R S AGGARWAL AND V AGGARWAL Solutions for Class 10 Maths Chapter 2 - Polynomials

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Chapter 2 - Polynomials Exercise Ex. 2A

Question 1

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

Solution 1

Question 2

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

x2 - 2x - 8

Solution 2

Question 3

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

x2 + 7x + 12

Solution 3

Question 5

Find the zeros of the quadratic polynomial (4x2 - 4x - 3) and verify the relation between its zeros and coefficients.

Solution 5

We have


Question 6

Find the zeros of the quadratic polynomial (5x2 - 4 - 8x) and verify the relationship between its zeros and coefficients of the given polynomial.

Solution 6

Question 7

Find the zeros of the quadratic polynomial (2x2 - 11x + 15) and verify the relation between its zeros and coefficients.

Solution 7

Question 8

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

Solution 8

Question 9

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

4x2 - 4x + 1

Solution 9

Question 10

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

3x2 - x - 4

Solution 10

Question 11

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

5y2 + 10y

Solution 11

Question 12

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

Solution 12

Let

 

 

Question 14

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

Solution 14

 

Question 15

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

Solution 15

Question 16

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

Solution 16

Question 18

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

Solution 18

Question 20

If x = and x = -3 are the roots of the quadratic equation ax2 + 7x + b = 0 then find the values of a and b.

Solution 20

Question 21

One zero of the polynomial 3x3 + 16x2 + 15x - 18 is  Find the other zeros of the polynomial.

Solution 21

Chapter 2 - Polynomials Exercise Ex. 2B

Question 1

Verify 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.

Solution 1
Question 2

Verify that are the zeros of the cubic polynomial p(x) = 3x3 - 10x2 - 27x + 10 and verify the relation between its zeros and coefficients.

Solution 2

Question 3

Find a cubic polynomial whose zeros are 2, -3 and 4.

Solution 3

Question 4

Find a cubic polynomial whose zeros are  1 and -3.

Solution 4

Question 5

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and the product of its zeros as 5, -2 and -24 respectively.

Solution 5

Question 6

Find the quotient and the remainder when:

f(x) = x3 - 3x2 + 5x - 3 is divided by g(x)= x2 - 2.

Solution 6

Question 7

Find the quotient and the remainder when:

f(x)= x4 -3x2 + 4x + 5 is divided by g(x)= x2 + 1 - x.

Solution 7

Question 8

Find the quotient and the remainder when:

f(x)= x4 - 5x + 6 is divided by g(x) = 2 - x2.

Solution 8

Question 9

By actual division, show that x2 - 3 is a factor of 2x4 + 3x3 - 2x2 - 9x - 12.

Solution 9

Question 11

On dividing 3x3 + x2 + 2x + 5 by a polynomial g(x), the quotient and remainder are (3x - 5) and (9x + 10) respectively. Find g(x).

  

Solution 11

Question 12

Verify division algorithm for the polynomials f(x) = 8 + 20x + x2 - 6x3 and g(x) = 2 + 5x - 3x2.

Solution 12

Question 13

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

Solution 13

1

Question 14

If 1 and -2 are two zeros of the polynomial, find its third zero.

Solution 14



Question 15

If 3 and -3 are two zeros of the polynomial, find all the zeroes of the given polynomial.

Solution 15

 

Question 16

If 2 and -2 are two zeros of the polynomial, find all the zeros of the given Polynomial.

Solution 16


Question 17

Find all the zeros of, if it is given that two of its zeros are

Solution 17


Question 18

Obtain all other zeros of , if two of its zeros are .

Solution 18


Question 23

Find all the zeros of the polynomial , it being given that two of its zeros are .

Solution 23



Chapter 2 - Polynomials Exercise Ex. 2C

Question 1

If one zero of the polynomial x2 - 4x + 1 is (2 + ), write the other zero.

Solution 1

Question 2

Find the zeros of the polynomial x2 + x - p(p + 1).

Solution 2

Question 3

Find the zeros of the polynomial x2 - 3x - m(m + 3).

Solution 3

Question 4

  

Solution 4

Question 5

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

Solution 5

Question 6

If 3 is a zero of the polynomial 2x2 + x + k, find the value of k.

Solution 6

Question 7

If -4 is a zero of the polynomial x2 - x - (2k + 2) then find the value of k.

Solution 7

Question 8

If 1 is a zero of the polynomial ax2 - 3(a - 1)x - 1 then find the value of a.

Solution 8

Question 9

If -2 is a zero of the polynomial 3x2 + 4x + 2k then find the value of k.

Solution 9

Question 10

Write the zeros of the polynomial x2 - x - 6.

Solution 10

Question 11

If the sum of the zeros of the quadratic polynomial kx2 - 3x + 5 is 1, write the value of k.

Solution 11

Question 12

If the product of the zeros of the quadratic polynomial x2 - 4x + k is 3 then write the value of k.

Solution 12

Question 13

If (x + a) is a factor of (2x2 + 2ax + 5x + 10), find the value of a.

Solution 13

Question 14

If (a - b), a and (a + b) are zeros of the polynomial 2x3 - 6x2 + 5x - 7, write the value of a.

Solution 14

Question 15

If x3 + x2 - ax + b is divisible by (x2 - x), write the values of a and b.

Solution 15

Question 16

  

Solution 16

Question 17

State division algorithm for polynomials.

Solution 17

If f(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can always find polynomials q(x) and r(x) such that f(x) = q(x)g(x) + r(x),

where r(x) = 0 or degree r(x) < degree g(x).

Question 18

The sum of the zeros and the product of zeros of a quadratic polynomial are  and -3 respectively. Write the polynomial.

Solution 18

Question 19

Write the zeros of the quadratic polynomial f(x) = 6x2 - 3.

Solution 19

Question 20

Solution 20

Question 21

  

Solution 21

Question 22

Solution 22

Question 23

  

Solution 23

Question 24

Solution 24

Question 25

If the zeros of the polynomial f(x) = x3 - 3x2 + x + 1 are (a - b), a and (a + b), find a and b.

Solution 25

Chapter 2 - Polynomials Exercise MCQ

Question 1

Which of the following is a polynomial?

Solution 1

Correct answer: (d)

An expression of the form p(x) = a0 + a1x + a2x2 + ….. + anxn, where an ≠ 0, is called a polynomial in x of degree n.

Here, a0, a1, a2, ……, an are real numbers and each power of x is a non-negative integer.

Question 2

Which of the following is not a polynomial?

Solution 2

Correct answer: (d)

An expression of the form p(x) = a0 + a1x + a2x2 + ….. + anxn, where an ≠ 0, is called a polynomial in x of degree n.

Here, a0, a1, a2, ……, an are real numbers and each power of x is a non-negative integer.

Question 3

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

(a)-3, 1

(b)-3, -1

(c) 3, -1

(d) 3, 1

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

The sum and the 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

Solution 8

Question 9

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

Solution 9

Question 10

(a) 10x2 +x + 3

(b) 10x2 + x - 3

(c) 10x2 - x + 3

(d) 10x2 - x - 3

Solution 10

Question 11

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

Solution 11

Question 12

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

(a) 5

(b) -5

(c) 8

(d) -8

Solution 12

Question 13

If 𝛼 and 𝛽 are the zeros of 2x2 + 5x - 9 then the value of 𝛼𝛽 is

Solution 13

Question 14

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

  

Solution 14

Question 15

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

Solution 15

Question 16

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

Solution 16

Question 17

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

(a) 3

(b) -3

(c)

(d)

Solution 17

Question 18

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

Solution 18

Question 19

(a) 3

(b) -3

(c) 12

(d)-12

Solution 19

Question 20

If 𝛼, 𝛽, 𝛾 are the zeros of the polynomial x3 - 6x2 - x + 30, then (𝛼𝛽 + 𝛽𝛾 + 𝛾𝛼) = ?

(a) -1

(b) 1

(c) -5

(d)30

Solution 20

Question 21

If 𝛼, 𝛽, 𝛾 be the zeros of the polynomial 2x3 + x2 - 13x + 6, then 𝛼𝛽𝛾 

(a) -3

(b) 3

(c) 

(d)   

Solution 21

Question 22

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

Solution 22

Question 23

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

Solution 23

Question 24

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

Solution 24

Question 25

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) a - b - 1

(b) b - a - 1

(c) 1 - a + b

(d) 1 + a - b

Solution 25

Question 26

(a) 3

(b) -3

(c) -2

(d) 2

Solution 26

Question 27

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, then 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)

Solution 27

Question 28

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

Solution 28

Chapter 2 - Polynomials Exercise FA

Question 1

Zeros of p(x) = x2 - 2x - 3 are

(a) 1, -3

(b) 3, -1

(c) -3, -1

(d)1, 3

Solution 1

Question 2

If 𝛼, 𝛽, 𝛾 are the zeros of the polynomial x3 - 6x2 - x + 30, then (𝛼𝛽 + 𝛽𝛾 + 𝛾𝛼) = ?

(a) -1

(b) 1

(c) -5

(d)30

Solution 2

Question 3

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

Solution 3

Question 4

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

Solution 4

Question 5

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

Solution 5

Question 6

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

Solution 6

Question 7

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

Solution 7

Question 8

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

Solution 8

Question 9

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

Solution 9

Question 10

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

Solution 10

Question 11

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

Solution 11

Question 12

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

Solution 12

Question 13

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

Solution 13

Question 14

Solution 14

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.

Solution 15

Question 16

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

Solution 16

Question 17

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

Solution 17

Question 18

Solution 18

Question 19

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

Solution 19

  

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.

Solution 20