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#### Question 1:

Find the zeros of the quadratic polynomial (x2 + 3x − 10) and verify the relation between its zeros and coefficients.

#### Question 2:

Find the zeros of the quadratic polynomial (6x2 − 7x − 3) and verify the relation between its zeros an coefficients.

#### Question 3:

Find the zeros of the quadratic polynomial 4x2 − 4x − 3 and verify the relation between the zeros and the coefficients.

#### Question 4:

Find the zeros of the quadratic polynomial 5x2 − 4 − 8x and verify the relationship between the zeros and the coefficients of the given polynomial.

#### Question 5:

Find the zeros of the quadratic polynomial 6x2 − 3 − 7x and verify the relationship between the zeros and the coefficients of the given polynomial.

#### Question 6:

Find the zeros of the quadratic polynomial 2x2 − 11x + 15 and verify the relation between the zeros and the coefficients.

#### Question 7:

Find the zeros of the quadratic polynomial (x2 − 5) and verify the relation between the zeros and the coefficients.

#### Question 8:

Find the zeros of the quadratic polynomial (8 x2 − 4) and verify the relation between the zeros and the coefficients.

#### Question 9:

Find the zeros of the quadratic polynomial (5u2 + 10u) and verify the relation between the zeros and the coefficients.

#### Question 10:

Find the quadratic polynomial whose zeros are 2 and −6. Verify the relation between the coefficients and the zeros of the polynomial.

#### Question 11:

Find the quadratic polynomial whose zeros are $\frac{2}{3}$ and $\frac{-1}{4}$. Verify the relation between the coefficients and the zeros of the polynomial.

#### Question 12:

Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Hence, find the zeros of the polynomial.

#### Question 13:

Find the quadratic polynomial, the sum of whose zeros is −5 and their product is 6. Hence, find the zeros of the polynomial.

#### Question 14:

Find the quadratic polynomial, the sum of whose zeros is $\left(\frac{5}{2}\right)$ and their product is 1. Hence, find the zeros of the polynomial.

#### Question 15:

Find the quadratic polynomial, the sum of whose zeros is 0 and their product is −1. Hence, find the zeros of the polynomial.

#### Question 16:

Find the quadratic polynomial, the sum of whose zeros is $\sqrt{2}$ and their product is −12. Hence, find the zeros of the polynomial.

#### Question 17:

If α, β are the zeros of a polynomial, such that α + β = 6 and αβ = 4, then write the polynomial.

#### Question 1:

Verity that 3, −2, 1 are the zeros of the cubic polynomial p(x) = x3 − 2x2 − 5x + 6 and verify the relation between its zeros and coefficients.

#### Question 2:

Verify that 5, −2 and $\frac{1}{3}$ are the zeros of the cubic polynomial p(x) = 3x3 − 10x2 − 27x + 10 and verify the relation between its zeros and coefficients.

#### Question 3:

Find a cubic polynomial whose zeros are −2, −3 and −1.

#### Question 4:

Find a cubic polynomial whose zeros are 3, $\frac{1}{2}$ and −1.

#### Question 5:

When f(x) = 4x3 − 8x2 + 8x + 1 is divided by a polynomial g(x), we get (2x − 1) as quotient and (x + 3) as remainder. Find g(x).

$\begin{array}{l}\\ \therefore g\left(x\right)=2{x}^{2}-3x+2\end{array}$

#### Question 6:

Divide (2x2 + x − 15) by (x + 3) and verify the division algorithm.

#### Question 7:

Divide (12 − 17x − 5x2) by (3 − 5x) and verify the division algorithm.

#### Question 8:

Divide (3x3 − 4x2 + 7x − 2) by (x2 x + 2) and verify the division algorithm.

#### Question 9:

Divide (6 + 19x + x2 − 6x3) by (2+ 5x − 3x2) and verify the division algorithm.

#### Question 10:

It being given that 2 is one of the zeros of the polynomial x3 − 4x2 + x + 6. Find its other zeros.

#### Question 11:

It is given that −1 is one of the zeros of the polynomial x3 + 2x2 − 11x − 12. Find all the given zeros of the given polynomial.

#### Question 12:

If 1 and −2 are two zeros of the polynomial (x3 − 4x2 − 7x + 10), find its third zero.

#### Question 13:

If 3 and −3 are two zeros of the polynomial (x4 + x3 − 11x2 − 9x + 18), find all the zeros of the given polynomial.

#### Question 14:

If 2 and −2 are two zeros of the polynomial (x4 + x3 − 34x2 − 4x + 120), find all the zeros of given polynomial.

#### Question 15:

Find all the zeros of (x4 + x3 − 23x2 − 3x + 60), if it is given that two of its zeros are $\sqrt{3}$ and $-\sqrt{3}$.

#### Question 16:

Find all the zeros of (2x4 − 3x3 − 5x2 + 9x − 3), it being given that two of its zeros are $\sqrt{3}$ and $-\sqrt{3}$.

#### Question 17:

Find all the zeros of the polynomial (2x4 − 11x3 + 7x2 + 13x), it being given that two if its zeros are $3+\sqrt{2}$ and $3-\sqrt{2}$.

#### Question 18:

Obtain all other zeros of (x4 + 4x3 − 2x2 − 20x − 15) if two of its zeros are $\sqrt{5}$ and $-\sqrt{5}$.

#### Question 1:

Which of the following is a polynomial?

(a) ${x}^{2}-5x+6\sqrt{x}+2$
(b) ${x}^{3/2}-x+{x}^{1/2}+1$
(c) $\sqrt{x}+\frac{1}{\sqrt{x}}$
(d) None of these

(d) none of these

A polynomial in x of degree n is an expression of the form p(x) =ao +a1x+a2x2 +...+an xn, where an $\ne$0.

#### Question 2:

Which of the following is not a polynomial?

(a) $\sqrt{3}{x}^{2}-2\sqrt{3}x+5$
(b) $9{x}^{2}-4x+\sqrt{2}$
(c) $\frac{3}{2}{x}^{3}+6{x}^{2}-\frac{1}{\sqrt{2}}x-8$
(d) $x+\frac{3}{x}$

It is because in the second term, the degree of x is −1 and an expression with a negative degree is not a polynomial.

#### Question 3:

The zeros of the polynomial x2 − 2x − 3 are

(a) −3, 1
(b) −3, −1
(c) 3, −1
(d) 3, 1

#### Question 4:

The zeros of the polynomial ${x}^{2}-\sqrt{2}x-12$ are
(a) $\sqrt{2},-\sqrt{2}$
(b)
(c)
(d)

#### Question 5:

The zeros of the polynomial $4{x}^{2}+5\sqrt{2}x-3$ are
(a) $-3\sqrt{2},\sqrt{2}$
(b) $-3\sqrt{2},\frac{\sqrt{2}}{2}$
(c) $\frac{-3\sqrt{2}}{2},\frac{\sqrt{2}}{4}$
(d) none of these

#### Question 6:

The zeros of the polynomial ${x}^{2}+\frac{1}{6}x-2$ are
(a) −3, 4
(b) $\frac{-3}{2},\frac{4}{3}$
(c) $\frac{-4}{3},\frac{3}{2}$
(d) none of these

#### Question 7:

The zeros of the polynomial $7{x}^{2}-\frac{11}{3}x-\frac{2}{3}$ are
(a) $\frac{2}{3},\frac{-1}{7}$
(b) $\frac{2}{7},\frac{-1}{3}$
(c) $\frac{-2}{3},\frac{1}{7}$
(d) none of these

#### Question 8:

A quadratic polynomial whose zeros are 5 and −3, is

(a) x2 + 2x − 15
(b) x2 − 2x + 15
(c) x2 − 2x − 15
(d) none of these

#### Question 9:

A quadratic polynomial whose zeros are $\frac{3}{5}$ and $\frac{-1}{2}$,is
(a) 10x2 + x + 3
(b) 10x2 + x − 3
(c) 10x2x + 3
(d) ${x}^{2}-\frac{1}{10}x-\frac{3}{10}$

#### Question 10:

The sum and product of the zeros of a quadratic polynomial are 3 and −10 respectively. The quadratic polynomial is

(a) x2 − 3x + 10
(b) x2 + 3x −10
(c) x2 − 3x −10
(d) x2 + 3x + 10

#### Question 11:

How many polynomials are there having 4 and −2 as zeros?

(a) One
(b) Two
(c) Three
(d) More than three

#### Question 12:

The zeros of the quadratic polynomial x2 + 88x + 125 are

(a) both positive
(b) both negative
(c) one positive and one negative
(d) both equal

#### Question 13:

If α and β are the zero of x2 + 5x + 8, then the value of (α + β) is

(a) 5
(b) −5
(c) 8
(d) −8

#### Question 14:

If α and β are the zero of 2x2 + 5x − 8, then the value of (αβ) is

(a) $\frac{-5}{2}$
(b) $\frac{5}{2}$
(c) $\frac{-9}{2}$
(d) $\frac{9}{2}$

#### Question 15:

If one zero of the quadratic polynomial kx2 + 3x + k is 2, then the value of k is

(a) $\frac{5}{6}$
(b) $\frac{-5}{6}$
(c) $\frac{6}{5}$
(d) $\frac{-6}{5}$

#### Question 16:

If one zero of the quadratic polynomial (k − 1) x2 + kx + 1 is −4, then the value of k is

(a) $\frac{-5}{4}$
(b) $\frac{5}{4}$
(c) $\frac{-4}{3}$
(d) $\frac{4}{3}$

#### Question 17:

If −2 and 3 are the zeros of the quadratic polynomial x2 + (a + 1) x + b, then

(a) a = −2, b = 6
(b) a = 2, b = −6
(c) a = −2, b = −6
(d) a = 2, b = 6

#### Question 18:

If one of the zeroes of the quadratic polynomial x2 + bx + c is negative of the other, then

(a) b = 0 and c is positive
(b) b = 0 and c is negative
(c) b ≠ 0 and c is positive
(d) b ≠ 0 and c is negative

#### Question 19:

If the zeros of the quadratic polynomial ax2 + bx + c, where a ≠ 0 and c ≠ 0, are equal then

(a) c and a have the same sign
(b) c and a have opposite signs
(c) c and b have the same sign
(d) c and b have opposite sign

#### Question 20:

The zeros of the quadratic polynomial x2 + kx + k, where k > 0

(a) are both positive
(b) are both negative
(c) are always equal
(d) are always unequal

#### Question 21:

If one zero of 3x2 + 8x + k be the reciprocal of the other, then k = ?

(a) 3
(b) −3
(c) $\frac{1}{3}$
(d) $\frac{-1}{3}$

#### Question 22:

If the sum of the zeros of the quadratic polynomial kx2 + 2x + 3k is equal to the product of its zeros, then k = ?

(a) $\frac{1}{3}$
(b) $\frac{-1}{3}$
(c) $\frac{2}{3}$
(d) $\frac{-2}{3}$

#### Question 23:

If α, β are the zeros of f (x) = 2x2 + 6x − 6, then

(a) α + β = αβ
(b) α + β > αβ
(c) α + β < αβ
(d) α + β + αβ = 0

#### Question 24:

If α, β are the zeros of the polynomial x2 − 5x + c and α − β = 1, then c = ?

(a) 0
(b) 1
(c) 4
(d) 6

#### Question 25:

If α, β are the zeros of the polynomial x2 + 6x + 2, then $\left(\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}\right)=?$
(a) 3
(b) −3
(c) 12
(d) −12

#### Question 26:

If α, β, γ are the zeros of the polynomial x3 − 6x2x + 30, then (αβ + βγ + γα) = ?

(a) −1
(b) 1
(c) −5
(d) 30

#### Question 27:

If α, β, γ are the zeros of the polynomial 2x3x2 − 13x + 6, then αβγ = ?

(a) −3
(b) 3
(c) $\frac{-1}{2}$
(d) $\frac{-13}{2}$

#### Question 28:

If α, β, γ be the zeros of the polynomial p(x) such that (α + β + γ) = 3, (αβ + βγ + γα) = −
10 and αβγ = −24, then p(x) = ?

(a) x3 + 3x2 − 10x + 24
(b) x3 + 3x2 + 10x −24
(c) x3 − 3x2 −10x + 24
(d) None of these

#### Question 29:

If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d is 0, then the third zeros is

(a) $\frac{\mathit{-}\mathit{b}}{\mathit{a}}$
(b) $\frac{b}{a}$
(c) $\frac{c}{a}$
(d) $\frac{-d}{a}$

#### Question 30:

If one of the zeros of the cubic polynomial ax3 + bx2 + cx + d is 0, then the product of the other two zeros is

(a) $\frac{-c}{a}$
(b) $\frac{c}{a}$
(c) 0
(d) $\frac{-b}{a}$

#### Question 31:

If one of the zeros of the cubic polynomial x3 + ax2 + bx + c is −1, then the product of the other two zeros is

(a) ab − 1
(b) b a − 1
(c) 1 − a + b
(d) 1 + a b

#### Question 32:

If the zeros of the polynomial x3 − 3x2 + x + 1 are ad, a and a + d, then a + d is

(a) a natural number
(b) an integer
(c) a rational number
(d) an irrational number

#### Question 33:

If α, β be the zeros of the polynomial x2 − 8x + k such that α2 + β2 = 40, then k = ?

(a) 6
(b) 9
(c) 12
(d) −12

#### Question 34:

If α, β be the zero of the polynomial 2x2 + 5x + k such that α2 + β2 + αβ = $\frac{21}{4}$, then k = ?
(a) 3
(b) −3
(c) −2
(d) 2

#### Question 35:

On dividing a polynomial p(x) by a non-zero polynomial q(x), let g(x) be the quotient and r(x) be the remainder, than p(x) = q(x)⋅g(x) + r(x), where

(a) r(x) = 0 always
(b) deg r (x) <deg g(x) always
(c) either r(x) = 0 or deg r(x) <deg g(x)
(d) r(x) = g(x)

#### Question 36:

Which of the following is a true statement?

(a) x2 + 5x − 3 is a linear polynomial.
(b) x2 + 4x − 1 is a binomial.
(c) x + 1 is a monomial.
(d) 5x3 is a monomial.

#### Question 37:

If α, β are the zeros of the polynomial ax2 + bx + c, then (α2 + β2) = ?

(a) $\frac{{a}^{2}-2bc}{{b}^{2}}$
(b) $\frac{{b}^{2}-2ac}{{a}^{2}}$
(c) $\frac{{a}^{2}+2bc}{{b}^{2}}$
(d) $\frac{{b}^{2}+2ac}{{a}^{2}}$

#### Question 38:

Which of the following is not a graph of a quadratic polynomial?

(a)
(b)
(c)
(d)

(c)

#### Question 39:

I. If α, β are the zeros of the polynomial x2p(x + 1) −c, then (a + 1)(β + 1) = 1 − c.

II. If α, β are the zeros of the polynomial x2 + px + q, then the polynomial having $\frac{1}{\mathrm{\alpha }},\frac{1}{\mathrm{\beta }}$ as zeros is qx2 + px + 1.
III. When x3 + 3x2 − 5x + 4 is divided by (x + 1), then the remainder is 9.

Which of the above statements is false?

(a) I only
(b) II only
(c) III only
(d) I and III both

#### Question 40:

I. If the polynomial p(x) = 2x3kx2 + 5x + 2 is exactly divisible by (x + 2), then k = −6.

II. If the polynomial q(x) = x3 − 7x + k when divided by (x − 1) leaves the remainder 2, then k = 6.

III. If two zeros of the polynomial f(x) = x3 − 5x2 − 16x + 80 are equal in magnitude and opposite in sigh, then the third zero is 5.

Which of the above statements is not true?

(a) I only
(b) II only
(c) III only
(d) I as well as II

#### Question 41:

Assertion (A)
If one zero of the polynomial p(x) = (k2 + 4) x2 + 9x + 4k is the reciprocal of the other zero, then k = 2.

Reason (R)
If (x − α) is a factor of the polynomial p(x), then α is a zero of p(x).

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

#### Question 42:

Assertion (A)
The polynomial p(x) = x3 + x has one real zero.

Reason (R)
A polynomial of nth degree has at most n zeros.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

#### Question 43:

Assertion (A)
If on dividing the polynomial p(x) = x2 − 3ax + 3a − 7 by (x + 1), we get 6 as remainder, then a= 3.

Reason (R)
When a polynomial p(x) is divided by (x − α), then the remainder is p(α).

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

#### Question 44:

Assertion (A)
A monic quadratic polynomial having 4 and −2 as zeroes is x2 − 2x − 8.

Reason (R)
The monic quadratic polynomial having α and β as zeroes is given by p(x) = x2 − (α + β) x + αβ.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

#### Question 45:

If the zeros of a quadratic polynomial ax2 + bx + c are both negative, then a, b, c will have the same sign.

(a) True
(b) False

#### Question 46:

Match the following columns:

 Column I Column II (a) If α and β be the zeros of the polynomial x2 − 5x + k such that (α − β) = 1, then k = ......... . (p) 10 (b) If one zero of 4x2 + 17x + p is the reciprocal of the other, then p = ......... . (q) −3 (c) If the zeros of x3 − 6x2 + 3x + m are (a − d), a and (a + d), then m = ......... . (r) 4 (d) If the zeros of x3 + 9x2 + 23x + 15 are (a − d), a and (a + d), then a = ......... . (s) 6

#### Question 47:

Match the following columns:

 Column I Column II (a) The polynomial whose zeros are 2 and −3 is ......... . (p) x2 − 4x + 1 (b) The polynomial whose zeros are is ......... . (q) ${x}^{2}-2\sqrt{3}x+2$ (c) The polynomial whose zeros are is ......... . (r) x2 + x − 6 (d) The polynomial whose zeros are (s) 4x2 − 4x − 3

#### Question 1:

Zeros of p(x) = x2 − 2x − 3 are

(a) 1, −3
(b) 3, −1
(c) −3, −1
(d) 1, 3

(b) 3,-1
Here, ${\mathrm{p}\left(\mathrm{x}\right)=x}^{2}-2x-3\phantom{\rule{0ex}{0ex}}$

#### Question 2:

If α, β, γ are the zeros of the polynomial x3 − 6x2x + 30, then the value of (αβ + βγ + γα) is

(a) −1
(b) 1
(c) −5
(d) 30

(a) −1
Here,

Comparing the given polynomial with , we get:

#### Question 3:

If α, β are the zeroes of kx2 − 2x + 3k such that α + β = αβ, then k = ?

(a) $\frac{1}{3}$
(b) $\frac{-1}{3}$
(c) $\frac{2}{3}$
(d) $\frac{-2}{3}$

(c) $\frac{2}{3}$
Here, $\mathrm{p}\left(x\right)={x}^{2}-2x+3k$
Comparing the given polynomial with $a{x}^{2}+bx+c$, we get:

It is given that
are the roots of the polynomial.

Also, =$\frac{c}{a}$

#### Question 4:

It is given that the difference between the zeroes of 4x2 − 8kx + 9 is 4 and k > 0. Then, k = ?

(a) $\frac{1}{2}$
(b) $\frac{3}{2}$
(c) $\frac{5}{2}$
(d) $\frac{7}{2}$

(c) $\frac{5}{2}$
Let the zeroes of the polynomial be .
Here,
p
Comparing the given polynomial with $a{x}^{2}+bx+c$, we get:
a = 4, b = −8k and c = 9
Now, sum of the roots$=-\frac{b}{a}$

#### Question 5:

Find the zeros of the polynomial x2 + 2x − 195.

Here, p

#### Question 6:

If one zero of the polynomial (a2 + 9)x2 + 13x + 6a is the reciprocal of the other, find the value of a.

#### Question 7:

Find a quadratic polynomial whose zeros are 2 and −5.

It is given that the two roots of the polynomial are 2 and −5.
Let
Now, sum of the zeroes, $\mathrm{\alpha }+\mathrm{\beta }$ = 2 + (5) = 3
Product of the zeroes, $\mathrm{\alpha \beta }$ = 2$×$5 = 10
∴ Required polynomial = ${x}^{2}-\left(\mathrm{\alpha }+\mathrm{\beta }\right)x+\mathrm{\alpha \beta }$
$={x}^{2}—\left(-3\right)x+\left(-10\right)\phantom{\rule{0ex}{0ex}}={x}^{2}+3x-10$

#### Question 8:

If the zeroes of the polynomial x3 − 3x2 + x + 1 are (ab), a and (a + b), find the values of a and b.

The given polynomial and its roots are .

#### Question 9:

Verify that 2 is a zero of the polynomial x3 + 4x2 − 3x − 18.

Let p$\left(\mathrm{x}\right)={x}^{3}+4{x}^{2}-3x-18$

#### Question 10:

Find the quadratic polynomial, the sum of whose zeroes is −5 and their product is 6.

Given:
Sum of the zeroes = −5
Product of the zeroes = 6
∴ Required polynomial =
$={x}^{2}-\left(-5\right)x+6\phantom{\rule{0ex}{0ex}}={x}^{2}+5x+6$

#### Question 11:

Find a cubic polynomial whose zeros are 3, 5 and −2.

#### Question 12:

Using remainder theorem, find the remainder when p(x) = x3 + 3x2 − 5x + 4 is divided by (x − 2).

#### Question 13:

Show that (x + 2) is a factor of f(x) = x3 + 4x2 + x − 6.

#### Question 14:

If α, β, γ are the zeroes of the polynomial p(x) = 6x3 + 3x2 − 5x + 1, find the value of $\left(\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}+\frac{1}{\mathrm{\gamma }}\right)$

Comparing the polynomial with ${x}^{3}-{x}^{2}\left(\alpha +\beta +\gamma \right)+x\left(\alpha \beta +\beta \gamma +\gamma \alpha \right)-\alpha \beta \gamma$, we get:

#### Question 15:

If α, β are the zeros of the polynomial f(x) = x2 − 5x + k such that α − β = 1, find the value of k.

#### Question 16:

Show that the polynomial f(x) = x4 + 4x2 + 6 has no zeroes.

#### Question 17:

If one zero of the polynomial p(x) = x3 − 6x2 + 11x − 6 is 3, find the other two zeroes.

#### Question 18:

If two zeroes of the polynomial p(x) = 2x4 − 3x3 − 3x2 + 6x − 2 are $\sqrt{2}$ and $-\sqrt{2}$, find its other two zeroes.

#### Question 19:

Find the quotient when p(x) = 3x4 + 5x3 − 7x2 + 2x + 2 is divided by (x2 + 3x + 1).

#### Question 20:

Use remainder theorem to find the value of k, it being given that when x3 + 2x2 + kx + 3 is divided by (x − 3), then the remainder is 21.