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#### Which of the following is not the graph of a quadratic polynomial? Solution.  Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 2.

If p(x) is quadratic polynomial then there are almost 2 values of x exists called roots.

Hence, the graph of p(x) cuts the x-axis almost 2 times.

But here we see that in option (D) the curve cuts the x-axis at 3 times

Hence, graph (D) is not a graph of a quadratic polynomial.

#### If one of the zeroes of a quadratic polynomial of the form is the negative of the other, then it(A) has no linear term and the constant term is negative.(B) has no linear term and the constant term is positive.(C) can have a linear term but the constant term is negative.(D) can have a linear term but the constant term is positive.

Solution. Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 2.

Here the given quadratic polynomial is                …..(1)

a = 1, b = a, c = b

Let x1, x2 are the zeroes of the equation (1)

According to question:

sum of zeroes =

( because x2 = - x1 )

( because b = a , a = 1)

Product of zeroes

( because c= b, a = 1)

Put value of a and b in (1)

Hence, it has no linear term and the constant term is negative.

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• Faculty Support #### If the zeroes of the quadratic polynomial are equal, then(A) c and a have opposite signs(B) c and b have opposite signs(C) c and a have the same sign(D) c and b have the same sign

Solution. Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 2.

Here the given polynomial is

We know that if both the zeroes are equal there

….(1)

(A) c and a have opposite sign

If c and a have opposite sign then R.H.S. of equation (1) is negative but L.H.S. is always positive. So (A) is not a correct one.

(B) c and b have opposite sign

If c is negative and b is positive L.H.S. is positive but R.H.S. of eqn. (1) is negative. Hence (B) is not correct one.

(C) c and a have same sign

If c and a have same sign R.H.S. of eqn. (1) is positive and L.H.S. is always positive hence it is a correct one.

(D) c and b have same sign

If c and b both have negative sign then R.H.S. of eqn. (1) is negative and L.H.S. is positive. So this is not correct.

Only one option i.e. (c) is correct one.

#### The zeroes of the quadratic polynomial ,(A) cannot both be positive(B) cannot both be negative(C) are always unequal(D) are always equal

Solution. Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 2.

Here the given quadratic polynomial is

must be grater then 0

Hence the value of k is either less than 0 or greater than 4.

If value of k is less than 0 only one zero is positive.

If value of k is greater than 4 only one zero is positive.

Hence both the zeroes can not be positive.

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• Faculty Support #### The zeroes of the quadratic polynomial are(A) both positive(B) both negative(C) one positive and one negative(D) both equal

Solution. Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 2.

Here the given quadratic polynomial is

The value of both the zeroes are negative.

#### If one of the zeroes of the cubic polynomial  is –1, then the product of the other two zeroes is(A) b – a + 1(B) b – a – 1(C) a – b + 1(D) a – b –1

Solution.  Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 3.

Here the given cubic polynomial is

Let are the zeroes of polynomial

(given)               …..(1)

put x = –1 in

We know that

Here, a = 1, b = a, c = b, d = c

So,

( using equation (1))

Hence the product of other two is b – a + 1.

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• Faculty Support #### Given that one of the zeroes of the cubic polynomial is zero, the product of the other two zeroes is(A) (B) (C) 0(D)

Solution. Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 3.

Here the given cubic polynomial is

Let three zeroes are

(given)    …..(1)

we know that

Put

( using equation (1))

Hence the product of other two zeroes is .

#### The number of polynomials having zeroes as –2 and 5 is(A) 1(B) 2(C) 3(D) more than 3

Solution. Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s).

Let the polynomial is                   ….(*)

We know that sum of zeroes

…..(1)

Multiplication of zeroes

…..(2)

Form equation (1) and (2) it is clear that

a = 1, b = –3, c = –10

put value of a, b and c in equation (*)

….(3)

But we can multiply of divide eqn. (3) by any real number except 0 and the zeroes remain same.

Hence, there are infinite number of polynomial exist with zeroes –2 and 5.

Hence the answer is more than 3.

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• Faculty Support #### If the zeroes of the quadratic polynomial are 2 and –3, then(A) a = –7, b = –1(B) a = 5, b = –1(C) a = 2, b = – 6(D) a = 0, b = – 6

Solution. Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 2.

If 2 and –3 are the zero of   then p(2) = p(–3) = 0.

….(1)

….(2)

2a + 3a = 0

5a = 0

a = 0

put a = 0 in (1)

2(0) + b = –6

b = –6

Hence a = 0, b = –6.

#### A quadratic polynomial, whose zeroes are –3 and 4, is(A) (B)(C) (D)

Solution.  Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 2.

(A)

put x = –3                                                                            put x = 4

(B)

put x = –3                                                                                    put x = 4

= 16 + 16 = 32 0

(C)

put x = –3                                                                    put x = 4

(D)

put x = –3                                                                    put x = 4

If –3, 4 is zeros of a polynomial p(x) then p(–3) = p(4) = 0

Here only (C) option satisfy p(–3) = p(4) = 0  