A quadratic polynomial, whose zeroes are -4 and -5, is
a) x² - 9x + 20
b) x² + 9x + 20
c) x² - 9x - 20
8. For p(x) = ax² + bx + c, sum of zeroes is equal to
a) zero
b) c/a
c) -b/a
9. Assertion: √2 is an irrational number.
Reason: A number is irrational if it cannot be expressed in the form p/q, where p and q are integers and q ≠ 0
a) Both A and R are true, and R is the correct explanation of A.
b) Both A and R are true, but R is not the correct explanation of A.
c) A is true, but R is false.
d) A is false, but R is true.
10. Assertion: If α & β are the zeroes of the quadratic polynomial x² - 7x + 10, then α + β = 7 and αβ = 10.
Reason: For a polynomial ax² + bx + c, sum of zeroes = -b/a, and product = c/a.
a) Both A and R are true, and R is the correct explanation of A.
b) Both A and R are true, but R is not the correct explanation of A.
c) A is true, but R is false.
d) A is false, but R is true.
SECTION - B
11. Find the HCF & LCM of 12, 15, 21 by applying the prime factorization method.
12. Given that HCF(306, 657) = 9, find LCM(306, 657).
13. Find the sum and product of the polynomial 2x² - 9x + 4 = 0.
14. Find the zeroes of the quadratic polynomial x² - 2x - 8, and verify the relationship between the zeroes and its coefficients.
15. If α & β are zeroes of the polynomial x² + 2x - 4, then find the value of 1/α + 1/β.
Answer (1)
Let's address each part of the questions:
7. A quadratic polynomial, whose zeroes are -4 and -5:
The zeroes of a quadratic polynomial, denoted as x 2 + b x + c , are the values for which the polynomial equals zero. If the zeroes are α = − 4 and β = − 5 , the sum of the zeroes is α + β = − 4 − 5 = − 9 and the product of the zeroes is α β = ( − 4 ) ( − 5 ) = 20 .
The quadratic polynomial based on these zeroes can be expressed using the following form: x 2 − ( sum of zeroes ) x + product of zeroes = x 2 − 9 x + 20
The correct answer is: a) x 2 − 9 x + 20
9. Assertion: 2 is an irrational number. Reason: A number is irrational if it cannot be expressed in the form q p , where p and q are integers and q = 0 .
Assertion (A) is true. 2 is an irrational number.
Reason (R) is true, and it correctly explains why 2 is irrational because it cannot be expressed as a fraction of two integers where the denominator is not zero.
The correct option is: a) Both A and R are true, and R is the correct explanation of A.
10. Assertion: If α & β are the zeroes of the quadratic polynomial x 2 − 7 x + 10 , then α + β = 7 and α β = 10 . Reason: For a polynomial a x 2 + b x + c , sum of zeroes = − b / a , and product = c / a .
Assertion (A) is true, as for the polynomial x 2 − 7 x + 10 , by Vieta's formulas, α + β = − ( − 7 ) /1 = 7 and α β = 10/1 = 10 .
Reason (R) is true and correctly explains (A).
The correct option is: a) Both A and R are true, and R is the correct explanation of A.
