Assertion & Reason Test: Polynomials - Grade 10 MCQ

The assertion is false because the quadratic equation x² + 7x + 12 can be factored as (x + 3)(x + 4), which means it has two real zeros (-3 and -4). The reason is true because a quadratic polynomial indeed can have at most two zeros, but it doesn't directly relate to the assertion being true or false.

The assertion is false because the degree of the polynomial is determined by the highest power of x, which is 4 in this case, making it a polynomial of degree 4. The reason is true because the highest power of x indeed determines the degree of a polynomial, but it incorrectly describes the polynomial in the assertion.

The assertion is false because x² +4x+5 has no real zeroes (its discriminant, b ² −4ac, is negative). However, it has two complex zeroes. The reason is true, as a quadratic polynomial can have at most two zeroes (real or complex).

Zeroes of f(x) = x² − 4x − 5 are obtained by solving:
x² − 4x − 5 = 0, which implies 
x² − 5x + x − 5 = 0 or, x(x − 5) + 1(x − 5) = 0 
which means x = 5 or x = −1 
Thus the assertion is correct. 
However, the reason is incorrect. 
The numbers given are not the zeroes  of x² − 4x + 7

Product of Zeroes

⇒ k² – 4k + 4 = 0

⇒ (k – 2)² = 0 ⇒ k = 2

Both the assertion and the reason are true, but the reason provided is related to the sum of the zeroes, not their product. The correct explanation involves the formula for the product of the zeroes of a quadratic polynomial, c/a, which would lead to 5k/−10, hence determining the value of k.

Assertion: Consider the given polynomial x² + 4x + 5
as its discriminant is negative b² -4ac = 16 - 20 = -4 so it dont have real roots ,
∴ x² + 4x + 5 has no zeroes.
∴ Assertion is false.
Reason: Clearly, Reason is true.
Since Assertion (A) is false but reason (R) is true.

Option A: Both A and R are true and R is the correct explanation for A.

The polynomial p(x) = (x - 1)(x - 2)(x - 3) \) is factored into three linear factors: (x - 1) , (x - 2) , and  (x - 3) .

The zeroes of a polynomial are the values of  x  for which p(x) = 0. Setting each factor equal to zero gives the zeroes as x = 1 , x = 2 , and x = 3 , which confirms Assertion (A) as correct.

The reason (R) correctly explains the concept that the zeroes of a polynomial are the x-coordinates where the polynomial graph intersects or touches the x-axis, or where p(x) = 0.

Therefore, both the assertion and reason are correct, and the reason provides the correct explanation for the assertion.