please answer the red mark questions
Answers
Very Short Answer Type Questions :
Question 1: The sum and product of zeroes of p(x) = 63x² - 7x - 9 are the S and P respectively. Find S and P.
Solution:
Given : p(x) = 63x² - 7x - 9
On comparing this with ax² + bx + c, we get
a = 63, b = - 7, c = - 9
Let α and β be the zeroes of the required polynomial.
Now,
Sum of zeroes, S = α + β = - b/a
→ - (- 7)/63
→ 7/63
→ 1/9
Product of zeroes, P = αβ = c/a
→ - 9/63
→ - 1/7
Hence, the sum(S) is 1/9 and product(P) is - 1/7.
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Question 2: Write a quadratic polynomial having zeroes 1 and - 2.
Solution:
Let α and β be the zeroes of the required polynomial.
♦ α = 1, β = - 2 (given)
• Sum of zeroes = α + β
→ (1) + (- 2)
→ 1 - 2
→ - 1
• Product of zeroes = αβ
→ (1)(- 2)
→ - 2
The required polynomial is :
→ p(x) = k [ x² - (α + β)x + αβ ]
- Putting known values.
→ p(x) = k [ x² - (- 1)x + (- 2) ]
→ p(x) = k [ x² + x - 2 ]
- Putting k = 1.
→ p(x) = x² + x - 2
Hence, the required polynomial is x² + x - 2.
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Short Answer Type Questions :
Question 1: Form a quadratic polynomial p(x) with sum and product of zeroes as 2 and - 3/5 respectively.
Solution:
Let α and β be the zeroes of the required polynomial.
It is given that :
• Sum of zeroes = α + β = 2
• Product of zeroes = αβ = - 3/5
The required polynomial is :
→ p(x) = k [ x² - (α + β)x + αβ ]
- Putting known value in it.
→ p(x) = k [ x² - (2)x + (- 3/5) ]
→ p(x) = k [ x² - 2x - 3/5 ]
→ p(x) = k [ (5x² - 10x - 3)/5 ]
→ p(x) = k/5 [ 5x² - 10x - 3 ]
- Putting k = 5.
→ p(x) = 5x² - 10x - 3
Hence, the required polynomial is 5x² - 10x - 3.
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Question 2: α and β are the zeroes of the quadratic polynomial p(x) = x² - (k + 6)x + 2(2k - 1). Find the value of k, if α + β = (1/2)αβ
Solution:
Given : p(x) = x² - (k + 6)x + 2(2k - 1)
On comparing this with ax² + bx + c, we get
a = 1, b = - (k + 6), c = 2(2k - 1)
It is given that α and β be the zeroes of the required polynomial.
Now,
• Sum of zeroes = α + β = - b/a
→ - [ - (k + 6) ] / 1
→ k + 6
• Product of zeroes = αβ = c/a
→ 2(2k - 1)/1
→ 2(2k - 1)
Now, it is given that,
→ α + β = (1/2)αβ
- Putting known values.
→ k + 6 = (1/2)[ 2(2k - 1) ]
→ k + 6 = 2k - 1
→ 2k - k = 6 + 1
→ k = 7
Hence, the value of k is 7.
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Question 3: Form a quadratic polynomial whose one zero is 8 and the product of the zeroes is - 56.
Solution:
Let α and β be the zeroes of the required polynomial.
One zero is given. Let it be α.
So, α = 8
Also, Product of zeroes = - 56
→ αβ = - 56
- Putting the value of α in it.
→ (8)(β) = - 56
→ β = - 56/8
→ β = - 7
Now,
• Sum of zeroes = α + β
→ (8) + (- 7)
→ 8 - 7
→ 1
The required polynomial is :
→ p(x) = k [ x² - (α + β)x + αβ ]
- Putting known values.
→ p(x) = k [ x² - (1)x + (- 56) ]
→ p(x) = k [ x² - x - 56 ]
- Putting k = 1.
→ p(x) = x² - x - 56
Hence, the required polynomial is x² - x - 56.
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