#### 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

#### RD Sharma 10 Mathematics

## Chapter 2: Polynomials

#### Chapter 2: Polynomials Exercise 2.10, 21.00 solutions [Pages 0 - 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}.

#### Chapter 2: Polynomials Exercise 2.20 solutions [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*.

#### Chapter 2: Polynomials Exercise 2.30 solutions [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`

#### Chapter 2: Polynomials solutions [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*?

#### Chapter 2: Polynomials solutions [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 10 Mathematics

#### Textbook solutions for Class 10

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

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