Class 10 Exam > Class 10 Notes > Mathematics (Maths) Class 10 > RD Sharma Solutions: Polynomials
Polynomials RD Sharma Solutions | Mathematics (Maths) Class 10 PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
Exercise 2.1
1. Find the zeroes of each of the following quadratic polynomials and verify the relationship
between the zeroes and their co efficient:
(i) f(x) = ?? 2
- 2?? - 8
(ii) g(s) = 4?? 2
- 4?? + 1
(iii) h(t) = ?? 2
- 15
(iv) p(x) = ?? 2
+ 2v2?? + 6
(v) q(x) = v3?? 2
+ 10?? + 7v3
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3
(vii) g(x) = ?? ( ?? 2
+ 1)- ?? ( ?? 2
+ 1)
(viii) 6?? 2
- 3 - 7?? Sol:
(i) f(x) = ?? 2
- 2?? - 8
?? ( ?? ) = ?? 2
- 2?? - 8 = ?? 2
- 4?? + 2?? - 8
= ?? ( ?? - 4)+ 2( ?? - 4)
= ( ?? + 2) ( ?? - 4)
Zeroes of the polynomials are -2 and 4
Sum of the zeroes =
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ??
-2 + 4 =
-( -2)
1
2 = 2
Product of the zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
= 24 =
-8
1
- 8 = -8
? Hence the relationship verified
(ii) 9( 5) = 45 - 45 + 1 = 45
2
- 25 - 25 + 1 = 25( 25 - 1)- 1( 25 - 1)
= ( 25 - 1) ( 25 - 1)
Zeroes of the polynomials are
1
2
?????? 1
2
Sum of zeroes =
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 2
1
2
+
1
2
=
-( -4)
4
1 = 1
Product of the zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
1
2
×
1
2
=
1
4
?
1
4
=
1
4
? Hence the relationship verified.
(iii) h(t) = ?? 2
- 15 = ( ?? 2
)- ( v15)
2
= ( ?? + v15) ( ?? - v15)
zeroes of the polynomials are -v15 ?????? v15
sum of zeroes = 0
-v15 + v15 = 0
0 = 0
Page 2
Exercise 2.1
1. Find the zeroes of each of the following quadratic polynomials and verify the relationship
between the zeroes and their co efficient:
(i) f(x) = ?? 2
- 2?? - 8
(ii) g(s) = 4?? 2
- 4?? + 1
(iii) h(t) = ?? 2
- 15
(iv) p(x) = ?? 2
+ 2v2?? + 6
(v) q(x) = v3?? 2
+ 10?? + 7v3
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3
(vii) g(x) = ?? ( ?? 2
+ 1)- ?? ( ?? 2
+ 1)
(viii) 6?? 2
- 3 - 7?? Sol:
(i) f(x) = ?? 2
- 2?? - 8
?? ( ?? ) = ?? 2
- 2?? - 8 = ?? 2
- 4?? + 2?? - 8
= ?? ( ?? - 4)+ 2( ?? - 4)
= ( ?? + 2) ( ?? - 4)
Zeroes of the polynomials are -2 and 4
Sum of the zeroes =
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ??
-2 + 4 =
-( -2)
1
2 = 2
Product of the zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
= 24 =
-8
1
- 8 = -8
? Hence the relationship verified
(ii) 9( 5) = 45 - 45 + 1 = 45
2
- 25 - 25 + 1 = 25( 25 - 1)- 1( 25 - 1)
= ( 25 - 1) ( 25 - 1)
Zeroes of the polynomials are
1
2
?????? 1
2
Sum of zeroes =
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 2
1
2
+
1
2
=
-( -4)
4
1 = 1
Product of the zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
1
2
×
1
2
=
1
4
?
1
4
=
1
4
? Hence the relationship verified.
(iii) h(t) = ?? 2
- 15 = ( ?? 2
)- ( v15)
2
= ( ?? + v15) ( ?? - v15)
zeroes of the polynomials are -v15 ?????? v15
sum of zeroes = 0
-v15 + v15 = 0
0 = 0
Product of zeroes =
-15
1
-v15 × v15 = -15
-15 = -15
? Hence the relationship verified.
(iv) p(x) = ?? 2
+ 2v2?? - 6 = ?? 2
+ 3v2?? + v2 × 3v2
= ?? ( ?? + 3v2)- v2( 2 + 3v2) = ( ?? - v2) ( ?? + 3v2)
Zeroes of the polynomial are 3v2 and -3v2
Sum of the zeroes =
-3v2
1
v2 - 3v2 = -2v2
-2v2 = -2v2
?????????????? ???? ???????????? ? v2 × -3v2 = -
6
1
-6 = -6
?????????? ?? h?? ???????????????? h???? ????????????????
(v) 2(x) = v3?? 2
+ 10?? + 7v3 = v3?? 2
+ 7?? + 3?? + 7v3
= v3?? ( ?? + v3)+ 7( ?? + v3)
= ( v3?? + 7) ( ?? + v3)
Zeroes of the polynomials are -v3,
-7
v3
Sum of zeroes =
-10
v3
? -v3 -
7
v3
=
-10
v3
?
-10
v3
=
-10
v3
Product of zeroes =
7v3
3
?
v3?? -7
v30
= 7
? 7 = 7
Hence, relationship verified.
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 = ?? 2
- v3?? - ?? + v3
= x (x - v3) – 1 (x - v3)
= (x – 1) (x - v3)
Zeroes of the polynomials are 1 and v3
Sum of zeroes =
-{?????????????????????? ???? ?? }
???? ?????????????????? ???? ?? 2
=
-[-v3-1]
1
1 + v3 = v3 + 1
Product of zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
=
v3
1
1 × v3 = v3 = v3 = v3
? Hence, relationship verified
(vii) g(x) = ?? [( ?? 2
+ 1)- ?? ( ?? 2
+ 1) ]
2
= ?? ?? 2
+ ?? - ?? 2
?? - ??
= ?? ?? 2
- [( ?? 2
+ 1)- ?? ] + 0 = ?? ?? 2
- ?? 2
?? - ?? + ??
Page 3
Exercise 2.1
1. Find the zeroes of each of the following quadratic polynomials and verify the relationship
between the zeroes and their co efficient:
(i) f(x) = ?? 2
- 2?? - 8
(ii) g(s) = 4?? 2
- 4?? + 1
(iii) h(t) = ?? 2
- 15
(iv) p(x) = ?? 2
+ 2v2?? + 6
(v) q(x) = v3?? 2
+ 10?? + 7v3
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3
(vii) g(x) = ?? ( ?? 2
+ 1)- ?? ( ?? 2
+ 1)
(viii) 6?? 2
- 3 - 7?? Sol:
(i) f(x) = ?? 2
- 2?? - 8
?? ( ?? ) = ?? 2
- 2?? - 8 = ?? 2
- 4?? + 2?? - 8
= ?? ( ?? - 4)+ 2( ?? - 4)
= ( ?? + 2) ( ?? - 4)
Zeroes of the polynomials are -2 and 4
Sum of the zeroes =
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ??
-2 + 4 =
-( -2)
1
2 = 2
Product of the zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
= 24 =
-8
1
- 8 = -8
? Hence the relationship verified
(ii) 9( 5) = 45 - 45 + 1 = 45
2
- 25 - 25 + 1 = 25( 25 - 1)- 1( 25 - 1)
= ( 25 - 1) ( 25 - 1)
Zeroes of the polynomials are
1
2
?????? 1
2
Sum of zeroes =
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 2
1
2
+
1
2
=
-( -4)
4
1 = 1
Product of the zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
1
2
×
1
2
=
1
4
?
1
4
=
1
4
? Hence the relationship verified.
(iii) h(t) = ?? 2
- 15 = ( ?? 2
)- ( v15)
2
= ( ?? + v15) ( ?? - v15)
zeroes of the polynomials are -v15 ?????? v15
sum of zeroes = 0
-v15 + v15 = 0
0 = 0
Product of zeroes =
-15
1
-v15 × v15 = -15
-15 = -15
? Hence the relationship verified.
(iv) p(x) = ?? 2
+ 2v2?? - 6 = ?? 2
+ 3v2?? + v2 × 3v2
= ?? ( ?? + 3v2)- v2( 2 + 3v2) = ( ?? - v2) ( ?? + 3v2)
Zeroes of the polynomial are 3v2 and -3v2
Sum of the zeroes =
-3v2
1
v2 - 3v2 = -2v2
-2v2 = -2v2
?????????????? ???? ???????????? ? v2 × -3v2 = -
6
1
-6 = -6
?????????? ?? h?? ???????????????? h???? ????????????????
(v) 2(x) = v3?? 2
+ 10?? + 7v3 = v3?? 2
+ 7?? + 3?? + 7v3
= v3?? ( ?? + v3)+ 7( ?? + v3)
= ( v3?? + 7) ( ?? + v3)
Zeroes of the polynomials are -v3,
-7
v3
Sum of zeroes =
-10
v3
? -v3 -
7
v3
=
-10
v3
?
-10
v3
=
-10
v3
Product of zeroes =
7v3
3
?
v3?? -7
v30
= 7
? 7 = 7
Hence, relationship verified.
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 = ?? 2
- v3?? - ?? + v3
= x (x - v3) – 1 (x - v3)
= (x – 1) (x - v3)
Zeroes of the polynomials are 1 and v3
Sum of zeroes =
-{?????????????????????? ???? ?? }
???? ?????????????????? ???? ?? 2
=
-[-v3-1]
1
1 + v3 = v3 + 1
Product of zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
=
v3
1
1 × v3 = v3 = v3 = v3
? Hence, relationship verified
(vii) g(x) = ?? [( ?? 2
+ 1)- ?? ( ?? 2
+ 1) ]
2
= ?? ?? 2
+ ?? - ?? 2
?? - ??
= ?? ?? 2
- [( ?? 2
+ 1)- ?? ] + 0 = ?? ?? 2
- ?? 2
?? - ?? + ??
= ???? ( ?? - ?? )- 1( ?? - ?? ) = ( ?? - ?? ) ( ???? - 1)
Zeroes of the polynomials =
1
?? ?????? ??
Sum of the zeroes =
-[-?? 2
-1]
??
?
1
?? + ?? =
?? 2
+1
?? ?
?? 2
+1
?? =
?? 2
+1
??
Product of zeroes =
?? ??
?
1
?? × ?? =
?? ?? ?
?? 2
+1
?? =
?? 2
+1
??
Product of zeroes =
?? ?? ? 1 = 1
Hence relationship verified
(viii) 6?? 2
- 3 - 7?? = 6?? 2
- 7?? - 3 = ( 3?? + 11) ( 2?? - 3)
Zeroes of polynomials are +
3
2
?????? -1
3
Sum of zeroes =
-1
3
+
3
2
=
7
6
=
-( -7)
6
=
-( ???? ?????????????????? ???? ?? )
???? ?????????????????? ???? ?? 2
Product of zeroes =
-1
3
×
3
2
=
-1
2
=
-3
6
=
???????????????? ???????? ???? ?????????????????? ???? ?? 2
? Hence, relationship verified.
2. If ?? and ?? are the zeros of the quadratic polynomial f(x) = ax
2
+ bx + c, then evaluate:
(i) ?? - ??
(ii)
1
?? -
1
??
(iii)
1
?? +
1
?? - 2?? ??
(iv) ?? 2
?? + ?? ?? 2
(v) ?? 4
+ ?? 4
(vi)
1
???? +?? +
1
???? +??
(vii)
?? ???? +?? +
?? ???? +??
(viii) ?? [
?? 2
?? +
?? 2
?? ] +
?? [
?? ?? +
?? ?? ]
Sol:
f(x) = ?? ?? 2
+ ???? + ??
?? + ?? =
-?? ??
???? =
?? ??
?????????? ?? + ?? ?????? ?? h?? ?????????? ( ???? ) ???????????? ???? ?? h?? ?????????? ??????????????????????
(i) ?? - ??
The two zeroes of the polynomials are
-?? +v?? 2
-4????
2?? - (?? -v?? 2
-4????
2?? ) = -?? +
v?? 2
-4???? +?? +v?? 2
-4????
2?? =
2v?? 2
-4????
2?? =
v?? 2
-4????
2??
(ii)
1
?? -
1
?? =
?? -?? ????
=
-( ?? - ?? )
????
… ( ?? )
From (i) we know that ?? - ?? =
v?? 2
-4????
2?? [???????? ( ?? ) ]???? =
?? ??
Putting the values in the (a) = - (
v?? 2
-4???? ×?? ?? ×?? ) =
-v?? 2
-4????
??
(iii)
1
?? +
1
?? - 2?? ??
Page 4
Exercise 2.1
1. Find the zeroes of each of the following quadratic polynomials and verify the relationship
between the zeroes and their co efficient:
(i) f(x) = ?? 2
- 2?? - 8
(ii) g(s) = 4?? 2
- 4?? + 1
(iii) h(t) = ?? 2
- 15
(iv) p(x) = ?? 2
+ 2v2?? + 6
(v) q(x) = v3?? 2
+ 10?? + 7v3
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3
(vii) g(x) = ?? ( ?? 2
+ 1)- ?? ( ?? 2
+ 1)
(viii) 6?? 2
- 3 - 7?? Sol:
(i) f(x) = ?? 2
- 2?? - 8
?? ( ?? ) = ?? 2
- 2?? - 8 = ?? 2
- 4?? + 2?? - 8
= ?? ( ?? - 4)+ 2( ?? - 4)
= ( ?? + 2) ( ?? - 4)
Zeroes of the polynomials are -2 and 4
Sum of the zeroes =
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ??
-2 + 4 =
-( -2)
1
2 = 2
Product of the zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
= 24 =
-8
1
- 8 = -8
? Hence the relationship verified
(ii) 9( 5) = 45 - 45 + 1 = 45
2
- 25 - 25 + 1 = 25( 25 - 1)- 1( 25 - 1)
= ( 25 - 1) ( 25 - 1)
Zeroes of the polynomials are
1
2
?????? 1
2
Sum of zeroes =
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 2
1
2
+
1
2
=
-( -4)
4
1 = 1
Product of the zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
1
2
×
1
2
=
1
4
?
1
4
=
1
4
? Hence the relationship verified.
(iii) h(t) = ?? 2
- 15 = ( ?? 2
)- ( v15)
2
= ( ?? + v15) ( ?? - v15)
zeroes of the polynomials are -v15 ?????? v15
sum of zeroes = 0
-v15 + v15 = 0
0 = 0
Product of zeroes =
-15
1
-v15 × v15 = -15
-15 = -15
? Hence the relationship verified.
(iv) p(x) = ?? 2
+ 2v2?? - 6 = ?? 2
+ 3v2?? + v2 × 3v2
= ?? ( ?? + 3v2)- v2( 2 + 3v2) = ( ?? - v2) ( ?? + 3v2)
Zeroes of the polynomial are 3v2 and -3v2
Sum of the zeroes =
-3v2
1
v2 - 3v2 = -2v2
-2v2 = -2v2
?????????????? ???? ???????????? ? v2 × -3v2 = -
6
1
-6 = -6
?????????? ?? h?? ???????????????? h???? ????????????????
(v) 2(x) = v3?? 2
+ 10?? + 7v3 = v3?? 2
+ 7?? + 3?? + 7v3
= v3?? ( ?? + v3)+ 7( ?? + v3)
= ( v3?? + 7) ( ?? + v3)
Zeroes of the polynomials are -v3,
-7
v3
Sum of zeroes =
-10
v3
? -v3 -
7
v3
=
-10
v3
?
-10
v3
=
-10
v3
Product of zeroes =
7v3
3
?
v3?? -7
v30
= 7
? 7 = 7
Hence, relationship verified.
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 = ?? 2
- v3?? - ?? + v3
= x (x - v3) – 1 (x - v3)
= (x – 1) (x - v3)
Zeroes of the polynomials are 1 and v3
Sum of zeroes =
-{?????????????????????? ???? ?? }
???? ?????????????????? ???? ?? 2
=
-[-v3-1]
1
1 + v3 = v3 + 1
Product of zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
=
v3
1
1 × v3 = v3 = v3 = v3
? Hence, relationship verified
(vii) g(x) = ?? [( ?? 2
+ 1)- ?? ( ?? 2
+ 1) ]
2
= ?? ?? 2
+ ?? - ?? 2
?? - ??
= ?? ?? 2
- [( ?? 2
+ 1)- ?? ] + 0 = ?? ?? 2
- ?? 2
?? - ?? + ??
= ???? ( ?? - ?? )- 1( ?? - ?? ) = ( ?? - ?? ) ( ???? - 1)
Zeroes of the polynomials =
1
?? ?????? ??
Sum of the zeroes =
-[-?? 2
-1]
??
?
1
?? + ?? =
?? 2
+1
?? ?
?? 2
+1
?? =
?? 2
+1
??
Product of zeroes =
?? ??
?
1
?? × ?? =
?? ?? ?
?? 2
+1
?? =
?? 2
+1
??
Product of zeroes =
?? ?? ? 1 = 1
Hence relationship verified
(viii) 6?? 2
- 3 - 7?? = 6?? 2
- 7?? - 3 = ( 3?? + 11) ( 2?? - 3)
Zeroes of polynomials are +
3
2
?????? -1
3
Sum of zeroes =
-1
3
+
3
2
=
7
6
=
-( -7)
6
=
-( ???? ?????????????????? ???? ?? )
???? ?????????????????? ???? ?? 2
Product of zeroes =
-1
3
×
3
2
=
-1
2
=
-3
6
=
???????????????? ???????? ???? ?????????????????? ???? ?? 2
? Hence, relationship verified.
2. If ?? and ?? are the zeros of the quadratic polynomial f(x) = ax
2
+ bx + c, then evaluate:
(i) ?? - ??
(ii)
1
?? -
1
??
(iii)
1
?? +
1
?? - 2?? ??
(iv) ?? 2
?? + ?? ?? 2
(v) ?? 4
+ ?? 4
(vi)
1
???? +?? +
1
???? +??
(vii)
?? ???? +?? +
?? ???? +??
(viii) ?? [
?? 2
?? +
?? 2
?? ] +
?? [
?? ?? +
?? ?? ]
Sol:
f(x) = ?? ?? 2
+ ???? + ??
?? + ?? =
-?? ??
???? =
?? ??
?????????? ?? + ?? ?????? ?? h?? ?????????? ( ???? ) ???????????? ???? ?? h?? ?????????? ??????????????????????
(i) ?? - ??
The two zeroes of the polynomials are
-?? +v?? 2
-4????
2?? - (?? -v?? 2
-4????
2?? ) = -?? +
v?? 2
-4???? +?? +v?? 2
-4????
2?? =
2v?? 2
-4????
2?? =
v?? 2
-4????
2??
(ii)
1
?? -
1
?? =
?? -?? ????
=
-( ?? - ?? )
????
… ( ?? )
From (i) we know that ?? - ?? =
v?? 2
-4????
2?? [???????? ( ?? ) ]???? =
?? ??
Putting the values in the (a) = - (
v?? 2
-4???? ×?? ?? ×?? ) =
-v?? 2
-4????
??
(iii)
1
?? +
1
?? - 2?? ??
? [
?? + ?? ????
] - 2????
?
-?? ?? ×
?? ?? - 2
?? ?? = -2
?? ?? -
?? ?? =
-???? -2?? 2
????
- [
?? ?? +
2?? ?? ]
(iv) ?? 2
?? + ?? ?? 2
?? ?? ( ?? + ?? )
=
?? ?? (
-?? ?? )
=
-????
?? 2
(v) ?? 4
+ ?? 4
= ( ?? 2
+ ?? 2
)
2
- 2?? 2
+ ?? 2
= ( ( ?? + ?? )
2
- 2???? )
2
- 2( ???? )
2
= [(-
?? ?? )
2
- 2
?? ?? ]
2
- [2 (
?? ?? )
2
]
= [
?? 2
-2????
?? 2
]
2
-
2?? 2
?? 2
=
( ?? 2
2???? )
2
-2?? 2
?? 2
?? 4
(vi)
1
???? +?? +
1
???? +??
?
???? +?? +???? +?? ( 3?? +?? ) ( ???? +?? )
=
?? ( ?? + ?? ) +2?? ?? 2
???? +???? ?? +?????? +?? 2
=
?? ( ?? +?? ) +?? ?? 2
???? +???? ( ?? 2
?? ) +?? 2
=
?? ×
?? +2?? ?? ?? ×
?? ?? +
?????? ( -?? ) +?? 2
?? =
?? ???? -?? 2
+?? 2
=
?? ????
(vii)
?? ???? +?? +
?? ???? +??
=
?? ( ???? +?? ) +?? ( ???? +?? )
( ???? +?? ) ( ???? +?? )
=
?? ?? 2
+???? +?? ?? 2
+????
?? 2
???? +?????? +?????? +?? 2
=
?? ?? 2
+?? ?? 2
+?? ?? 2
+????
?? ×
?? ?? +???? ( ?? +?? ) +?? 2
=
?? [( ?? 2
+?? 2
) +?? ( ?? +?? ) ]
???? +???? +?? (
-?? ?? )+?? 2
=
?? [( ?? + ?? )
2
-2???? ]+???? -
?? ?? ????
=
?? [
?? 2
?? -
2?? ?? ]-
?? 2
?? ????
=
?? ×[
?? 2
-2?? ?? ]-?? 2
????
=
-2
??
(viii) ?? [
?? 2
?? +
?? 2
?? ] + ?? [
?? ?? +
?? ?? ]
= ?? [
?? 3
+?? 3
????
] + ?? (
?? 2
+?? 2
????
)
Page 5
Exercise 2.1
1. Find the zeroes of each of the following quadratic polynomials and verify the relationship
between the zeroes and their co efficient:
(i) f(x) = ?? 2
- 2?? - 8
(ii) g(s) = 4?? 2
- 4?? + 1
(iii) h(t) = ?? 2
- 15
(iv) p(x) = ?? 2
+ 2v2?? + 6
(v) q(x) = v3?? 2
+ 10?? + 7v3
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3
(vii) g(x) = ?? ( ?? 2
+ 1)- ?? ( ?? 2
+ 1)
(viii) 6?? 2
- 3 - 7?? Sol:
(i) f(x) = ?? 2
- 2?? - 8
?? ( ?? ) = ?? 2
- 2?? - 8 = ?? 2
- 4?? + 2?? - 8
= ?? ( ?? - 4)+ 2( ?? - 4)
= ( ?? + 2) ( ?? - 4)
Zeroes of the polynomials are -2 and 4
Sum of the zeroes =
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ??
-2 + 4 =
-( -2)
1
2 = 2
Product of the zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
= 24 =
-8
1
- 8 = -8
? Hence the relationship verified
(ii) 9( 5) = 45 - 45 + 1 = 45
2
- 25 - 25 + 1 = 25( 25 - 1)- 1( 25 - 1)
= ( 25 - 1) ( 25 - 1)
Zeroes of the polynomials are
1
2
?????? 1
2
Sum of zeroes =
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 2
1
2
+
1
2
=
-( -4)
4
1 = 1
Product of the zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
1
2
×
1
2
=
1
4
?
1
4
=
1
4
? Hence the relationship verified.
(iii) h(t) = ?? 2
- 15 = ( ?? 2
)- ( v15)
2
= ( ?? + v15) ( ?? - v15)
zeroes of the polynomials are -v15 ?????? v15
sum of zeroes = 0
-v15 + v15 = 0
0 = 0
Product of zeroes =
-15
1
-v15 × v15 = -15
-15 = -15
? Hence the relationship verified.
(iv) p(x) = ?? 2
+ 2v2?? - 6 = ?? 2
+ 3v2?? + v2 × 3v2
= ?? ( ?? + 3v2)- v2( 2 + 3v2) = ( ?? - v2) ( ?? + 3v2)
Zeroes of the polynomial are 3v2 and -3v2
Sum of the zeroes =
-3v2
1
v2 - 3v2 = -2v2
-2v2 = -2v2
?????????????? ???? ???????????? ? v2 × -3v2 = -
6
1
-6 = -6
?????????? ?? h?? ???????????????? h???? ????????????????
(v) 2(x) = v3?? 2
+ 10?? + 7v3 = v3?? 2
+ 7?? + 3?? + 7v3
= v3?? ( ?? + v3)+ 7( ?? + v3)
= ( v3?? + 7) ( ?? + v3)
Zeroes of the polynomials are -v3,
-7
v3
Sum of zeroes =
-10
v3
? -v3 -
7
v3
=
-10
v3
?
-10
v3
=
-10
v3
Product of zeroes =
7v3
3
?
v3?? -7
v30
= 7
? 7 = 7
Hence, relationship verified.
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 = ?? 2
- v3?? - ?? + v3
= x (x - v3) – 1 (x - v3)
= (x – 1) (x - v3)
Zeroes of the polynomials are 1 and v3
Sum of zeroes =
-{?????????????????????? ???? ?? }
???? ?????????????????? ???? ?? 2
=
-[-v3-1]
1
1 + v3 = v3 + 1
Product of zeroes =
???????????????? ???????? ???? ?????????????????? ???? ?? 2
=
v3
1
1 × v3 = v3 = v3 = v3
? Hence, relationship verified
(vii) g(x) = ?? [( ?? 2
+ 1)- ?? ( ?? 2
+ 1) ]
2
= ?? ?? 2
+ ?? - ?? 2
?? - ??
= ?? ?? 2
- [( ?? 2
+ 1)- ?? ] + 0 = ?? ?? 2
- ?? 2
?? - ?? + ??
= ???? ( ?? - ?? )- 1( ?? - ?? ) = ( ?? - ?? ) ( ???? - 1)
Zeroes of the polynomials =
1
?? ?????? ??
Sum of the zeroes =
-[-?? 2
-1]
??
?
1
?? + ?? =
?? 2
+1
?? ?
?? 2
+1
?? =
?? 2
+1
??
Product of zeroes =
?? ??
?
1
?? × ?? =
?? ?? ?
?? 2
+1
?? =
?? 2
+1
??
Product of zeroes =
?? ?? ? 1 = 1
Hence relationship verified
(viii) 6?? 2
- 3 - 7?? = 6?? 2
- 7?? - 3 = ( 3?? + 11) ( 2?? - 3)
Zeroes of polynomials are +
3
2
?????? -1
3
Sum of zeroes =
-1
3
+
3
2
=
7
6
=
-( -7)
6
=
-( ???? ?????????????????? ???? ?? )
???? ?????????????????? ???? ?? 2
Product of zeroes =
-1
3
×
3
2
=
-1
2
=
-3
6
=
???????????????? ???????? ???? ?????????????????? ???? ?? 2
? Hence, relationship verified.
2. If ?? and ?? are the zeros of the quadratic polynomial f(x) = ax
2
+ bx + c, then evaluate:
(i) ?? - ??
(ii)
1
?? -
1
??
(iii)
1
?? +
1
?? - 2?? ??
(iv) ?? 2
?? + ?? ?? 2
(v) ?? 4
+ ?? 4
(vi)
1
???? +?? +
1
???? +??
(vii)
?? ???? +?? +
?? ???? +??
(viii) ?? [
?? 2
?? +
?? 2
?? ] +
?? [
?? ?? +
?? ?? ]
Sol:
f(x) = ?? ?? 2
+ ???? + ??
?? + ?? =
-?? ??
???? =
?? ??
?????????? ?? + ?? ?????? ?? h?? ?????????? ( ???? ) ???????????? ???? ?? h?? ?????????? ??????????????????????
(i) ?? - ??
The two zeroes of the polynomials are
-?? +v?? 2
-4????
2?? - (?? -v?? 2
-4????
2?? ) = -?? +
v?? 2
-4???? +?? +v?? 2
-4????
2?? =
2v?? 2
-4????
2?? =
v?? 2
-4????
2??
(ii)
1
?? -
1
?? =
?? -?? ????
=
-( ?? - ?? )
????
… ( ?? )
From (i) we know that ?? - ?? =
v?? 2
-4????
2?? [???????? ( ?? ) ]???? =
?? ??
Putting the values in the (a) = - (
v?? 2
-4???? ×?? ?? ×?? ) =
-v?? 2
-4????
??
(iii)
1
?? +
1
?? - 2?? ??
? [
?? + ?? ????
] - 2????
?
-?? ?? ×
?? ?? - 2
?? ?? = -2
?? ?? -
?? ?? =
-???? -2?? 2
????
- [
?? ?? +
2?? ?? ]
(iv) ?? 2
?? + ?? ?? 2
?? ?? ( ?? + ?? )
=
?? ?? (
-?? ?? )
=
-????
?? 2
(v) ?? 4
+ ?? 4
= ( ?? 2
+ ?? 2
)
2
- 2?? 2
+ ?? 2
= ( ( ?? + ?? )
2
- 2???? )
2
- 2( ???? )
2
= [(-
?? ?? )
2
- 2
?? ?? ]
2
- [2 (
?? ?? )
2
]
= [
?? 2
-2????
?? 2
]
2
-
2?? 2
?? 2
=
( ?? 2
2???? )
2
-2?? 2
?? 2
?? 4
(vi)
1
???? +?? +
1
???? +??
?
???? +?? +???? +?? ( 3?? +?? ) ( ???? +?? )
=
?? ( ?? + ?? ) +2?? ?? 2
???? +???? ?? +?????? +?? 2
=
?? ( ?? +?? ) +?? ?? 2
???? +???? ( ?? 2
?? ) +?? 2
=
?? ×
?? +2?? ?? ?? ×
?? ?? +
?????? ( -?? ) +?? 2
?? =
?? ???? -?? 2
+?? 2
=
?? ????
(vii)
?? ???? +?? +
?? ???? +??
=
?? ( ???? +?? ) +?? ( ???? +?? )
( ???? +?? ) ( ???? +?? )
=
?? ?? 2
+???? +?? ?? 2
+????
?? 2
???? +?????? +?????? +?? 2
=
?? ?? 2
+?? ?? 2
+?? ?? 2
+????
?? ×
?? ?? +???? ( ?? +?? ) +?? 2
=
?? [( ?? 2
+?? 2
) +?? ( ?? +?? ) ]
???? +???? +?? (
-?? ?? )+?? 2
=
?? [( ?? + ?? )
2
-2???? ]+???? -
?? ?? ????
=
?? [
?? 2
?? -
2?? ?? ]-
?? 2
?? ????
=
?? ×[
?? 2
-2?? ?? ]-?? 2
????
=
-2
??
(viii) ?? [
?? 2
?? +
?? 2
?? ] + ?? [
?? ?? +
?? ?? ]
= ?? [
?? 3
+?? 3
????
] + ?? (
?? 2
+?? 2
????
)
=
?? [( ?? +?? )
3
-3???? ( ?? +?? ) ]
?? ?? + ?? ( ?? + ?? )
2
- 2?? ??
=
?? [(
-?? 3
?? 3
)+
3?? ?? .
?? ?? +?? (
?? 2
?? 2
-
2?? ?? )]
?? ??
=
?? 2
?? [
-?? 3
?? 3
+
3????
?? 2
+
?? 3
?? 2
-
2????
?? ]
=
-?? 2
?? 3
?? ?? 3
+
3?? 2
????
?? ?? 2
+
?? 3
?? 2
?? 2
?? -
2???? ?? 2
????
=
-?? 3
????
+ 3?? +
?? 3
????
- 2??
= b
3. If ?? and ?? are the zeros of the quadratic polynomial f(x) = 6x
2
+ x - 2, find the value of
?? ?? +
?? ??
Sol:
f(x) = 6?? 2
- ?? - 2
Since ?? and ?? are the zeroes of the given polynomial
? Sum of zeroes [?? + ?? ] =
-1
6
Product of zeroes (???? ) =
-1
3
=
?? ?? +
?? ?? =
?? 2
+?? 2
????
=
( ?? + ?? )
2
-2????
????
=
(
1
6
)
2
-2×(
-1
3
)
-
1
3
=
1
6
-
2
3
-1
3
=
1+24
36
-1
3
=
2
36
1
3
=
-25
12
4. If a and are the zeros of the quadratic polynomial f(x) = ?? 2
- ?? - 4, find the value of
1
?? +
1
?? - ????
Sol:
Since ?? + ?? are the zeroes of the polynomial: ?? 2
- ?? - 4
Sum of the roots (?? + ?? ) = 1
Product of the roots (???? ) = -4
1
?? +
1
?? - ???? =
?? +?? ????
- ????
=
1
-4
+ 4 =
-1
4
+ 4 =
-1+16
4
=
15
4
5. If ?? and ?? are the zeros of the quadratic polynomial p(x) = 4x
2
- 5x -1, find the value of
?? 2
?? + ?? ?? 2
.
Sol:
Since ?? ?????? ?? are the roots of the polynomial: 4?? 2
- 5?? - 1
Read More
|
126 videos|457 docs|75 tests
|
About this Document
|
4.86/5 Rating |
|
Dec 23, 2024 Last updated |
Document Description: RD Sharma Solutions: Polynomials for Class 10 2024 is part of Mathematics (Maths) Class 10 preparation.
The notes and questions for RD Sharma Solutions: Polynomials have been prepared according to the Class 10 exam syllabus. Information about RD Sharma Solutions: Polynomials covers topics
like and RD Sharma Solutions: Polynomials Example, for Class 10 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for RD Sharma Solutions: Polynomials.
Introduction of RD Sharma Solutions: Polynomials in English is available as part of our Mathematics (Maths) Class 10
for Class 10 & RD Sharma Solutions: Polynomials in Hindi for Mathematics (Maths) Class 10 course.
Download more important topics related with notes, lectures and mock test series for Class 10
Exam by signing up for free. Class 10: Polynomials RD Sharma Solutions | Mathematics (Maths) Class 10
Description
Full syllabus notes, lecture & questions for Polynomials RD Sharma Solutions | Mathematics (Maths) Class 10 - Class 10 | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) Class 10 | Best notes, free PDF download
Information about RD Sharma Solutions: Polynomials
In this doc you can find the meaning of RD Sharma Solutions: Polynomials defined & explained in the simplest way possible. Besides explaining types of
RD Sharma Solutions: Polynomials theory, EduRev gives you an ample number of questions to practice RD Sharma Solutions: Polynomials tests, examples and also practice Class 10
tests
|
Explore Courses for Class 10 exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Semester Notes
,shortcuts and tricks
,video lectures
,Summary
,Objective type Questions
,Free
,Polynomials RD Sharma Solutions | Mathematics (Maths) Class 10
,ppt
,practice quizzes
,study material
,Exam
,Polynomials RD Sharma Solutions | Mathematics (Maths) Class 10
,Viva Questions
,mock tests for examination
,Extra Questions
,MCQs
,Sample Paper
,Polynomials RD Sharma Solutions | Mathematics (Maths) Class 10
,Important questions
,Previous Year Questions with Solutions
,past year papers
;
Additional Information about RD Sharma Solutions: Polynomials for Class 10 Preparation
RD Sharma Solutions: Polynomials Free PDF Download
The RD Sharma Solutions: Polynomials is an invaluable resource that delves deep into the core of the Class 10 exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the RD Sharma Solutions: Polynomials now and kickstart your journey towards success in the Class 10 exam.
Importance of RD Sharma Solutions: Polynomials
The importance of RD Sharma Solutions: Polynomials cannot be overstated, especially for Class 10 aspirants.
This document holds the key to success in the Class 10 exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
RD Sharma Solutions: Polynomials Notes
RD Sharma Solutions: Polynomials Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to RD Sharma Solutions: Polynomials.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, RD Sharma Solutions: Polynomials Notes on EduRev are your ultimate resource for success.
RD Sharma Solutions: Polynomials Class 10 Questions
The "RD Sharma Solutions: Polynomials Class 10 Questions" guide is a valuable resource for all aspiring students preparing for the
Class 10 exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study RD Sharma Solutions: Polynomials on the App
Students of Class 10 can study RD Sharma Solutions: Polynomials alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the RD Sharma Solutions: Polynomials,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of RD Sharma Solutions: Polynomials is prepared as per the latest Class 10 syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup