RD SHARMA Solutions for Class 10 Maths Chapter 2 - Polynomials

Chapter 2 - Polynomials Exercise Ex. 2.1

Question 1(i)

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

f(x) = x2 - 2x - 8 

Solution 1(i)

x2 - 2x - 8 = x2 - 4x + 2x - 8 = x(x - 4) + 2(x - 4) = (x - 4)(x + 2) The zeroes of the quadratic equation are 4 and -2. Let = 4 and β = -2 Consider f(x) = x2 - 2x - 8 Sum of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(i) Also, + β = 4 - 2 = 2 …(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = -8 …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients isverified.

Question 1(ii)

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

g(s) = 4s2 - 4s + 1 

Solution 1(ii)

4s2 - 4s + 1 = 4s2 - 2s - 2s + 1 = 2s(2s - 1) - (2s - 1) = (2s - 1)(2s - 1) The zeroes of the quadratic equation are Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomialsand Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials. Let = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials Consider4s2 - 4s + 1 Sum of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(i) Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 1(iii)

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

h (t) = t2 - 15 

Solution 1(iii)

h (t) = t2 - 15 = (t + √15)(t - √15)  The zeroes of the quadratic equation areRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomialsand Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials. Let = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials Considert2 - 15 = t2 - 0t - 15 Sum of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(i) Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 1(iv)

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

f(s) = 6x2 - 3 - 7x 

Solution 1(iv)

f(s) = 0 6x2 - 3 - 7x =0 6x2 - 9x + 2x - 3 = 0 3x (2x - 3) + (2x - 3) = 0 (3x + 1) (2x - 3) = 0 The zeroes of a quadratic equation are Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials. Let = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials Consider6x2 - 7x - 3 = 0 Sum of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(i) Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 1(v)

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials 

Solution 1(v)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials The zeroes of a quadratic equation are Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomialsand Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials. Let = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials ConsiderRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials Sum of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(i) Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 1(vi)

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials 

Solution 1(vi)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials The zeroes of a quadratic equation are Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomialsand Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials. Let = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials ConsiderRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials Sum of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(i) Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 1(vii)

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials 

Solution 1(vii)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials The zeroes of a quadratic equation are Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and 1. Let = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and β = 1 ConsiderRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials Sum of the zeroes =Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials …(i)  Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 1(viii)

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials 

Solution 1(viii)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - PolynomialsRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials The zeroes of a quadratic equation are a andRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials. Let = a and β =Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials ConsiderRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials Sum of the zeroes =Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials …(i)  Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 1(ix)

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials 

Solution 1(ix)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials The zeroes of a quadratic equation are Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomialsand Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials. Let = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials ConsiderRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials Sum of the zeroes =Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials …(i)  Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 1(x)

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials 

Solution 1(x)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials The zeroes of a quadratic equation are Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomialsand Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials. Let = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials ConsiderRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials=0 Sum of the zeroes =Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials …(i)  Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 1(xi)

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials 

Solution 1(xi)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials The zeroes of a quadratic equation are Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials andRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials. Let = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and β =Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials ConsiderRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials=0 Sum of the zeroes =Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials …(i)  Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 1(xii)

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials 

Solution 1(xii)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials The zeroes of a quadratic equation are Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials andRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials. Let = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials and β =Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials ConsiderRd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials=0 Sum of the zeroes =Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials …(i)  Also, + β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(ii) Product of the zeroes = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iii) Also, β = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials…(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Question 2(i)

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorization.

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 2(i)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 2(ii)

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorization.

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 2(ii)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 2(iii)

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorization.

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 2(iii)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 2(iv)

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorization.

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 2(iv)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 3

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 3

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 4

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 4

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 5

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 5

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 6

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 6

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 7

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 7

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 8

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 8

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 9

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 9

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 10

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 10

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 11

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 11

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 12

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 12

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 13

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 13

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 14

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 14

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 15

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 15

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 16

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 16

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 17

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 17

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 18

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 18

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 19

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 19

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 20

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 20

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 21

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 21

(i)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

(ii)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

(iii)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

(iv)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

(v)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

(vi)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

(vii)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

(viii)

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials 
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Chapter 2 - Polynomials Exercise Ex. 2.2

Question 1
Verify that the numbers given along side of the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and coefficients in each case:

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 1
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

On comparing the given polynomial with the polynomial ax3 + bx2 + cx + d, we obtain a = 2, b = 1, c = -5, d = 2

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Thus, the relationship between the zeroes and the coefficients is verified.

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

On comparing the given polynomial with the polynomial ax3 + bx2 + cx + d, we obtain a = 1, b = -4, c = 5, d = -2.

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Thus, the relationship between the zeroes and the coefficients is verified.

Concept insight: The zero of a polynomial is that value of the variable which makes the polynomial 0. Remember that there are three  relationships between the zeroes of a cubic polynomial and its coefficients which involve the sum of zeroes, product of all zeroes and the product of zeroes taken two at a time.

Question 2
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 2
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 3
If the zeros of the polynomial f(x) = 2x3 - 15x2 + 37x - 30 are in A.P., find them.
Solution 3

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 4
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 4
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 5
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 5
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 6
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 6
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Chapter 2 - Polynomials Exercise Ex. 2.3

Question 1 (i)
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 1 (i)
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 1 (ii)
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 1 (ii)
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 1 (iii)
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 1 (iii)
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 1 (iv)
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 1 (iv)
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 2
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 2
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 3
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 3
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 4
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 4
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 5
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 5
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 6
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Solution 6
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials
Question 7

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 7

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 8

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 8

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 9

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 9

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 10

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 10

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 11

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 11

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 12

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 12

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 13

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 13

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 14

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 14

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Chapter 2 - Polynomials Exercise 2.61

Question 1

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

(a) 1

(b) -1

(c) 0

(d) None of these

Solution 1

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 2

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 2

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

So, the correct option is (d).

Question 3

If one zero of the polynomial f(x) = (k2 + 4)x+ 13x + 4k is reciprocal of the other, then k =

(a) 2

(b) -2

(c) 1

(d) -1

Solution 3

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Chapter 2 - Polynomials Exercise 2.62

Question 4

If the sum of the zeros of the polynomial f(x) = 2x3 - 3kx2 + 4x - 5 is 6, then value of k is

(a) 2

(b) 4

(c) -2

(d) -4

Solution 4

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

So, the correct option is (b).

Question 5

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

(a) x+ qx + p

(b) x2 - px + q

(c) qx2 + px + 1

(d) px2 + qx + 1

Solution 5

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 6

If α, β are the zeros of polynomial f(x) = x- p(x + 1) - c, then (α + 1) (β + 1) =

(a) c - 1

(b) 1 - c

(c) c

(d) 1 + c

Solution 6

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 7

If α, β are the zeros of the polynomial f(x) = x- p(x + 1) - c such that (α + 1) (β + 1) = 0 then c =

(a) 1

(b) 0

(c) -1

(d) 2

Solution 7

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Given that (α + 1) (β + 1) = 0

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

So, the correct option is (a).

Question 8

If f(x) = ax2 + bx + c has no real zeros and a + b + c < 0, then

(a) c = 0

(b) c > 0

(c) c < 0

(d) None of these

Solution 8

We know that, if the quadratic equation ax2 + bx + c = 0 has no real zeros

then

Case 1:

a > 0, the graph of quadratic equation should not intersect x - axis

It must be of the type

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Case 2 :

a < 0, the graph will not intersect x - axis and it must be of type

 Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

According to the question,

a + b + c < 0

This means,

f(1) = a + b + c

f(1) < 0

Hence, f(0) < 0 [as Case 2 will be applicable]

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

 So, the correct option is (c).

Question 9

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

If the diagram in figure show the graph of the polynomial f(x) = ax2 + bx + c then

(a) a > 0, b < 0 and c > 0

(b) a < 0, b < 0 and c < 0

(c) a < 0, b > 0 and c > 0

(d) a < 0, b > 0 and c < 0

Solution 9

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 10

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Figure shows the graph of the polynomial f(x) = ax2 + bx + c for which

(a) a < 0, b > 0 and c > 0

(b) a < 0, b < 0 and c > 0

(c) a < 0, b < 0 and c < 0

(d) a > 0, b > 0 and c < 0

Solution 10

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 11

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 11

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 12

If zeros of the polynomial f(x) = x3 - 3px2 + qx - r are in A.P, then

(a) 2p3 = pq - r

(b) 2p3 = pq + r

(c) p3 = pq - r

(d) None of these

Solution 12

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Chapter 2 - Polynomials Exercise 2.63

Question 13

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 13

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

So, the correct option is (b).

Question 14

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 14

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 15

In Q. No. 14, c =

(a) b

(b) 2b

(c) 2b2

(d) -2b

Solution 15

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

So, the correct option is (c).

Question 16

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 16

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 17

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 17

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 18

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 18

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 19

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 19

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 20

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 20

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 21

If two of the zeros of the cubic polynomial ax + bx2 + cx + d are each equal to zero then the third zero is

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 21

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 22

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 22

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 23

The product of the zeros of x+ 4x2 + x - 6 is 

(a) -4

(b) 4

(c) 6

(d) -6

Solution 23

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Chapter 2 - Polynomials Exercise 2.64

Question 24

What should be added to the polynomial x2 - 5x + 4, so that 3 is the zero of the resulting polynomial ?

(a) 1

(b) 2

(c) 4

(d) 5

Solution 24

We know that, if α and β are roots of ax+ bx + c = 0 then they must satisfy the equation.

According to the question, the equation is

x- 5x + 4 = 0

If 3 is the root of equation it must satisfy equation.

x- 5x + 4 = 0

but f(3) = 3- 5(3) + 4 = -2

so, 2 has to be added in the equation.

So, the correct option is (b).

Question 25

What should be subtracted to the polynomial x2 - 16x + 30, so that 15 is the zero of resulting polynomial?

(a) 30

(b) 14

(c) 15

(d) 16

Solution 25

We know that, if α and β are roots of ax + bx + c = 0, then α and β must satisfy the equation.

According to the question, the equation is

x- 16x + 30 = 0

If 15 is a root, then it must satisfy the equation x- 16x + 30 = 0, 

But f(15) = 15- 16(15) + 30 = 225 - 240 + 30 = 15

and so 15 should be subtracted from the equation.

So, the correct option is (c).

Question 26

A quadratic polynomial, the sum of whose zeros is 0 and one zero is 3, is 

(a) x- 9

(b) x2 + 9

(c) x + 3

(d) x- 3

Solution 26

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 27

If two zeros of the polynomial x + x- 9x - 9 are 3 and -3, then its third zero is

(a) -1

(b) 1

(c) -9

(d) 9

Solution 27

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 28

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 28

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 29

If x + 2 is a factor of x + ax + 2b and a + b = 4, then

(a) a = 1, b = 3

(b) a = 3, b = 1

(c) a = -1, b = 5

(d) a = 5, b = -1

Solution 29

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 30

The polynomial which when divided by -x+ x - 1 gives a quotient x - 2 and remainder is 3, is

(a) x3 - 3x2 + 3x - 5

(b) -x3 - 3x2 - 3x - 5

(c) -x3 + 3x2  - 3x + 5

(d) x3 - 3x2 - 3x + 5

Solution 30

We know that 

Dividend = Divisor × quotient  + remainder

Then according to question,

Required polynomial

= (-x2 + x - 1) (x - 2) + 3

= -x3 + 2x2 + x2 -2x - x + 2 + 3

= -x3 + 3x2 - 3x + 5

So, the correct option is (c).

Question 31

The number of polynomials having zeroes -2 and 5 is

 

a. 1

b. 2

c. 3

d. more than 3

Solution 31

Correct option: (d)

The polynomials having -2 and 5 as the zeroes can be written in the form 

k(x + 2)(x - 5), where k is a constant. 

Thus, number of polynomials with roots -2 and 5 are infinitely many, since k can take infinitely many values.

Question 32

If one of the zeroes of the quadratic polynomial (k - 1)x2 + kx + 1 is - 3, then the value of k is

 

a. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

b. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

c. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

d. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 32

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 33

The zeroes of the quadratic polynomial x2 + 99x + 127 are

 

a. Both positive

b. Both negative

c. both equal

d. One positive and one negative

Solution 33

The zeroes of the quadratic polynomial x2 + 99x + 127 are both negative since all terms are positive.

Hence, correct option is (b).

Question 34

If the zeroes of the quadratic polynomial x2 + (a + 1)x + b 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 34

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 35

Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is

 

a. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

b. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

c. 0

d. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 35

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 36

The zeroes of the quadratic polynomial x2 + ax + a, a 0,

 

a. cannot both be positive

b. cannot both be negative

c. are always unequal

d. are always equal

Solution 36

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 37

If one of the zeros of the cubic polynomial x3 + ax2 + bx + x is -1, then the product of other two zeroes is

 

a. b - a + 1

b. b - a - 1

c. a - b + 1

d. a - b - 1

Solution 37

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Chapter 2 - Polynomials Exercise 2.65

Question 38

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

 

a. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

b. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

c. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

d. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Solution 38

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 39

If one zero of the quadratic polynomial x2 + 3x + k is 2, then the value of k is

 

a. 10

b. -10

c. 5

d. -5

Solution 39

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 40

If the zeros of the quadratic polynomial ax2 + bx + c, c 0 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 40

It is given that the zeros of the quadratic polynomial ax2 + bx + c, c ≠ 0 are equal.

Discriminant = 0

b2 - 4ac = 0

b2 = 4ac

Now, b2 can never be negative,

Hence, 4ac also can never be negative.

a and c should have same sign.

Hence, correct option is (c).

Question 41

If one of the zeros of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it

 

a. has no linear term and 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 41

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Question 42

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

 

Solution 42

The graph of a quadratic polynomial crosses X-axis at atmost two points.

Hence, correct option is (d).