Which is true‚ but does not make the assumption(1 = 2) even one bit less false. As we know‚ falsity implies anything‚ truth in particular. Proof 2 This proof is by E. S. Loomis (Am Math Monthly‚ v. 8‚ n. 11 (1901)‚ 233.) Let ABC be a right triangle whose sides are tangent to the circle O. Since CD = CF‚ BE = BF‚ and AE = AD = r = radius of circle‚ it is easily shown that (CB = a) + 2r = (AC + AB = b + c). And if (1) a + 2r = b + c then (1)² = (2): (2) a² + 4ra + 4r² = b² + 2bc + c².
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questions Real-life money questions Generate a sequence from the nth term Substitution Alternate angles Angle sum of a triangle Properties of special triangles Finding angles of regular polygons Area of circle Circumference of circle Area of compound shapes Rotations Reflections Enlargements Translations Find the mid-point of a line Measuring and drawing angles Drawing triangles Plans and elevations Nets Symmetries Questionnaires and data collection Two-way tables Pie charts Scatter graphs Frequency
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Introduction: “You can learn more from solving one problem in many different ways than you can from solving many different problems‚ each in only one way.” Islamic civilization in the middle ages‚ like all of Europe‚ had a dichotomy between theoretical and practical mathematics. Practical mathematics was the common subject‚ “whereas theoretical and argumentative mathematics were reserved for specialists” (Abedljaouad‚ 2006‚ p. 629). Between the eighth and the fifteenth centuries‚ Islamic civilization
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respectively. (i) Find the co-ordinates of C and D. (ii) Name the figure ACBD and find its area. [3] b) PAQ is a tangent at A to the circumcircle of Δ ABC such that PAQ is parallel to BC‚ prove that ABC is an isosceles triangle. [3] c) A rectangular piece of paper 30 cm long and 21 cm wide is taken. Find the area of the biggest circle that can be cut out from this paper. Also find the area of the paper left after cutting out the
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Castle Rock‚ walk x paces to the north‚ and then walk 2x + 4 paces to the east. If they share their information‚ then they can find x and save a lot of digging. What is x? The Pythagorean Theorem states to find the missing side of a right triangle you can square to know lengths and add the two together. The result will be the distance of the missing length squared. A^2+b^2=C^2 We know that Ahmed has a map with a distance to the treasure of 2x+6. We know that Vanessa has a map with
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section × length 1 Area of trapezium = 2 (a + b)h a cross section h length b Volume of cone = 1 r 2h 3 Curved surface area of cone = rl Volume of sphere = 4 r 3 3 Surface area of sphere = 4 r 2 r l h r In any triangle ABC b A Sine Rule a B c a sin A The Quadratic Equation The solutions of ax 2 + bx + c = 0 where a 0‚ are given by C b
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Find the width of the room. 4. A rectangular park is 45 m long and 30 m width. A path 2‚5 m wide is constructed outside the park. Find the area of the path and the cost of constructing it at Rs. 125 per m2. 5. The area of a right angled triangle is 400 m2. If one of its legs measures 8 cm. Find the length of the other leg. 6. The diameter of the wheel of a car is 10 cm. How many revolutions will it make to travel 99 km? 7. A race track is in the form of a ring whose inner circumference
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Answers Chapter 1 c i No x-intercept‚ y = 4 iii y Pre-test 1 a 2x + 1 1 3 c x+ x= x 2 2 f 3x − 7 b 5(x − 1) 1 (x + 4) 3 -5ab -21x -4x + 4 3 ii 0 y=4 d 2x − 3 e a a a a 7x 8y 3x + 3 4m b b b b 6a 7 6 b 7 12 9 a 3 b -2 c 1 10 a y = 2x + 1 b y = -x + 5 c 4 7 d 19 72 Exercise 1A f 21 20 2 3 4 5 4 e 7 c c c c -5x 2 15a 2 -10x + 2x 2 a (0‚ 4) d g c 10 ii
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b)h h 2 b Volume of prism = area of cross section × length crosssection h lengt r 4 πr3 Volume of sphere = – 3 Surface area of sphere = 4 π r 2 1 Volume of cone = – π r2h l 3 h Curved surface area of cone = π rl r In any triangle ABC Area of triangle = 1 – 2 C ab sin C a b b a c Sine rule ––––– = ––––– = ––––– sin A sin B sin C A c B Cosine rule a2 = b2 + c2 – 2bc cos A The Quadratic Equation The solutions of ax2 + bx + c = 0‚ where a ≠ 0‚ are given by – b ± √ (b2 – 4ac)
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l3 in fig.‚ are parallel. The value of x is : [Marks:1] A. 50o B. 80o C. 40o D. 140o 6] [Marks:1] A. BC = EF B. AC = EF C. BC = DE D. AC = DE 7] The area of an equilateral triangle with perimeter 18x is: [Marks:1] A. B. C. D. 8] The area of triangle‚ whose sides are 15 cm‚ 25 cm and 14 cm is [Marks:1] A. B. C. D. 9] Simplify: . [Marks:2] 10] [Marks:2] 11] Without actually calculating the cubes‚ find the value
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