 2 + 4x 2y 11 = 0. Find (a) the coordinates of the centre of C, (b) the radius of C, (c) the coordinates of the points where C crosses the y-axis, giving your answers as simplified surds. (4) (C2, May2011 Q4) Core Mathematics 2 Coordinate Geometry 29

30 15. A circle C has centre ( 1, 7) and passes through the point (0, 0). Find an equation for C. (4) (C2, Jan 2012 Q2) 16. Figure 1 The circle C with centre T and radius r has equation x 2 + y 2 20x 16y = 0. (a) Find the coordinates of the centre of C. (b) Show that r = 5 (3) The line L has equation x = 13 and crosses C at the points P and Q as shown in Figure 1. (c) Find the y coordinate of P and the y coordinate of Q. (3) Given that, to 3 decimal places, the angle PTQ is radians, (d) find the perimeter of the sector PTQ. (3) (C2, May 2012 Q3) Core Mathematics 2 Coordinate Geometry 30

31 17. The circle C has equation x 2 + y 2 20x 24y = 0. The centre of C is at the point M. (a) Find (i) the coordinates of the point M, (ii) the radius of the circle C. (5) N is the point with coordinates (25, 32). (b) Find the length of the line MN. The tangent to C at a point P on the circle passes through point N. (c) Find the length of the line NP. 18. (C2, Jan 2013 Q5) Figure 4 The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4. (a) Write down an equation for the circle C, that is shown in Figure 4. (3) Core Mathematics 2 Coordinate Geometry 31

32 A line through the point P(8, 7) is a tangent to the circle C at the point T. (b) Find the length of PT. 19. The circle C, with centre A, passes through the point P with coordinates ( 9, 8) and the point Q with coordinates (15, 10). (3) (C2, May 2013 Q10) Given that PQ is a diameter of the circle C, (a) find the coordinates of A, (b) find an equation for C. A point R also lies on the circle C. Given that the length of the chord PR is 20 units, (3) 20. (c) find the length of the shortest distance from A to the chord PR. Give your answer as a surd in its simplest form. (d) Find the size of the angle ARQ, giving your answer to the nearest 0.1 of a degree. (C2, May 20134_R Q10) Figure 3 Figure 3 shows a circle C with centre Q and radius 4 and the point T which lies on C. The tangent to C at the point T passes through the origin O and OT = 6 5. Given that the coordinates of Q are (11, k), where k is a positive constant, Core Mathematics 2 Coordinate Geometry 32

33 (a) find the exact value of k, (b) find an equation for C. 21. A circle C with centre at the point (2, 1) passes through the point A at (4, 5). (a) Find an equation for the circle C. (3) (C2, May 2014 Q9) (3) (b) Find an equation of the tangent to the circle C at the point A, giving your answer in the form ax + by + c = 0, where a, b and c are integers. (4) (C2, May 2015 Q2) Core Mathematics 2 Coordinate Geometry 33

Edexcel New GCE A Level Maths workbook Circle.

Edexcel New GCE A Level Maths workbook Circle. Edexcel New GCE A Level Maths workbook Circle. Edited by: K V Kumaran kumarmaths.weebly.com 1 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint

More information

Circles - Edexcel Past Exam Questions. (a) the coordinates of A, (b) the radius of C,

Circles - Edexcel Past Exam Questions. (a) the coordinates of A, (b) the radius of C, - Edecel Past Eam Questions 1. The circle C, with centre at the point A, has equation 2 + 2 10 + 9 = 0. Find (a) the coordinates of A, (b) the radius of C, (2) (2) (c) the coordinates of the points at

More information

Circles, Mixed Exercise 6

Circles, Mixed Exercise 6 Circles, Mixed Exercise 6 a QR is the diameter of the circle so the centre, C, is the midpoint of QR ( 5) 0 Midpoint = +, + = (, 6) C(, 6) b Radius = of diameter = of QR = of ( x x ) + ( y y ) = of ( 5

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com 1. The diagram above shows the sector OA of a circle with centre O, radius 9 cm and angle 0.7 radians. Find the length of the arc A. Find the area of the sector OA. The line AC shown in the diagram above

More information

5 Find an equation of the circle in which AB is a diameter in each case. a A (1, 2) B (3, 2) b A ( 7, 2) B (1, 8) c A (1, 1) B (4, 0)

5 Find an equation of the circle in which AB is a diameter in each case. a A (1, 2) B (3, 2) b A ( 7, 2) B (1, 8) c A (1, 1) B (4, 0) C2 CRDINATE GEMETRY Worksheet A 1 Write down an equation of the circle with the given centre and radius in each case. a centre (0, 0) radius 5 b centre (1, 3) radius 2 c centre (4, 6) radius 1 1 d centre

More information

CIRCLES, CHORDS AND TANGENTS

CIRCLES, CHORDS AND TANGENTS NAME SCHOOL INDEX NUMBER DATE CIRCLES, CHORDS AND TANGENTS KCSE 1989 2012 Form 3 Mathematics Working Space 1. 1989 Q24 P2 The figure below represents the cross section of a metal bar. C A 4cm M 4cm B The

More information

AQA IGCSE FM "Full Coverage": Equations of Circles

AQA IGCSE FM Full Coverage: Equations of Circles AQA IGCSE FM "Full Coverage": Equations of Circles This worksheet is designed to cover one question of each type seen in past papers, for each AQA IGCSE Further Maths topic. This worksheet was automatically

More information

b UVW is a right-angled triangle, therefore VW is the diameter of the circle. Centre of circle = Midpoint of VW = (8 2) + ( 2 6) = 100

b UVW is a right-angled triangle, therefore VW is the diameter of the circle. Centre of circle = Midpoint of VW = (8 2) + ( 2 6) = 100 Circles 6F a U(, 8), V(7, 7) and W(, ) UV = ( x x ) ( y y ) = (7 ) (7 8) = 8 VW = ( 7) ( 7) = 64 UW = ( ) ( 8) = 8 Use Pythagoras' theorem to show UV UW = VW 8 8 = 64 = VW Therefore, UVW is a right-angled

More information

Edexcel New GCE A Level Maths workbook

Edexcel New GCE A Level Maths workbook Edexcel New GCE A Level Maths workbook Straight line graphs Parallel and Perpendicular lines. Edited by: K V Kumaran kumarmaths.weebly.com Straight line graphs A LEVEL LINKS Scheme of work: a. Straight-line

More information

DEPARTMENT OF MATHEMATICS

DEPARTMENT OF MATHEMATICS DEPARTMENT OF MATHEMATICS AS level Mathematics Core mathematics 1 C1 2015-2016 Name: Page C1 workbook contents Indices and Surds Simultaneous equations Quadratics Inequalities Graphs Arithmetic series

More information

"Full Coverage": Pythagoras Theorem

Full Coverage: Pythagoras Theorem "Full Coverage": Pythagoras Theorem This worksheet is designed to cover one question of each type seen in past papers, for each GCSE Higher Tier topic. This worksheet was automatically generated by the

More information

NAME: Date: HOMEWORK: C1. Question Obtained. Total/100 A 80 B 70 C 60 D 50 E 40 U 39

NAME: Date: HOMEWORK: C1. Question Obtained. Total/100 A 80 B 70 C 60 D 50 E 40 U 39 NAME: Date: HOMEWORK: C1 Question Obtained 1 2 3 4 5 6 7 8 9 10 Total/100 A 80 B 70 C 60 D 50 E 40 U 39 1. Figure 2 y A(1, 7) B(20, 7) D(8, 2) O x C(p, q) The points A(1, 7), B(20, 7) and C(p, q) form

More information

y hsn.uk.net Straight Line Paper 1 Section A Each correct answer in this section is worth two marks.

y hsn.uk.net Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. Straight Line Paper 1 Section Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of a?.

More information

Core Mathematics C12

Core Mathematics C12 Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C12 Advanced Subsidiary Tuesday 10 January 2017 Morning Time: 2 hours

More information

Possible C2 questions from past papers P1 P3

Possible C2 questions from past papers P1 P3 Possible C2 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P1 January 2001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.

More information

The gradient of the radius from the centre of the circle ( 1, 6) to (2, 3) is: ( 6)

The gradient of the radius from the centre of the circle ( 1, 6) to (2, 3) is: ( 6) Circles 6E a (x + ) + (y + 6) = r, (, ) Substitute x = and y = into the equation (x + ) + (y + 6) = r + + + 6 = r ( ) ( ) 9 + 8 = r r = 90 = 0 b The line has equation x + y = 0 y = x + y = x + The gradient

More information

CIRCLES MODULE - 3 OBJECTIVES EXPECTED BACKGROUND KNOWLEDGE. Circles. Geometry. Notes

CIRCLES MODULE - 3 OBJECTIVES EXPECTED BACKGROUND KNOWLEDGE. Circles. Geometry. Notes Circles MODULE - 3 15 CIRCLES You are already familiar with geometrical figures such as a line segment, an angle, a triangle, a quadrilateral and a circle. Common examples of a circle are a wheel, a bangle,

More information

81-E If set A = { 2, 3, 4, 5 } and set B = { 4, 5 }, then which of the following is a null set? (A) A B (B) B A (C) A U B (D) A I B.

81-E If set A = { 2, 3, 4, 5 } and set B = { 4, 5 }, then which of the following is a null set? (A) A B (B) B A (C) A U B (D) A I B. 81-E 2 General Instructions : i) The question-cum-answer booklet contains two Parts, Part A & Part B. ii) iii) iv) Part A consists of 60 questions and Part B consists of 16 questions. Space has been provided

More information

Recognise the Equation of a Circle. Solve Problems about Circles Centred at O. Co-Ordinate Geometry of the Circle - Outcomes

Recognise the Equation of a Circle. Solve Problems about Circles Centred at O. Co-Ordinate Geometry of the Circle - Outcomes 1 Co-Ordinate Geometry of the Circle - Outcomes Recognise the equation of a circle. Solve problems about circles centred at the origin. Solve problems about circles not centred at the origin. Determine

More information

x n+1 = ( x n + ) converges, then it converges to α. [2]

x n+1 = ( x n + ) converges, then it converges to α. [2] 1 A Level - Mathematics P 3 ITERATION ( With references and answers) [ Numerical Solution of Equation] Q1. The equation x 3 - x 2 6 = 0 has one real root, denoted by α. i) Find by calculation the pair

More information

Mark scheme Pure Mathematics Year 1 (AS) Unit Test 2: Coordinate geometry in the (x, y) plane

Mark scheme Pure Mathematics Year 1 (AS) Unit Test 2: Coordinate geometry in the (x, y) plane Mark scheme Pure Mathematics Year 1 (AS) Unit Test : Coordinate in the (x, y) plane Q Scheme Marks AOs Pearson 1a Use of the gradient formula to begin attempt to find k. k 1 ( ) or 1 (k 4) ( k 1) (i.e.

More information

2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW

2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Find the values of k for which the line y 6 is a tangent to the curve k 7 y. Find also the coordinates of the point at which this tangent touches the curve.

More information

Additional Mathematics Lines and circles

Additional Mathematics Lines and circles Additional Mathematics Lines and circles Topic assessment 1 The points A and B have coordinates ( ) and (4 respectively. Calculate (i) The gradient of the line AB [1] The length of the line AB [] (iii)

More information

Math & 8.7 Circle Properties 8.6 #1 AND #2 TANGENTS AND CHORDS

Math & 8.7 Circle Properties 8.6 #1 AND #2 TANGENTS AND CHORDS Math 9 8.6 & 8.7 Circle Properties 8.6 #1 AND #2 TANGENTS AND CHORDS Property #1 Tangent Line A line that touches a circle only once is called a line. Tangent lines always meet the radius of a circle at

More information

Straight Line. SPTA Mathematics Higher Notes

Straight Line. SPTA Mathematics Higher Notes H Straight Line SPTA Mathematics Higher Notes Gradient From National 5: Gradient is a measure of a lines slope the greater the gradient the more steep its slope and vice versa. We use the letter m to represent

More information

P1 Chapter 6 :: Circles

P1 Chapter 6 :: Circles P1 Chapter 6 :: Circles jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 11 th August 2017 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework

More information

Core Mathematics C2 (R) Advanced Subsidiary

Core Mathematics C2 (R) Advanced Subsidiary Paper Reference(s) 6664/01R Edexcel GCE Core Mathematics C2 (R) Advanced Subsidiary Thursday 22 May 2014 Morning Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae (Pink)

More information

9.7 Extension: Writing and Graphing the Equations

9.7 Extension: Writing and Graphing the Equations www.ck12.org Chapter 9. Circles 9.7 Extension: Writing and Graphing the Equations of Circles Learning Objectives Graph a circle. Find the equation of a circle in the coordinate plane. Find the radius and

More information

+ 2gx + 2fy + c = 0 if S

+ 2gx + 2fy + c = 0 if S CIRCLE DEFINITIONS A circle is the locus of a point which moves in such a way that its distance from a fixed point, called the centre, is always a constant. The distance r from the centre is called the

More information

Core Mathematics 2 Radian Measures

Core Mathematics 2 Radian Measures Core Mathematics 2 Radian Measures Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Radian Measures 1 Radian Measures Radian measure, including use for arc length and area of sector.

More information

Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours

Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours 1. Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Mark scheme Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question

More information

(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2

(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2 CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5

More information

Circle geometry investigation: Student worksheet

Circle geometry investigation: Student worksheet Circle geometry investigation: Student worksheet http://topdrawer.aamt.edu.au/geometric-reasoning/good-teaching/exploringcircles/explore-predict-confirm/circle-geometry-investigations About these activities

More information

Practice Test Geometry 1. Which of the following points is the greatest distance from the y-axis? A. (1,10) B. (2,7) C. (3,5) D. (4,3) E.

Practice Test Geometry 1. Which of the following points is the greatest distance from the y-axis? A. (1,10) B. (2,7) C. (3,5) D. (4,3) E. April 9, 01 Standards: MM1Ga, MM1G1b Practice Test Geometry 1. Which of the following points is the greatest distance from the y-axis? (1,10) B. (,7) C. (,) (,) (,1). Points P, Q, R, and S lie on a line

More information

Ch 10 Review. Multiple Choice Identify the choice that best completes the statement or answers the question.

Ch 10 Review. Multiple Choice Identify the choice that best completes the statement or answers the question. Ch 10 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In the diagram shown, the measure of ADC is a. 55 b. 70 c. 90 d. 180 2. What is the measure

More information

Candidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.

Candidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required. Revision Checklist Unit C2: Core Mathematics 2 Unit description Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; differentiation;

More information

C=2πr C=πd. Chapter 10 Circles Circles and Circumference. Circumference: the distance around the circle

C=2πr C=πd. Chapter 10 Circles Circles and Circumference. Circumference: the distance around the circle 10.1 Circles and Circumference Chapter 10 Circles Circle the locus or set of all points in a plane that are A equidistant from a given point, called the center When naming a circle you always name it by

More information

DISCRIMINANT EXAM QUESTIONS

DISCRIMINANT EXAM QUESTIONS DISCRIMINANT EXAM QUESTIONS Question 1 (**) Show by using the discriminant that the graph of the curve with equation y = x 4x + 10, does not cross the x axis. proof Question (**) Show that the quadratic

More information

chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?

chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true? chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "

More information

DEPARTMENT OF MATHEMATICS

DEPARTMENT OF MATHEMATICS DEPARTMENT OF MATHEMATICS AS level Mathematics Core mathematics 2 - C2 2015-2016 Name: Page C2 workbook contents Algebra Differentiation Integration Coordinate Geometry Logarithms Geometric series Series

More information

ANALYTICAL GEOMETRY Revision of Grade 10 Analytical Geometry

ANALYTICAL GEOMETRY Revision of Grade 10 Analytical Geometry ANALYTICAL GEOMETRY Revision of Grade 10 Analtical Geometr Let s quickl have a look at the analtical geometr ou learnt in Grade 10. 8 LESSON Midpoint formula (_ + 1 ;_ + 1 The midpoint formula is used

More information

Part (1) Second : Trigonometry. Tan

Part (1) Second : Trigonometry. Tan Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,

More information

S56 (5.3) Recurrence Relations.notebook September 09, 2015

S56 (5.3) Recurrence Relations.notebook September 09, 2015 Daily Practice 31.8.2015 Q1. Write down the equation of a circle with centre (-1, 4) and radius 5 Q2. Given the circle with equation (x 4) 2 + (y + 5) 2 = 40. Find the equation of the tangent to this circle

More information

1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3.

1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3. Higher Maths Non Calculator Practice Practice Paper A. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient

More information

MATHEMATICS Unit Pure Core 2

MATHEMATICS Unit Pure Core 2 General Certificate of Education June 2008 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Thursday 15 May 2008 9.00 am to 10.30 am For this paper you must have: an 8-page answer book

More information

You must have: Mathematical Formulae and Statistical Tables, calculator

You must have: Mathematical Formulae and Statistical Tables, calculator Write your name here Surname Other names Pearson Edexcel Level 3 GCE Centre Number Mathematics Advanced Paper 2: Pure Mathematics 2 Candidate Number Specimen Paper Time: 2 hours You must have: Mathematical

More information

Test Corrections for Unit 1 Test

Test Corrections for Unit 1 Test MUST READ DIRECTIONS: Read the directions located on www.koltymath.weebly.com to understand how to properly do test corrections. Ask for clarification from your teacher if there are parts that you are

More information

Chapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in

Chapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in Chapter - 10 (Circle) Key Concept * Circle - circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle - Circle having same centre called concentric circle.

More information

UNIT 6. BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle. The Circle

UNIT 6. BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle. The Circle UNIT 6 BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle The Circle 1 Questions How are perimeter and area related? How are the areas of polygons and circles

More information

10. Circles. Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Marks (2) Marks (2)

10. Circles. Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Marks (2) Marks (2) 10. Circles Q 1 True or False: It is possible to draw two circles passing through three given non-collinear points. Mark (1) Q 2 State the following statement as true or false. Give reasons also.the perpendicular

More information

A-Level Notes CORE 1

A-Level Notes CORE 1 A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is

More information

Circle. Paper 1 Section A. Each correct answer in this section is worth two marks. 5. A circle has equation. 4. The point P( 2, 4) lies on the circle

Circle. Paper 1 Section A. Each correct answer in this section is worth two marks. 5. A circle has equation. 4. The point P( 2, 4) lies on the circle PSf Circle Paper 1 Section A Each correct answer in this section is worth two marks. 1. A circle has equation ( 3) 2 + ( + 4) 2 = 20. Find the gradient of the tangent to the circle at the point (1, 0).

More information

SM2H Unit 6 Circle Notes

SM2H Unit 6 Circle Notes Name: Period: SM2H Unit 6 Circle Notes 6.1 Circle Vocabulary, Arc and Angle Measures Circle: All points in a plane that are the same distance from a given point, called the center of the circle. Chord:

More information

Chapter 10. Properties of Circles

Chapter 10. Properties of Circles Chapter 10 Properties of Circles 10.1 Use Properties of Tangents Objective: Use properties of a tangent to a circle. Essential Question: how can you verify that a segment is tangent to a circle? Terminology:

More information

Core Mathematics 2 Unit C2 AS

Core Mathematics 2 Unit C2 AS Core Mathematics 2 Unit C2 AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics C2.1 Unit description Algebra and functions; coordinate geometry in the (, y) plane; sequences

More information

0114ge. Geometry Regents Exam 0114

0114ge. Geometry Regents Exam 0114 0114ge 1 The midpoint of AB is M(4, 2). If the coordinates of A are (6, 4), what are the coordinates of B? 1) (1, 3) 2) (2, 8) 3) (5, 1) 4) (14, 0) 2 Which diagram shows the construction of a 45 angle?

More information

Trans Web Educational Services Pvt. Ltd B 147,1st Floor, Sec-6, NOIDA, UP

Trans Web Educational Services Pvt. Ltd B 147,1st Floor, Sec-6, NOIDA, UP Solved Examples Example 1: Find the equation of the circle circumscribing the triangle formed by the lines x + y = 6, 2x + y = 4, x + 2y = 5. Method 1. Consider the equation (x + y 6) (2x + y 4) + λ 1

More information

MEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions

MEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions MEI Core Basic Algebra Section : Basic algebraic manipulation and solving simple equations Notes and Examples These notes contain subsections on Manipulating algebraic expressions Collecting like terms

More information

LLT Education Services

LLT Education Services 8. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle. (a) 4 cm (b) 3 cm (c) 6 cm (d) 5 cm 9. From a point P, 10 cm away from the

More information

physicsandmathstutor.com

physicsandmathstutor.com physicsadmathstutor.com 8. The circle C, with cetre at the poit A, has equatio x 2 + y 2 10x + 9 = 0. Fid (a) the coordiates of A, (b) the radius of C, (c) the coordiates of the poits at which C crosses

More information

Verulam School Mathematics. Year 9 Revision Material (with answers) Page 1

Verulam School Mathematics. Year 9 Revision Material (with answers) Page 1 Verulam School Mathematics Year 9 Revision Material (with answers) Page 1 Q1. (a) Simplify a 2 a 4 Answer... (b) Simplify b 9 b 3 Answer... (c) Simplify c 5 c c 5 Answer... (Total 3 marks) Q2. (a) Expand

More information

Higher Mathematics Course Notes

Higher Mathematics Course Notes Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that

More information

MATHEMATICS: PAPER II

MATHEMATICS: PAPER II NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 2014 MATHEMATICS: PAPER II EXAMINATION NUMBER Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of

More information

Cambridge International Examinations Cambridge Ordinary Level

Cambridge International Examinations Cambridge Ordinary Level Cambridge International Examinations Cambridge Ordinary Level *8790810596* ADDITIONAL MATHEMATICS 4037/13 Paper 1 October/November 2017 2 hours Candidates answer on the Question Paper. No Additional Materials

More information

Circles Unit Test. Secondary Math II

Circles Unit Test. Secondary Math II Circles Unit Test Secondary Math II 1. Which pair of circles described are congruent to each other? Circle M has a radius of 6 m; Circle N has a diameter of 10 m. Circle J has a circumference of in; Circle

More information

1. Peter cuts a square out of a rectangular piece of metal. accurately drawn. x + 2. x + 4. x + 2

1. Peter cuts a square out of a rectangular piece of metal. accurately drawn. x + 2. x + 4. x + 2 1. Peter cuts a square out of a rectangular piece of metal. 2 x + 3 Diagram NOT accurately drawn x + 2 x + 4 x + 2 The length of the rectangle is 2x + 3. The width of the rectangle is x + 4. The length

More information

CBSE MATHEMATICS (SET-2)_2019

CBSE MATHEMATICS (SET-2)_2019 CBSE 09 MATHEMATICS (SET-) (Solutions). OC AB (AB is tangent to the smaller circle) In OBC a b CB CB a b CB a b AB CB (Perpendicular from the centre bisects the chord) AB a b. In PQS PQ 4 (By Pythagoras

More information

VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER)

VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER) BY:Prof. RAHUL MISHRA Class :- X QNo. VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER) CIRCLES Subject :- Maths General Instructions Questions M:9999907099,9818932244 1 In the adjoining figures, PQ

More information

This section will help you revise previous learning which is required in this topic.

This section will help you revise previous learning which is required in this topic. Higher Portfolio Circle Higher 10. Circle Revision Section This section will help you revise previous learning which is required in this topic. R1 I can use the distance formula to find the distance between

More information

National Quali cations

National Quali cations H 2018 X747/76/11 National Quali cations Mathematics Paper 1 (Non-Calculator) THURSDAY, 3 MAY 9:00 AM 10:10 AM Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given

More information

Class IX : Math Chapter 11: Geometric Constructions Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A.

Class IX : Math Chapter 11: Geometric Constructions Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A. 1 Class IX : Math Chapter 11: Geometric Constructions Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A. Required : To construct an angle at A equal to POQ. 1.

More information

CFE National 5 Resource

CFE National 5 Resource Pegasys Educational Publishing CFE National 5 Resource Unit Expressions and Formulae Homework Exercises Homework exercises covering all the Unit topics + Answers + Marking Schemes Pegasys 0 National 5

More information

Topic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths

Topic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is

More information

1. Draw and label a diagram to illustrate the property of a tangent to a circle.

1. Draw and label a diagram to illustrate the property of a tangent to a circle. Master 8.17 Extra Practice 1 Lesson 8.1 Properties of Tangents to a Circle 1. Draw and label a diagram to illustrate the property of a tangent to a circle. 2. Point O is the centre of the circle. Points

More information

Full Question Paper Maths, X Class

Full Question Paper Maths, X Class Full Question Paper Maths, X Class Time: 3 Hrs MM: 80 Instructions: 1. All questions are compulsory. 2. The questions paper consists of 34 questions divided into four sections A,B,C and D. Section A comprises

More information

Solutionbank C2 Edexcel Modular Mathematics for AS and A-Level

Solutionbank C2 Edexcel Modular Mathematics for AS and A-Level file://c:\users\buba\kaz\ouba\c_rev_a_.html Eercise A, Question Epand and simplify ( ) 5. ( ) 5 = + 5 ( ) + 0 ( ) + 0 ( ) + 5 ( ) + ( ) 5 = 5 + 0 0 + 5 5 Compare ( + ) n with ( ) n. Replace n by 5 and

More information

St. Anne s Diocesan College. Grade 12 Core Mathematics: Paper II September Time: 3 hours Marks: 150

St. Anne s Diocesan College. Grade 12 Core Mathematics: Paper II September Time: 3 hours Marks: 150 St. Anne s Diocesan College Grade 12 Core Mathematics: Paper II September 2018 Time: 3 hours Marks: 150 Please read the following instructions carefully: 1. This question paper consists of 21 pages and

More information

Sample Question Paper Mathematics First Term (SA - I) Class IX. Time: 3 to 3 ½ hours

Sample Question Paper Mathematics First Term (SA - I) Class IX. Time: 3 to 3 ½ hours Sample Question Paper Mathematics First Term (SA - I) Class IX Time: 3 to 3 ½ hours M.M.:90 General Instructions (i) All questions are compulsory. (ii) The question paper consists of 34 questions divided

More information

Key competencies (student abilities)

Key competencies (student abilities) Year 9 Mathematics Cambridge IGCSE Mathematics is accepted by universities and employers as proof of mathematical knowledge and understanding. Successful Cambridge IGCSE Mathematics candidates gain lifelong

More information

Unit 3: Number, Algebra, Geometry 2

Unit 3: Number, Algebra, Geometry 2 Unit 3: Number, Algebra, Geometry 2 Number Use standard form, expressed in standard notation and on a calculator display Calculate with standard form Convert between ordinary and standard form representations

More information

MEMO MATHEMATICS: PAPER II

MEMO MATHEMATICS: PAPER II MEMO CLUSTER PAPER 2016 MATHEMATICS: PAPER II Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 28 pages and an Information Sheet of 2 pages(i-ii).

More information

Secondary School Certificate Examination Syllabus MATHEMATICS. Class X examination in 2011 and onwards. SSC Part-II (Class X)

Secondary School Certificate Examination Syllabus MATHEMATICS. Class X examination in 2011 and onwards. SSC Part-II (Class X) Secondary School Certificate Examination Syllabus MATHEMATICS Class X examination in 2011 and onwards SSC Part-II (Class X) 15. Algebraic Manipulation: 15.1.1 Find highest common factor (H.C.F) and least

More information

International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL EXAMINATIONS PAPER 2 MAY/JUNE SESSION 2002

International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL EXAMINATIONS PAPER 2 MAY/JUNE SESSION 2002 International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL EXAMINATIONS ADDITIONAL MATHEMATICS 0606/2 PAPER 2 MAY/JUNE SESSION 2002 2 hours Additional materials: Answer paper Electronic

More information

Example 1: Finding angle measures: I ll do one: We ll do one together: You try one: ML and MN are tangent to circle O. Find the value of x

Example 1: Finding angle measures: I ll do one: We ll do one together: You try one: ML and MN are tangent to circle O. Find the value of x Ch 1: Circles 1 1 Tangent Lines 1 Chords and Arcs 1 3 Inscribed Angles 1 4 Angle Measures and Segment Lengths 1 5 Circles in the coordinate plane 1 1 Tangent Lines Focused Learning Target: I will be able

More information

AS Mathematics Assignment 8 Due Date: Friday 15 th February 2013

AS Mathematics Assignment 8 Due Date: Friday 15 th February 2013 AS Mathematics Assignment 8 Due Date: Friday 15 th February 2013 NAME GROUP: MECHANICS/STATS Instructions to Students All questions must be attempted. You should present your solutions on file paper and

More information

MATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment -II. Revision CLASS X Prepared by

MATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment -II. Revision CLASS X Prepared by MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Summative Assessment -II Revision CLASS X 06 7 Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed. Kendriya Vidyalaya GaCHiBOWli

More information

PRMO Solution

PRMO Solution PRMO Solution 0.08.07. How many positive integers less than 000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3?. Suppose a, b

More information

CBSE Sample Paper-03 (Unsolved) SUMMATIVE ASSESSMENT II MATHEMATICS Class IX. Time allowed: 3 hours Maximum Marks: 90

CBSE Sample Paper-03 (Unsolved) SUMMATIVE ASSESSMENT II MATHEMATICS Class IX. Time allowed: 3 hours Maximum Marks: 90 CBSE Sample Paper-3 (Unsolved) SUMMATIVE ASSESSMENT II MATHEMATICS Class IX Time allowed: 3 hours Maximum Marks: 9 General Instructions: a) All questions are compulsory. b) The question paper consists

More information

Circles. hsn.uk.net. Contents. Circles 1

Circles. hsn.uk.net. Contents. Circles 1 hsn.uk.net Circles Contents Circles 1 1 Representing a Circle A 1 Testing a Point A 3 The General Equation of a Circle A 4 Intersection of a Line and a Circle A 4 5 Tangents to Circles A 5 6 Equations

More information

Circles EOC Assessment 15%

Circles EOC Assessment 15% MGSE9-12.G.C.1 1. Which of the following is false about circles? A. All circles are similar but not necessarily congruent. B. All circles have a common ratio of 3.14 C. If a circle is dilated with a scale

More information

Yes zero is a rational number as it can be represented in the

Yes zero is a rational number as it can be represented in the 1 REAL NUMBERS EXERCISE 1.1 Q: 1 Is zero a rational number? Can you write it in the form 0?, where p and q are integers and q Yes zero is a rational number as it can be represented in the form, where p

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS /3 UNIT (COMMON) Time allowed Three hours (Plus minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL

More information

Domino Servite School

Domino Servite School Domino Servite School Accreditation Number 13SCH0100008 Registration Number 122581 Mathematics Paper II Grade 12 2017 Trial Examination Name: Time: 3 hours Total: 150 Examiner: H Pretorius Moderators:

More information

Practice Papers Set D Higher Tier A*

Practice Papers Set D Higher Tier A* Practice Papers Set D Higher Tier A* 1380 / 2381 Instructions Information Use black ink or ball-point pen. Fill in the boxes at the top of this page with your name, centre number and candidate number.

More information

Topic 3 Part 1 [449 marks]

Topic 3 Part 1 [449 marks] Topic 3 Part [449 marks] a. Find all values of x for 0. x such that sin( x ) = 0. b. Find n n+ x sin( x )dx, showing that it takes different integer values when n is even and when n is odd. c. Evaluate

More information

PRACTICE TEST 1 Math Level IC

PRACTICE TEST 1 Math Level IC SOLID VOLUME OTHER REFERENCE DATA Right circular cone L = cl V = volume L = lateral area r = radius c = circumference of base h = height l = slant height Sphere S = 4 r 2 V = volume r = radius S = surface

More information

Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions

Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions Pure Mathematics Year (AS) Unit Test : Algebra and Functions Simplify 6 4, giving your answer in the form p 8 q, where p and q are positive rational numbers. f( x) x ( k 8) x (8k ) a Find the discriminant

More information

QUESTION BANK ON STRAIGHT LINE AND CIRCLE

QUESTION BANK ON STRAIGHT LINE AND CIRCLE QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,

More information

Practice Assessment Task SET 3

Practice Assessment Task SET 3 PRACTICE ASSESSMENT TASK 3 655 Practice Assessment Task SET 3 Solve m - 5m + 6 $ 0 0 Find the locus of point P that moves so that it is equidistant from the points A^-3, h and B ^57, h 3 Write x = 4t,

More information

COMMON UNITS OF PERIMITER ARE METRE

COMMON UNITS OF PERIMITER ARE METRE MENSURATION BASIC CONCEPTS: 1.1 PERIMETERS AND AREAS OF PLANE FIGURES: PERIMETER AND AREA The perimeter of a plane figure is the total length of its boundary. The area of a plane figure is the amount of

More information