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Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : The sum and product of the zeros of a quadratic polynomial are respectively.
Then the quadratic polynomial is 4x^{2} +x+1.
Reason : The quadratic polynomial whose sum and product of zeros are given is x^{2}(sum of zeros). x + product of zeros.
Sum of zeros
product of zeros = 1/4
Quadratic polynomial be
Quadratic polynomial be 4x^{2} +x + 1. So, both A and R are correct and R explains A.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : If (x + 1) is a factor of f(x) = x^{2} + ax + 2, then a =  3.
Reason : If (x  a) is a factor of p (x), if p (a) = 0.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : If both zeros of the quadratic polynomial x^{2}  2kx + 2 are equal in magnitude but opposite in sign then value of k is 1/2.
Reason : Sum of zeros of a quadratic polynomial ax^{2} + bx + c is b/a
sum of zeros = 0
Assertion (A) is false but reason (R) is true.
Thus (d) is correct option.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : x^{3} + x has only one real zero.
Reason : A polynomial of nth degree must have n real zeroes.
Again, x^{3} +x = x(x^{2} + 1)
which has only one real zero
(x = 0) [x^{2} + 1 ≠ 0 for all x ε R]
Assertion is true.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : The remainder when p(x) = x^{3}  6x^{2} + 2x  4 is divided by (3x  1) is 107/27.
Reason : If a polynomial p(x) is divided by ax  b , the remainder is the value of p(x) at x = b/a.
p(x) = x^{3 } 6x^{2} + 2x  4
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : P(x) = 14x^{3}  2x^{2} + 8x^{4} + 7x  8 is a polynomial of degree 3.
Reason : The highest power of x in the polynomial p(x) is the degree of the polynomial.
= 14x^{3}  2x^{2} + 8x^{4} + 7x  8 is 4.
Degree of p(x) is 4. So, A is incorrect but R is correct.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : (x + 2) is a factor of x^{3} + 3x^{2} + 5x + 6 and of 2x + 4.
Reason : If p(x) be a polynomial of degree greater than or equal to one, then (x  a) is a factor of p(x), if p(a) = 0
g(x) = x + 2
x = 2
put x = 2 in p(x)
p(2) = (2)^{3} + 3(2)^{2} + 5(2) + 6
= 8 + 12  10 + 6
= 0
since p(x) is zero
X + 2 is a factor of x^{3} + 3x^{2} + 5x + 6
Checking
let x = 2 q(x) = 2x + 4
q(2) = 2(2) + 4
= 4 + 4
= 0
since reminder is zero
x + 2 is a factor of 2x + 4
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : If one zero of polynominal p(x) = (k^{2} + 4)x^{2 }+ 13x + 4k is reciprocal of other, then k = 2.
Reason : If (x  α) is a factor of p(x), then p(α) = 0
i.e. α is a zero of p(x).
Let α, 1/α be the zeroes of p(x), then
k^{2}  4k + 4 = 0
(k  2)^{2} = 0
k = 2
Assertion is true Since, Reason is not correct for Assertion.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : The expression 3x^{4}  4x^{3/2} + x^{2} = 2 is not a polynomial because the term 4x^{3/2} contains a rational power of x.
Reason : The highest exponent in various terms of an algebraic expression in one variable is called its degree.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : x^{2} + 4x + 5 has two zeroes.
Reason : A quadratic polynomial can have at the most two zeroes.
Assertion
Consider the given polynomial x^{2} + 4x + 5
Because the degree of the polynomial is 2. It is a quadratic polynomial.
We know that the quadratic polynomial has at the most two zeroes.
∴ x^{2} + 4x + 5 has two zeroes.
∴ Assertion is true.
Reason Clearly, Reason is true.
Since both the Assertion and Reason are true and Reason is a correct explanation of Assertion.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : 3x^{2} + x  1 = (x + 1) (3x  2) + 1.
Reason : If p(x) and g(x) are two polynomials such that degree of p(x) ≥ degree of g(x) and g(x) ≥ 0 then we can find polynomials q(x) and r(x) such that p(x) = g(x) q(x) + r (x), where r(x) = 0 of degree of r(x) < degree of g(x).
If g(x) is any polynomial then it can divide p(x) by q(x) where 0 < q(x) and may get a remainder say r(x).
If g(x) perfectly divides p(x) by q(x), then r(x) = 0.
It is obvious that deg r(x) < deg g(x).
∴ we can find polynomial q(x) and r(x) such that
p(x) = q(x)q(x) + r(x), where r(x) = 0 or deg r(x) < deg g(x)
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : (x + 2) and (x  1) are factors of the polynomial x^{4} + x^{3} + 2x^{2} + 4x  8.
Reason : For a polynomial p(x) of degree ≥ 1, x  a is a factor of the polynomial p(x) if and only if p(a) ≥ 1.
p(x) = x4 + x3 + 2x2 + 4x  8
p(2) = (2)^{4} + (2)^{3} + 2(2)^{2} + 4 (2)  8
= 16  8 + 8  8  8 = 0
So, (x + 2) is a factor of p(x).
p(1) = (1)^{4} + (1)^{3} + 2(1)^{2} + 4(1)  8
= 1 + 1 + 2 + 4  8 = 0
(x  1) is a factor of p(x).
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : (2  √3) is one zero of the quadratic polynomial then other zero will be (2 + √3).
Reason : Irrational zeros (roots) always occurs in pairs.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : If f(x) = 3x^{7}  4x^{6} + x + 9 is a polynomial, then its degree is 7.
Reason : Degree of a polynomial is the highest power of the variable in it.
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