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Algebraic Identities

Class 9th Mathematics RD Sharma Solution
Exercise 4.1
  1. (i) (2x - 1/x)^2 (ii) (2x+y) (2x-y) (iii) (a^2b-b^2a)^2 (iv) (a - 0.1) (a +0.1)…
  2. (i) (399)^2 (ii) (0.98)^2 (iii) 9911009 (iv) 11783 Evaluate each of the…
  3. Simplify each of the following: (i) 175 175 + 2 175 25 + 25 25 (ii) 322 322 - 2…
  4. If x + 1/x = 11, find the value of x^2 + 1/x^2
  5. If x - 1/x = -1, find the value of x^2 + 1/x^2
  6. If x + 1/x = root 5 , find the value of x^2 + 1/x^2 and x^4 + 1/x^4…
  7. If x^2 + 1/x^2 = 66, find the value of x - 1/x
  8. If x^2 + 1/x^2 = 79, find the value of x + 1/x
  9. If 9x^2 +25y^2 = 181 and xy = -6, find the value of 3x +5y
  10. If 2x +3y = 8 and xy = 2, find the value of 4x^2 +9y^2
  11. If 3x - 7y = 10 and xy = -1, find the value of 9x^2 + 49y^2
  12. Simplify each of the following products: (i) (1/2 a-3b) (3b + 1/2 a) (1/4 a^2 +…
  13. Prove that a^2 + b^2 +c^2 -ab-bc-ca is always non-negative for all values of a,…
Exercise 4.2
  1. Write the following in the expanded form: (i) (a + 2b + c)^2 (ii) (2a - 3b -…
  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…
  3. If a+b+c =0 and a^2 +b^2 +c^2 =16, find the value of ab + bc + ca.…
  4. If a^2 +b^2 +c^2 =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 a^2 +b^2 +c^2 .
  6. Find the value of 4x^2 +y^2 +25z^2 +4xy-10yz-20zx when x = 4, y = 3 and z = 2.…
  7. Simplify each of the following expressions : (i) (x+y+z)^2 + (x + y/2 + z/3)^2 -…
Exercise 4.3
  1. Find the cube of each of the following binomial expressions: (i) (1/x + y/3)^3…
  2. Simplify each of the following : (i) (x+3)^3 + (x-3)^3 (ii) (x/2 + y/3)^3 - (x/2…
  3. If a+b=10 and ab=21, find the value of a^3 +b^3 .
  4. If a-b=4 and ab=21, find the value of a^3 -b^3 .
  5. If x+ 1/x =5, find the value of x^3 + 1/x^3 .
  6. If x- 1/x = 7, find the value of x^3 - 1/x^3 .
  7. If x- 1/x =5, find the value of x^3 - 1/x^3 .
  8. If x^2 + 1/x^2 =51, find the value of x^3 - 1/x^3 .
  9. If x^2 + 1/x^2 =98, find the value of x^3 + 1/x^3 .
  10. If 2x+3y=13 and xy=6, find the value of 8x^3 +27y^3 .
  11. If 3x-2y=11 and xy=12, find the value of 27x^3 -8y^3 .
  12. If x^4 + 1/x^4 =119, find the value of x^3 - 1/x^3 .
  13. Evaluate each of the following: (i) (103)^3 (ii) (98)^3 (iii) (9.9)^3 (iv)…
  14. Evaluate each of the following : (i) 111^3 -89^3 (ii) 46^3 +34^3 (iii) 104^3…
  15. If x+ 1/x = 3, Calculate x^2 + 1/x^2 , x^3 + 1/x^3 and x^4 + 1/x^4 .…
  16. If x^4 + 1/x^4 =194, find x^3 + 1/x^3 , x^2 + 1/x^2 and x+ 1/x
  17. Find the value of 27x^3 + 8y^3 , if (i) 3x + 2y = 14 and xy=8 (ii) 3x + 2y = 20…
  18. Find the value of 64x^3 - 125z^3 , if 4x - 5z = 16 and xz=12.
  19. If x- 1/x = 3+2 root 2 , find the value of x^3 - 1/x^3 .
Exercise 4.4
  1. Find the following products: (i) (3x+2y)(9x^2 -6xy+4y^2) (ii) (4x-5y)(16x^2…
  2. If x = 3 and y = -1, find the values of each of the following using in identity:…
  3. If a+b=10 and ab=16, find the value of a^2 - ab+b^2 and a^2 +ab+b^2 .…
  4. If a+b=8 and ab=6, find the value of a^3 +b^3 .
  5. If a-b=6 and ab=20, find the value of a^3 -b^3 .
  6. If x = -2 and y = 1, by using an identity find the value of the following: (i)…
Exercise 4.5
  1. Find the following product: (i) (3x+2y+2z) (9x^2 +4y^2 +4z^2 -6xy-4yz-6zx) (ii)…
  2. If x+y+z = 8 and xy+yz+zx =20, find the value of x^3 +y^3 +z^3 -3xyz.…
  3. If a+b+c=9 and ab+bc+ca=26, find the value of a^3 +b^3 +c^3 -3abc.…
  4. If a+b+c=9 and a^2 +b^2 +c^2 =35, find the value of a^3 +b^3 +c^3 -3abc.…
  5. Evaluate : (i) 25^3 - 75^3 + 50^3 (ii) 48^3 - 30^3 - 18^3 (iii) (1/2)^3 +…
Cce - Formative Assessment
  1. If x+ 1/x =3, then find the value of x^2 + 1/x^2 .
  2. If x+ 1/x =5, then x^2 + 1/x^2 =A. 25 B. 10 C. 23 D. 27
  3. If x+ 1/x =3, then find the value of x^6 + 1/x^6 .
  4. If x+ 1/x =2, then x^3 + 1/x^3 =A. 64 B. 14 C. 8 D. 2
  5. If a+b=7 and ab=12, find the value of a^2 +b^2 .
  6. If x+ 1/x =4, then x^4 + 1/x^4 =A. 196 B. 194 C. 192 D. 190
  7. If a-b=5 and ab=12, find the value of a^2 +b^2 .
  8. If x+ 1/x =3, then x^6 + 1/x^6 =A. 927 B. 414 C. 364 D. 322
  9. If x- 1/x = 1/2 , then write the value of 4x^2 + 4/x^2 .
  10. If x^4 + 1/x^4 = 623, then x+ 1/x =A. 27 B. 25 C. 3 root 3 B. -3 root 3…
  11. If x^2 + 1/x^2 = 102, then x- 1/x =A. 8 B. 10 C. 12 D. 13
  12. If a^2 + 1/a^2 =102, find the value of a- 1/a .
  13. If a+b+c=0 then write the value of a^2/bc + b^2/ca + c^2/ab .
  14. If x^3 + 1/x^3 =110, then x + 1/x =A. 5 B. 10 C. 15 D. none of these…
  15. If x^3 - 1/x^3 =14, then x - 1/x =A. 5 B. 4 C. 3 D. 2
  16. If x^4 + 1/x^4 = 194, then x^3 - 1/x^3 =A. 76 B. 52 C. 64 D. none of these…
  17. If x- 1/x = 15/4 , then x + 1/x =A. 4 B. 17/4 C. 13/4 D. 1/4
  18. If 3x+ 2/x =7, then (9x^2 - 4/x^2) =A. 25 B. 35 C. 49 D. 30
  19. If a^2 +b^2 + c^2 -ab-bc-ca =0, thenA. a + b =c B. b + c = a C. c + a = b D. a = b = c…
  20. If a + b + c =0, then, a^2/bc + b^2/ca + c^2/ab =A. 0B. 1C. -1D. 3…
  21. If a1/3+b1/3 + c1/3 =0, thenA. a+b+ c =0 B. (a+b+ c)^3 =27abc C. a+b+ c =3abc D. a^3…
  22. If a+b+ c =9 and ab + bc + ca = 23, then a^2 +b^2 + c^2 =A. 35 B. 58 B. 127 D. none of…
  23. If a+b+ c =9, then ab+bc+ca=23, then a^3 +b^3 + c^3 - 3abc =A. 108 B. 207 C. 669 D.…
  24. (a-b)^3 +(b-c)^3 + (c-a)^3 =A. (a+b+ c) (a^2 +b^2 + c^2 -ab-bc-ca) B. (a-b)(b-c) (c-a)…
  25. Solve the equation and choose the correct answer: (a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2…
  26. The product (a+b) (a-b) (a^2 -ab+b^2) (a^2 +ab+b^2) is equal toA. a^6 +b^6 B. a^6 -b^6…
  27. If a/b + b/a = -1, then a^3 -b^3 =A. 1 B. -1 C. 1/2 D. 0
  28. The product (x^2 -1) (x^4 +x^2 +1) is equal to=A. x^8 -1 B. x^8 +1 C. x^6 -1 D. x^6 +1…
  29. If a-b=-8, and ab =-12, then a^3 - b^3 =A. -244 B. -240 C. -224 D. -260…
  30. If the volume of a cuboid is 3x^2 -27, then its possible dimensions areA. 3, x^2 ,…
  31. If a/b + b/a = 1, then a^3 +b^3 =A. 1 B. -1 C. 1/2 D. 0
  32. 7575+27525+2525 equal toA. 10000 B. 6250 C. 7500 D. 3750
  33. (x-y)(x+y)(x^2 +y^2)(x^4 +y^4) is equal toA. x^16 - y^16 B. x^8 -y^8 C. x^8 +y^8 D.…
  34. If 48a^2 -b= (7a + 1/2) (7a - 1/2) , then the value of b isA. 0 B. 1/4 C. 1/root 2 D.…
Exercise 4.1
Question 1.

Evaluate each of the following using identities:

(i)

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

(iii)

(iv) (a – 0.1) (a +0.1)

(v)


Answer:

(i) We know, (a-b)2 = a2 + b2 -2ab


(ii) We know, (a-b)2 = (a+b) (a-b)


(2x+y) (2x-y) = (2x)2– y2 = 4x2 – y2


(iii) We know, (a-b)2 = a2 + b2 -2ab



(iv) We know, (a-b)2 = (a+b) (a-b)


(a – 0.1) (a +0.1) = (a)2– (0.1)2


= a2 – 0.01


(v) We know, (a-b)2 = a2 - b2
Question 2.

Evaluate each of the following using identities:

(i) (399)2

(ii) (0.98)2

(iii) 991×1009

(iv) 117×83
Answer:

(i) We will use the identity, (a-b)2 = a2 + b2 – 2ab


(ii) We will use the identity, (a-b)2 = a2 + b2 – 2ab



(iii) We will use the identity, (a-b)(a+b)= a2 - b2



(iv) We will use the identity, (a-b) (a+b)= a2 - b2
Question 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)


Answer:

(i) We know, (a+b)2 = a2 + b2 +2ab



(ii) We know, (a-b)2 = a2 + b2 -2ab




(iii) We know, (a+b)2 = a2 + b2 +2ab




(iv) We know, (a-b)2 = (a-b) (a+b)
Question 4.

If x + = 11, find the value of x2 +
Answer:

Here, we will use (a+b)2 = a2 + b2 +2ab
Question 5.

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

Here, we will use (a-b)2 = a2 + b2 -2ab
Question 6.

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

Here, we will use (a+b)2 = a2 + b2 +2ab



Now,
Question 7.

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


Answer:

Here, we will use (a-b)2 = a2 + b2 -2ab.
Question 8.

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


Answer:

Here, we will use (a+b)2 = a2 + b2 + 2ab
Question 9.

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

Here, we will use (a+b)2 = a2 + b2 + 2ab
Question 10.

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

Here, we will use (a+b)2 = a2 + b2 + 2ab
Question 11.

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

Here, we will use (a+b)2 = a2 + b2 + 2ab
Question 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)
Answer:

(i) On regarranging we get,


(ii) On regarranging we get,



(iii) On rearranging we get, -x2 + 2x


=


Using, (a-b)2 = a2 + b2 – 2ab



(iv) Using the idendity, (a+b)(a-b) = a2-b2


On rearranging we get,


(x2 + x – 2) (x2 - x + 2) = {x2 + (x – 2)} {(x2 – (x - 2)}


= (x2)2 – (x – 2)2 = x4-(x2 - 4x + 4)


= x4 – x2 + 4x – 4


(v) Taking x as common factor, we write,


= x (x2 - 3x – 1) (x2 - 3x + 1)


= {x (x2 - 3x – 1)} (x2 - 3x + 1)


= x [{(x2 - 3x) – 1)} {(x2 - 3x)+1)}]


= x {(x2 - 3x)2 – 12}


= x (x4 - 6x3+9x2-1)


= x5 – 6x4 + 9x3 -x


(vi) On Reaaranging we get,


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


= {(2x4 - 4x2) + 1} {(2x4 - 4x2)- 1)}


= (2x4 - 4x2)2 – 12


= 4x8 + 16x4 -2 × 2x4 × 4x2 – 1


= 4x8 + 16x4 -16x6 -1
Question 13.

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

We have to prove that a2+ b2+c2–ab–bc–ca 0

Lets us consider,
Exercise 4.2
Question 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
Answer:

(i) Using idendity,


(ii) Using idendity,



(iii) Using idendity,



(iv) Using idendity,



(v) Using idendity,



(vi) Using idendity,



(vii) Using idendity,



(viii) Using idendity,



(ix) Using idendity,



(x) Using idendity,



(xi) Using idendity,



(xii) Using idendity,
Question 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
Answer:

(i) Using idendity,


(ii) Using idendity,



(iii) Using idendity,



(iv) Using idendity,



(v) Using identity: a2 - b2 = (a + b)(a - b)


(x2 + y2 - z2)2 - (x2 -y2 +z2)2
=(x2 + y2 - z2 +(x2 -y2 +z2)) (x2 + y2 - z2 -(x2 -y2 +z2))
= 2x2(2y2 - 2z2)
= 4x2y2 -4x2z2
Question 3.

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

Using idendity,


Given: a+b+c = 0 and a2 + b2 + c2 =16


Squaring the equation, a+b+c = 0 on both the sides, we get,
Question 4.

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

Using the identity,
Question 5.

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

Usint the identity,
Question 6.

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

Using the identity,
Question 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
Answer:

(i) Using identity,


(ii) Using the identity,




(iii) Using the identity,
Exercise 4.3
Question 1.

Find the cube of each of the following binomial expressions:

(i)

(ii)

(iii)

(iv)


Answer:

(i) Using the identity, (a+b)3 = a3 + b3 + 3a2 b + 3ab2

We will write the bionomial expression,



(ii) Using the identity, (a-b)3 = a3 - b3 - 3a2 b + 3ab2


We will write the bionomial expression,



(iii) Using the identity, (a-b)3 = a3 - b3 - 3a2 b + 3ab2


We will write the bionomial expression,



(iv) Using the identity, (a-b)3 = a3 - b3 - 3a2 b + 3ab2


We will write the bionomial expression,
Question 2.

Simplify each of the following :
(i) (x+3)3 + (x-3)3

(ii)

(iii)

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

(i) Using the identity,


(ii) Using the identity, a3 - b3 = (a-b)(a2 +ab + b2)



(iii) Using the identity, a3 + b3 = (a+b)(a2 - ab + b2)



(iv) Using the identity, a3 - b3 = (a-b)(a2 + ab + b2)
Question 3.

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

Given (a+b)=10 and ab = 21

Using, (a+b)3 = a3 + b3 + 3ab(a+b), we get,
Question 4.

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

Given (a - b)=4 and ab = 21

Using, (a-b)3 = a3 - b3 - 3ab(a-b), we get,
Question 5.

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

Given: x+=5,

Using, (a+b)3 = a3 + b3 + 3ab(a+b), we get
Question 6.

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

Given: x-= 7

Using, (a-b)3 = a3 - b3 - 3ab(a-b), we get
Question 7.

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

Given: x-=5

Using, (a+b)3 = a3 + b3 + 3ab(a+b), we get
Question 8.

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

Using the identity, (x+y)2=x2 + y2 + 2xy
Question 9.

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

Using the identity, (x+y)2=x2 + y2 + 2xy
Question 10.

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

Given :- 2x+3y=13 and xy=6.

Using, (a+b)3 = a3 + b3 + 3ab(a+b), we get
Question 11.

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

Given :- 3x-2y=11 and xy=12.


Using, (a-b)3 = a3 - b3 - 3ab(a-b), we get
Question 12.

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

Given :- x4+=119


Using, (a+b)2 = a2 + b2 +2ab, we get
Question 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
Answer:

(i) Using the identity,


(ii) Using the identity,



(iii) Using the identity,



(iv) Using the identity,



(v) Using the identity,



(vi) Using the identity,
Question 14.

Evaluate each of the following :

(i) 1113-893

(ii) 463+343

(iii) 1043+963

(iv) 933-1073
Answer:

(i) Uisng the identity:


(ii) Uisng the identity:



(iii) Using the identity:



(iv) Uisng the identity:
Question 15.

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

Given x+= 3
Question 16.

If x4+=194, find x3 +, x2 +and x+
Answer:

Given x4+=194
Question 17.

Find the value of 27x3 + 8y3, if

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

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

(i) Using,


(ii) Using the identity, we get,
Question 18.

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

Using the identity, we write,
Question 19.

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

Using the identity, we write,
Exercise 4.4
Question 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)
Answer:

(i)


(ii)



(iii)



(iv)



(v)



(vi)



(vii)



(viii)



(ix)



(x)



(xi)



(xii)
Question 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)


Answer:

(i)


(ii)



(iii)



(iv)



(v)
Question 3.

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

Given: a+b=10 and ab =16

To find: a2– ab+b2



To find: a2+ ab+b2
Question 4.

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

Given: a+b=8 and ab =6

To find: a3+b3
Question 5.

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

Given: a-b=6 and ab =20

To find: a3-b3
Question 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)


Answer:

(i)


(ii)



(iii)
Exercise 4.5
Question 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)
Answer:

(i) Using the identity,


(ii) Using the identity,



(iii) Using the identity,



(iv) Using the identity,
Question 2.

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

In x3 +y3 +z3 -3xyz,

Using the identity

a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca) we get,

x3 +y3 +z3 -3xyz = ( x + y +z) (x2+y2+z2-xy-yz-zx)

x3 +y3 +z3 -3xyz = ( x + y +z) [x2+y2+z2-(xy+yz+zx)] …. (1)

We also know,

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

For x2+y2+z2

(x + y + z)2 = x2 + y2 + z2 + 2(xy + yz+ zx)

⇒ 82 = x2 + y2 + z2 + 2(20)

⇒ 64 = x2 + y2 + z2 + 40

⇒ x2 + y2 + z2 = 24

From (1) we get,

x3 +y3 +z3 -3xyz = 8 [24-20]

= 8(4)

= 32
Question 3.

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

Using the identity,
Question 4.

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

Using the identity,
Question 5.

Evaluate :

(i) 253 – 753 + 503

(ii) 483 – 303 - 183

(iii)

(iv) (0.2)3 – (0.3)3 + (0.1)3
Answer:

(i)


(ii)



(iii)



(iv)
Cce - Formative Assessment
Question 1.

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

Given: x+ =3
Question 2.

If x+ =5, then x2+=
A. 25

B. 10

C. 23

D. 27
Answer:


Question 3.

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

We are given that x+ =3
Question 4.

If x+ =2, then x3+=
A. 64

B. 14

C. 8

D. 2
Answer:


Question 5.

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

Using the identity,
Question 6.

If x+ =4, then x4+ =
A. 196

B. 194

C. 192

D. 190
Answer:


Question 7.

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

Using the identity,
Question 8.

If x+ =3, then x6+ =
A. 927

B. 414

C. 364

D. 322
Answer:


Question 9.

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

Given: x- =
Question 10.

If x4+ = 623, then x+=
A. 27

B. 25

C. 3

B. -3


Answer:


Question 11.

If x2+= 102, then x-=
A. 8

B. 10

C. 12

D. 13
Answer:


Question 12.

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

Here, we will use (a+b)2 = a2 + b2 + 2ab
Question 13.

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



Question 14.

If x3+=110, then x +=?
A. 5

B. 10

C. 15

D. none of these
Answer:


Question 15.

If x3-=14, then x -=
A. 5

B. 4

C. 3

D. 2
Answer:


Question 16.

If x4+ = 194, then x3-=
A. 76

B. 52

C. 64

D. none of these
Answer:


Question 17.

If x- = , then x +=
A. 4

B.

C.

D.


Answer:


Question 18.

If 3x+ =7, then =
A. 25

B. 35

C. 49

D. 30
Answer:


Question 19.

If a2+b2 + c2-ab-bc-ca =0, then
A. a + b =c

B. b + c = a

C. c + a = b

D. a = b = c
Answer:


Question 20.

If a + b + c =0, then, =
A. 0
B. 1
C. -1
D. 3
Answer:


Question 21.

If a1/3+b1/3 + c1/3 =0, then
A. a+b+ c =0

B. (a+b+ c)3 =27abc

C. a+b+ c =3abc

D. a3+b3+ c3 =0
Answer:


Question 22.

If a+b+ c =9 and ab + bc + ca = 23, then a2+b2 + c2=
A. 35

B. 58

B. 127

D. none of these
Answer:


Hence, 92 = a2 + b2 + c2 + 2 × 23


⇒ a2 + b2 + c2 = 35
Question 23.

If a+b+ c =9, then ab+bc+ca=23, then a3+b3 + c3 – 3abc =
A. 108

B. 207

C. 669

D. 729
Answer:


Question 24.

(a-b)3+(b-c)3+ (c-a)3=
A. (a+b+ c) (a2+b2 + c2-ab-bc-ca)

B. (a-b)(b-c) (c-a)

C. 3 (a-b)(b-c) (c-a)

D. none of these
Answer:


Question 25.

Solve the equation and choose the correct answer:

=
A. 3(a+b)(b+c) (c+a)

B. 3 (a-b)(b-c) (c-a)

C. (a-b)(b-c) (c-a)

D. None of these
Answer:


Question 26.

The product (a+b) (a-b) (a2-ab+b2) (a2+ab+b2) is equal to
A. a6+b6

B. a6-b6

C. a3-b3

D. a3+b3
Answer:


Question 27.

If = -1, then a3-b3 =
A. 1

B. -1

C.

D. 0
Answer:


Question 28.

The product (x2-1) (x4+x2+1) is equal to=
A. x8-1

B. x8+1

C. x6-1

D. x6+1
Answer:


Question 29.

If a-b=-8, and ab =-12, then a3 – b3 =
A. -244

B. -240

C. -224

D. -260
Answer:


Question 30.

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

B. 3, x-3, x+3

C. 3, x2, 27x

D. 3, 3, 3
Answer:


Thus, the possible dimensions are 3, (x+3)(x-3)
Question 31.

If = 1, then a3+b3 =
A. 1

B. -1

C.

D. 0
Answer:


Question 32.

75×75+2×75×25+25×25 equal to
A. 10000

B. 6250

C. 7500

D. 3750
Answer:

We know, (a+b)2 = a2 + b2 +2ab
Question 33.

(x-y)(x+y)(x2+y2)(x4+y4) is equal to
A. x16 – y16

B. x8 –y8

C. x8+y8

D. x16 + y16
Answer:


Question 34.

If 48a2-b= , then the value of b is
A. 0

B.

C.

D.


Answer: