Answer:
- 23) 4x² - 8k²x + k⁴ = 0
- 24) (i) 2, (ii) 7
- 25) (i) 11/8, (ii) -81/8
- 26) (a³ + b³ + c³ + 3abc)/a³
Step-by-step explanation:
23..............................
<h3>Given</h3>
- Equation 2x(x - k) = k² with the roots α and β
<h3>To find</h3>
- The equation with roots α² and β²
<h3>Solution</h3>
- 2x(x - k) = k²
- 2x² - 2kx - k² = 0
<u>The sum and the product of the roots </u>
- α + β = - (-2k)/2 = k
- αβ = - k²/2
<u>The equation with roots α² and β² is:</u>
- (x - α²)(x - β²) = 0
- x² - (α² + β²)x + α²β²= 0
- x² - ((α + β)² - 2αβ)x + (αβ)² = 0
- x² - (k² - 2( -k²/2))x + (- k²/2)² = 0
- x² - (k² + k²)x + k⁴/4 = 0
- 4x² - 8k²x + k⁴ = 0
24..............................
<h3>Given</h3>
- Equation 3x² - 9x + 2 = 0 with the roots α and β
<h3>To find</h3>
<u>The values of </u>
<h3>Solution</h3>
<u>The sum and the product of the roots </u>
- α + β = - (-9)/3 = 3
- αβ = 2/3
(i)
- αβ² + α²β =
- αβ(α + β) =
- 2/3(3) =
- 2
(ii)
- α² - αβ + β² =
- α² + 2αβ + β² - 3αβ =
- (α + β)² - 3αβ =
- 3² - 3(2/3) =
- 9 - 2 =
- 7
25..............................
<h3>Given</h3>
- Equation 2x² + 9x + 12 = 0 with the roots α and β
<h3>To find</h3>
- (a) show that the quadratic equation whose roots are (α - 1/α) and (β - 1/β) is 24x² + 90x + 115 = 0
The values of
- (i) αβ (1/α² + 1/β²)
- (ii) α³ + β³
<h3>Solution</h3>
<u>The sum and the product of the roots </u>
- α + β = - 9/2
- αβ = 12/2 = 6
a) The quadratic equation whose roots are (α - 1/α) and (β - 1/β) is:
- (x - (α - 1/α))(x - (β - 1/β)) = 0
- x² - (α - 1/α + β - 1/β)x + (α - 1/α)(β - 1/β) = 0
- x² - ((α + β) - (α + β)/αβ)x + αβ + 1/(αβ) - (α/β + β/α) = 0
- x² - ((α + β) - (α + β)/αβ)x + αβ + 1/(αβ) - ((α+β)² - 2αβ)/(αβ)= 0
- x² - (-9/2 - (-9/2)/6)x + 6 + 1/6 - ((-9/2)² - 2(6))/6 = 0
- x² - ( -9/2 + 3/4)x + 37/6 - (81/4 - 12)/6 = 0
- x² + 15/4x + 37/6 - 33/24 = 0
- x² + 90/24x + 148/24 - 33/24 = 0
- 24x² + 90x + 115 = 0
- Proven
(i)
- αβ (1/α² + 1/β²) =
- αβ(α² + β²)/(α²β²) =
- ((α + β)² -2αβ)/(αβ) =
- ((-9/2)² - 2(6))/6 =
- (81/4 - 12)/6 =
- 81/24 - 2 =
- 33/24 =
- 11/8
(ii)
- α³ + β³ =
- (α + β)(α² - αβ + β²) =
- (α + β)(α² + 2αβ + β² - 3αβ) =
- (α + β)((α + β)² - 3αβ) =
- (-9/2)((-9/2)² - 3(6)) =
- -9/2(81/4 - 18) =
- -9/2(9/4) =
- -81/8
26..............................
<h3>Given</h3>
- Equation ax² + bx + c = 0 with the roots α and β
<h3>To find</h3>
- Express (1 - α³)(1 - β³) in terms of a, b and c
<h3>Solution</h3>
<u>The sum and the product of the roots </u>
<u>The expression is evaluated as follows:</u>
- (1 - α³)(1 - β³) =
- 1 - (α³ + β³) + α³β³ =
- 1 - (α + β)((α + β)² - 3αβ) + (αβ)³ =
- 1 - (-b/a)((-b/a)² - 3c/a) + (c/a)³ =
- 1 + (b/a)³ + 3bc/a² + c³/a³ =
- 1 + (b³ + c³ + 3abc)/a³ =
- (a³ + b³ + c³ + 3abc)/a³
Answer:
![P(Not\ Blue) = \frac{2}{3}](https://tex.z-dn.net/?f=P%28Not%5C%20Blue%29%20%3D%20%5Cfrac%7B2%7D%7B3%7D)
Step-by-step explanation:
Given
![Red = 6](https://tex.z-dn.net/?f=Red%20%3D%206)
![Blue =5](https://tex.z-dn.net/?f=Blue%20%20%3D5)
![Green = 4](https://tex.z-dn.net/?f=Green%20%3D%204)
Required
The probability of not selecting blue.
This means that any of the other colors is selected.
So, we have:
![P(Not\ Blue) = P(Red) + P(Green)](https://tex.z-dn.net/?f=P%28Not%5C%20Blue%29%20%3D%20P%28Red%29%20%2B%20P%28Green%29)
![P(Not\ Blue) = \frac{Red}{Total} + \frac{Green}{Total}](https://tex.z-dn.net/?f=P%28Not%5C%20Blue%29%20%3D%20%5Cfrac%7BRed%7D%7BTotal%7D%20%2B%20%5Cfrac%7BGreen%7D%7BTotal%7D)
![P(Not\ Blue) = \frac{6}{15} + \frac{4}{15}](https://tex.z-dn.net/?f=P%28Not%5C%20Blue%29%20%3D%20%5Cfrac%7B6%7D%7B15%7D%20%2B%20%5Cfrac%7B4%7D%7B15%7D)
Take LCM
![P(Not\ Blue) = \frac{6+4}{15}](https://tex.z-dn.net/?f=P%28Not%5C%20Blue%29%20%3D%20%5Cfrac%7B6%2B4%7D%7B15%7D)
![P(Not\ Blue) = \frac{10}{15}](https://tex.z-dn.net/?f=P%28Not%5C%20Blue%29%20%3D%20%5Cfrac%7B10%7D%7B15%7D)
Divide by 5/5
![P(Not\ Blue) = \frac{2}{3}](https://tex.z-dn.net/?f=P%28Not%5C%20Blue%29%20%3D%20%5Cfrac%7B2%7D%7B3%7D)
The answer is 94sq inches :D
The answer would be(-4,8)...i think☺
Answer:
Step-by-step explanation:
B) -8/3 < -2 < -1 < 0 < 3/2 < 8
Positive numbers has more value
Negative numbers has less value
-2 is less than 2
-2 is less than -1