Mathematics Multiple Choice Questions & Answers on “Zeros and Coefficients of Polynomial – 1”.
1. The zeros of the polynomial 18x2-27x+7 are ___________
a) (frac {7}{6}, frac {1}{3})
b) (frac {-7}{6}, frac {1}{3})
c) (frac {7}{6}, frac {-1}{3})
d) (frac {7}{3}, frac {1}{3})
Answer: a
Clarification: 18x2-27x+7=0
18x2-21x-6x+7=0
3x(6x-7)-1(6x-7)=0
(6x-7)(3x-1)=0
x=(frac {7}{6}, frac {1}{3})
The zeros are (frac {7}{6}) and (frac {1}{3}).
2. What will be the polynomial if its zeros are 3, -3, 9 and -9?
a) x4-80x2+729
b) x4-90x2+729
c) x4-90x2+79
d) x4-100x2+729
Answer: b
Clarification: The zeros of the polynomial are 3, -3, 9 and -9.
Then, (x-3), (x+3), (x-9) and (x+9) are the factors of the polynomial.
Multiplying the factors, we have
(x-3) (x+3) (x-9) (x+9)
(x2-9) (x2-81) (By identity (x-a)(x+a)=x2-a2)
(x4-9x2-81x2+729)
x4-90x2+729
3. The sum and product of zeros of a quadratic polynomial are 10 and (frac {5}{2}) respectively. What will be the quadratic polynomial?
a) 2x2-20x+10
b) 2x2-x+5
c) 2x2-20x+5
d) x2-20x+5
Answer: c
Clarification: The sum of the polynomial is 10, that is, α+β = 10
The product of the polynomial is (frac {5}{2}) i.e. αβ = (frac {5}{2})
∴ f(x)=x2-(α+β)x+αβ
f(x)=x2-10x+(frac {5}{2})
f(x)=2x2-20x+5
4. If α and β are the zeros of x2+20x-80, then the value of α+β is _______
a) -15
b) -5
c) -10
d) -20
Answer: d
Clarification: α and β are the zeros of x2+20x-80.
Sum of zeros or α+β = (frac {-coefficient , of , x}{coefficient , of , x^2} = frac {-20}{1}) = -20
5. If α and β are the zeros of 3x2-5x-15, then the value of αβ is _______
a) -5
b) -10
c) -15
d) -20
Answer: a
Clarification: α and β are the zeros of 3x2-5x-15.
Product of zeros or αβ = (frac {constant , term}{coefficient , of , x^2} = frac {-15}{3}) = -5
6. What will be the value of other zero, if one zero of the quadratic polynomial is 5 and the sum of the zeros is 10?
a) 10
b) 5
c) -5
d) -10
Answer: b
Clarification: One zero of the quadratic polynomial is 5. ∴ the factor of the polynomial is (x-5)
Let us assume the other zero to be b. ∴ the other factor of the polynomial is (x-b)
Multiplying the factors, we have (x-5)(x-b)
x2-5x-bx+5b
x2-(5+b)x+5b
The sum of zeros is 10.
∴ (frac {-coefficient , of , x}{coefficient , of , x^2})=10
(frac {-(-5-b)}{1})=10
5+b=10
b=5
The equation becomes x2-10x+25.
Therefore, the other zero is 5.
7. The value of a and b, if the zeros of x2+(a+5)x-(b-4) are -5 and 9 will be _________
a) 47, -5
b) -5, 47
c) -9, 49
d) -4, 45
Answer: c
Clarification: The zeros of the polynomial are -5 and 9.
Hence, α=-5, β=9
The polynomial is x2+(a+5)x-(b-4).
Sum of zeros or α+β=-5+9 = (frac {-coefficient , of , x}{coefficient , of , x^2} = frac {a+5}{1})
-4=a+5
a = -9
Product of zeros or αβ = -45 = (frac {constant , term}{coefficient , of , x^2} = frac {-(b-4)}{1})
-45=-b+4
b=49
8. What will be the value of k, if one zero of x2+(k-3)x-16=0 is additive inverse of other?
a) 4
b) -4
c) -3
d) 3
Answer: d
Clarification: Since, one zero of the polynomial is the additive inverse of the other.
Hence, the sum of roots will be zero.
The polynomial is x2+(k-3)x-16=0
Sum of zeros or α+β=(frac {-coefficient , of , x}{coefficient , of , x^2} = frac {k-3}{1})=0
k-3=0
k=3
9. If α and β are the zeros of 10x2+20x-80, then the value of (frac {1}{alpha } + frac {1}{beta }) is _______
a) (frac {5}{4})
b) (frac {1}{5})
c) (frac {3}{4})
d) (frac {1}{4})
Answer: d
Clarification: (frac {1}{alpha } + frac {1}{beta } = frac {alpha +beta }{alpha beta })
α+β=(frac {-20}{10})=-2
αβ=(frac {-80}{10})=-8
∴ (frac {alpha +beta }{alpha beta } = frac {-2}{-8} = frac {1}{4})
10. If α and β are the zeros of x2+35x-75, then _______
a) α+β<αβ
b) α+β>αβ
c) α+β=αβ
d) α+β≠αβ
Answer: b
Clarification: The given polynomial is x2+35x-75.
The sum of zeros, α + β = (frac {-coefficient , of , x}{coefficient , of , x^2} = frac {-35}{1}) = -35
The product of zeros, αβ = (frac {constant , term}{coefficient , of , x^2}) = -75
Clearly, sum of zeros is greater than product of zeros.