# Class 10

Explore popular questions from Polynomials for Class 10. This collection covers Polynomials previous year Class 10 questions hand picked by experienced teachers.

## Mathematics

Polynomials

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Q 1. Which of the following is a zero of the polynomial {tex} \mathrm { p } ( \mathrm { x } ) = \mathrm { x } ^ { 2 } - 2 \mathrm { x } - 3 {/tex}

A

{tex} 0 {/tex}

B

{tex} 1 {/tex}

{tex} - 1 {/tex}

D

{tex} - 3 {/tex}

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Q 2. If one root of the polynomial {tex} 5 \mathrm { x } ^ { 2 } + 13 \mathrm { x } + \mathrm { K } {/tex} is reciprocal of the other, then the value of {tex} \mathrm { k } {/tex} is

A

{tex} 0 {/tex}

{tex} 5 {/tex}

C

{tex} 6 {/tex}

D

{tex} \frac { 1 } { 6 } {/tex}

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Q 3. If {tex} \alpha , \beta {/tex} are the zeroes of the polynomial {tex} f ( x ) = x ^ { 2 } - p ( x + 1 ) - c {/tex} then {tex} ( \alpha + 1 ) ( \beta + 1 ) {/tex} is equal to :

A

{tex} 1 + c {/tex}

{tex} 1 - c {/tex}

C

{tex} c - 1 {/tex}

D

{tex} { c } {/tex}

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Q 4. Quadratic polynomial having zeroes {tex}1 {/tex} and {tex} - 2 {/tex} is

A

{tex} x ^ { 2 } - x + 2 {/tex}

B

{tex} x ^ { 2 } - x - 2 {/tex}

{tex} x ^ { 2 } + x - 2 {/tex}

D

{tex} x ^ { 2 } + x + 2 {/tex}

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Q 5. If {tex} \operatorname { deg } p ( x ) = m {/tex} and {tex} \operatorname { deg } q ( x ) = n , {/tex} then {tex} \operatorname { deg } [ p ( x ) - q ( x ) ] {/tex} equal to:

{tex} \max \{ m , n \} {/tex}

B

{tex} \min \{ m , n \} {/tex}

C

{tex} m + n {/tex}

D

{tex} m - n {/tex}

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Q 6. {tex} p ( x ) {/tex} and {tex} q ( x ) {/tex} are two reducible (factoriable) unequal polymonial with real coefficient and neither of them is a factor of the other. If {tex} \ell {/tex} and {tex} h {/tex} are their LCM and HCF respectively, then {tex} \ell {/tex} and {tex} h {/tex} must satisfy the equality:

A

{tex} \ell p ( x ) = h q ( x ) {/tex}

B

{tex} h p ( x ) = \ell q ( x ) {/tex}

{tex} p ( x ) q ( x ) = \ell h {/tex}

D

{tex} \ell h = 1 {/tex}

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Q 7. The G.C.D. of {tex} ( x + a ) ^ { 2 } {/tex} and {tex} x ^ { 3 } + a ^ { 3 } {/tex} is

A

{tex} ( x + a ) ^ { 2 } {/tex}

B

{tex} x ^ { 2 } - a ^ { 2 } {/tex}

{tex} x + a {/tex}

D

{tex} x ^ { 2 } + a ^ { 2 } {/tex}

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Q 8. The graph of the polynomial {tex} f ( x ) = a x ^ { 2 } + b x + c {/tex} is as shown below, then which of the following is are true:-

{tex} \mathrm { a } > 0 , \mathrm { b } < 0 , \mathrm { D } > 0 {/tex}

B

{tex} \mathrm { a } > 0 , \mathrm { b } > 0 , \mathrm { c } > 0 {/tex}

C

{tex} \mathrm { a } < 0 , \mathrm { b } < 0 , \mathrm { c } > 0 {/tex}

D

{tex} \mathrm { a } < 0 , \mathrm { b } > 0 , \mathrm { c } < 0 {/tex}

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Q 9. If {tex} \alpha {/tex} and {tex} \beta {/tex} are the zeroes of the polynomials {tex} f ( x ) = x ^ { 2 } - 5 x + k {/tex} such that {tex} \alpha - \beta = 1 , {/tex} then value of {tex} \mathrm { k } {/tex} is:-

A

{tex} 4 {/tex}

B

{tex} - 6 {/tex}

{tex} 6 {/tex}

D

{tex} 12 {/tex}

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Q 10. If zeroes of the polynomial {tex} f ( x ) = x ^ { 3 } - 3 p x ^ { 2 } + q x - r {/tex} are in A.P, then

{tex} 2 p ^ { 3 } = p q - r {/tex}

B

{tex} 2 p ^ { 3 } = p q + r {/tex}

C

{tex} p ^ { 3 } = p q - r {/tex}

D

None of these

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Q 11. If the sum of two roots of the polynomial {tex} f ( x ) = x ^ { 3 } - p x ^ { 2 } + q x - r {/tex} is zeroes, then which of the following condition holds good:-

A

{tex} p ^ { 3 } = r ^ { 3 } {/tex}

{tex} p q = r {/tex}

C

{tex} p ^ { 3 } - p r + r = 0 {/tex}

D

{tex} p ^ { 2 } q ^ { 2 } = r_3 {/tex}

##### Explanation

Given polynomial of f(x) = x3 - px2 + qx - r when the sum of its two zeroes is zero. Let the three roots be α , - α and β Sum of zeroes can be expressed as α - α + β = p β = p ​ ---------------------- ( 1 ) Products of roots taken product of two roots = (α)(β) + (-α)(β) + (-α)(α) = q α2 = - q ----------------------- ( 2 ) Product of three roots = (-α )(α)(β) = r = -α2 β = r From (1) and (2) pq = r The correct answer is option (B)

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Q 12. The product of the zeroes of the polynomial {tex} b x ( a x + b ) - c ( a x + b ) {/tex} is

A

{tex} c / b {/tex}

B

{tex} \frac { a c - b ^ { 2 } } { a b } {/tex}

{tex}- c / a {/tex}

D

{tex} - b / a {/tex}

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Q 13. If the zeroes of a quadratic polynomial {tex} a x ^ { 2 } + b x + c {/tex} are equal but opposite in sign then

A

{tex} a = b - c {/tex}

B

{tex} c = 0 {/tex}

C

{tex} b = a {/tex}

{tex} b = 0 {/tex}

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Q 14. If the zeroes of a quadratic polynomial {tex} \alpha x ^ { 2 } + \beta x + \gamma {/tex} are reciprocals of each other then

A

{tex} \beta = \alpha {/tex}

B

{tex} \gamma = \alpha {/tex}

C

{tex} \gamma - \alpha = 0 {/tex}

Both (B) and (C)

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Q 15. The following polynomials of degree 2 is/are

A

{tex} x y + 1 {/tex}

B

{tex} x ^ { 2 } + a x + c {/tex}

C

{tex} x y + y z + x z {/tex}

All of the above

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Q 16. If {tex} ( x + a ) {/tex} is the factor of the polynomials:
{tex} x ^ { 2 } + p x + q {/tex} and {tex} x ^ { 2 } + m x + n , {/tex} then which of the following is/are true:-

{tex} a m - ap - n + q = 0 {/tex}

B

{tex} a m - p m = a n + q {/tex}

C

{tex} a = \frac { m - q } { n - p } {/tex}

D

All of the above

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Q 17. If {tex} \left( x ^ { 2 } - 1 \right) {/tex} is a factor of {tex} a x ^ { 4 } + b x ^ { 3 } + c x ^ { 2 } + d x + e , {/tex} then which of the following is/are correct:

A

{tex} a + c + e = b + d {/tex}

B

{tex} a + c + e = 0 {/tex}

C

{tex} b+d = 0 {/tex}

All of the above

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Q 18. For which of the following values of {tex} a{/tex} will {tex} 3 x ^ { 5 } + 9 x ^ { 4 } - 7 x ^ { 3 } - 5 x ^ { 2 } - 3 a x + 3 a ^ { 2 } {/tex} be divisible by {tex} x - 1 ? {/tex}

{tex} 0 {/tex}

B

{tex} 2 {/tex}

C

{tex} - 1 {/tex}

D

{tex} 3 {/tex}

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Q 19. What is the remainder when {tex} 3 x ^ { 3 } - 2 x ^ { 2 } y - 13 x y ^ { 2 } + 10 y ^ { 3 } {/tex} is divided by {tex} x - 2 y ? {/tex}

A

{tex} x + y {/tex}

B

{tex} 10{/tex}

{tex} 0{/tex}

D

{tex} x + 2 y {/tex}

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Q 20. If {tex} x - y = 1 {/tex} and {tex} x ^ { 2 } + y ^ { 2 } = 41 , {/tex} then the value of {tex} x + y {/tex} will be

A

5

{tex} 9 {/tex}

C

{tex} 4 {/tex}

D

{tex} - 5 {/tex}

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Q 21. If {tex} ( x + a ) {/tex} is the HCF of {tex} x ^ { 2 }+ px + q {/tex} and {tex} x ^ { 2 } + \ell x + m , {/tex} then the value of {tex} ^ { \prime } a ^ { \prime } {/tex} is given by:-

{tex} \frac { q - m } { p - \ell } {/tex}

B

{tex} \frac { p - \ell } { q - m } {/tex}

C

{tex} \frac { q + m } { p + \ell } {/tex}

D

{tex} \frac { ( m - q ) } { p - \ell } {/tex}

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Q 22. If the polynomial {tex} f ( x ) = a x ^ { 3 } + b x - c {/tex} is divisible by the polynomial {tex} g ( x ) = x ^ { 2 } + b x + c , {/tex} then

{tex} a b = 1 {/tex}

B

{tex} c - 2 b ^ { 2 } = 0 {/tex}

C

{tex} a c = 2 b ^ { 2 } {/tex}

D

{tex} a b + 1 = 0 {/tex}

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Q 23. The number of polynomials having zeroes as {tex} - 2 {/tex} and {tex} 5 {/tex} is :

A

1

B

2

C

3

more than 3

##### Explanation

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Q 24. Given that one of the zeroes of the cubic polynomial {tex} a x ^ { 3 } + b x ^ { 2 } + c x + d {/tex} is zero, the product of the other two zeroes is:

A

{tex} - \frac { c } { a } {/tex}

{tex} \frac { c } { a } {/tex}

C

0

D

{tex} - \frac { b } { a } {/tex}

##### Explanation

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Q 25. If one of the zeroes of the cubic polynomial {tex} x ^ { 3 } + a x ^ { 2 } + b x + c {/tex} is {tex} - 1 {/tex}, then the product of the other two zeroes is:

{tex} b - a + 1 {/tex}

B

{tex} b - a - 1 {/tex}

C

{tex} a - b + 1 {/tex}

D

{tex} a - b - 1 {/tex}