RD Sharma - Mathematics

Book: RD Sharma - Mathematics

Chapter: 4. Algebraic Identities

Subject: Mathematics - Class 9th

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Chapter Exercises

Exercise 4.1
Exercise 4.2
Exercise 4.3
Exercise 4.4
Exercise 4.5
CCE - Formative Assessment

Exercise 4.1

1

Evaluate each of the following using identities:

(i)


(ii) (2x+y) (2x-y)


(iii)


(iv) (a – 0.1) (a +0.1)


(v)

2

Evaluate each of the following using identities:

(i) (399)2


(ii) (0.98)2


(iii) 991×1009


(iv) 117×83
3

Simplify each of the following:

(i) 175 × 175 + 2 × 175 × 25 + 25 ×25


(ii) 322 × 322 – 2 × 322 × 22 + 22 × 22


(iii) 0.76 ×0.76+2×0.76×0.24+0.24×0.24


(iv)

4

If x + = 11, find the value of x2 +
5

If x - = -1, find the value of x2 +
6

If x + = , find the value of x2 + and x4+
7

If x2 + = 66, find the value of x -

8

If x2 + = 79, find the value of x +

9

If 9x2 +25y2 = 181 and xy = -6, find the value of 3x +5y
10

If 2x +3y = 8 and xy = 2, find the value of 4x2 +9y2
11

If 3x - 7y = 10 and xy = -1, find the value of 9x2 + 49y2
12

Simplify each of the following products:

(i)


(ii)


(iii) -x2 + 2x


(iv) (x2 + x – 2) (x2 - x + 2)


(v) (x3 - 3x2 – x) (x2 - 3x + 1)


(vi) (2x4 - 4x2 + 1) (2x4 - 4x2 - 1)
13

Prove that a2+ b2+c2–ab–bc–ca is always non-negative for all values of a, b and c.

Exercise 4.2

1

Write the following in the expanded form:

(i) (a + 2b + c)2


(ii) (2a - 3b - c)2


(iii) (-3x + y + z)2


(iv) (m + 2n – 5p)2


(v) (2 + x – 2y)2


(vi) (a2 + b2 + c2)2


(vii) (ab + bc + ca)2


(viii)


(ix)


(x) (x + 2y + 4z)2


(xi) (2x - y + z)2


(xii) (-2x + 3y + 2z)2
2

Simplify:

(i) (a+b+c)2 +(a-b+c)2


(ii) (a+b+c)2 -(a-b+c)2


(iii) (a+b+c)2 +(a-b+c)2+(a+b-c)2


(iv) (2x+p+c)2-(2x-p+c)2


(v) (x2+y2-z2)2-(x2-y2+z2)2
3

If a+b+c =0 and a2+b2+c2 =16, find the value of ab + bc + ca.
4

If a2+b2+c2 =16, and ab + bc + ca=10, find the value of a+b+c.
5

If a+b+c =9 and ab+bc+ca=23, find the value of a2+b2+c2.
6

Find the value of 4x2+y2+25z2+4xy-10yz-20zx when x = 4, y = 3 and z = 2.
7

Simplify each of the following expressions :

(i) (x+y+z)2+


(ii) (x+y-2z)2-x2 – y2 - 3z2+4xy


(iii) (x2 – x + 1)2 - (x2 + x + 1)2

Exercise 4.3

1

Find the cube of each of the following binomial expressions:

(i)


(ii)


(iii)


(iv)

2

Simplify each of the following :

(i) (x+3)3 + (x-3)3


(ii)


(iii)


(iv) (2x-5y)3 - (2x+5y)3
3

If a+b=10 and ab=21, find the value of a3+b3.
4

If a-b=4 and ab=21, find the value of a3-b3.
5

If x+=5, find the value of x3+.
6

If x-= 7, find the value of x3-.
7

If x-=5, find the value of x3-.
8

If x2+=51, find the value of x3-.
9

If x2+=98, find the value of x3+.
10

If 2x+3y=13 and xy=6, find the value of 8x3+27y3.
11

If 3x-2y=11 and xy=12, find the value of 27x3-8y3.
12

If x4+=119, find the value of x3-.
13

Evaluate each of the following:

(i) (103)3


(ii) (98)3


(iii) (9.9)3


(iv) (10.4)3


(v) (598)3


(vi) (99)3
14

Evaluate each of the following :

(i) 1113-893


(ii) 463+343


(iii) 1043+963


(iv) 933-1073
15

If x+= 3, Calculate x2 +, x3 +and x4+.
16

If x4+=194, find x3 +, x2 +and x+
17

Find the value of 27x3 + 8y3, if

(i) 3x + 2y = 14 and xy=8


(ii) 3x + 2y = 20 and xy=
18

Find the value of 64x3 – 125z3, if 4x – 5z = 16 and xz=12.
19

If x-= 3+2 , find the value of x3 -.

Exercise 4.4

1

Find the following products:

(i) (3x+2y)(9x2-6xy+4y2)


(ii) (4x-5y)(16x2+20xy+25y2)


(iii) (7p4+q)(49p8-7p4q+q2)


(iv)


(v)


(vi)


(vii)


(viii)


(ix) (1+x)(1+x+x2)


(x) (1+x)(1-x+x2)


(xi) (x2-1)(x4+x2+1)


(xii) (x3+1)(x6-x3+1)
2

If x = 3 and y = -1, find the values of each of the following using in identity:

(i) (9y2 -4x2) (81y4 + 36x2y2 +16x4)


(ii)


(iii)


(iv)


(v)

3

If a+b=10 and ab=16, find the value of a2– ab+b2 and a2+ab+b2.
4

If a+b=8 and ab=6, find the value of a3+b3.
5

If a-b=6 and ab=20, find the value of a3-b3.
6

If x = -2 and y = 1, by using an identity find the value of the following:

(i) (4y2–9x2) (16y4+36x2y2+81x4)


(ii)


(iii)

Exercise 4.5

1

Find the following product:

(i) (3x+2y+2z) (9x2+4y2+4z2-6xy-4yz-6zx)


(ii) (4x-3y+2z) (16x2+9y2+4z2+12xy+6yz-8zx)


(iii) (2a-3b-2c) (4a2+9b2+4c2+6ab-6bc+4ca)


(iv) (3x-4y+5z)(9x2+16y2+25z2+12xy-15zx+20yz)
2

If x+y+z = 8 and xy+yz+zx =20, find the value of x3 +y3 +z3 -3xyz.
3

If a+b+c=9 and ab+bc+ca=26, find the value of a3 +b3 +c3 -3abc.
4

If a+b+c=9 and a2+b2+c2=35, find the value of a3 +b3 +c3 -3abc.
5

Evaluate :

(i) 253 – 753 + 503


(ii) 483 – 303 - 183


(iii)


(iv) (0.2)3 – (0.3)3 + (0.1)3

CCE - Formative Assessment

1

If x+ =3, then find the value of x2+.
2

If x+ =3, then find the value of x6+.
3

If a+b=7 and ab=12, find the value of a2+b2.
4

If a-b=5 and ab=12, find the value of a2+b2.
5

If x- =, then write the value of 4x2+.
6

If a2+ =102, find the value of a-.
7

If a+b+c=0 then write the value of.
1

If x+ =5, then x2+=
2

If x+ =2, then x3+=
3

If x+ =4, then x4+ =
4

If x+ =3, then x6+ =
5

If x4+ = 623, then x+=
6

If x2+= 102, then x-=
7

If x3+=110, then x +=
8

If x3-=14, then x -=
9

If x4+ = 194, then x3-=
10

If x- = , then x +=
11

If 3x+ =7, then =
12

If a2+b2 + c2-ab-bc-ca =0, then
13

If a + b + c =0, then,   =
14

If a1/3+b1/3 + c1/3 =0, then
15

If a+b+ c =9 and ab + bc + ca = 23, then a2+b2 + c2=
16

If a+b+ c =9, then ab+bc+ca=23, then a3+b3 + c3 – 3abc =
17

(a-b)3+(b-c)3+ (c-a)3=
18

=
19

The product (a+b) (a-b) (a2-ab+b2) (a2+ab+b2) is equal to
20

If = -1, then a3-b3 =
21

The product (x2-1) (x4+x2+1) is equal to=
22

If a-b=-8, and ab =-12, then a3 – b3 =
23

If the volume of a cuboid is 3x2-27, then its possible dimensions are
24

If = 1, then a3+b3 =
25

75×75+2×75×25+25×25 equal to
26

(x-y)(x+y)(x2+y2)(x4+y4) is equal to
27

If 48a2-b= , then the value of b is