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RD Sharma solutions for Class 10 Mathematics chapter 2 - Polynomials

10 Mathematics

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Chapters

RD Sharma 10 Mathematics

10 Mathematics

Chapter 2: Polynomials

Ex. 2.10OthersEx. 21.00Ex. 2.20Ex. 2.30

Chapter 2: Polynomials Exercise 2.10, 21.00 solutions [Pages 0 - 35]

find the zeroes of the quadratic polynomial x2 – 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

4s2 – 4s + 1

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

t2 – 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`

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

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

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

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

 

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

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

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

`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 α2β + αβ2

If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α4 + β4 

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) = 6x2 + x − 2, find the value of `alpha/beta+beta/alpha`

If a and are the zeros of the quadratic polynomial f(x) = 𝑥2 − 𝑥 − 4, find the value of `1/alpha+1/beta-alphabeta`

If 𝛼 and 𝛽 are the zeros of the quadratic polynomial p(x) = 4x2 − 5x −1, find the value of α + αβ2.

If a and 3 are the zeros of the quadratic polynomial f(x) = x2 + x − 2, find the value of `1/alpha-1/beta`

If 𝛼 and 𝛽 are the zeros of the quadratic polynomial f(x) = x2 − 5x + 4, find the value of `1/alpha+1/beta-2alphabeta`

If 𝛼 and 𝛽 are the zeros of the quadratic polynomial f(t) = t2 − 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) = 5y2 − 7y + 1, find the value of `1/alpha+1/beta`

If α and β are the zeros of the quadratic polynomial p(s) = 3s2 − 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) = x2 − 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) = x2 + px + 45 is equal to 144, find the value of p.

If the sum of the zeros of the quadratic polynomial f(t) = kt2 + 2t + 3k is equal to their product, find the value of k.

If one zero of the quadratic polynomial f(x) = 4x2 − 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) = x2 − 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) = x2 − 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) = x2 − 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) = x2 – 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) = x2 – 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) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α − β)2.

Chapter 2: Polynomials Exercise 2.20 solutions [Page 43]

Ex. 2.20 | Q 1.1 | 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`

Ex. 2.20 | Q 1.2 | Page 43

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

x3 – 4x2 + 5x – 2; 2, 1, 1

Ex. 2.20 | Q 2 | Page 43

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.

Ex. 2.20 | Q 3 | Page 43

If the zeros of the polynomial f(x) = 2x3 − 15x2 + 37x − 30 are in A.P., find them.

Ex. 2.20 | Q 4 | Page 43

Find the condition that the zeros of the polynomial f(x) = x3 + 3px2 + 3qx + r may be in A.P.

Ex. 2.20 | Q 5 | Page 43

If the zeros of the polynomial f(x) = ax3 + 3bx2 + 3cx + d are in A.P., prove that 2b3 − 3abc + a2d = 0.

Ex. 2.20 | Q 6 | Page 43

If the zeros of the polynomial f(x) = x3 − 12x2 + 39x + k are in A.P., find the value of k.

Chapter 2: Polynomials Exercise 2.30 solutions [Pages 57 - 58]

Ex. 2.30 | Q 1.1 | Page 57

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) = x3 − 6x2 + 11x − 6, g(x) = x2 + x + 1

Ex. 2.30 | Q 1.2 | Page 57

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) = 10x4 + 17x3 − 62x2 + 30x − 3, g(x) = 2x2 + 7x + 1

Ex. 2.30 | Q 1.3 | Page 57

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) = 4x3 + 8x2 + 8x + 7, g(x) = 2x2 − x + 1

Ex. 2.30 | Q 1.4 | Page 57

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) = 15x3 − 20x2 + 13x − 12; g(x) = x2 − 2x + 2

Ex. 2.30 | Q 2.1 | Page 57

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

t2 – 3, 2t4 + 3t3 – 2t2 – 9t – 12

Ex. 2.30 | Q 2.2 | Page 57

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

x3 – 3x + 1, x5 – 4x3 + x2 + 3x + 1

Ex. 2.30 | Q 2.3 | Page 57

Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm g(x) = 2x2 − x + 3, f(x) = 6x5 − x4 + 4x3 − 5x2 − x − 15

Ex. 2.30 | Q 4 | Page 57

Obtain all zeros of f(x) = x3 + 13x2 + 32x + 20, if one of its zeros is −2.

Ex. 2.30 | Q 5 | Page 57

Obtain all zeros of the polynomial f(x) = x4 − 3x3 − x2 + 9x − 6, if two of its zeros are `-sqrt3` and `sqrt3`

Ex. 2.30 | Q 6 | Page 57

Find all zeros of the polynomial f(x) = 2x4 − 2x3 − 7x2 + 3x + 6, if its two zeros are `-sqrt(3/2)` and `sqrt(3/2)`

Ex. 2.30 | Q 7 | Page 58

What must be added to the polynomial f(x) = x4 + 2x3 − 2x2 + x − 1 so that the resulting polynomial is exactly divisible by x2 + 2x − 3 ?

Ex. 2.30 | Q 8 | Page 58

What must be subtracted from the polynomial f(x) = x4 + 2x3 − 13x2 − 12x + 21 so that the resulting polynomial is exactly divisible by x2 − 4x + 3 ?

Ex. 2.30 | Q 9 | Page 57

Find all the zeros of the polynomial x4 + x3 − 34x2 − 4x + 120, if two of its zeros are 2 and −2.

Ex. 2.30 | Q 10 | Page 57

Find all zeros of the polynomial 2x4 + 7x3 − 19x2 − 14x + 30, if two of its zeros are `sqrt2` and `-sqrt2`.

Ex. 2.30 | Q 11 | Page 58

Find all the zeros of the polynomial 2x3 + x2 − 6x − 3, if two of its zeros are `-sqrt3` and `sqrt3`

Ex. 2.30 | Q 12 | Page 58

Find all the zeros of the polynomial x3 + 3x2 − 2x − 6, if two of its zeros are `-sqrt2` and `sqrt2`

Chapter 2: Polynomials solutions [Pages 58 - 61]

Q 1 | Page 58

Define a polynomial with real coefficients.

Q 2 | Page 58

Define degree of a polynomial.

Q 3 | Page 58

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

Q 4 | Page 58

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

Q 5 | Page 58

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

Q 6 | Page 58

Define value of polynomial at a point.

Q 7 | Page 58

Define the zero of a polynomial.

Q 8 | Page 58

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

Q 9 | Page 58

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

Q 10 | Page 58

If the product of zeros of the quadratic polynomial f(x) = x2 − 4x + k is 3, find the value of k.

Q 11 | Page 58

If the sum of the zeros of the quadratic polynomial f(x) = kx2 − 3x + 5 is 1, write the value of k.

Q 12 | Page 59

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

Q 13 | Page 59

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

Q 14 | Page 59

The graph of the polynomial f(x) = ax2 + bx + c is as shown below (Fig. 2.19). Write the signs of 'a' and b2 − 4ac.

Q 15 | Page 59

The graph of the polynomial f(x) = ax2 + bx + c is as shown in Fig. 2.20. Write the value of b2 − 4ac and the number of real zeros of f(x).

Q 16 | Page 59

In Q. No. 14, write the sign of c.

Q 17 | Page 59

In Q. No. 15, write the sign of c.

Q 18 | Page 59

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

Q 19 | Page 59

If x = 1 is a zero of the polynomial f(x) = x3 − 2x2 + 4x + k, write the value of k.

Q 20 | Page 59

State division algorithm for polynomials.

Q 21 | Page 59

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.

Q 22 | Page 60

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

Q 23 | Page 60

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

Q 24 | Page 60

If f(x) = x3 + x2 − ax + b is divisible by x2 − x write the value of a and b.

Q 25 | Page 60

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

Q 26 | Page 60

Write the coefficient of the polynomial p(z) = z5 − 2z2 + 4.

Q 27 | Page 60

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

Q 28 | Page 60

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

Q 29 | Page 60

For what value of k, −4 is a zero of the polynomial x2 − x − (2k + 2)?

Q 30 | Page 60

If 1 is a zero of the polynomial p(x) = ax2 − 3(a − 1) x − 1, then find the value of a.

Q 31 | Page 60

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

Q 32 | Page 60

If α, β are the zeros of the polynomial 2y2 + 7y + 5, write the value of α + β + αβ.

Q 33 | Page 60

For what value of k, is 3 a zero of the polynomial 2x2 + x + k?

Q 34 | Page 60

For what value of k, is −3 a zero of the polynomial x2 + 11x + k?

Q 35 | Page 60

For what value of k, is −2 a zero of the polynomial 3x2 + 4x + 2k?

Q 36 | Page 60

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)?

Q 37 | Page 61

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

Q 38 | Page 61

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

Q 39 | Page 61

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

Q 40 | Page 61

If graph of quadratic polynomial ax2 + bx + c cuts positive direction of y-axis, then what is the sign of c?

Q 41 | Page 61

If the graph of quadratic polynomial ax2 + bx + c cuts negative direction of y-axis, then what is the sign of c?

Chapter 2: Polynomials solutions [Pages 61 - 64]

Q 1 | Page 61

If α, β are the zeros of the polynomial f(x) = x2 + x + 1, then \[\frac{1}{\alpha} + \frac{1}{\beta} =\]

  • 1

  • -1

  • 0

  • None of these

Q 2 | Page 61

If α, β are the zeros of the polynomial p(x) = 4x2 + 3x + 7, then \[\frac{1}{\alpha} + \frac{1}{\beta}\]  is equal to

  • \[\frac{7}{3}\]
  • \[- \frac{7}{3}\]
  • \[\frac{3}{7}\]
  • \[- \frac{3}{7}\]
Q 3 | Page 61

If one zero of the polynomial f(x) = (k2 + 4)x2 + 13x + 4k is reciprocal of the other, then k=

  • 2

  • -2

  • 1

  • -1

Q 4 | Page 62

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

  • 2

  • 4

  •  −2

  • −4

Q 5 | Page 61

If α and β are the zeros of the polynomial f(x) = x2 + px + q, then a polynomial having \[\frac{1}{\alpha} \text{and}\frac{1}{\beta}\]  is its zero is 

  •  x2 + qx + p

  • x2 − px + q

  • qx2 + px + 1

  • px2 + qx + 1

Q 6 | Page 61

If α, β are the zeros of polynomial f(x) = x2 − p (x + 1) − c, then (α + 1) (β + 1) =

  • c − 1

  • 1 − c

  • c

  • 1 + c

Q 7 | Page 61

If α, β are the zeros of the polynomial f(x) = x2 − p(x + 1) − c such that (α +1) (β + 1) = 0, then c =

  • 1

  • 0

  • -1

  • 2

Q 8 | Page 61

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

  • c = 0

  • c > 0

  • < 0

  • None of these

Q 9 | Page 61

If the diagram in Fig. 2.22 shows the graph of the polynomial f(x) = ax2 + 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

Q 10 | Page 61

Figure 2.23 show the graph of the polynomial f(x) = ax2 + 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

Q 11 | Page 62

If the product of zeros of the polynomial f(xax3 − 6x2 + 11x − 6 is 4, then a =

  • \[\frac{3}{2}\]

  • \[- \frac{3}{2}\]

  • \[\frac{2}{3}\]

  • \[- \frac{2}{3}\]

  • \[- \frac{2}{3}\]

Q 12 | Page 62

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

  • 2p3 = pq − r

  • 2p3 = pq + r

  •  p3 = pq − r

  • None of these

Q 13 | Page 63

If the product of two zeros of the polynomial f(x) = 2x3 + 6x2 − 4x + 9 is 3, then its third zero is

  • \[\frac{3}{2}\]

     

  • \[- \frac{3}{2}\]
  • \[\frac{9}{2}\]
  • \[- \frac{9}{2}\]
Q 14 | Page 63

If the polynomial f(x) = ax3 + bx − c is divisible  by the polynomial g(x) = x2 + bx + c, then ab =

  •  1

  • \[\frac{1}{c}\]
  • −1

  • \[- \frac{1}{c}\]
Q 15 | Page 63

If Q.No. 14, c =

  •  b

  • 2b

  • 2b2

  • −2b

Q 16 | Page 63

If one root of the polynomial f(x) = 5x2 + 13x + k is reciprocal of the other, then the value of k is

  • 0

  • 5

  • \[\frac{1}{6}\]
  • 6

Q 17 | Page 63

If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, the\[\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} =\]

  • \[- \frac{b}{d}\]
  • \[\frac{c}{d}\]
  • \[- \frac{c}{d}\]
  • \[- \frac{c}{a}\]
Q 18 | Page 63

If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 cx + d, then α2 + β2 + γ2 =

  • \[\frac{b^2 - ac}{a^2}\]
  • \[\frac{b^2 - 2ac}{a}\]
  • \[\frac{b^2 + 2ac}{b^2}\]
  • \[\frac{b^2 - 2ac}{a^2}\]
Q 19 | Page 63

If α, β, γ are are the zeros of the polynomial f(x) = x3 − px2 + qx − r, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]

Q 20 | Page 63

If α, β are the zeros of the polynomial f(x) = ax2 + bx + c, then\[\frac{1}{\alpha^2} + \frac{1}{\beta^2} =\]

  • \[\frac{b^2 - 2ac}{a^2}\]
  • \[\frac{b^2 - 2ac}{c^2}\]
  • \[\frac{b^2 + 2ac}{a^2}\]
  • \[\frac{b^2 + 2ac}{c^2}\]
Q 21 | Page 63

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

  • \[\frac{- d}{a}\]
  • \[\frac{c}{a}\]
  • \[\frac{- b}{a}\]
  • \[\frac{b}{a}\]
Q 22 | Page 63

If two zeros x3 + x2 − 5x − 5 are \[\sqrt{5}\ \text{and} - \sqrt{5}\], then its third zero is

  •  1

  • −1

  • 2

  • −2

Q 23 | Page 63

The product of the zeros of x3 + 4x2 + x − 6 is

  • −4

  • 4

  • 6

  • −6

Q 24 | Page 64

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

  • 1

  • 2

  • 4

  • 5

Q 25 | Page 64

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

  • 30

  • 14

  • 15

  • 16

Q 26 | Page 64

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

  • x2 − 9

  • x2 + 9

  • x2 + 3

  • x2 − 3

Q 27 | Page 64

If two zeroes of the polynomial x3 + x2 − 9x − 9 are 3 and −3, then its third zero is

  • -1

  • 1

  • -9

  • 9

Q 28 | Page 64

If \[\sqrt{5}\ \text{and} - \sqrt{5}\]   are two zeroes of the polynomial x3 + 3x2 − 5x − 15, then its third zero is

  •  3

  • -3

  • 5

  • -5

Q 29 | Page 63

If x + 2 is a factor of x2 + ax + 2b and a + b = 4, then

  • a= 1, b = 3

  • a = 3, b = 1

  • a = −1, b = 5

  • a = 5, b = −1

Q 30 | Page 64

The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, is

  • x3 − 3x2 + 3x − 5

  • x3 − 3x2 − 3x − 5

  • x3 + 3x2 − 3x + 5

  • x3 − 3x2 − 3x + 5

Chapter 2: Polynomials

Ex. 2.10OthersEx. 21.00Ex. 2.20Ex. 2.30

RD Sharma 10 Mathematics

10 Mathematics

RD Sharma solutions for Class 10 Mathematics chapter 2 - Polynomials

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

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