RD SHARMA Solutions for Class 10 Maths Chapter 2 - Polynomials

Chapter 2 - Polynomials Exercise Ex. 2.1

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 =  …(i) Also, + β = 4 - 2 = 2 …(ii) Product of the zeroes =  …(iii) Also, β = -8 …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients isverified.

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  and  . Let =   and β =  Consider4s2 - 4s + 1 Sum of the zeroes =  …(i) Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Solution 1(iii)

h (t) = t2 - 15 = (t + √15)(t - √15)  The zeroes of the quadratic equation are and  . Let =   and β =  Considert2 - 15 = t2 - 0t - 15 Sum of the zeroes =  …(i) Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

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   and  . Let =   and β =  Consider6x2 - 7x - 3 = 0 Sum of the zeroes =  …(i) Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Solution 1(v)

 The zeroes of a quadratic equation are  and  . Let =   and β =  Consider Sum of the zeroes =  …(i) Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Solution 1(vi)

 The zeroes of a quadratic equation are  and  . Let =   and β =  Consider Sum of the zeroes =  …(i) Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Solution 1(vii)

 The zeroes of a quadratic equation are   and 1. Let =   and β = 1 Consider Sum of the zeroes =  …(i)  Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Solution 1(viii)

  The zeroes of a quadratic equation are a and . Let = a and β = Consider Sum of the zeroes =  …(i)  Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Solution 1(ix)

 The zeroes of a quadratic equation are  and  . Let =   and β =  Consider Sum of the zeroes =  …(i)  Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Solution 1(x)

 The zeroes of a quadratic equation are  and  . Let =   and β =  Consider =0 Sum of the zeroes =  …(i)  Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Solution 1(xi)

 The zeroes of a quadratic equation are   and . Let =   and β = Consider =0 Sum of the zeroes =  …(i)  Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Solution 1(xii)

 The zeroes of a quadratic equation are   and . Let =   and β = Consider =0 Sum of the zeroes =  …(i)  Also, + β =  …(ii) Product of the zeroes =  …(iii) Also, β =  …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.

Solution 2(i)

Solution 2(ii)

Solution 2(iii)

Solution 2(iv)

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

(i)



(ii)



(iii)



(iv)



(v)



(vi)



(vii)



(viii)

 

Chapter 2 - Polynomials Exercise Ex. 2.2

Solution 1


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



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



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



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.

Solution 2
Solution 3

Let f(x) = 3x3 + 10x2 - 9x - 4 

As 1 is one of the zeroes of the polynomial, so (x - 1) becomes the factor of f(x).

Dividing f(x) by (x - 1), we have

Hence, the zeroes are   

Solution 4

Let f(x) = x3 - 3x2 - 10x + 24 

As 4 is one of the zeroes of the polynomial, so (x - 4) becomes the factor of f(x).

Dividing f(x) by (x - 4), we have

Hence, the zeroes are 4, -3 and 2.

Solution 5
Solution 6

Chapter 2 - Polynomials Exercise Ex. 2.3

Solution 1 (i)
Solution 1 (ii)
Solution 1 (iii)
Solution 1 (iv)
Solution 2

Solution 3
Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Let f(x) = 2x4 - 9x3 + 5x2 + 3x - 1 

As   are two of the zeroes of the polynomial, so   becomes the factor of f(x).

Dividing f(x) by   we have

Hence, the other zeros   

Solution 12

Let f(x) = 3x4 - 9x3 + x2 + 15x + k 

As f(x) is completely divisible by 3x2 - 5, it becomes one of the factors of f(x).

Dividing f(x) by 3x2 - 5, we have

As (3x2 - 5) is one of the factors, the remainder will be 0.

Therefore, k + 10 = 0

Thus, k = -10.

Solution 13

Solution 14

Chapter 2 - Polynomials Exercise 2.61

Solution 1

begin mathsize 12px style We space know comma space for space straight a space quadratic space equation comma space having space roots space straight alpha space and space straight beta
ax squared space plus space bx space plus space straight c space equals space 0 space space space space space space space space space space space space space space space space space........ open parentheses 1 close parentheses
straight alpha space plus space straight beta space equals space fraction numerator negative space straight b over denominator straight a end fraction space and space αβ space equals space straight c over straight a
Now comma space according space to space the space question
straight x squared space plus space straight x space plus space 1 space equals space 0
On space comparison space with space open parentheses 1 close parentheses
box enclose straight a space equals space 1 end enclose space space space space space space box enclose straight b space equals space 1 end enclose space space space space space space space box enclose straight c space equals space 1 end enclose
Hence space box enclose straight alpha space plus space straight beta space equals space minus 1 end enclose space space space space space space....... open parentheses 2 close parentheses space and space box enclose αβ space equals space 1 end enclose space space space space space space space space space space space space....... open parentheses 3 close parentheses
To space find space colon space 1 over straight alpha space plus space 1 over straight beta
space space space space space space space space space space space space space space space rightwards double arrow space fraction numerator straight alpha space plus space straight beta over denominator αβ end fraction
from space eq to the power of straight n space space open parentheses 2 close parentheses space & space open parentheses 3 close parentheses
box enclose 1 over straight alpha space plus space 1 over straight beta space equals space minus 1 end enclose
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 2

begin mathsize 12px style We space know space that comma space quadratic space equations space having space roots space straight alpha space and space straight beta
ax squared space plus space bx space plus space straight c space equals space 0 space space space space space space space space space....... open parentheses 1 close parentheses
straight alpha space plus space straight beta space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space αβ space equals space straight c over straight a
Now comma space according space to space the space question comma space the space equation space is
4 straight x squared space plus space 3 straight x space plus space 7 space equals space 0
On space comparing space with space equation space left parenthesis 1 right parenthesis comma
box enclose straight a space equals space 4 end enclose space space space space space space space space space space box enclose straight b space equals space 3 end enclose space space space space space space space space space space space space box enclose straight c space equals space 7 end enclose
Hence space box enclose straight alpha space plus space straight beta space equals space fraction numerator negative 3 over denominator 4 end fraction end enclose space space........ left parenthesis 2 right parenthesis space space space space space space space space and space space space space space box enclose αβ space equals space 7 over 4 end enclose space space space....... open parentheses 3 close parentheses
To space find space colon space 1 over straight alpha space plus space 1 over straight beta
space space space space space space space space space space space space rightwards double arrow space fraction numerator straight alpha space plus space straight beta over denominator αβ end fraction
from space eq space open parentheses 2 close parentheses space and space open parentheses 3 close parentheses
1 over straight alpha space plus space 1 over straight beta space equals space fraction numerator negative 3 divided by 4 over denominator 7 divided by 4 end fraction
space space space space space space space space space space space space space space space space space space space space equals space fraction numerator negative 3 over denominator 7 end fraction end style

So, the correct option is (d).

Solution 3

begin mathsize 12px style We space know space that comma space for space straight a space quadratic space equation space having space roots space straight alpha space and space straight beta
ax squared space plus space bx space plus space straight c space equals space 0 space space space space space space space space space....... open parentheses 1 close parentheses
straight alpha space plus space straight beta space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space space space space space space space space space space space space space space αβ space equals space straight c over straight a space space space space space space space space space...... open parentheses 2 close parentheses
Now comma space according space to space the space question comma space the space equation space is
open parentheses straight k squared space plus space 4 close parentheses space straight x squared space plus space 13 straight x space plus space 4 straight k space equals space 0 space space space space space space....... open parentheses 3 close parentheses
Let space one space root space of space the space equation space is space straight alpha comma space then
according space to space the space question space another space root space of space equation space given space above space is space 1 over straight alpha.
On space comparing space eq space open parentheses 3 close parentheses space with space eq space open parentheses 1 close parentheses
box enclose straight a space equals straight k squared plus space 4 end enclose space space space space space space space space space space space space space box enclose straight b space equals space 13 end enclose space space space space space space space space space box enclose straight c space equals space 4 straight k end enclose
Hence comma
straight alpha space cross times space 1 over straight alpha space equals space fraction numerator 4 straight k over denominator straight k squared space plus space 4 end fraction
rightwards double arrow space straight k squared space plus space 4 space equals space 4 straight k
rightwards double arrow straight k squared space minus space 4 straight k space plus space 4 space equals space 0
rightwards double arrow space open parentheses straight k space minus space 2 close parentheses squared space equals space 0
rightwards double arrow space straight k space equals space 2
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Chapter 2 - Polynomials Exercise 2.62

Solution 4

begin mathsize 12px style We space know space that comma space for space straight a space quadratic space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space space space space space space...... open parentheses 1 close parentheses
straight alpha space plus space straight beta space plus space straight gamma space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space space space space....... open parentheses 2 close parentheses
Now comma space according space to space the space question comma space the space equation space is
2 straight x cubed space minus space 3 kx squared space plus space 4 straight x space minus space 5 space equals space 0 space space space space space space space space space space space space space space....... left parenthesis 3 right parenthesis
On space comparing space equation space open parentheses 3 close parentheses space with space equation space open parentheses 1 close parentheses
box enclose straight a space equals space 2 end enclose space space space space space space space space space space box enclose straight b space equals space minus 3 straight k end enclose space space space space space space space space space space space space box enclose straight c space equals space 4 end enclose space space space space space space space box enclose straight d space equals space minus 5 end enclose
According space to space the space question comma space from space equation space open parentheses 2 close parentheses
straight alpha space plus space straight beta space plus space straight gamma space equals space 6
therefore space fraction numerator negative open parentheses negative 3 straight k close parentheses over denominator 2 end fraction equals space 6
space space space space space space space space space space space space space space space space space 3 straight k space equals space 12
space space space space space space space space space space space space space space space space space space space straight k space equals space 4 end style

So, the correct option is (b).

Solution 5

begin mathsize 12px style We space know space that comma space quadratic space equation space having space roots space straight alpha space and space straight beta
ax squared space plus space bx space plus space straight c space equals space 0 space space space space space...... open parentheses 1 close parentheses

straight alpha space plus space straight beta space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space space space space space...... open parentheses 2 close parentheses
αβ space equals space straight c over straight a space space space space space space space space space space..... open parentheses 3 close parentheses
Eq. space open parentheses 1 close parentheses space can space also space be space written space as
straight x squared space plus space open parentheses straight b over straight a close parentheses straight x space plus space straight c over straight a space equals space 0
straight x to the power of 2 space end exponent space minus space open parentheses fraction numerator negative straight b over denominator straight a end fraction close parentheses straight x space plus space straight c over straight a space equals space 0
straight x squared space minus space open parentheses sum space of space roots close parentheses space straight x space plus space open parentheses product space of space roots close parentheses space equals space 0 space space space space space space space space space..... open parentheses 4 close parentheses
Now comma space according space to space the space question comma space the space equation space is
straight x squared space plus space px space plus space straight q space equals space 0
on space comparing space with space eq. space open parentheses 1 close parentheses
box enclose straight a space equals space 1 end enclose space space space space space space space space box enclose straight b space equals space straight p end enclose space space space space space space space space box enclose straight c space equals space straight q end enclose
Hence comma
straight alpha space plus space straight beta space equals space minus straight p space space space space space space space space space space space space..... open parentheses 5 close parentheses
αβ space equals space straight q space space space space space space space....... open parentheses 6 close parentheses
According space to space eq space open parentheses 4 close parentheses space equation space having space roots space 1 over straight alpha space and space 1 over straight beta
rightwards double arrow space straight x squared space minus space open parentheses 1 over straight alpha space plus space 1 over straight beta close parentheses straight x space plus space open parentheses 1 over αβ close parentheses space equals space 0
rightwards double arrow straight x squared space minus space open parentheses fraction numerator straight alpha space plus space straight beta over denominator αβ end fraction close parentheses straight x space plus space open parentheses 1 over αβ close parentheses space equals space 0
rightwards double arrow According space to space eq. space open parentheses 5 close parentheses space and space eq. space open parentheses 6 close parentheses
straight x squared space minus space open parentheses fraction numerator negative straight p over denominator straight q end fraction close parentheses straight x space plus space open parentheses 1 over straight q close parentheses space equals space 0
qx squared space plus space px space plus space 1 space equals space 0
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 6

begin mathsize 12px style We space know space that comma space for space straight a space space quadratic space equation space having space roots space straight alpha space and space straight beta
ax squared space plus space bx space plus space straight c space equals space 0 space space space space space space space space space space space........ open parentheses 1 close parentheses
straight alpha space plus space straight beta space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space space space space...... open parentheses 2 close parentheses
αβ space equals space straight c over straight a space space space space space space space space space space space....... open parentheses 3 close parentheses
Now comma space according space to space the space question comma space the space equation space is
straight x squared space minus space straight p open parentheses straight x space plus space 1 close parentheses space minus space straight c space equals space 0
This space equation space can space also space written space as
straight x squared space minus space px space minus space straight p space minus space straight c space equals space 0
straight x squared space minus space px space minus space open parentheses straight p space plus space straight c close parentheses space equals space 0 space space space space space space space space..... open parentheses 4 close parentheses
Comparing space equation space open parentheses 4 close parentheses space with space equation space open parentheses 1 close parentheses
straight alpha space plus space straight beta space equals space straight p space space space space space space space space space space space.... open parentheses 5 close parentheses
αβ space equals space minus open parentheses straight p space plus space straight c close parentheses space space space space space space space space space space space..... open parentheses 6 close parentheses
Now comma space the space value space of space open parentheses 1 space plus space straight alpha close parentheses space open parentheses 1 space plus space straight beta close parentheses
open parentheses 1 space plus space straight alpha close parentheses space open parentheses 1 space plus space straight beta close parentheses
rightwards double arrow 1 space plus space straight alpha space plus space straight beta space plus space αβ
rightwards double arrow 1 space plus space open parentheses straight alpha space plus space straight beta close parentheses space plus space αβ
According space to space equation space open parentheses 5 close parentheses space & space equation space open parentheses 6 close parentheses
rightwards double arrow 1 space plus space straight p space minus space straight p space minus space straight c
rightwards double arrow space 1 space minus space straight c
Hence comma space value space of space open parentheses 1 space plus space straight alpha close parentheses space open parentheses 1 space plus space straight beta close parentheses space is space 1 space minus space straight c.
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 7

begin mathsize 12px style We space know space that comma space for space straight a space space quadratic space equation space having space roots space straight alpha space and space straight beta
ax squared space plus space bx space plus space straight c space equals space 0 space space space space space space space space space space space........ open parentheses 1 close parentheses
straight alpha space plus space straight beta space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space space space space...... open parentheses 2 close parentheses
αβ space equals space straight c over straight a space space space space space space space space space space space....... open parentheses 3 close parentheses
Now comma space according space to space the space question comma space the space equation space is
straight x squared space minus space straight p open parentheses straight x space plus space 1 close parentheses space minus space straight c space equals space 0
This space equation space can space also space written space as
straight x squared space minus space px space minus space straight p space minus space straight c space equals space 0
straight x squared space minus space px space minus space open parentheses straight p space plus space straight c close parentheses space equals space 0 space space space space space space space space..... open parentheses 4 close parentheses
Comparing space equation space open parentheses 4 close parentheses space with space equation space open parentheses 1 close parentheses
straight alpha space plus space straight beta space equals space straight p space space space space space space space space space space space.... open parentheses 5 close parentheses
αβ space equals space minus open parentheses straight p space plus space straight c close parentheses space space space space space space space space space space space..... open parentheses 6 close parentheses
Now comma space the space value space of space open parentheses 1 space plus space straight alpha close parentheses space open parentheses 1 space plus space straight beta close parentheses
open parentheses 1 space plus space straight alpha close parentheses space open parentheses 1 space plus space straight beta close parentheses
rightwards double arrow 1 space plus space straight alpha space plus space straight beta space plus space αβ
rightwards double arrow 1 space plus space open parentheses straight alpha space plus space straight beta close parentheses space plus space αβ
According space to space equation space open parentheses 5 close parentheses space & space equation space open parentheses 6 close parentheses
rightwards double arrow 1 space plus space straight p space minus space straight p space minus space straight c
rightwards double arrow space 1 space minus space straight c
Hence comma space value space of space open parentheses 1 space plus space straight alpha close parentheses space open parentheses 1 space plus space straight beta close parentheses space is space 1 space minus space straight c. end style

Given that (α + 1) (β + 1) = 0

begin mathsize 12px style therefore space 1 space minus space straight c space equals space 0
space space space space space space space space space space space space space space straight c space equals space 1 end style

So, the correct option is (a).

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

Case 2 :

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

 

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]

begin mathsize 12px style therefore space straight c space less than space 0 end style

 So, the correct option is (c).

Solution 9

begin mathsize 12px style We space know space that comma space if space the space graph space of space the space quadratic space equation space is space concave space upward comma space then space the
space box enclose straight a space greater than space 0 end enclose
rightwards double arrow from space the space graph space given space above comma
straight f left parenthesis 0 right parenthesis space greater than space 0 space open square brackets as space graph space intersect space at space positive space straight y minus space axis close square brackets
Hence space box enclose straight c space greater than thin space 0 end enclose
rightwards double arrow As space the space straight x minus coordinate space of space point space Plies space on space the space positive space side space of space the space straight x space minus space axis comma
therefore space fraction numerator negative straight b over denominator 2 straight a end fraction space greater than space 0
Hence space space box enclose straight b space less than space 0 end enclose
therefore space Conditions space from space the space graph space given space above space must space be
rightwards double arrow space straight a space greater than space 0 comma space straight b space less than space 0 space and space straight c space greater than space 0
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 10

Error converting from MathML to accessible text.

Solution 11

begin mathsize 12px style We space know space that comma space for space straight a space cubic space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space space space....... left parenthesis 1 right parenthesis
box enclose αβγ space equals space fraction numerator negative straight d over denominator straight a end fraction end enclose space space space space space space space space space space....... left parenthesis 2 right parenthesis
Now comma space according space to space the space question comma space the space equation space is
ax cubed space minus space 6 straight x squared space plus space 11 straight x space minus space 6 space equals space 0 space space space space space space space space space..... left parenthesis 3 right parenthesis
straight O straight n space comparing space equation space left parenthesis 3 right parenthesis space with space equation space left parenthesis 1 right parenthesis
box enclose straight b space equals space minus 6 end enclose space space space space space space space space space space box enclose straight c space equals space 11 end enclose space space space space space space space space space space space box enclose straight d space equals space minus 6 end enclose
Now comma space according space to space equation space left parenthesis 2 right parenthesis
6 over straight a space equals space 4
straight a space equals space 3 over 2
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 12

begin mathsize 12px style We space know space that comma space for space straight a space space cubic space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space space space space..... left parenthesis 1 right parenthesis
box enclose straight alpha space plus space straight beta space plus straight gamma space equals space fraction numerator negative straight b over denominator straight a end fraction end enclose space space space space space space...... left parenthesis 2 right parenthesis space space space space space space space space space space space space space space space space space space space space space box enclose αβ space plus space βγ space plus space αγ space equals space straight c over straight a end enclose space space space space space space space....... open parentheses 3 close parentheses space space space space space space space space space space space space space space space box enclose αβγ space equals space fraction numerator negative straight d over denominator straight a end fraction space end enclose space space space space space space space........ left parenthesis 4 right parenthesis space space
According space to space the space question comma space roots space are space in space straight A. straight P
Thus comma space let space the space roots space are space space straight s space plus space straight t comma space straight s comma space straight s space minus space straight t
Given space equation space is
straight x cubed space minus space 3px squared space plus space qx space minus space straight r space equals space 0 space space space space space..... left parenthesis 5 right parenthesis
On space comparing space equation space open parentheses 5 close parentheses space with space equation space open parentheses 1 close parentheses
box enclose straight a space equals space 1 end enclose space space space space space space box enclose straight b equals space minus 3 straight p end enclose space space space space space space space space space box enclose straight c space equals space straight q end enclose space space space space space space space space space space box enclose straight d space equals space minus straight r end enclose
Then comma space according space to space the space question
open parentheses straight s space plus space straight t close parentheses space plus space straight s space plus space open parentheses straight s space minus space straight t close parentheses space equals space 3 straight p
3 straight s space equals space 3 straight p
box enclose straight s space equals space straight p end enclose space space space space space space space space space space space space space space space.... open parentheses 6 close parentheses
straight s space open parentheses straight s space plus space straight t close parentheses space left parenthesis straight s space minus space straight t right parenthesis space equals space straight r
straight s space open parentheses straight s squared space minus space straight t squared close parentheses space equals space straight r
straight s cubed space minus space st squared space equals space straight r
from space equation space open parentheses 6 close parentheses
box enclose straight p cubed space minus space pt squared space equals space straight r end enclose space space space space space space space space space space space space space space space space space space space...... open parentheses 7 close parentheses
straight s space open parentheses straight s space plus space straight t close parentheses space plus space open parentheses straight s squared space minus space straight t squared close parentheses space plus space straight s space open parentheses straight s space minus space straight t close parentheses space equals space minus straight q
straight s squared space plus space st space plus space straight s squared space minus space straight t squared space plus space straight s squared space minus space st space equals space minus straight q
3 straight s squared space minus space straight t squared space equals negative straight q
box enclose straight t squared space equals space minus straight q space plus space 3 straight s squared end enclose space space space space space space space space space space space space...... open parentheses 8 close parentheses
Putting space value space of space straight t squared space in space equation space open parentheses 7 close parentheses
straight p cubed space minus space straight p space open parentheses negative straight q space plus space 3 straight p squared close parentheses space equals space straight r
straight p cubed space plus space pq space minus space 3 straight p squared space equals space straight r
box enclose 2 straight p cubed space equals space pq space minus space straight r end enclose
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Chapter 2 - Polynomials Exercise 2.63

Solution 13

begin mathsize 12px style We space know space that comma space for space straight a space cubic space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space space space..... open parentheses 1 close parentheses
box enclose αβγ space equals fraction numerator negative straight d over denominator straight a end fraction end enclose space space space space space space space space.... open parentheses 2 close parentheses
According space to space the space question comma space the space equation space is
2 straight x cubed space plus space 6 straight x squared space minus space 4 straight x space plus space 9 space equals space 0 space space space space space space space space..... open parentheses 3 close parentheses
On space comparing space equation space open parentheses 3 close parentheses space with space equation space open parentheses 2 close parentheses
box enclose straight a space equals space 2 end enclose space space space space space box enclose straight b space equals space 6 end enclose space space space space space box enclose straight c space equals space minus 4 end enclose space space space space space space space space box enclose straight d space equals space 9 end enclose
According space to space equation space open parentheses 2 close parentheses
αβγ space equals space fraction numerator negative 9 over denominator 2 end fraction
Given space that space product space of space straight beta space and space straight alpha space is space 3
3 space cross times space straight gamma space equals space fraction numerator negative 9 over denominator 2 end fraction
straight gamma space equals space fraction numerator negative 3 over denominator 2 end fraction end style

So, the correct option is (b).

Solution 14

Error converting from MathML to accessible text.

Solution 15

Error converting from MathML to accessible text.

So, the correct option is (c).

Solution 16

begin mathsize 12px style We space know space that comma space for space straight a space quadratic space equation space having space roots space straight alpha space and space straight beta
ax squared space plus space bx space plus space straight c space equals space 0 space space space space space space.... open parentheses 1 close parentheses
box enclose straight alpha space plus space straight beta space equals space fraction numerator negative straight b over denominator straight a end fraction end enclose space space space space space space space space space.... open parentheses 2 close parentheses
box enclose αβ space equals space straight c over straight a end enclose space space space space space.... open parentheses 3 close parentheses
According space to space the space question comma space the space equation space is
5 straight x squared space plus space 13 straight x space plus space straight k space equals space 0 space space space space...... left parenthesis 4 right parenthesis
comparing space equation space open parentheses 4 close parentheses space with space equation space open parentheses 1 close parentheses
box enclose straight a space equals space 5 end enclose space space space space space space space space space box enclose straight b space equals space 13 end enclose space space space space space space space space box enclose space straight c space equals space straight k end enclose
Given space that space one space root space of space the space polynomial space is space the space reciprocal space of space the space other space root
Let space the space roots space of space equation space be space straight alpha space and space 1 over straight alpha
Hence space according space to space equation space left parenthesis 3 right parenthesis
straight alpha space cross times space 1 over straight alpha space equals space straight k over 5
therefore space straight k space equals space 5
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 17

begin mathsize 12px style We space know space that comma space for space straight a space cubic space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space space space space space....... open parentheses 1 close parentheses
box enclose straight alpha space plus space straight beta space plus space straight gamma space equals space fraction numerator negative straight b over denominator straight a end fraction end enclose space space space space space space space space space space space..... open parentheses 2 close parentheses
box enclose αβ space plus space βγ space plus space αγ space equals space straight c over straight a end enclose space space space space space space space..... open parentheses 3 close parentheses
box enclose αβγ space equals space fraction numerator negative straight d over denominator straight a end fraction end enclose space space space space space space space..... open parentheses 4 close parentheses
value space of space 1 over straight alpha space plus space 1 over straight beta space plus space 1 over straight gamma
rightwards double arrow space fraction numerator αγ space plus space βγ space plus space αβ over denominator αβγ end fraction
According space to space equation space open parentheses 3 close parentheses space and space space equation space open parentheses 4 close parentheses
equals space fraction numerator begin display style bevelled straight c over straight a end style over denominator begin display style bevelled fraction numerator negative straight d over denominator straight a end fraction end style end fraction
equals space fraction numerator negative straight c over denominator straight d end fraction
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 18

begin mathsize 12px style We space know space that comma space for space straight a space cubic space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space space space..... open parentheses 1 close parentheses
box enclose straight alpha space plus space straight beta space plus space straight gamma space equals space fraction numerator negative straight b over denominator straight a end fraction end enclose space space space space space space space space space space space space space..... open parentheses 2 close parentheses
box enclose αβ space plus space βγ space plus space αγ space equals space straight c over straight a end enclose space space space space space space...... open parentheses 3 close parentheses
box enclose αβγ space equals space fraction numerator negative straight d over denominator straight a end fraction end enclose space space space space space space space space space..... open parentheses 4 close parentheses
We space also space know space that comma
open parentheses straight alpha space plus space straight beta space plus space straight gamma close parentheses squared space equals space straight alpha squared space plus space straight beta squared space plus space straight gamma squared space plus space 2 αβ space plus space 2 space βγ space plus space 2 γα
therefore space straight alpha squared space plus space straight beta squared space plus space straight gamma squared space equals space open parentheses straight alpha space plus space straight beta space plus space straight gamma close parentheses squared space minus space 2 open parentheses αβ space plus space βγ space plus space γα close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses fraction numerator negative straight b over denominator straight a end fraction close parentheses squared space minus space 2 open parentheses straight c over straight a close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator straight b squared space minus space 2 ac over denominator straight a squared end fraction space
So comma space the space correct space option space is space left parenthesis straight d right parenthesis.
end style

Solution 19

begin mathsize 12px style We space know space that comma space for space straight a space cubic space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space..... open parentheses 1 close parentheses
straight alpha space plus space straight beta space plus space straight gamma space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space space.... open parentheses 2 close parentheses
αβγ space equals space fraction numerator negative straight d over denominator straight a end fraction space space space space space space space space space..... open parentheses 3 close parentheses
αβ space plus space βγ space plus space αγ space equals space straight c over straight a space space space space space space space...... open parentheses 4 close parentheses
According space to space the space question comma space the space equation space is space
straight x cubed space minus space px squared space plus space qx space minus space straight r space equals space 0 space space space space space space space space space space space.... open parentheses 5 close parentheses
comparing space eq. space open parentheses 5 close parentheses space with space eq. space open parentheses 1 close parentheses
box enclose straight a space equals space 1 end enclose space space space space space box enclose straight b space equals space minus straight p end enclose space space space space space space space space space box enclose straight c equals straight q end enclose space space space space space space space box enclose straight d space equals space minus straight r end enclose
Hence space 1 over αβ space plus space 1 over βγ space plus space 1 over γα
equals space fraction numerator straight alpha space plus space straight beta space plus space straight gamma over denominator αβγ end fraction
equals space straight p over straight r
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 20

begin mathsize 12px style We space know space that space for space straight a space quadratic space equation space having space roots space straight alpha space and space straight beta
ax squared space plus space bx space plus space straight c space equals space 0 space space space space space space space...... open parentheses 1 close parentheses
box enclose straight alpha space plus space straight beta space equals space fraction numerator negative straight b over denominator straight a end fraction end enclose space space space space space space space space space..... open parentheses 2 close parentheses
box enclose αβ space equals space straight c over straight a end enclose space space space space space..... open parentheses 3 close parentheses
Then comma space value space of space
1 over straight alpha squared plus space 1 over straight beta squared
rightwards double arrow space fraction numerator straight alpha squared space plus space straight beta squared over denominator straight alpha squared straight beta squared end fraction
equals space fraction numerator open parentheses straight alpha space plus space straight beta close parentheses squared space minus space 2 αβ over denominator straight alpha squared straight beta squared end fraction
equals space fraction numerator begin display style straight b squared over straight a squared minus fraction numerator 2 straight c over denominator straight a end fraction end style over denominator begin display style straight c squared over straight a squared end style end fraction
equals space fraction numerator straight b squared space minus space 2 ac over denominator straight c squared end fraction
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 21

begin mathsize 12px style We space know space that comma space for space straight a space space cubic space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0
box enclose straight alpha space plus space straight beta space plus space straight gamma space equals space fraction numerator negative straight b over denominator straight a end fraction end enclose space space space space space space space space space..... open parentheses 1 close parentheses
Given space that comma space two space of space the space zeros space of space polynomial space are space equal space to space zero
Hence comma space according space to space eq. space open parentheses 1 close parentheses
box enclose straight gamma space equals space fraction numerator negative straight b over denominator straight a end fraction end enclose
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 22

begin mathsize 12px style We space know space that comma space for space straight a space cubi c space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space space space space.... open parentheses 1 close parentheses
straight alpha space plus space straight beta space plus space straight gamma space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space.... open parentheses 2 close parentheses
According space to space the space question comma space the space equation space is space
straight x cubed space plus space straight x squared space minus space 5 straight x space minus space 5 space equals space 0 space space space space space space.... open parentheses 3 close parentheses
On space comparing space eq. space open parentheses 3 close parentheses space with space eq. space open parentheses 1 close parentheses
box enclose straight a space equals space 1 end enclose space space space space space space space space space space space space space box enclose straight b space equals space 1 end enclose space space space space space space space space space space space space space box enclose straight c space equals space minus 5 end enclose space space space space space space space space space space space space box enclose straight d space equals space minus 5 end enclose
According space to space equation space open parentheses 2 close parentheses
straight alpha space plus space straight beta space plus space straight gamma space equals space minus 1
Given space that space two space roots space are space square root of 5 space and space minus square root of 5
Hence comma space space square root of 5 space minus space square root of 5 space plus space straight gamma space equals space minus 1
box enclose straight gamma space equals space minus 1 end enclose
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 23

begin mathsize 12px style We space know space that comma space for space straight a space cubic space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space space..... open parentheses 1 close parentheses
αβγ space equals space fraction numerator negative straight d over denominator straight a end fraction space space space space space space.... open parentheses 2 close parentheses
According space to space the space question comma space the space equation space is
straight x cubed space plus space 4 straight x squared space plus space straight x space minus space 6 space equals space 0 space space space space space space space space...... open parentheses 3 close parentheses
On space comparing space eq. space open parentheses 3 close parentheses space with space eq. space open parentheses 1 close parentheses
straight a equals space 1 comma space straight b space equals space 4 comma space straight c space equals space 1 comma space straight d space equals space minus 6
therefore space product space of space roots space equals space 6
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Chapter 2 - Polynomials Exercise 2.64

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).

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).

Solution 26

begin mathsize 12px style We space know space that comma space quadratic space equation space having space roots space straight alpha space and space straight beta
ax squared space plus space bx space plus space straight c space equals space 0 space space space space.... open parentheses 1 close parentheses
straight alpha space plus space straight beta space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space space space space space space..... open parentheses 2 close parentheses
αβ space equals space straight c over straight a space space space space space space space..... open parentheses 3 close parentheses
According space to space question
straight alpha space plus space straight beta space equals space 0
if space straight alpha space equals space 3
then space straight beta space equals space minus 3
So comma space straight alpha space plus space straight beta space equals space 0 space and space αβ equals left parenthesis 3 right parenthesis left parenthesis negative 3 right parenthesis equals negative 9
The space quadratic space equation space in space general space is space straight x squared minus left parenthesis sum space of space the space roots right parenthesis space straight x space plus space product space of space the space roots equals 0
So comma space the space quadratic space equation space is space straight x squared minus left parenthesis 0 right parenthesis straight x space plus space left parenthesis negative 9 right parenthesis equals 0 space rightwards double arrow straight x squared minus 9 equals 0.
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 27

begin mathsize 12px style We space kno w space that comma space for space straight a space cubi c space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space space..... open parentheses 1 close parentheses
straight alpha space plus space straight beta space plus space straight gamma space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space space space space..... open parentheses 2 close parentheses
Acoording space to space the space question comma space the space equation space is space
straight x to the power of 3 space end exponent space plus space straight x squared space minus space 9 straight x space minus space 9 space equals space 0 space space space space space space space space..... open parentheses 3 close parentheses
On space comparing space eq. space open parentheses 3 close parentheses space with space eq. space open parentheses 1 close parentheses
box enclose straight a space equals space 1 end enclose space space space space space space space space space space space space space box enclose straight b space equals space 1 end enclose space space space space space space space space space space space space box enclose straight c space equals space minus 9 end enclose space space space space space space space space space space space box enclose straight d space equals space minus 9 end enclose
Hence comma
straight alpha space plus space straight beta space plus space straight gamma space equals space minus 1
3 space minus space 3 space plus space straight gamma space equals space minus 1
straight gamma space equals space minus 1
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 28

begin mathsize 12px style We space know space that comma space for space straight a space cubic space equation space having space roots space straight alpha comma space straight beta comma space straight gamma
ax cubed space plus space bx squared space plus space cx space plus space straight d space equals space 0 space space space space space....... open parentheses 1 close parentheses
straight alpha space plus space straight beta space plus space straight gamma space equals space fraction numerator negative straight b over denominator straight a end fraction space space space space space space space space space..... open parentheses 2 close parentheses
According space to space the space question comma space the space equation space is space
straight x cubed space plus space 3 straight x to the power of 2 space end exponent minus space 5 straight x space minus space 15 space equals space 0 space space space space space..... open parentheses 3 close parentheses
On space comparing space eq. space open parentheses 3 close parentheses space with space eq. space open parentheses 1 close parentheses
box enclose straight a space equals space 1 end enclose space space space space space space space space space space space space box enclose straight b space equals space 3 end enclose space space space space space space space space space space space space space space box enclose straight c equals space minus 5 end enclose space space space space space space space space space space space space space box enclose straight d space equals space minus 15 end enclose
Hence comma space straight alpha space plus space straight beta space plus space straight gamma space equals space minus 3
Given space that space square root of 5 space and space minus square root of 5 space are space two space roots space of space equation
square root of 5 space minus space square root of 5 space plus space straight gamma space equals space minus 3
straight gamma space equals space minus 3
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 29

Error converting from MathML to accessible text.

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).

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.

Solution 32

Solution 33

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

Hence, correct option is (b).

Solution 34

Solution 35

Solution 36

Solution 37

Chapter 2 - Polynomials Exercise 2.65

Solution 38

Solution 39

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).

Solution 41

Solution 42

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

Hence, correct option is (d).