Prove `sqrt5 + sqrt3 > sqrt14` by contradiction. (2 marks)
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Proof, EXT2 P1 SM-Bank 10
Prove that `sqrt11 - sqrt5 < sqrt2` by contradiction. (2 marks)
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Prove `sqrt5 + sqrt3 > sqrt14` by contradiction. (2 marks)
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`text(See Worked Solutions)`
`text(Proof by contradiction:)`
`text(Assume)\ \ sqrt5 + sqrt3 <= sqrt14`
| `(sqrt5 + sqrt3)^2` | `<= (sqrt14)^2` |
| `5 + 2sqrt15 + 3` | `<= 14` |
| `2sqrt15` | `<= 6` |
| `sqrt15` | `<= 3` |
| `15` | `<= 9\ \ \ text{(incorrect)}` |
`:. text(By contradiction,)\ sqrt5 + sqrt3 > sqrt14`
Prove that `sqrt11 - sqrt5 < sqrt2` by contradiction. (2 marks)
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`text{Proof (See Worked Solutions)}`
`text(Proof by contradiction:)`
`text(Assume that)\ \ sqrt11 – sqrt5 >= sqrt2`
| `( sqrt11 – sqrt5)^2` | `>= 2` |
| `11 – 2 sqrt55 + 5` | `>= 2` |
| `2 sqrt55` | `<= 14` |
| `sqrt55` | `<= 7` |
| `55` | `<= 49 \ \ text{(which is incorrect)}` |
`:.\ text(By contradiction,) \ sqrt11 – sqrt5 < sqrt2`