#### Chapters

Chapter 2: Polynomials

Chapter 3: Pair of Linear Equations in Two Variables

Chapter 4: Quadratic Equations

Chapter 5: Arithmetic Progression

Chapter 6: Co-Ordinate Geometry

Chapter 7: Triangles

Chapter 8: Circles

Chapter 9: Constructions

Chapter 10: Trigonometric Ratios

Chapter 11: Trigonometric Identities

Chapter 12: Trigonometry

Chapter 13: Areas Related to Circles

Chapter 14: Surface Areas and Volumes

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 2: Polynomials

#### RD Sharma solutions for Class 10 Mathematics Chapter 2 Polynomials Exercise 2.1, 21.00 [Pages 33 - 35]

find the zeroes of the quadratic polynomial x^{2} – 2x – 8 and verify a relationship between zeroes and its coefficients

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients

4s^{2} – 4s + 1

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients

t^{2} – 15

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients

`p(x) = x^2 + 2sqrt2x + 6`

`q(x)=sqrt3x^2+10x+7sqrt3`

`f(x)=x^2-(sqrt3+1)x+sqrt3`

`g(x)=a(x^2+1)-x(a^2+1)`

`6x^2-3-7x`

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α - β

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/alpha-1/beta`

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/alpha+1/beta-2alphabeta`

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α^{2}β + αβ^{2}

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α^{4} + β^{4}

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/(aalpha+b)+1/(abeta+b)`

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `beta/(aalpha+b)+alpha/(abeta+b)`

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate

`a[alpha^2/beta+beta^2/alpha]+b[alpha/beta+beta/alpha]`

If α and β are the zeros of the quadratic polynomial f(x) = 6x^{2} + x − 2, find the value of `alpha/beta+beta/alpha`

If a and are the zeros of the quadratic polynomial f(x) = 𝑥^{2} − 𝑥 − 4, find the value of `1/alpha+1/beta-alphabeta`

If 𝛼 and 𝛽 are the zeros of the quadratic polynomial p(x) = 4x^{2} − 5x −1, find the value of α^{2β} + αβ^{2}.

If a and 3 are the zeros of the quadratic polynomial f(x) = x^{2} + x − 2, find the value of `1/alpha-1/beta`

If 𝛼 and 𝛽 are the zeros of the quadratic polynomial f(x) = x^{2} − 5x + 4, find the value of `1/alpha+1/beta-2alphabeta`

If 𝛼 and 𝛽 are the zeros of the quadratic polynomial f(t) = t^{2} − 4t + 3, find the value of `alpha^4beta^3+alpha^3beta^4`

If α and β are the zeros of the quadratic polynomial p(y) = 5y^{2} − 7y + 1, find the value of `1/alpha+1/beta`

If α and β are the zeros of the quadratic polynomial p(s) = 3s^{2} − 6s + 4, find the value of `alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta`

If α and β are the zeros of the quadratic polynomial f(x) = x^{2} − px + q, prove that `alpha^2/beta^2+beta^2/alpha^2=p^4/q^2-(4p^2)/q+2`

If the squared difference of the zeros of the quadratic polynomial f(x) = x^{2} + px + 45 is equal to 144, find the value of p.

If the sum of the zeros of the quadratic polynomial f(t) = kt^{2} + 2t + 3k is equal to their product, find the value of k.

If one zero of the quadratic polynomial f(x) = 4x^{2} − 8kx − 9 is negative of the other, find the value of k.

If α and β are the zeros of the quadratic polynomial f(x) = x^{2} − 1, find a quadratic polynomial whose zeroes are `(2alpha)/beta" and "(2beta)/alpha`

If α and β are the zeros of the quadratic polynomial f(x) = x^{2} − 3x − 2, find a quadratic polynomial whose zeroes are `1/(2alpha+beta)+1/(2beta+alpha)`

If α and β are the zeros of a quadratic polynomial such that α + β = 24 and α − β = 8, find a quadratic polynomial having α and β as its zeros.

If α and β are the zeros of the quadratic polynomial f(x) = x^{2} − p (x + 1) — c, show that (α + 1)(β +1) = 1− c.

If If α and β are the zeros of the quadratic polynomial f(x) = x^{2} – 2x + 3, find a polynomial whose roots are α + 2, β + 2.

If If α and β are the zeros of the quadratic polynomial f(x) = x^{2} – 2x + 3, find a polynomial whose roots are `(alpha-1)/(alpha+1)` , `(beta-1)/(beta+1)`

If α and β are the zeroes of the polynomial f(x) = x^{2} + px + q, form a polynomial whose zeroes are (α + β)^{2} and (α − β)^{2}.

#### RD Sharma solutions for Class 10 Mathematics Chapter 2 Polynomials Exercise 2.2 [Page 43]

Verify that the numbers given along side of the cubic polynomials are their zeroes. Also verify the relationship between the zeroes and the coefficients.

`2x^3 + x^2 – 5x + 2 ; 1/2, 1, – 2`

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case

x^{3} – 4x^{2} + 5x – 2; 2, 1, 1

Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and product of its zeros as 3, −1 and −3 respectively.

If the zeros of the polynomial f(x) = 2x^{3} − 15x^{2} + 37x − 30 are in A.P., find them.

Find the condition that the zeros of the polynomial f(x) = x^{3} + 3px^{2} + 3qx + r may be in A.P.

If the zeros of the polynomial f(x) = ax^{3} + 3bx^{2} + 3cx + d are in A.P., prove that 2*b*^{3} − 3*abc* + *a*^{2}*d* = 0.

If the zeros of the polynomial f(x) = x^{3} − 12x^{2} + 39x + k are in A.P., find the value of *k*.

#### RD Sharma solutions for Class 10 Mathematics Chapter 2 Polynomials Exercise 2.3 [Pages 57 - 58]

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following f(x) = x^{3} − 6x^{2} + 11x − 6, g(x) = x^{2} + x + 1

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following f(x) = 10x^{4} + 17x^{3} − 62x^{2} + 30x − 3, g(x) = 2x^{2} + 7x + 1

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following f(x) = 4x^{3}^{ }+ 8x^{2} + 8x + 7, g(x) = 2x^{2} − x + 1

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following f(x) = 15x^{3} − 20x^{2} + 13x − 12; g(x) = x^{2} − 2x + 2

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial

t^{2} – 3, 2t^{4} + 3t^{3} – 2t^{2} – 9t – 12

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial

x^{3} – 3x + 1, x^{5} – 4x^{3} + x^{2} + 3x + 1

Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm g(x) = 2x^{2} − x + 3, f(x) = 6x^{5} − x^{4} + 4x^{3} − 5x^{2} − x − 15

Obtain all zeros of f(x) = x^{3} + 13x^{2} + 32x + 20, if one of its zeros is −2.

Obtain all zeros of the polynomial f(x) = x^{4} − 3x^{3} − x^{2} + 9x − 6, if two of its zeros are `-sqrt3` and `sqrt3`

Find all zeros of the polynomial *f*(*x*) = 2*x*^{4} − 2*x*^{3} − 7*x*^{2} + 3*x* + 6, if its two zeros are `-sqrt(3/2)` and `sqrt(3/2)`

What must be added to the polynomial f(x) = x^{4} + 2x^{3} − 2x^{2} + x − 1 so that the resulting polynomial is exactly divisible by x^{2} + 2x − 3 ?

What must be subtracted from the polynomial f(x) = x^{4} + 2x^{3} − 13x^{2} − 12x + 21 so that the resulting polynomial is exactly divisible by x^{2} − 4x + 3 ?

Find all the zeros of the polynomial x^{4} + x^{3} − 34x^{2} − 4x + 120, if two of its zeros are 2 and −2.

Find all zeros of the polynomial 2x^{4} + 7x^{3} − 19x^{2} − 14x + 30, if two of its zeros are `sqrt2` and `-sqrt2`.

Find all the zeros of the polynomial 2x^{3} + x^{2} − 6x − 3, if two of its zeros are `-sqrt3` and `sqrt3`

Find all the zeros of the polynomial x^{3} + 3x^{2} − 2x − 6, if two of its zeros are `-sqrt2` and `sqrt2`

#### RD Sharma solutions for Class 10 Mathematics Chapter 2 Polynomials [Pages 58 - 61]

Define a polynomial with real coefficients.

Define degree of a polynomial.

Write the standard form of a linear polynomial with real coefficients.

Write the standard form of a quadratic polynomial with real coefficients.

Write the standard form of a cubic polynomial with real coefficients.

Define value of polynomial at a point.

Define the zero of a polynomial.

The sum and product of the zeros of a quadratic polynomial are \[- \frac{1}{2}\] and −3 respectively. What is the quadratic polynomial.

Write the family of quadratic polynomials having \[- \frac{1}{4}\] and 1 as its zeros.

If the product of zeros of the quadratic polynomial *f*(*x*) = *x*^{2} − 4*x* + *k* is 3, find the value of *k*.

If the sum of the zeros of the quadratic polynomial *f*(*x*) = *kx*^{2} − 3*x* + 5 is 1, write the value of *k*.

In Fig. 2.17, the graph of a polynomial *p*(*x*) is given. Find the zeros of the polynomial.

The graph of a polynomial *y* = *f*(*x*), shown in Fig. 2.18. Find the number of real zeros of *f*(*x*).

The graph of the polynomial *f*(*x*) = *ax*^{2} + *bx* + *c* is as shown below (Fig. 2.19). Write the signs of '*a*' and *b*^{2} − 4*ac*.

The graph of the polynomial *f*(*x*) = *ax*^{2} + *bx* + *c* is as shown in Fig. 2.20. Write the value of *b*^{2} − 4*ac* and the number of real zeros of *f*(*x*).

In Q. No. 14, write the sign of *c*.

In Q. No. 15, write the sign of *c*.

The graph of a polynomial *f*(*x*) is as shown in Fig. 2.21. Write the number of real zeros of *f*(*x*).

If *x* = 1 is a zero of the polynomial *f*(*x*) = *x*^{3} − 2*x*^{2} + 4*x* + *k*, write the value of *k*.

State division algorithm for polynomials.

Give an example of polynomials *f*(*x*), *g*(*x*), *q*(*x*) and *r*(*x*) satisfying *f*(*x*) = *g*(*x*), *q*(*x*) + *r*(*x*), where degree *r*(*x*) = 0.

Write a quadratic polynomial, sum of whose zeros is \[2\sqrt{3}\] and their product is 2.

If fourth degree polynomial is divided by a quadratic polynomial, write the degree of the remainder.

If *f*(*x*) = *x*^{3} + *x*^{2} − *ax* + *b* is divisible by *x*^{2} −* x* write the value of *a* and *b*.

If* a* − *b*, *a* and *b* are zeros of the polynomial *f*(*x*) = 2*x*^{3}^{ }− 6*x*^{2} + 5*x* − 7, write the value of *a*.

Write the coefficient of the polynomial *p*(*z*) = *z*^{5} − 2*z*^{2} + 4.

Write the zeros of the polynomial *x*^{2} − *x* − 6.

If (*x* + *a*) is a factor of 2*x*^{2} + 2*ax* + 5*x* + 10, find *a*.

For what value of *k*, −4 is a zero of the polynomial *x*^{2} − *x* − (2*k* + 2)?

If 1 is a zero of the polynomial *p*(*x*) = *ax*^{2} − 3(*a* − 1) *x* − 1, then find the value of *a*.

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

If α, β are the zeros of the polynomial 2*y*^{2} + 7*y* + 5, write the value of α + β + αβ.

For what value of *k*, is 3 a zero of the polynomial 2*x*^{2} + *x* + *k*?

For what value of *k*, is −3 a zero of the polynomial *x*^{2} + 11*x* + *k*?

For what value of *k*, is −2 a zero of the polynomial 3*x*^{2} + 4*x* + 2*k*?

If a quadratic polynomial* f*(*x*) is factorizable into linear distinct factors, then what is the total number of real and distinct zeros of *f*(*x*)?

If a quadratic polynomial *f*(*x*) is a square of a linear polynomial, then its two zeros are coincident. (True/False).

If a quadratic polynomial *f*(*x*) is not factorizable into linear factors, then it has no real zero. (True/False)

If *f*(*x*) is a polynomial such that *f*(*a*) *f*(*b*) < 0, then what is the number of zeros lying between *a* and *b*?

If graph of quadratic polynomial *ax*^{2} + *bx* + *c* cuts positive direction of *y*-axis, then what is the sign of *c*?

If the graph of quadratic polynomial *ax*^{2} + *bx* + *c* cuts negative direction of *y*-axis, then what is the sign of *c*?

#### RD Sharma solutions for Class 10 Mathematics Chapter 2 Polynomials [Pages 61 - 64]

If α, β are the zeros of the polynomial *f*(*x*) = *x*^{2} + *x* + 1, then \[\frac{1}{\alpha} + \frac{1}{\beta} =\]

1

-1

0

None of these

If α, β are the zeros of the polynomial *p*(*x*) = 4*x*^{2} + 3*x* + 7, then \[\frac{1}{\alpha} + \frac{1}{\beta}\] is equal to

- \[\frac{7}{3}\]
- \[- \frac{7}{3}\]
- \[\frac{3}{7}\]
- \[- \frac{3}{7}\]

If one zero of the polynomial *f*(*x*) = (*k*^{2} + 4)*x*^{2} + 13*x* + 4*k* is reciprocal of the other, then *k*=

2

-2

1

-1

If the sum of the zeros of the polynomial *f*(x) = 2*x*^{3} − 3*kx*^{2} + 4*x* − 5 is 6, then the value of*k* is

2

4

−2

−4

If α and β are the zeros of the polynomial *f*(*x*) = *x*^{2} + *px* + *q*, then a polynomial having \[\frac{1}{\alpha} \text{and}\frac{1}{\beta}\] is its zero is

*x*^{2}+*qx*+*p**x*^{2}−*px*+*q**qx*^{2}+*px*+ 1*px*^{2}+*qx*+ 1

If α, β are the zeros of polynomial *f*(*x*) = *x*^{2} − *p* (*x* + 1) − *c*, then (α + 1) (β + 1) =

*c*− 11 −

*c**c*1 +

*c*

If α, β are the zeros of the polynomial *f*(*x*) = *x*^{2} − *p*(*x* + 1) − c such that (α +1) (β + 1) = 0, then *c* =

1

0

-1

2

If *f*(*x*) = *ax*^{2} + *bx* + *c* has no real zeros and *a* + *b* + *c* = 0, then

*c*= 0*c*> 0*c*< 0None of these

If the diagram in Fig. 2.22 shows the graph of the polynomial *f*(*x*) = *ax*^{2} + *bx* + *c*, then

*a*> 0,*b*< 0 and*c*> 0*a*< 0,*b*< 0 and*c*< 0*a*< 0,*b*> 0 and*c*> 0*a*< 0,*b*> 0 and*c*< 0

Figure 2.23 show the graph of the polynomial *f*(*x*) = *ax*^{2} + *bx* + *c* for which

*a*< 0,*b*> 0 and c > 0*a*< 0,*b*< 0 and c > 0*a*< 0,*b*< 0 and c < 0*a*> 0,*b*> 0 and c < 0

If the product of zeros of the polynomial *f*(*x*) *ax*^{3} − 6*x*^{2} + 11*x* − 6 is 4, then *a* =

\[\frac{3}{2}\]

\[- \frac{3}{2}\]

\[\frac{2}{3}\]

\[- \frac{2}{3}\]

\[- \frac{2}{3}\]

If zeros of the polynomial *f*(*x*) = *x*^{3} − 3*px*^{2} + *qx* − *r* are in A.P., then

2

*p*^{3}=*pq*−*r*2

*p*^{3}=*pq*+*r**p*^{3}=*pq*−*r*None of these

If the product of two zeros of the polynomial *f*(*x*) = 2*x*^{3} + 6*x*^{2} − 4*x* + 9 is 3, then its third zero is

- \[\frac{3}{2}\]
- \[- \frac{3}{2}\]
- \[\frac{9}{2}\]
- \[- \frac{9}{2}\]

If the polynomial *f*(*x*) = *ax*^{3} + *bx* − *c* is divisible by the polynomial *g*(*x*) = *x*^{2} + *bx* + *c*, then *ab* =

1

- \[\frac{1}{c}\]
−1

- \[- \frac{1}{c}\]

If Q.No. 14, *c* =

*b*2

*b*2

*b*^{2}−2

*b*

If one root of the polynomial *f*(*x*) = 5*x*^{2} + 13*x* + *k* is reciprocal of the other, then the value of *k* is

0

5

- \[\frac{1}{6}\]
6

If α, β, γ are the zeros of the polynomial *f*(*x*) = *ax*^{3} + *bx*^{2} +* **cx* + *d*, the\[\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} =\]

- \[- \frac{b}{d}\]
- \[\frac{c}{d}\]
- \[- \frac{c}{d}\]
- \[- \frac{c}{a}\]

If α, β, γ are the zeros of the polynomial *f*(*x*) = *ax*^{3} + *bx*^{2}^{ }+ *cx* + *d*, then α^{2} + β^{2} + γ^{2} =

- \[\frac{b^2 - ac}{a^2}\]
- \[\frac{b^2 - 2ac}{a}\]
- \[\frac{b^2 + 2ac}{b^2}\]
- \[\frac{b^2 - 2ac}{a^2}\]

If α, β, γ are are the zeros of the polynomial *f*(*x*) = *x*^{3} − *px*^{2} + *qx* − *r*, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]

If α, β are the zeros of the polynomial *f*(*x*) = *ax*^{2} + *bx* + *c*, then\[\frac{1}{\alpha^2} + \frac{1}{\beta^2} =\]

- \[\frac{b^2 - 2ac}{a^2}\]
- \[\frac{b^2 - 2ac}{c^2}\]
- \[\frac{b^2 + 2ac}{a^2}\]
- \[\frac{b^2 + 2ac}{c^2}\]

If two of the zeros of the cubic polynomial* **ax*^{3} + *bx*^{2} + *cx* + *d* are each equal to zero, then the third zero is

- \[\frac{- d}{a}\]
- \[\frac{c}{a}\]
- \[\frac{- b}{a}\]
- \[\frac{b}{a}\]

If two zeros *x*^{3} +* **x*^{2} − 5*x* − 5 are \[\sqrt{5}\ \text{and} - \sqrt{5}\], then its third zero is

1

−1

2

−2

The product of the zeros of *x*^{3} + 4*x*^{2} + *x* − 6 is

−4

4

6

−6

What should be added to the polynomial *x*^{2} − 5*x* + 4, so that 3 is the zero of the resulting polynomial?

1

2

4

5

What should be subtracted to the polynomial *x*^{2} − 16*x* + 30, so that 15 is the zero of the resulting polynomial?

30

14

15

16

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

*x*^{2}− 9*x*^{2}+ 9*x*^{2}+ 3*x*^{2}− 3

If two zeroes of the polynomial *x*^{3} + *x*^{2} − 9*x* − 9 are 3 and −3, then its third zero is

-1

1

-9

9

If \[\sqrt{5}\ \text{and} - \sqrt{5}\] are two zeroes of the polynomial *x*^{3} + 3*x*^{2} − 5*x* − 15, then its third zero is

3

-3

5

-5

If *x* + 2 is a factor of *x*^{2} + *ax* + 2*b* and *a* + *b* = 4, then

*a*= 1,*b*= 3*a*= 3,*b*= 1*a*= −1,*b*= 5*a*= 5,*b*= −1

The polynomial which when divided by −*x*^{2} + *x* − 1 gives a quotient *x* − 2 and remainder 3, is

*x*^{3}− 3*x*^{2}+ 3*x*− 5−

*x*^{3}− 3*x*^{2}− 3*x*− 5−

*x*^{3}+ 3*x*^{2}− 3*x*+ 5*x*^{3}− 3*x*^{2}− 3*x*+ 5

## Chapter 2: Polynomials

## RD Sharma solutions for Class 10 Mathematics chapter 2 - Polynomials

RD Sharma solutions for Class 10 Mathematics chapter 2 (Polynomials) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10 Mathematics chapter 2 Polynomials are Introduction to Polynomials, Geometrical Meaning of the Zeroes of a Polynomial, Relationship Between Zeroes and Coefficients of a Polynomial, Division Algorithm for Polynomials, Polynomials Examples and Solutions.

Using RD Sharma Class 10 solutions Polynomials exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 10 prefer RD Sharma Textbook Solutions to score more in exam.

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