Chapter 10 Exercise 10.1
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1 Chapter 0 Exercise 0. Q.. A(, ), B(,), C(, ), D(, ), E(0,), F(,), G(,0), H(, ) Q.. (i) nd (vi) st (ii) th (iii) nd (iv) rd (v) st (vii) th (viii) rd (ix) st (viii) rd Q.. (i) Y (v) X (ii) Y (vi) X (iii) Y (vii) Y (iv) X Q.. (i) (,0) (iii) (0,) (ii) (,0) (iv) (0, ) Q.. (i) ( + ) + ( ) (ii) (iii) ( ) + ( ) = = (iv) 0 + = Q.. (i) ( + ) + ( ) (ii) ( ) + () + = (iii) ( 7) + () 9 + (iv) ( + ) + ( + ) + Q. 7. (i) ( ) + ( ) 9 + (ii) ( + ) + ( + ) (iii) ( ) + ( ) + (iv) ( 9) + ( 7) + 0 Q.. (ii) ( 0) + ( 0) + 9 (iii) (7 0) + ( 7 0) (iv) BD ( ) + ( + ) Yes. Q. 9. (ii) AB ( + ) + ( + ) + DC ( ) + ( 0) + (iii) BC ( ) + (0 ) + =. Active Maths Book Strands Ch 0 Solutions
2 (iv) AC A(,) C(,0) ( + ) + (0 ) + 9 (v) BD ( + ) + ( + ) +. Q. 0. AB ( 0) + ( + ) 9 + = BC ( ) + ( ) 9 + = AC ( + 0) + ( + ) + _ 00 = 0 + = 0 Q.. AB ( ) + ( ) + = BC ( ) + ( ) AC ( ) + ( ) + = Q.. WX + ( ) = = WY + 0 = = XY ( ) + ( 0 ) + = = Active Maths Book Strands Ch 0 Solutions Q.. ( 0) + ( 0) Inside, as <. Q.. ( + ) + ( ) + 0 Q.. (i) (,) (ii) Graph (iii) Both points are distance of from origin. Q.. ( 7) + ( k) = ( ) + ( k) = + 0k + k = 0 k 0k + = 0 (k )(k ) = 0 k = or k = Q. 7. ( + ) + (k ) + (k ) = k k + + = 0 k k = 0 k(k ) = 0 k = 0 or k = Exercise 0. Q.. (i) (,) (iii) (,) (ii) (,) (iv) (,) Q.. (i) (,.) (iii) (., ) (ii) (,0) (iv) (0., ) Q.. (i) (.,.) (iii) (.,) (ii) (, ) (iv) (., ) Q.. C(,) Q.. Midpoint = (,0) Q.. (i) C(, ) (ii) ( ) + ( + ) +
3 Q. 7. (i) M(, ) (ii) AM ( ) + ( + ) 9 MB ( ) + ( 7 + ) 9 Q.. + x y + 7 = = + x = y + 7 = 0 x = y = Q. 9. (i) + x + = y = + x = + y = 0 x = y = B(, ) (ii) B(9, ) (iii) B(,) (iv) B(, ) Q x y + = = + x = y + = x = 0 y = Q.. (i) 7 0 = (ii) 0 + = (iii) = (iv) + + = 9 Q.. (i) 0 = (ii) + + = = (iii) 7 = 7 = 7 (iv) = = Q.. a = b = 0 c = d = e = f = Q.. q = t = Exercise 0. r = s = u = v = 0 Q.. (i) = (ii) (iii) = 0 = (iv) 7 = Q.. (i) c (ii) slope b =, slope d =, slope e = Q. 7. (i) Active Maths Book Strands Ch 0 Solutions
4 (ii) (iii) (iv) Active Maths Book Strands Ch 0 Solutions
5 (v) (vi) Q.. Two lines are parallel Active Maths Book Strands Ch 0 Solutions
6 Q. 9. Q. 0. (i) Two lines are perpendicular W X Z Y W(,), X(,), Y(, ), Z(, ) (ii) Slope of WX = ( ) = ( ) Slope of ZY = = = WX ZY Slope of WZ = ( ) ( ) = Slope of XY = ( ) = WZ XY WXYZ is a parallelogram. Active Maths Book Strands Ch 0 Solutions
7 Exercise 0. Q.. Which lines pass through (,)? Line ü or û x + y = ü x + y = ü x y = ü x y + = 0 û x + y = ü (iii) = (iv) Is (0) =? 00 Not on line Q. 9. () + ( ) = k = k k = Q.. Which lines pass through (,)? Line x + y = x + y = 0 x + y = 000x + 000y = 0 x y = Q.. () = 0 = 0 0 = 0 On line Q.. (, ) Is ( ) ( ) = 7? Not on the line Q.. Is + (7) =? Not on the line ü or û û ü ü ü û Q. 7. (i) Any three points (0,) (,) (,0) (ii) Sketch (iii) = (iv) + (0) = On line Q.. (i) (0, ) (, ) (,0) (ii) Sketch = Q. 0. ( ) + k = 0 + k = 0 k = 9 Q.. () + ( ) = c 0 = c c = 7 Q.. k + = k = k = Q.. (c) = 0 c = c = 7 Q.. (i) x = (iii) x = (ii) x = (iv) x = Q.. (i) y = (iii) y = (ii) y = (iv) y = Exercise 0. Q.. (i) (iii) 7 (ii) (iv) Q.. (i) (iii) (ii) (iv) Q.. (i) (0, ) (iii) (0,) (ii) (0, ) (iv) ( 0, ) Q.. (i) (0, ) (iii) (0, ) (ii) (0, ) (iv) ( 0, ) Active Maths Book Strands Ch 0 Solutions 7
8 Q.. (i) y = x + 0 (0,) (,0) (ii) y = x (.,0) 0 (0, 7) 0 (iii) y = x 0 (,0) 0 0 (0, ) 0 Active Maths Book Strands Ch 0 Solutions
9 ) (iv) y = x,0 ) 0, ) ) (v) y = x (,0) 0 0 (0, 0) 0 (vi) y = x 0 ),0 ) 0 0 (0, ) 0 Active Maths Book Strands Ch 0 Solutions 9
10 ) (vii) y = x,0 ) (0, ) (viii) y = x y x Q.. (i) y = x + (ii) y = x + y (,7) x Point of intersection (,7) 0 Active Maths Book Strands Ch 0 Solutions
11 Q. 7. (i) y = x + (ii) y = x + y x (, ) Q.. Point of intersection (, ) (i) y = x + (ii) y = x 0 Meet at (,) Q. 9. (i) (0, ) (iii) (0,) (ii) (0,) (iv) (0, ) Q. 0. (i) Let y = 0 0 = x + x = x = (,0) (ii) 0 = x x = x = (,0) (iii) 0 = x x = x = 9 (9,0) (iv) 0 = 7x 7x = x = (,0) Active Maths Book Strands Ch 0 Solutions
12 Q.. (i) x-intercept y-intercept x + 0 = y = x + x = (,0) (0,) (ii) x + (0) = y = x + (,0) (0,) (iii) x + 0 = (0,) x = (,0) (iv) x = y = x x = (,0) (0, ) (v) x + y = 0 y = x + 0 y = 0 x = (,0) (0,0) Exercise 0. Q.. (i) (y ) = (x ) x y + = 0 x y = 0 (ii) (y ) = (x + ) x + + y = 0 x + y + = 0 (iii) (y + ) = (x + ) x y + = 0 x + y + = 0 (iv) (y + ) = (x + ) (y + ) = x + x y + = 0 x y = 0 (v) (y + ) = (x ) y + = x + x + y + = 0 Q.. (i) (y ) = (x + ) y = x + 9 x y = 0 x y + = 0 (ii) (y ) = (x + ) y = x x + y = 0 (iii) (y ) = (x + ) y 0 = x + x y + = 0 (iv) (v) (y ) = (x + ) 7 7y = x x + 7y 0 = 0 (y + ) = (x ) y + 0 = x + x + y + = 0 Q.. (i) y = x (ii) + = (y ) = (x + ) y 9 = x + x y + 0 = 0 (iii) (iv) Q.. (i) + + = = (y + ) = (x + ) x y + 0 = 0 x y + 9 = 0 + = (y ) = (x + ) x + y = 0 (v) 0 + = 7 (y + ) = 7(x ) 7x + y + 7 = 0 7x + y = 0 = + (y ) = (x + ) (y ) = x x + y + = 0 x + y = 0 Active Maths Book Strands Ch 0 Solutions
13 (ii) 7 = y 7 = x x y + = 0 (iii) 9 + = 9 (y + ) = (x ) 9 9y = x x + 9y 9 = 0 (iv) = y = (x ) y = x x y + = 0 (v) + = 9 (y ) = 9(x + ) y = 9x 7 9x + y + = 0 Q.. l contains (,) and (0, ) = 0 (y ) = (x ) x y + = 0 x y = 0 k (,0) (0, ) = (y 0) = (x + ) y = x x + y + = 0 j y = Q.. j x = k (0, ) (,0) = (y + ) = (x 0) x y = 0 l (0,) (,0) = (y ) = (x 0) x + y = 0 Q. 7. (i) x = (ii) x = (iii) x = (iv) y = (v) y = Q.. m = (y ) = (x ) y = x + x + y = 0 + () = 0 + = 0 0 = 0 Q. 9. = (y ) = (x ) y = x x y = 0 () () = 0 = 0 0 = 0 Q. 0. Midpoint (, + ) (,.) m =. = 0. = y = (x ) y = x + x + y = 0 Active Maths Book Strands Ch 0 Solutions
14 Exercise 0.7 Q.. (i) + (ii) I: y x = 9, k: y = x Q.. (i) + (ii) m: x + y = 0, n: x y + 7 = 0 (,0) l 0 (0,9) 0 0 (0, ) (, 9) 0 (,0) k 0 (0,7) (,) m (,0) ( 7,0) (,0) n (iii) x y + 9 = 0 x + y + = 0 x + = 0 x = ( ) y + 9 = 0 y + 9 = 0 y = 9 (, 9) (iii) x + y = 0 x + y + 7 = 0 y = 0 y = x + 7 = 0 x + = 0 x = (,) Q.. (i) + (ii) r: x y + = 0, s: x + y 0 = 0 s (0,0) 0 r (0,), (,0) ) ) (iii) x y + = 0 ( ) y + = 0 9 x + y 0 = 0 y + = 0 x = 0 y = x = (, ) Active Maths Book Strands Ch 0 Solutions
15 Q.. (i) + (ii) I: x y = 0, k: x + y = 0 (iii) x y = 0 0 k (,) l + x + y = 0 y + = 0 y = x = 0 (0, ) (,0) 0 (,) x = 0 Q.. (i) + (ii) p: x + y = 9, q: y = x p 0 q 0, ) ) (7,) ),0 ) (iii) x + (x) = 9 7x = 9 x = 7 y = (7,) Q.. (i) + (ii) a: x + y =, b: y = x (0,) b 0 (,) (iii) x + x = x = y = (,) a 0 0 (,0) 0 Active Maths Book Strands Ch 0 Solutions
16 Q. 7. (i) x + y = x y = (ii) x = x = y = y = Q.. (i) x y = x y = 9 (ii) x + y = x y = 9 x = y = y = Exercise 0. Q.. Slope Slope of parallel line Q.. Q.. Q.. Q.. Q.. Q. 7. Slope of perpendicular line Q.. (i) = (ii) Q. 9. (i) y = x (ii) y = x (iii) y = 9 x + (iv) y = x + (v) y = 9 x (vi) y = x (vii) y = x + (viii) y = x + (ix) y = 7 x + Q. 0. (i) y = x (ii) y = x + (iii) y = x + (iv) y = x (v) y = 9 x + 0 (vi) y = x (vii) y = 7 x + (viii) y = x (ix) y = x + Q.. (i) (ii) (iii) (iv) (v) 9 (vi) 7 Active Maths Book Strands Ch 0 Solutions
17 Q.. m = (y + ) = (x ) y + = x x y = 0 Q.. l m = n m = (y ) = (x + ) x + y + = 0 x + y + = 0 Q.. m = (y ) = (x + ) x y + + = 0 x y + = 0 Q.. (y ) = (x ) x + y = 0 Exercise 0.9 x + y = 0 Q.. (i) 000 (ii) 00 (iii) (0, 000) (0000, 7000) = 0000 (y 000) = (x 0) y 000 = x y = 0.x (iv) 000 = 0.x = 0.x x = 000. Q.. (i) 0 Cost Time (ii) 9 (iii) (iv) minutes Active Maths Book Strands Ch 0 Solutions 7
18 Q.. (i) 0 0 Option Money 0 Option Number of chores (ii) I think John will prefer Option, as it is the more lucrative option. It would take chores for Option to catch up with Option. (iii) y = x y = + 0.x 0.x + 0 = 0.x + 0.x = x = Q.. (i) 97 7 Fahrenheit Celsius (ii) Fahrenheit (iii) 0 Celsius (iv) y = 9 x + y = 9(00) + = F Active Maths Book Strands Ch 0 Solutions
19 Q.. (i) 0 + () = km/h (ii) 0 + () = km/h (iii) Time (s) 0 Speed (km/h) 0 (iv) 0 Speed Time (v) km/h. Q.. S = D T (i) So x + y = Time x + y = 9. (ii) x + y = 7. (iii) x + y = 9 x + y = 7. x + y = 0 x + y = 0 x + y = 70 x + y = 00 x = 0 0 metres. A B Q. 7. (i) 7 (ii) 7 (iii) 90 units (iv) y = 0.(00) +.. Exercise 0.0 Q.. (i) 0() () 0 sq units (ii) () () = = sq units Active Maths Book Strands Ch 0 Solutions 9
20 (iii) 0() () sq unit (iv) () () 0 sq units Q.. (i) (iii) (ii). (iv) Q.. (i) (,) (0,) (, ) (, ) (,0) (0,0) () 0( ) = sq units (ii) (,) (0,0) + (,) (, ) (, 7) (, 0) ( ) ( 0) =. sq units. (iii) (,) (,+) (,) (0,0) + (,0) (, ) () ( )( ) 0 sq units (iv) (, ) (0,0). (,) (,7) (, ) (,) (7) () 0 sq units Q.. (i) T(,), U(,), V(,) move T to (0,0) (,) (0,0) (,) (,) = (x,y ) (,) (,) = (x,y ) Area of ΔTUV =. ( ) = + = 0 Collinear (ii) T(,), U(,), V(,) (,) (0,0) (,) (,0) (,) (,0) area =.0.0 = 0 Collinear. (iii) T(,), U(,), V(,) (,) (0,0) (,) (,) (,) (,) Area = (. ( ).) = + = 0 Collinear (iv) T(,), U(,), V(,) (,) (0,0) (,) (, ) (,) (, ) area = ( ) ( ) = 0 Collinear 0 Active Maths Book Strands Ch 0 Solutions
21 Q.. (i) P Q R (ii) P (,) (,) Q (,) (,) S (, ) (0,0) () ( )() + 0 sq units Area PQRS = sq units. S Q.. (i) C(,) B(, ) D(,0) A(0, ) (ii) (0, ) (0,0) (,0) (,) (,) (,) Area Δ = ()() ()() = 0 = = (, ) (,) (,) (,) Area Δ = ()() ( )() = + = =. Area Δ BCD = +. =. units Active Maths Book Strands Ch 0 Solutions
22 Q. 7. (i) ()() = 9 sq units (ii) ()() sq units (iii) ()() sq units (iv) ()() sq units. Q.. (i) ( +, ) C(, ) (iii) + = + = (iv) M = C = (, ) y + = (x ) x + y + = 0 x + y + = 0 (v) + + = 0 0 = 0. D is on k (vii) AB ( + ) + ( + ) + CD ( + ) + ( ) 0 (viii) ABD = ( 0 ) Q. 9. (i) Y(,0) = 0 units (iii) 0 = (iv) (Y 0) = (X ) Y = X + X + Y = (v) () + () = Z is on l. = Q. 0. Q.. Q.. (vii) WX 0 + _ YZ ( ) + (0 ) + 9 (viii) 9 9 sq units. t(t) (t)(t) = 7 t t = t = t = 9 t = ± k(k) 7k(k) = 0 k k = 0 0k = 0 k = k = ± () x() = 7. x = x = or x = x = 9 x = x = x = 7 Q.. (i) M(,) (ii) A (, ) (0,0) B (,) (,) M (,) (,) () ( )() sq units A (, ) (, ) C (7,0) (0,0) M (,) (,) ( )( ) ( )() = + = 9 sq units. Active Maths Book Strands Ch 0 Solutions
23 Revision Exercises Q.. (i) A (iv) Diagonal [AC] splits parallelogram into two triangles: Δ ABC and Δ ACD. area Δ ABC = _ ().() = Q.. (i) (ii) (, ) (iii) ü (iv) + = (v) (y ) = (x + ) x + y + = 0 x + y + = 0 (vi) y = 0 x = 0 (,0) x = B Q.. (i) area Δ ACD = _ ().() = [AC] bisects the area of the parallelogram. P(0,0) (ii) (.,0) (iii). = M R(,) Q(,0) (iv) (y ) = (x ) y = x x y = 0 (v) x = 0 y = y = (0, ) C(0,) B(,) (vi) (0) () = sq units. D(, ) A(0, ) (ii) D = (, ), C = (0,) (iii) Slope CB = 0 = 0 Slope DA = CB CA ( ) = 0 0 Slope CD = 0 = Slope AB = ( ) 0 = CD AB Parallelogram. Q.. (i) M(-,) N(,) (ii) Slope MN = ( ) = 7 (iii) Equation of MN = y = (x ) 7 7y = x 0 eqn 7y = x + Active Maths Book Strands Ch 0 Solutions
24 (iv) x = 0 7y = y = 7 (v) 7y = x + y = 7 x + 7 Q.. (i) () + 0 = = on line (ii) (0) + k = k = (iii) l 0 (iv) 0 = (v) y = x + M = Q.. (,0) (0,.) M = =. (y 0) =.(x + ).x + y +. = 0 Q. 7. (i) P(,) Q(,0) (ii) PR + 0 PQ + ( ) 0 R(,) Q.. (i) (iii) RQ ( ) + = + = 0 (iv) 0 _ = = 0 Tablespoons 0 Cup (ii) tablespoons = cup (iii) _ = = (iv) ( y ) = (x ) y = x x y = 0 y = x (v) y = (0) y =. Q. 9. (i) 0 km/h (ii) s (iii) (0,0) (0,0) 0 0 = y 0 = x 0 x y + 0 = 0 (iv) 0 (v) 0 (vi) y + 0 = 0 y = km/h Active Maths Book Strands Ch 0 Solutions
25 Q. 0. (i) 0 (ii) 90 (iii) 0 0 = 0 (iv) The slope represents the change in cost for every minute spent on the job. (v) y 0 = x 0 x y + 0 = 0 (vi) 0 y + 0 = 0 y = Q.. (i) 0 km (iii). kms (ii) 0 mins (iv) 0 = 0 km/hr 0. Q.. (ii) (.) + 0 =. mins (iii) 00 = x = x. kgs Active Maths Book Strands Ch 0 Solutions
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