Polynomials
Previous Year Solved Questions 2023
1. Which of the following is a quadratic polynomial with zeroes 5/3 and 0? [CBSE 2023]
(a) 3x(3x-5) (b) 3x(x-5) (c) x² – 5/3 (d) (5/3)x²
(a) 3x(3x-5) (b) 3x(x-5) (c) x² – 5/3 (d) (5/3)x²
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Ans: (a) 3x(3x-5)
Zeroes: 5/3 and 0
Sum = 5/3, Product = 0
Polynomial form: k[x² – (5/3)x]
3x(3x-5) = 9x² – 15x = 9[x² – (5/3)x] ✓
Matches with k=9
Zeroes: 5/3 and 0
Sum = 5/3, Product = 0
Polynomial form: k[x² – (5/3)x]
3x(3x-5) = 9x² – 15x = 9[x² – (5/3)x] ✓
Matches with k=9
2. If α,β are the zeroes of polynomial p(x) = x² + x – 1, then 1/α + 1/β equals: [CBSE 2023]
(a) 1 (b) 2 (c) -1 (d) -1/2
(a) 1 (b) 2 (c) -1 (d) -1/2
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Ans: (a) 1
For x² + x – 1:
α+β = -1
αβ = -1
1/α + 1/β = (α+β)/(αβ) = (-1)/(-1) = 1
For x² + x – 1:
α+β = -1
αβ = -1
1/α + 1/β = (α+β)/(αβ) = (-1)/(-1) = 1
3. If α,β are zeroes of the polynomial x² – 1, then value of (α+β) is: [CBSE 2023]
(a) 2 (b) 1 (c) -1 (d) 0
(a) 2 (b) 1 (c) -1 (d) 0
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Ans: (d) 0
x² – 1 = x² + 0x – 1
Sum of zeroes = -0/1 = 0
Or: x²-1 = (x+1)(x-1)
Zeroes: 1 and -1, Sum = 1+(-1)=0
x² – 1 = x² + 0x – 1
Sum of zeroes = -0/1 = 0
Or: x²-1 = (x+1)(x-1)
Zeroes: 1 and -1, Sum = 1+(-1)=0
4. If α,β are zeroes of p(x) = 4x² – 3x – 7, then 1/α + 1/β is equal to: [CBSE 2023]
(a) 7/3 (b) -7/3 (c) 3/7 (d) -3/7
(a) 7/3 (b) -7/3 (c) 3/7 (d) -3/7
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Ans: (d) -3/7
For 4x² – 3x – 7:
α+β = -(-3)/4 = 3/4
αβ = -7/4
1/α + 1/β = (α+β)/(αβ) = (3/4)/(-7/4) = 3/4 × (-4/7) = -3/7
For 4x² – 3x – 7:
α+β = -(-3)/4 = 3/4
αβ = -7/4
1/α + 1/β = (α+β)/(αβ) = (3/4)/(-7/4) = 3/4 × (-4/7) = -3/7
5. If one zero of polynomial p(x) = 6x² + 37x – (k-2) is reciprocal of the other, find k. [CBSE 2023]
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Ans: k = -4
Let zeroes be α and 1/α
Product = α × (1/α) = 1
For given polynomial:
Product = constant term/coefficient of x²
= [-(k-2)]/6
So [-(k-2)]/6 = 1
-(k-2) = 6
-k + 2 = 6
-k = 4
k = -4
Let zeroes be α and 1/α
Product = α × (1/α) = 1
For given polynomial:
Product = constant term/coefficient of x²
= [-(k-2)]/6
So [-(k-2)]/6 = 1
-(k-2) = 6
-k + 2 = 6
-k = 4
k = -4