Questions

1. Find the angle θ in degrees from the figure below
   
2. In the diagram below, determine the equation of the line XY in the form y = mx + c
   
3. Find the equation of a line which passes through the point (2, 3) and is perpendicular to y – 3x+ 1 = 0, giving your answer in the form y = mx + c
4. T is the mid-point of line XY where X is point (1,4) and Y is the point (-5, 10). Find the equation of a line, L₂ which is perpendicular to line XY and goes through point T
5.  
   1. On the grid provided below, plot points A(2,1) B(-4,3) and C(2,5)
   2. Given that the gradient of CD = -1 and CD =AD locate D and complete the quadrilateral ABCD
   3. What name is given to quadrilateral ABCD?
6.  In the figure below (not drawn to scale), PQRS is a rectangle and P and Q are the points (3, 2) and (1,4) respectively.
   
   Given that the equation of the line PQ is y =3x -7, find:
   1. The equation of line QR
   2. The coordinates of point R
   3. The coordinates of point S
7. OABC is a trapezium such that the coordinates of O, A , B and C are (0, 0), (2, -1), (4, 3) and (0, y)
   1. Find the value of y
   2. M is the mid-point of AB and N is the mid-point of OM. Find in column form
      1. the vector AN
      2. the vector 
      3. Vector AC NC
   3. Hence show that A, N and C are collinear
8. Use ruler and a pair of compasses only in this question.
   1. Construct triangle ABC in which AB = 7 cm, BC = 8 cm and ∠ABC = 600.
   2. Measure
      1. side AC
      2. ∠ ACB
   3. Construct a circle passing through the three points A, B and C. Measure the radius of the circle.
   4. Construct ∆ PBC such that P is on the same side of BC as point A and ∠ PCB = ½ ∠ ACB, ∠ BPC = ∠ BAC measure ∠ PBC.
9. ABCD is a parallelogram with vertices A (1,1) and C(8,10). AB has the equation 4x -5y = -1 and BC has the equation 5x – 2y = 20. Determine by calculation;
   1.  the co-ordinates of the point M where the diagonals meet
   2. The co-ordinates of the vertices B and D
   3. the length of AB correct to 4 significant figures
10. The table shows corresponding values of x and y for a certain curve; 
    

    x	1.0	1.2	1.4	1.6	1.8	2.0	2.3
    y	6.5	6.2	5.2	4.3	4.0	2.6	2.4

    
    Using 3 strips and mid-ordinate rule estimate the area between the curve, x-axis

Answers

1. 2x - 3y + 6 =0
   -3y = -2x – 6
   y = ^2x/₃ + 2
   When y = 0 x = -3
   x = 0 y = 2
   
   ∴ Co-ordinate of y – intercept is (0,2) 
   Coordinate of x – intercept is (-3,0)
   ∴ ∠CAO = tan^{-1 2}/₃
   = 33.69^o
   ∴∠θ = 180 – 33.69^o
   = 146.31^o
   
2. Point y(^{4 + -2}/₂, ^{7 + -1}/₂) = (1, 3)
   grad AB = ^{7 + 1}/_{4 + 2} = ⁸/₆
   grad xy = - ¾
   grad xy = - ¾
   ^{y – 3}/_{x – 2} = - ¾
   y = - ¾ x + ¹⁵/₄
3.  Y = 3x – 1
   M =3
   M₁M₂ = -1
   M₂ = -¹/₃
   ^{y – 3}/_{x -2}= -¹/₃
   3y – 9 =-x +2
   ^3y/₃ = -^x/₃ + ^11/₃
   Y = -^x/₃ + ¹¹/₃
4.  Pt T is ^{1 + 5}/₂, ^{4 + 10}/₂ = (-2, 7)
   grad. of grid xy = ^{10 - 4}/_{-5 -1} = ¹⁴/_-6 = ^-7/_-3
   ∴ grad of L₂ = ³/₇
   Take a general pt P(x,y) on L2
   →^{y - 7}/_{x - 2} = ³/₇
   →7y – 49 = 3x + 6
   7y = 3x + 55
   Or y = 3x + 55 (equation of L₂
5.  a, b
   
   (c) Name : a kite
6. 1. Grad of line QP = ^4-2/_1-3= ²/_-2= -1
      Grad of line QR = 1
      Take a pt Q(1,4) and T(x,y) on line QR
      ^{y- 4}/_{x – 1}= 1
      y – 4 = x -1
      y = x + 3 .....equ. of QR
   2. y= x+3 …(i) Equ of QR
      y = 3x -7 ...(ii) Equ. of Pr
      Solving simultaneously ;:
      x +3 = 3x -7
      2x = 10
      x = 5
      Substituting ; y = 8
      ∴R is the pt (5,8)
   3.  
      
7. 1. Gradient OA = Gradient of CB
      ^{-1 – 0}/_{2 – 0} = - ½
      Gradient of CB
      ^{y – 3}/_{0 – 4}= - ½
      2y – 6 = 4
      2y = 10
      y = 5
   2.  
      
   3.  
      
      4AN = AC And A is a common point hence A, N, C lie on a straight line.
8.  
   
   
   1. ∆ ABC line AB = 7 cm and BC = 8 cm ∡ ABC = 60^o
      Construction of ∡60^o
   2. AC = 7.6 + 0.1 and
      ∡ ACB = 53 + 1^o
   3. 2 sides bisector 1
      Circle drawn radius 4.4. ± 0.1
   4. Bisect ∡ ACB
      Bisection line to cut the circle to identify P
      ∡BPC = ∡BAC = 67^o
      ∡ PBC = 88 ± 0.1^o
9. 1.  
      
      M (¹⁺⁸/₂, ¹⁺¹⁰/₂) = M (4.5, 5.5)
      
   2. AB: 4x – 5y = -1 x 2
      BC: 5x – 2y = 20 x 5
      8x – 10y = -2
      25x – 10y = 100
      -17x = -102
      x=¹⁰²/₁₇ = 6.0
      24 – 5y = -1
      5y = -25
      Y = 5
      ∴B(6,5)
      ^{x+ 6.0}/₂ = 4.5 x = 3
      ^{y + 5}/₂ = 5.5 y = 6
      ∴D (3,6)
   3. AB = √(16 – 1)² + (5-1)²
      √25 + 16
      √41 = 6.40 (units)
10. Mid ordinate
    Area = 1.2 (6.2 + 4.3 + 2.6)
    = 15.72