Algebra Class 8th Mathematics Term 2 Tamilnadu Board Solution

Class 8th Mathematics Term 2 Tamilnadu Board Solution
Exercise 1.1
  1. The coefficient of x^4 in -5x^7 + 3/7 x^4 - 3x^3 + 7x^2 - 1 is Choose the…
  2. The coefficient of xy^2 in 7x^2 - 14x^2 y + 14xy^2 - 5 is Choose the correct…
  3. The power of the term x^3 y^2 z^2 is _____ Choose the correct answer for the…
  4. The degree of the polynomial x^2 - 5x^4 + 3/4 x^7 - 73x+5 is ______ Choose the…
  5. The degree of the polynomial x^2 - 5x^2 y^3 + 30x^3 y^4 - 576xy is Choose the…
  6. x^2 + y^2 - 2z^2 + 5x - 7 is a Choose the correct answer for the following:A.…
  7. The constant term of 0.4x^7 - 75y^2 - 0.75 is Choose the correct answer for the…
  8. Identify the terms and their coefficients for the following expressions: i. 3abc…
  9. Classify the following polynomials as monomials, binomials and trinomials: 3x^2…
  10. Add the following algebraic expressions: i. 2x^2 + 3x + 5 and 3x^2 - 4x -7 ii.…
  11. Subtract (i) Subtract 2 a - b from 3a - b (ii) Subtract -3x + 8y from -7x - 10y…
  12. Find out the degree of the polynomials and the leading coefficients of the…
Exercise 1.2
  1. Find the product of the following pairs of monomials: i. 3, 7x ii. - 7x, 3y iii.…
  2. Complete the following table of products:
  3. Find out the product : (i) 2a, 3a^2 , 5a^4 (ii) 2x, 4y, 9z (iii) ab, bc, ca (iv)…
  4. Find the product : (i) (a^3) × (2a^5) × (4a^15) (ii) (5 - 2x) (4 + x) (iii) (x +…
  5. (a + b) (2a^2 - 5ab + 3b^2) Find the product of the following :
  6. (2x + 3y) (x^2 - xy + y^2) Find the product of the following :
  7. (x + y + z) (x + y - z) Find the product of the following :
  8. (a + b) (a^2 + 2ab + b^2) Find the product of the following :
  9. (m - n) (m^2 + mn + n^2) Find the product of the following :
  10. Add 2x(x - y - z) and 2y(z - y - x)
  11. Subtract 3a(a-2b + 3c) from 4a(5a + 2b - 3c)
Exercise 1.3
  1. (a + b)^2 = (a + b) × ________ Choose the correct answer for the following:A.…
  2. (a - b)^2 = (a - b) × ________ Choose the correct answer for the following:A.…
  3. (a^2 - b^2) = (a - b) × ________ Choose the correct answer for the following:A.…
  4. 9.6^2 = __________ Choose the correct answer for the following:A. 9216 B. 93.6…
  5. (a + b)^2 - (a - b)^2 = ________ Choose the correct answer for the following:A.…
  6. m^2 + (c + d) m + cd = ______ Choose the correct answer for the following:A. (m…
  7. (x + 3) (x + 3) Using a suitable identity, find each of the following products:…
  8. (2m + 3) (2m + 3) Using a suitable identity, find each of the following…
  9. (2x - 5) (2x - 5) Using a suitable identity, find each of the following…
  10. Using a suitable identity, find each of the following products: (a - 1/a) (a -…
  11. (3x + 2) (3x - 2) Using a suitable identity, find each of the following…
  12. (5a - 3b) (5a - 3b) Using a suitable identity, find each of the following…
  13. (2l - 3m) (2l + 3m) Using a suitable identity, find each of the following…
  14. (3/4 - x) (3/4 + x) Using a suitable identity, find each of the following…
  15. (1/x + 1/y) (1/x - 1/y) Using a suitable identity, find each of the following…
  16. (100 + 3) (100 - 3) Using a suitable identity, find each of the following…
  17. (x + 4)(x + 7) Using the identity (x + a)(x + b) = x^2 + (a + b)x + ab, find…
  18. (5x + 3)(5x + 4) Using the identity (x + a)(x + b) = x^2 + (a + b)x + ab, find…
  19. (7x + 3y) (7x - 3y) Using the identity (x + a)(x + b) = x^2 + (a + b)x + ab,…
  20. (8x - 5) (8x - 2) Using the identity (x + a)(x + b) = x^2 + (a + b)x + ab, find…
  21. (2m + 3n) (2m + 4n) Using the identity (x + a)(x + b) = x^2 + (a + b)x + ab,…
  22. (xy - 3) (xy - 2) Using the identity (x + a)(x + b) = x^2 + (a + b)x + ab, find…
  23. (a + 1/x) (a + 1/y) Using the identity (x + a)(x + b) = x^2 + (a + b)x + ab,…
  24. (2 + x) (2 - y) Using the identity (x + a)(x + b) = x^2 + (a + b)x + ab, find…
  25. (p - q)^2 Find out the following squares by using the identities:…
  26. (a - 5)^2 Find out the following squares by using the identities:…
  27. (3x + 5)^2 Find out the following squares by using the identities:…
  28. (5x - 4)^2 Find out the following squares by using the identities:…
  29. (7x + 3y)^2 Find out the following squares by using the identities:…
  30. (10m - 9n)^2 Find out the following squares by using the identities:…
  31. (0.4a - 0.5b)^2 Find out the following squares by using the identities:…
  32. (x - 1/x)^2 Find out the following squares by using the identities:…
  33. (x/2 - y/3)^2 Find out the following squares by using the identities:…
  34. 0.54 × 0.54 - 0.46 × 0.46 Find out the following squares by using the…
  35. 103^2 Evaluate the following by using the identities:
  36. 48^2 Evaluate the following by using the identities:
  37. 54^2 Evaluate the following by using the identities:
  38. 92^2 Evaluate the following by using the identities:
  39. 998^2 Evaluate the following by using the identities:
  40. 53 × 47 Evaluate the following by using the identities:
  41. 96 × 104 Evaluate the following by using the identities:
  42. 28 × 32 Answer: 896 Evaluate the following by using the identities:…
  43. 81 × 79 Evaluate the following by using the identities:
  44. 2.8^2 Evaluate the following by using the identities:
  45. 12.1^2 - 7.9^2 Evaluate the following by using the identities:
  46. 9.7 × 9.8 Evaluate the following by using the identities:
  47. (3x + 7)^2 - 84x = (3x - 7)^2 Show that
  48. (a - b) (a + b) + (b - c) (b + c) + (c - a)(c + a) = 0 Show that
  49. If a + b = 5 and a - b = 4, find a^2 + b^2 and ab.
  50. i. If the values of a + b and ab are 12 and 32 respectively, find the values of…
  51. If (x + a) (x + b) = x^2 - 5x - 300, find the values of a^2 + b^2 .…
  52. Deduce the Algebraic identity for (x + a) (x + b) (x + c) by using the product…
Exercise 1.4
  1. The factors of 3a + 21ab are _______ Choose the correct answer for the…
  2. The factors of x^2 - x - 12 are ______ Choose the correct answer for the…
  3. The factors of 6x^2 - x - 15 are(2x + 3) and ___________ Choose the correct…
  4. The factors of 169l^2 - 441m^2 are ________ Choose the correct answer for the…
  5. The product of (x - 1) (2x - 3) is ____ Choose the correct answer for the…
  6. 3x - 45 Factorize the following expressions :
  7. 7x - 14y Factorize the following expressions :
  8. 5a^2 + 35a Factorize the following expressions :
  9. - 12y + 20y^3 Factorize the following expressions :
  10. 15a^2 b + 35ab Factorize the following expressions :
  11. pq - prq Factorize the following expressions :
  12. 18m^3 - 45mn^2 Factorize the following expressions :
  13. 17 l^2 + 85m^2 Factorize the following expressions :
  14. 6x^3 y - 12x^2 y + 15x^4 Factorize the following expressions :
  15. 2a^5 b^3 - 14a^2 b^2 + 4a^3 b Factorize the following expressions :…
  16. 2ab + 2b + 3a Factorize:
  17. 6xy - 4y + 6 - 9x Factorize:
  18. 2x + 3xy + 2y + 3y^2 Factorize:
  19. 15b^2 - 3bx2 - 5b + x^2 Factorize:
  20. a^2 x^2 + axy + abx + by Factorize:
  21. a^2 x + abx + ac + aby + b^2 y + bc Factorize:
  22. ax^3 - bx^2 + ax - b Factorize:
  23. mx - my - nx + ny Factorize:
  24. 2m^3 + 3m - 2m^2 - 3 Factorize:
  25. a^2 + 11b + 11ab + a Factorize:
  26. a^2 + 14a + 49 Factorize :
  27. x^2 - 12x + 36 Factorize :
  28. 4p^2 - 25q^2 Factorize :
  29. 25x^2 - 20xy + 4y^2 Factorize :
  30. 169m^2 - 625n^2 Factorize :
  31. x^2 + 2/3 x + 1/9 Factorize :
  32. 121a^2 + 154ab + 49b^2 Factorize :
  33. 3x^3 - 75x Factorize :
  34. 36 - 49x^2 Factorize :
  35. 1 - 6x + 9x^2 Factorize :
  36. x^2 + 7x + 12 Factorize :
  37. p^2 - 6p + 8 Factorize :
  38. m^2 - 4m - 21 Factorize :
  39. x^2 - 14x + 45 Factorize :
  40. x^2 - 24x + 108 Factorize :
  41. a^2 + 13a + 12 Factorize :
  42. x^2 - 5x + 6 Factorize :
  43. x^2 - 14xy + 24y^2 Factorize :
  44. m^2 - 21m - 72 Factorize :
  45. x^2 - 28x + 132 Factorize :
Exercise 1.5
  1. 16x^4 ÷ 32x Simplify:
  2. -42y^3 ÷ 7y^2 Simplify:
  3. 30a^3 b^3 c^3 ÷ 45abc Simplify:
  4. (7m^2 - 6m) ÷ m Simplify:
  5. 25x^3 y^2 ÷ 15x^2 y Simplify:
  6. (-72l^4 m^5 n^8) ÷ (-8l^2 m^2 n^3) Simplify:
  7. 5y^3 - 4y^2 + 3y ÷ y Work out the following divisions:
  8. (9x^5 - 15x^4 - 21 x^2) ÷ (3x^2) Work out the following divisions:…
  9. (5x3 - 4x2 +3x) ÷ (2x) Work out the following divisions:
  10. 4x^2 y - 28xy + 4xy^2 ÷ (4xy) Work out the following divisions:
  11. (8x^4 yz - 4xy^3 z +3x^2 yz^4) ÷ (xyz) Work out the following divisions:…
  12. (x^2 +7x + 10) ÷ (x + 2) Simplify the following expressions:
  13. (a^2 + 24a + 144) ÷ (a + 12) Simplify the following expressions:
  14. (m^2 + 5m - 14) ÷ (m + 7) Simplify the following expressions:
  15. (25m^2 - 4n^2)÷ (5m + 2n) Simplify the following expressions:
  16. (4a^2 - 4ab - 15b^2) ÷ (2a - 5b) Simplify the following expressions:…
  17. (a^4 - b^4) ÷ (a - b) Simplify the following expressions:
Exercise 1.6
  1. 3x + 5 = 23 Solve the following equations:
  2. 17 = 10 - y Solve the following equations:
  3. 2y - 7 = 1 Solve the following equations:
  4. 6x = 72 Solve the following equations:
  5. y/11 = - 7 Solve the following equations:
  6. 3(3x - 7) = 5(2x - 3) Solve the following equations:
  7. 4(2x - 3) + 5(3x - 4) = 14 Solve the following equations:
  8. 7/x-5 = 5/x-7 Solve the following equations:
  9. 2x+3/3x+7 = 3/5 Solve the following equations:
  10. m/3 + m/4 = 1/2 Solve the following equations:
  11. Half of a certain number added to its one third gives 15. Find the number.…
  12. Sum of three consecutive numbers is 90. Find the numbers. Frame and solve the…
  13. The breadth of a rectangle is 8 cm less than its length. If the perimeter is 60…
  14. Sum of two numbers is 60. The bigger number is 4 times the smaller one. Find…
  15. The sum of the two numbers is 21 and their difference is 3. Find the numbers.…
  16. Two numbers are in the ratio 5 : 3. If they differ by 18, what are the numbers?…
  17. A number decreased by 5% of it is 3800. What is the number? Frame and solve the…
  18. The denominator of a fraction is 2 more than its numerator. If one is added to…
  19. Mary is 3 times older than Nandhini. After 10 years the sum of their ages will…
  20. Murali gives half of his savings to his wife, two third of the remainder to his…

Exercise 1.1
Question 1.

Choose the correct answer for the following:

The coefficient of x4 in  is
A. – 5

B. – 3

C. 

D. 7


Answer:

The coefficient of a variable is a multiplicative factor or factors.


In the above question coefficient of x4 is its factor i.e.  .


Question 2.

Choose the correct answer for the following:

The coefficient of xy2 in 7x2 - 14x2y + 14xy2 – 5 is
A. 7

B. 14

C. – 14

D. – 5


Answer:

The coefficient of a variable is a multiplicative factor or factors.


In the above question coefficient of xy2 is its factor i.e. + 14.


Question 3.

Choose the correct answer for the following:

The power of the term x3 y2 z2 is _____
A. 3

B. 2

C. 12

D. 7


Answer:

The monomials in the polynomial are called the terms. The highest power of the terms is the degree of the polynomial.


It is a polynomial in variables x,y and z


The power of  is 3, power of is 2 and the power of is 2


So the power of x3 y2 z2 is 3 + 2 + 2 = 7


Question 4.

Choose the correct answer for the following:

The degree of the polynomial  is ______
A. 7

B. 

C. 4

D. -73


Answer:

The monomials in the polynomial are called the terms. The highest power of the terms is the degree of the polynomial.


x2 – 5x4 +  x7 – 73x + 5 is a polynomial in variable x. Here we have 5 monomials x2, – 5x4, + , x7, –73x, and + 5 which are called the terms of the polynomial.


The highest power is 7 so the degree of the polynomial is 7


Question 5.

Choose the correct answer for the following:

The degree of the polynomial x2 – 5x2y3 + 30x3 y4 – 576xy is
A. –576

B. 4

C. 5

D. 7


Answer:

x2 – 5x2y3 + 30x3 y4 – 576xy is a polynomial in variable x and y.


Term 1: x2 variable x, power of x is 2. Hence the power of the term is 2.


Term 2: – 5x2y3 the variables are x and y; the power of x is 2 and the power of y is 3.


Hence the power of the term –5x2y3 is 2 + 3 = 5 [Sum of the exponents of variables x and y ].


Term 3: 30x3y4 the variables are x and y; the power of x is 3 and the power of y is 4. Hence the power of the term 30x3y4 is 3 + 4 = 7 [Sum of the exponents of variables x and y ].


Term 4: – 576xy the variables are x and y; the power of x is 1 and the power of y is 1. Hence the power of the term -576xy is 1 + 1 = 2 [Sum of the exponents of variables x and y ].


So the highest power is 7,hence the degree of the polynomial is 7.


Question 6.

Choose the correct answer for the following:

x2 + y2 – 2z2 + 5x – 7 is a
A. monomial

B. binomial

C. trinomial

D. polynomial


Answer:

Expression that contains only one term is called a monomial.


Expression that contains two terms is called a binomial.


Expression that contains three terms is called a trinomial.


Expression that contains one or more terms with non-zero coefficient is called a polynomial.


Question 7.

Choose the correct answer for the following:

The constant term of 0.4x7 – 75y2 – 0.75 is
A. 0.4

B. 0.75

C. – 0.75

D. – 75


Answer:

.


Question 8.

Identify the terms and their coefficients for the following expressions:

i. 3abc - 5ca

ii. 1 + x + y2

iii. 3x2 y2 - 3xyz + z3

iv. 

v. 


Answer:

i. 3abc - 5ca



The terms are 3ab and -5ca


Coefficient of 3ab = 3


Coefficient of -5ca = -5


ii. 1 + x + y2



There are 3 terms 1,x and y2


Coefficient of 1 = 1


Coefficient of x = 1


Coefficient of y2 = 1


iii. 3x2 y2 - 3xyz + z3



There are 3 terms 3x2 y2, - 3xyz, and + z3


Coefficient of 3x2 y2 = 3


Coefficient of- 3xyz = -3


Coefficient of + z3 = 1


iv. -7 + 2pq –  qr + rp



v. 0.3xy



Thus, we can conclude




Question 9.

Classify the following polynomials as monomials, binomials and trinomials:

3x2, 3x + 2, x2 – 4x + 2, x5 – 7, x2 + 3xy + y2,

s2 + 3st – 2t2, xy + yz + zx, a2b + b2c, 2l + 2m


Answer:

3x2


Expression that contains only one term is called a monomial.


The above question has only one term i.e. 3, so it is monomial.


3x + 2


Expression that contains two terms is called a binomial.


The above question has two terms i.e. 3x and + 2 , so it is binomial.


x2 – 4x + 2


Expression that contains three terms is called a trinomial.


The above question has 3 terms i.e. , -4x and + 2, so it is trinomial.


x5 – 7


Expression that contains two terms is called a binomial.


The above question has two terms i.e.  and -7, so it is a binomial.


x2 + 3xy + y2


Expression that contains three terms is called a trinomial.


The above question has 3 terms i.e., + 3xy and , so it is trinomial.


s2 + 3st – 2t2


Expression that contains three terms is called a trinomial.


The above question has 3 terms i.e., + 3st and -, so it is trinomial.


xy + yz + zx


Expression that contains three terms is called a trinomial.


The above question has 3 terms i.e. xy, + yz and + zx, so it is trinomial.


a2b + b2c


Expression that contains two terms is called a binomial.


The above question has 2 terms i.e.  and +  , so it is a binomial.


2l + 2m


Expression that contains two terms is called a binomial.


The above question has 2 terms i.e. 2l and + 2m, so it is a binomial.



Question 10.

Add the following algebraic expressions:

i. 2x2 + 3x + 5 and 3x2 - 4x -7

ii. x2 - 2 x -3 and x2 + 3x + 1

iii. 2t2 + t -4 and 1 – 3t – 5t2

iv. xy - yz, yz - xz and zx - xy

v. a2 + b2, b2 + c2, c2 + a2 and 2ab + 2bc + 2ca


Answer:

i. 2x2 + 3x + 5 and 3x2 - 4x -7


Column method of addition



5 - x -2


ii.  -2 x -3 and  + 3x + 1


Column method of addition



 + x -2


iii.  + t -4 and 


Row method of addition


( + t -4) + ()


Now combine the like terms


= () + (t-3t) + (-4 + 1)


- 2t - 3


iv. xy - yz, yz - xz and zx - xy


(xy-yz) + (yz-xz) + ( zx-xy)


Now combine the like terms


= (xy-xy) + (-yz + yz) + ( -xz + zx)


= 0 + 0 + 0


= 0


V.  +  ,  +  +  and 2ab + 2bc + 2ca



Observe we have written the term-  of the second polynomial below the corresponding term of the first polynomial. Since the term  in the second polynomial and the term 2ab + 2bc + 2ca in the fourth polynomial do not exist, so their respective places have been left blank to facilitate the process of addition.



Question 11.

Subtract

(i) Subtract 2 a – b from 3a - b

(ii) Subtract -3x + 8y from -7x - 10y

(iii) Subtract 2ab + 5bc - 3ca from 7ab - 2bc + 10ca

(iv) Subtract x5 – 2x2 – 3x from x3 + 3x2 + 1

(v) Subtract 3x2y – 2xy + 2xy2 + 5x – 7y – 10 from 15 – 2x + 5y – 11 xy + 2xy2 + 8x2y


Answer:

i. Subtract 2 a – b from 3a – b



Answer = a


ii. Subtract -3x + 8y from -7x - 10y



Answer = -4x-18y


iii. Subtract 2ab + 5bc - 3ca from 7ab - 2bc + 10ca


Solution:



iv. Subtract x5 - 2x2 - 3x from x3 - 3x2 + 1



Answer = - x5 + x3 - x2 + 3x + 1


v. Subtract 3x2y - 2xy + 2xy2 + 5x -7y-10 from 15-2x + 5y-11xy + 2xy2 + 8x2y


Row method of subtraction


= (15-2x + 5y-11xy + 2xy2 + 8x2y) – (3x2y - 2xy + 2xy2 + 5x -7y-10)


= 15-2x + 5y-11xy + 2xy2 + 8x2y - 3x2y + 2xy - 2xy2 - 5x + 7y + 10


Now combining the like terms


= (2xy2-2xy2) + (8x2y- 3x2y) + (-11xy + 2xy) + (-2x-5x) + (5y + 7y) + (15 + 10)


= 5 x2y + (-9xy) + (-7x) + 12y + 25


= 5 x2y – 9xy – 7x + 12y + 25


Answer = 5 x2y – 9xy – 7x + 12y + 25



Question 12.

Find out the degree of the polynomials and the leading coefficients of the polynomials given below:

(i) 

(ii) 13x3 – x13 – 113

(iii) -77 + 7x2 – x7

(iv) -181 + 0.8y – 8y2 + 115y3 + y8

(v) x7 – 2x3y5 + 3xy4 – 10xy + 10


Answer:

(i) The monomials in the polynomial are called the terms. The highest power of the terms is the degree of the polynomial.


x2 – 2x3 + 5x7 – x3 – 70x – 8 is a polynomial in x. Here we have 6 monomials x2, – 2x3, + 5x7, –x3, –70x and –8 which are called the terms of the polynomial.


The highest power is 7 so the degree of the polynomial is 7.


(ii) 13x3 – x13 – 113 is a polynomial in x. Here we have 3 monomials and the highest power is 13 so the degree of the polynomial is 13.


(iii) -77 + 7x2 – x7 is a polynomial in x. Here we have 3 monomials and the highest power is 7 so the degree of the polynomial is 7.


(iv) -181 + 0.8y – 8y2 + 115y3 + y8 is a polynomial in x. Here we have 5 monomials and the highest power is 8 so the degree of the polynomial is 8.


(v) x7 – 2x3y5 + 3xy4 – 10xy + 10 is a polynomial in x and y, Here we have 5 monomials.


Term 1: x7 variable x, power of x is 7. Hence the power of the term is 7.


Term 2: – 2x3y5 the variables are x and y; the power of x is 3 and the power of y is 5.


Hence the power of the term – 2x3y5 is 3 + 5 = 8 [Sum of the exponents of variables x and y ].


Term 3: 3xy4 the variables are x and y; the power of x is 1 and the power of y is 4.


Hence the power of the term 3xy4 is 1 + 4 = 5 [Sum of the exponents of variables x and y].


Term 4: – 10xy the variables are x and y; the power of x is 1 and the power of y is 1.


Hence the power of the term -10xy is 1 + 1 = 2 [Sum of the exponents of variables x and y].


Term 5: 10 the constant term and it can be written as 10x0y0. The power of the variables x0y0 is zero. Hence the power of the term 10 is 0.


So the highest power is 8, hence the degree of the polynomial is 8.




Exercise 1.2
Question 1.

Find the product of the following pairs of monomials:

i. 3, 7x

ii. – 7x, 3y

iii. – 3a, 5ab

iv. 5a2, – 4a

v. 

vi. Xy2, x2y

vii. x3y5, xy2

viii. abc, abc

ix. xyz. x2yz

x. a2b2c3, abc2


Answer:

i. 3, 7x


= 3×7x


= 21x


ii. – 7x, 3y


= -7 ×3×x ×y


= -21xy


iii. – 3a, 5ab


= -3 ×5 ×a ×a ×b


= -15 b


= -15


= -15 b


iv. 5a2, – 4a


= (5


= -20)


= -20


v. 


 )


 ( )


 ( )



vi. Xy2, x2y


 y2 + 1



vii. x3y5, xy2




viii. abc, abc


= a× b× c × a × b × c


= (a × a) × (b × b) x ( c × c)


 ×  × 



ix. xyz. X2yz


= ( x × x2 ) x (y × y) × (z× z)


= x3


x. a2b2c3, abc2


= (a2 × a) x ( b2 × b)x ( c3 × c)


 ×  × 



Question 2.

Complete the following table of products:



Answer:


For finding B1A2


2X x -3y


= -6xy


B1A4


2X x -5xy


= -10  y


= -10  y


B2A1


-3Y x 2x


= -6xy


B2A6


-3Y x -6x2y2


= 18 x (y(1 + 2)) x x2


= 18 y3 x2


B3A3


4x2 x 4x2


= 16x(2 + 2)


= 16x4


B3A5


4x2 x 7x2y


= (4x7) x ( x2 + 2) x y


= 28 x4y


B4A1


-5xy X 2x


= (-5x2) x ( ) x y


= -10 y


= -10  y


So the final table will be




Question 3.

Find out the product :

(i) 2a, 3a2, 5a4

(ii) 2x, 4y, 9z

(iii) ab, bc, ca

(iv) m, 4m, 3m2, - 6m3

(v) xyz, y2z, yx2

(vi) lm2, mn2, ln2

(vii) -2p, -3q, -5p2


Answer:

i. 2a, 3a2, 5a4


= (2 3 5) (aa2a4)


= (30) (a1 + 2 + 4)


= 30 a7


ii. 2x, 4y, 9z


= (24(x


= 72xyz


iii. ab, bc, ca


= abbc


= axaxbxbxcxc


= a2 b2 c2


iv. m, 4m, 3m2, - 6m3


= mx4mx3 m2x- 6m


= (43 m2 m3)


= -72( m1 + 1 + 2 + 3)


= -72 m7


v. xyz, y2z, yx2


= (x x2


= (x1 + 2 (  (  )


= (x3 (y4 ()


= x3y4


vi. lm2, mn2, ln2


= ( l




vii. -2p, -3q, -5p2


= (-2 p2)


= (-30)  p1 + 2


= -30 p3q



Question 4.

Find the product :

(i) (a3) × (2a5) × (4a15)

(ii) (5 – 2x) (4 + x)

(iii) (x + 3y) (3x – y)

(iv) (3x + 2) (4x – 3)

(v) 


Answer:

(i) (2× 4) (a3 × a5 × a15)


⇒ 8 a3 + 5 + 15


⇒ 8 a23 (∵ an + m = an × am)


ii. 5 × (4 + x) -2x × (4 + x)


 (5× 4 + 5x) – (2x × 4 + 2x × x)


⇒ (20 + 5x) – (8x + 2x1 + 1 ) (∵ an + m = an × am)


⇒ (20 + 5x) – (8x + 2x2 )


⇒ 20 + 5x – 8x - 2x2


⇒ -2x2 -3x + 20


iii. x × (3x -y) + 3y × (3x -y)


 (x × 3x + x × (-y)) + (3y × 3x + 3y × (-y))


⇒ (3x1 + 1 - xy) + ((3× 3)xy – 3y1 + 1 ) (∵ an + m = an × am)


⇒ (3x2 - xy ) + (9xy – 3y2 )


⇒ 3x2 - xy + 9xy - 3y2


⇒ 3x2 + 8xy - 3y2


iv. 3x × (4x-3) + 2 × (4x-3)


⇒ (3x × 4x + 3x × (-3)) + (2× 4x + 2 × (-3))


⇒ ((3× 4)x1 + 1 -9x) + (8x -6) (∵ an + m = an × am)


⇒ 12x2 -9x + 8x -6


⇒ 12x2 -x -6



v. ()x( abX 


 x()




Question 5.

Find the product of the following :

(a + b) (2a2 – 5ab + 3b2)


Answer:

a × (2a2 – 5ab + 3b2) + b × (2a2 – 5ab + 3b2)


( a × 2a2 - a × 5ab + a × 3b2) + (b × 2a2 – b × 5ab + b × 3b2)


( 2a2 + 1 – 5a1 + 1b + 3ab2) + (2a2b – 5ab2 + 3b2 + 1)


(∵ an + m = an × am)


( 2a3 – 5a2b + 3ab2) + (2a2b – 5ab2 + 3b3)


 2a3 – 5a2b + 2a2b + 3ab2– 5ab2 + 3b3


 2a3 + (– 5 + 2)a2b + (3-5)ab2 + 3b3


 2a3 - 3a2b - 2ab2 + 3b3



Question 6.

Find the product of the following :

(2x + 3y) (x2 – xy + y2)


Answer:

2x × (x2 – xy + y2) + 3y × (x2 – xy + y2)


⇒ (2x × x2 – 2x × xy + 2x × y2) + (3y × x2 -3y × xy + 3y× y2)


⇒ (2x2 + 1 – 2x1 + 1y + 2xy2) + (3yx2 -3xy1 + 1 + 3y2 + 1)


(∵ an + m = an × am)


⇒ (2x2 + 1 – 2x1 + 1y + 2xy2) + (3yx2 -3xy1 + 1 + 3y2 + 1)


⇒ (2x3 – 2x2y + 2xy2) + (3yx2 -3xy2 + 3y3)


⇒ 2x3 – 2x2y + 3x2y + 2xy2 -3xy2 + 3y3


⇒ 2x3 (- 2 + 3)x2y + (2-3)xy2 + 3y3


⇒ 2x3 + x2y - xy2 + 3y3



Question 7.

Find the product of the following :

(x + y + z) (x + y – z)


Answer:

x(x + y-z) + y(x + y-z) + z(x + y-z)


⇒ (x× x + xy -xz ) + (yx + y × y -yz) + (zx + zy -z × z)


⇒ (x1 + 1 + xy -xz ) + (yx + y1 + 1 -yz) + (zx + zy -z1 + 1)


(∵ an + m = an × am)


⇒ x2 + xy -xz + xy + y2 -yz + xz + yz -z2


⇒ x2 + y2 -z2 + xy + xy -xz + xz -yz + yz


⇒ x2 + y2 -z2 + (1 + 1) xy + (1-1)xz + (1-1)yz


⇒ x2 + y2 -z2 + 2xy



Question 8.

Find the product of the following :

(a + b) (a2 + 2ab + b2)


Answer:

a(a2 + 2ab + b2) + b(a2 + 2ab + b2)


⇒ (a × a2 + a × 2ab + a × b2) + (b × a2 + b × 2ab + b× b2)


⇒ (a2 + 1 + 2a1 + 1b + ab2) + (ba2 + 2ab1 + 1 + b2 + 1)


(∵ an + m = an × am)


⇒ (a3 + 2a2b + ab2) + (ba2 + 2ab2 + b3)


⇒ a3 + 2a2b + a2b + ab2 + 2ab2 + b3


⇒ a3 + (2 + 1)a2b + (1 + 2)ab2 + b3


⇒ a3 + 3a2b + 3ab2 + b3



Question 9.

Find the product of the following :

(m – n) (m2 + mn + n2)


Answer:

m(m2 + mn + n2) -n(m2 + mn + n2)


⇒ (m× m2 + m× mn + mn2) –( nm2 + n× mn + n× n2)


⇒ (m2 + 1 + m1 + 1n + mn2) –( nm2 + mn1 + 1 + n2 + 1)


(∵ an + m = an × am)


⇒ (m3 + m2n + mn2) –( nm2 + mn2 + n3)


⇒ m3 + m2n + mn2 – nm2 - mn2 - n3


⇒ m3 + m2n– m2n + mn2 - mn2 - n3


⇒ m3 + (1-1)m2n + (1-1)mn2 - n3


= m3 - n3



Question 10.

Add 2x(x – y – z) and 2y(z – y – x)


Answer:

2xXx – 2xy – 2xz and z-y-x




Question 11.

Subtract 3a(a-2b + 3c) from 4a(5a + 2b - 3c)


Answer:

(3a X a) + ( 3ax -2b) + (3ax3c) from (4ax5a) + (4ax2b) + (4ax(-3c))


= 3 – 6ab + 9ac from 20 + 8ab -12ac





Exercise 1.3
Question 1.

Choose the correct answer for the following:

(a + b)2 = (a + b) × ________
A. ab

B. – 2ab

C. (a + b)

D. (a – b)


Answer:

Squaring a term means multiplying it with itself.


(a + b)2 = (a + b) × (a + b)


Question 2.

Choose the correct answer for the following:

(a – b)2 = (a – b) × ________
A. (a + b)

B. – 2ab

C. ab

D. (a – b)


Answer:

Squaring a term means multiplying it with itself.


(a – b)2 = (a – b) × (a – b)


Question 3.

Choose the correct answer for the following:

(a2 – b2) = (a – b) × ________
A. (a – b)

B. (a + b)

C. a2 + 2ab + b2

D. a2 – 2ab + b2


Answer:

(a – b) × (a + b) = a2 + ab – ab – b2


⇒ (a – b) × (a + b) = a2 – b2


So, (a2 – b2) = (a – b) × (a + b)


Question 4.

Choose the correct answer for the following:

9.62 = __________
A. 9216

B. 93.6

C. 9.216

D. 92.16


Answer:

Given 9.62 = (10 – 0.4)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ 9.62 = 102 + 0.42 – 2(10)×(0.4)


⇒ 9.62 = 100 + 0.16 – 8


⇒ 9.62 = 92.16


Question 5.

Choose the correct answer for the following:

(a + b)2 – (a – b)2 = ________
A. 4ab

B. 2ab

C. a2 + 2ab + b2

D. 2(a2 + b2)


Answer:

We know that


(a– b)2 = a2 + b2 – 2a×b


and (a+ b)2 = a2 + b2 + 2a×b


⇒ (a+ b)2 – (a– b)2 = (a2 + b2 + 2a×b) – (a2 + b2 – 2a×b)


⇒ (a+ b)2 – (a– b)2 = 4ab


Question 6.

Choose the correct answer for the following:

m2 + (c + d) m + cd = ______
A. (m + c)2

B. (m + c) (m + d)

C. (m + d)2

D. (m + c) (m – d)


Answer:

Given m2 + (c + d) m + cd = m2 + cm + dm + cd


⇒ m2 + (c + d) m + cd =m(m + c) + d(m + c)


⇒ m2 + (c + d) m + cd = (m + d)(m + c)


Question 7.

Using a suitable identity, find each of the following products:

(x + 3) (x + 3)


Answer:

Given (x + 3) (x + 3) = (x + 3)2


We know that


(a+ b)2 = a2 + b2 + 2a×b


⇒ (x + 3) (x + 3) = (x2 + 32 + 2x×3)


⇒ (x + 3) (x + 3) = x2 + 9 + 6x



Question 8.

Using a suitable identity, find each of the following products:

(2m + 3) (2m + 3)


Answer:

Given (2m + 3) (2m + 3) = (2m + 3)2


We know that


(a+ b)2 = a2 + b2 + 2a×b


⇒ (2m + 3) (2m + 3) = ((2m)2 + 32 + 2× 2m×3)


⇒ (2m + 3) (2m + 3)= 4m2 + 9 + 12m



Question 9.

Using a suitable identity, find each of the following products:

(2x – 5) (2x – 5)


Answer:

Given (2x – 5) (2x – 5) = (2x – 5)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (2x – 5) (2x – 5) = ((2x)2 + 52 – 2× 2x×5)


⇒ (2x – 5) (2x – 5) = 4x2 + 25 – 20x



Question 10.

Using a suitable identity, find each of the following products:


Answer:

Given 


We know that


(a– b)2 = a2 + b2 – 2a×b





Question 11.

Using a suitable identity, find each of the following products:

(3x + 2) (3x – 2)


Answer:

∵ (a2 – b2) = (a – b) × (a + b)


∴ (3x + 2) (3x – 2) = (3x)2 – (22)


⇒ (3x + 2) (3x – 2) = (9x2 – 4)



Question 12.

Using a suitable identity, find each of the following products:

(5a – 3b) (5a – 3b)


Answer:

Given (5a – 3b) (5a – 3b) = (5a – 3b)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (5a – 3b) (5a – 3b) = ((5a)2 + (3b)2 – 2×5a×3b)


⇒ (5a – 3b) (5a – 3b) = 25a2 + 9b2 – 30ab



Question 13.

Using a suitable identity, find each of the following products:

(2l – 3m) (2l + 3m)


Answer:

∵ (a2 – b2) = (a – b) × (a + b)


∴ (2l – 3m) (2l + 3m) = (2l)2 – (3m)2


⇒ (2l – 3m) (2l + 3m) = 4l2 – 9m2



Question 14.

Using a suitable identity, find each of the following products:



Answer:

∵ (a2 – b2) = (a – b) × (a + b)





Question 15.

Using a suitable identity, find each of the following products:



Answer:

∵ (a2 – b2) = (a – b) × (a + b)





Question 16.

Using a suitable identity, find each of the following products:

(100 + 3) (100 – 3)


Answer:

∵ (a2 – b2) = (a – b) × (a + b)


∴ (100 + 3) (100 – 3) = (100)2 – (3)2


⇒ (100 + 3) (100 – 3) = 10000 – 9 = 9991



Question 17.

Using the identity (x + a)(x + b) = x2 + (a + b)x + ab, find out the following products:

(x + 4)(x + 7)


Answer:

Given (x + 4)(x + 7)


Using (x + a)(x + b) = x2 + (a + b)x + ab


(x + 4)(x + 7) = x2 + (4 + 7)x + 4× 7


⇒ (x + 4)(x + 7) = x2 + 11x + 28



Question 18.

Using the identity (x + a)(x + b) = x2 + (a + b)x + ab, find out the following products:

(5x + 3)(5x + 4)


Answer:

Given (5x + 3)(5x + 4)


Using (x + a)(x + b) = x2 + (a + b)x + ab


(5x + 3)(5x + 4)= (5x)2 + (3 + 4)5x + 3×4


⇒ (5x + 3)(5x + 4)= 25x2 + 35x + 12



Question 19.

Using the identity (x + a)(x + b) = x2 + (a + b)x + ab, find out the following products:

(7x + 3y) (7x – 3y)


Answer:

Given (7x + 3y) (7x – 3y)


Using (x + a)(x + b) = x2 + (a + b)x + ab


(7x + 3y) (7x – 3y)= (7x)2 + (3y – 3y)7x + 3y×3y


⇒ (7x + 3y) (7x – 3y) = 49x2 – 9y2



Question 20.

Using the identity (x + a)(x + b) = x2 + (a + b)x + ab, find out the following products:

(8x – 5) (8x – 2)


Answer:

Given (8x – 5) (8x – 2)


Using (x + a)(x + b) = x2 + (a + b)x + ab


(8x – 5) (8x – 2) = (8x)2 + ( – 5 – 2)8x + ( – 5)×( – 2)


⇒ (8x – 5) (8x – 2) = 64x2 – 56x + 10



Question 21.

Using the identity (x + a)(x + b) = x2 + (a + b)x + ab, find out the following products:

(2m + 3n) (2m + 4n)


Answer:

Given (2m + 3n) (2m + 4n)


Using (x + a)(x + b) = x2 + (a + b)x + ab


(2m + 3n) (2m + 4n)= (2m)2 + (3n + 4n)(2m) + 3n×4n


⇒ (2m + 3n) (2m + 4n)= 4m2 + 14mn + 12n2



Question 22.

Using the identity (x + a)(x + b) = x2 + (a + b)x + ab, find out the following products:

(xy – 3) (xy – 2)


Answer:

Given (xy – 3) (xy – 2)


Using (x + a)(x + b) = x2 + (a + b)x + ab


(xy – 3) (xy – 2)= (xy)2 + ( – 3 – 2)xy + ( – 3)×( – 2)


⇒ (xy – 3) (xy – 2)= x2y2 – 5xy + 6



Question 23.

Using the identity (x + a)(x + b) = x2 + (a + b)x + ab, find out the following products:



Answer:

Given 


Using (x + a)(x + b) = x2 + (a + b)x + ab





Question 24.

Using the identity (x + a)(x + b) = x2 + (a + b)x + ab, find out the following products:

(2 + x) (2 – y)


Answer:

Given (2 + x) (2 – y)


Using (x + a)(x + b) = x2 + (a + b)x + ab


(2 + x) (2 – y) = (2)2 + (x – y)2 – x×y


⇒ (2 + x) (2 – y) = 4 + 2(x – y) – xy



Question 25.

Find out the following squares by using the identities:

(p – q)2


Answer:

Given (p – q)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (p – q)2 = ((p)2 + (q)2 – 2×p×q)


⇒ (p – q)2 = p2 + q2 – 2pq



Question 26.

Find out the following squares by using the identities:

(a – 5)2


Answer:

Given (a – 5)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (a – 5)2 = ((a)2 + (5)2 – 2×a×5)


⇒ (a – 5)2 = a2 + 25 – 10a



Question 27.

Find out the following squares by using the identities:

(3x + 5)2


Answer:

Given (3x + 5)2


We know that


(a+ b)2 = a2 + b2 + 2a×b


⇒ (3x + 5)2 = ((3x)2 + (5)2 + 2×3x×5)


⇒ (3x + 5)2 = 9x2 + 25 + 30x



Question 28.

Find out the following squares by using the identities:

(5x – 4)2


Answer:

Given (5x – 4)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (5x – 4)2 = ((5x)2 + (4)2 – 2×5x×4)


⇒ (5x – 4)2 = 25x2 + 16 – 40x



Question 29.

Find out the following squares by using the identities:

(7x + 3y)2


Answer:

Given (7x + 3y)2


We know that


(a+ b)2 = a2 + b2 + 2a×b


⇒ (7x + 3y)2 = ((7x)2 + (3y)2 + 2×7x×3y)


⇒ (7x + 3y)2 = 49x2 + 9y2 + 42xy



Question 30.

Find out the following squares by using the identities:

(10m – 9n)2


Answer:

Given (10m – 9n)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (10m – 9n)2 = ((10m)2 + (9n)2 – 2×10m×9n)


⇒ (10m – 9n)2 = 100m2 + 81n2 – 180mn



Question 31.

Find out the following squares by using the identities:

(0.4a – 0.5b)2


Answer:

Given (0.4a – 0.5b)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (0.4a – 0.5b)2 = ((0.4a)2 + (0.5b)2 – 2×0.4a×0.5b)


⇒ (0.4a – 0.5b)2 = 0.16a2 + 0.25b2 – 0.4ab



Question 32.

Find out the following squares by using the identities:



Answer:

Given 


We know that


(a– b)2 = a2 + b2 – 2a×b





Question 33.

Find out the following squares by using the identities:



Answer:

Given 


We know that


(a– b)2 = a2 + b2 – 2a×b





Question 34.

Find out the following squares by using the identities:

0.54 × 0.54 – 0.46 × 0.46


Answer:

∵ (a2 – b2) = (a – b) × (a + b)


∴ (0.54 × 0.54 – 0.46 × 0.46) = (0.54)2 – (0.46)2


⇒ 0.54 × 0.54 – 0.46 × 0.46 = (0.54 – 0.46) (0.56 + 0.46)


⇒ 0.54 × 0.54 – 0.46 × 0.46 = 0.08×1.02


⇒ 0.54 × 0.54 – 0.46 × 0.46 = 0.0816



Question 35.

Evaluate the following by using the identities:

1032


Answer:

∵ 1032 = (100 + 3)2


We know that


(a+ b)2 = a2 + b2 + 2a×b


⇒ (100 + 3)2 = ((100)2 + (3)2 + 2×100×3)


⇒ (100 + 3)2 = 10000 + 9 + 600


⇒ 1032 = 10609



Question 36.

Evaluate the following by using the identities:

482


Answer:

∵ 482 = (50 – 2)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (50 – 2)2 = ((50)2 + (2)2 – 2×50×2)


⇒ (50 – 2)2 = 2500 + 4 – 200


⇒ 482 = 2304



Question 37.

Evaluate the following by using the identities:

542


Answer:

∵ 542 = (50 + 4)2


We know that


(a+ b)2 = a2 + b2 + 2a×b


⇒ (50 + 4)2 = ((50)2 + (4)2 + 2×50×4)


⇒ (50 + 4)2 = 2500 + 16 + 400


⇒ 542 = 2916



Question 38.

Evaluate the following by using the identities:

922


Answer:

∵ 922 = (100 – 8)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (100 – 8)2= ((100)2 + (8)2 – 2×100×8)


⇒ (100 – 8)2 = 10000 + 64 – 1600


⇒ 922 = 8464



Question 39.

Evaluate the following by using the identities:

9982


Answer:

∵ 9982 = (1000 – 2)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (1000 – 2)2 = ((1000)2 + (2)2 – 2×1000×2)


⇒ (1000 – 2)2 = 1000000 + 4 – 4000


⇒ 9982 = 996004



Question 40.

Evaluate the following by using the identities:

53 × 47


Answer:

∵ 53 × 47 = (50 + 3) (50 – 3)


We know that


∵ (a2 – b2) = (a – b) × (a + b)


∴ (50 + 3) (50 – 3) = (50)2 – (3)2


⇒ (50 + 3) (50 – 3) = 2500 – 9 = 2491



Question 41.

Evaluate the following by using the identities:

96 × 104


Answer:

∵ 96 × 104= (100 – 4) (100 + 4)


We know that


∵ (a2 – b2) = (a – b) × (a + b)


∴ (100 – 4) (100 + 4) = (100)2 – (4)2


⇒ (100 – 4) (100 + 4)= 10000 – 16 = 9984



Question 42.

Evaluate the following by using the identities:

28 × 32

Answer: 896


Answer:

∵ 28 × 32= (30 – 2) (30 + 2)


We know that


∵ (a2 – b2) = (a – b) × (a + b)


∴ (30 – 2) (30 + 2) = (30)2 – (2)2


⇒ (30 – 2) (30 + 2) = 900 – 4 = 896



Question 43.

Evaluate the following by using the identities:

81 × 79


Answer:

∵ 81 × 79= (80 + 1) (80 – 1)


We know that


∵ (a2 – b2) = (a – b) × (a + b)


∴ (80 + 1) (80 – 1) = (80)2 – (1)2


⇒ (80 + 1) (80 – 1) = 6400 – 1 = 6399



Question 44.

Evaluate the following by using the identities:

2.82


Answer:

∵ 2.82 = (3 – 0.2)2


We know that


(a– b)2 = a2 + b2 – 2a×b


⇒ (3 – 0.2)2= ((3)2 + (0.2)2 – 2×3×0.2)


⇒ (3 – 0.2)2 = 9 + 0.04 – 1.2


⇒ 2.82 = 7.84



Question 45.

Evaluate the following by using the identities:

12.12 – 7.92


Answer:

We know that


∵ (a2 – b2) = (a – b) × (a + b)


∴ 12.12 – 7.92 = (12.1 + 7.9) (12.1 – 7.9)


⇒ 12.12 – 7.92 = 20 × 4.2 = 84



Question 46.

Evaluate the following by using the identities:

9.7 × 9.8


Answer:

Given 9.7 × 9.8 = (9 + 0.7)(9 + 0.8)


Using (x + a)(x + b) = x2 + (a + b)x + ab


(9 + 0.7)(9 + 0.8) = (9)2 + (0.7 + 0.8)9 + (0.7)×(0.8)


⇒ 9.7 × 9.8 = 81 + 13.5 + 0.56


⇒ 9.7 × 9.8 =95.06



Question 47.

Show that

(3x + 7)2 – 84x = (3x – 7)2


Answer:

Solving L.H.S. first,


(3x + 7)2 – 84x


We know that


(a+ b)2 = a2 + b2 + 2a×b


⇒ (3x + 7)2 – 84x = ((3x)2 + (7)2 + 2×3x×7) – 84x


⇒ (3x + 7)2 – 84x = 9x2 + 49 + 42x – 84x


⇒ (3x + 7)2 – 84x = 9x2 + 49 – 42x


⇒ (3x + 7)2 – 84x = ((3x)2 + (7)2 – 2×3x×7)


Using (a– b)2 = a2 + b2 – 2a×b


⇒ (3x + 7)2 – 84x = (3x – 7)2


∵ L.H.S. = R.H.S.


Hence, proved.



Question 48.

Show that
(a – b) (a + b) + (b – c) (b + c) + (c – a)(c + a) = 0


Answer:

To Prove: (a – b) (a + b) + (b – c) (b + c) + (c – a)(c + a) = 0
Proof:
Solving L.H.S. first,


(a – b)(a + b) + (b – c)(b + c) + (c – a)(c + a)


We know that


∵ (a2 – b2) = (a – b) × (a + b)


⇒ (a – b)(a + b) + (b – c)(b + c) + (c – a)(c + a) = (a2 – b2) + (b2 – c2) + (c2 – a2)


⇒ (a – b)(a + b) + (b – c)(b + c) + (c – a)(c + a) = 0


∵ L.H.S. = R.H.S.


Hence, proved.


Question 49.

If a + b = 5 and a – b = 4, find a2 + b2 and ab.


Answer:

Given: a + b = 5 and a + b = 5


Using (a– b)2 = a2 + b2 – 2a×b and (a+ b)2 = a2 + b2 + 2a×b






Similarly,






Question 50.

i. If the values of a + b and ab are 12 and 32 respectively, find the values of a2 + b2 and (a – b)2.

ii. If the values of (a – b) and ab are 6 and 40 respectively, find the values of a2 + b2 and (a + b)2.


Answer:

(i) Given (a + b) = 12 and ab = 32


a2 + b2 = (a+ b)2 – 2ab


⇒ a2 + b2 = (12)2 – 2(32)


⇒ a2 + b2 = 144 – 64


⇒ a2 + b2 = 80


(a– b)2 = (a+ b)2 – 4ab


⇒ (a– b)2 = (12)2 – 4(32)


⇒ (a– b)2 = 144 – 128


⇒ (a– b)2 = 16


(ii) Given (a – b) = 6 and ab = 40


a2 + b2 = (a– b)2 + 2ab


⇒ a2 + b2 = (6)2 + 2(40)


⇒ a2 + b2 = 36 + 80


⇒ a2 + b2 = 116


(a+ b)2 = (a– b)2 + 4ab


⇒ (a+ b)2 = (6)2 + 4(40)


⇒ (a+ b)2 = 36 + 160


⇒ (a+ b)2 = 196



Question 51.

If (x + a) (x + b) = x2 – 5x – 300, find the values of a2 + b2.


Answer:

Given: (x + a) (x + b) = x2 – 5x – 300


Using (x + a)(x + b) = x2 + (a + b)x + ab


⇒ x2 + (a + b)x + ab = x2 – 5x – 300


⇒ (a + b) = – 5 and ab = – 300


Also, a2 + b2 = (a+ b)2 – 2ab


⇒ a2 + b2 = ( – 5)2 – 2( – 300)


⇒ a2 + b2 = 25 + 600


⇒ a2 + b2 = 625



Question 52.

Deduce the Algebraic identity for (x + a) (x + b) (x + c) by using the product formula. [Hint: (x + a) (x + b)(x + c) = (x + a) [(x + b)(x + c)]]


Answer:

Given (x + a)(x + b)(x + c) = (x + a)[(x + b)(x + c)]


⇒ (x + a)(x + b)(x + c) = (x + a)[x (x + c) + b(x + c)]


⇒ (x + a)(x + b)(x + c) = (x + a)[x2 + cx + bx + bc]


⇒ (x + a)(x + b)(x + c) = (x + a)[x2 + x(c + b) + bc]


⇒ (x + a)(x + b)(x + c) = x[x2 + x(c + b) + bc] + a[x2 + x(c + b) + bc]


⇒ (x + a)(x + b)(x + c) = x3 + x2(c + b) + xbc + ax2 + xa(c + b) + abc


⇒ (x + a)(x + b)(x + c) = x3 + x2(a + b + c) + x(ab + bc + ca) + abc




Exercise 1.4
Question 1.

Choose the correct answer for the following:

The factors of 3a + 21ab are _______
A. ab, (3 + 21)

B. 3,(a + 7b)

C. 3a, (1 + 7b)

D. 3ab, (a + b)


Answer:

Given 3a + 21ab can be written as 3× a× (1 + 7b)


Question 2.

Choose the correct answer for the following:

The factors of x2 – x – 12 are ______
A. (x + 4), (x – 3)

B. (x – 4), (x – 3)

C. (x + 2), (x – 6)

D. (x + 3), (x – 4)


Answer:

Given x2 – x – 12


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ x2 – x – 12 = x2 + ( – 4 + 3)x + ( – 4)×3


On comparing with (I),


a = – 4 and b = 3


So, x2 – x – 12 = (x – 4)(x + 3)


The factors of x2 – x – 12 are (x – 4) and (x + 3)


Question 3.

Choose the correct answer for the following:

The factors of 6x2 – x – 15 are(2x + 3) and ___________
A. (3x – 5)

B. (3x + 5)

C. (5x – 3)

D. (2x – 3)


Answer:

Given 6x2 – x – 15


⇒ 6x2 – x – 15 = 6x2 – (10 – 9)x – 15


⇒ 6x2 – x – 15 = 6x2 – 10x + 9x – 15


⇒ 6x2 – x – 15 = 2x(3x – 5) + 3(3x – 5)


⇒ 6x2 – x – 15 = (3x – 5)(2x + 3)


The factors of 6x2 – x – 15 are(2x + 3) and (3x – 5)


Question 4.

Choose the correct answer for the following:

The factors of 169l2 – 441m2 are ________
A. (13l – 21 m), (13l – 21m)

B. (13l + 21 m), (13l + 21m)

C. (13l – 21 m), (13l + 21m)

D. 13(l + 21 m), 13(l – 21m)


Answer:

Given 169l2 – 441m2


∵ (a2 – b2) = (a – b) × (a + b)


⇒ 169l2 – 441m2 = (13l)2 – (21m)2


⇒ 169l2 – 441m2 = (13l – 21m)(13l + 21m)


The factors of 169l2 – 441m2 are (13l – 21m) and (13l + 21m).


Question 5.

Choose the correct answer for the following:

The product of (x – 1) (2x – 3) is ____
A. 2x2 – 5x – 3

B. 2x2 – 5x + 3

C. 2x2 + 5x – 3

D. 2x2 + 5x + 3


Answer:

Given (x – 1) (2x – 3)


⇒ (x – 1) (2x – 3) = x(2x – 3) + ( – 1)(2x – 3)


⇒ (x – 1) (2x – 3) = 2x2 – 3x – 2x + 3


⇒ (x – 1) (2x – 3) = 2x2 – 5x + 3


Question 6.

Factorize the following expressions :

3x – 45


Answer:

Given 3x – 45 = (3x – 3×15)


Taking 3 common,


⇒ 3x – 45 = 3(x – 15)



Question 7.

Factorize the following expressions :

7x – 14y


Answer:

Given 7x – 14y = (7x – 7×2)


Taking 7 common,


⇒ 7x – 14y = 7(x – 2)



Question 8.

Factorize the following expressions :

5a2 + 35a


Answer:

Given 5a2 + 35a = (5a2 + 5a×7)


Taking 5a common,


⇒ 5a2 + 35a = 5a(a + 7)



Question 9.

Factorize the following expressions :

– 12y + 20y3


Answer:

Given – 12y + 20y3= (4y×( – 3) + 4y×5y2)


Taking 4y common,


⇒ – 12y + 20y3= 4y( – 3 + 5y2)



Question 10.

Factorize the following expressions :

15a2b + 35ab


Answer:

Given 15a2b + 35ab = (5ab×3a + 5ab×7)


Taking 5ab common,


⇒ 15a2b + 35ab = 5ab(3a + 7)



Question 11.

Factorize the following expressions :

pq – prq


Answer:

Given (pq – prq) = (pq×1 – pq×r)


Taking pq common,


(pq – prq) = pq(1 – r)



Question 12.

Factorize the following expressions :

18m3 – 45mn2


Answer:

Given 18m3 – 45mn2= (9m×2m2 – 9m×5n2)


Taking 9m common,


18m3 – 45mn2= 9m×(2m2 – 5n2)



Question 13.

Factorize the following expressions :

17 l2 + 85m2


Answer:

Given 17 l2 + 85m2= (17×l2 + 17×5m2)


Taking 17 common,


17 l2 + 85m2= 17× (l2 + 5m2)



Question 14.

Factorize the following expressions :

6x3y – 12x2y + 15x4


Answer:

Given 6x3y – 12x2y + 15x4= (3x2×2xy – 3x2×4y + 3x2×5x2)


Taking 3x2 common,


6x3y – 12x2y + 15x4 = 3x2 (2xy – 4y + 5x2)



Question 15.

Factorize the following expressions :

2a5b3 – 14a2b2 + 4a3b


Answer:

Given 2a5b3 – 14a2b2 + 4a3b = (2a2b×a3b2 – 2a2b×7b + 2a2b×2a)


Taking 2a2b common,


2a5b3 – 14a2b2 + 4a3b= 2a2b (a3b2 – 7b + 2a)



Question 16.

Factorize:

2ab + 2b + 3a


Answer:

Given 2ab + 2b + 3a = (a×2b + 2b + 3a)


Taking 2b common from 1st and 2nd term,


2ab + 2b + 3a = 2b (a + 1) + 3a



Question 17.

Factorize:

6xy – 4y + 6 – 9x


Answer:

Given 6xy – 4y + 6 – 9x = (2y×3x – 2y×2 + 3×2 – 3×3x)


Taking 2y common from 1st and 2nd term and – 3 from 3rd and 4th,


6xy – 4y + 6 – 9x = 2y(3x – 2) + ( – 3)(3x – 2)


⇒ 6xy – 4y + 6 – 9x = (3x – 2) (2y – 3)



Question 18.

Factorize:

2x + 3xy + 2y + 3y2


Answer:

Given 2x + 3xy + 2y + 3y2 = (x×2 + x×3y + y×2 + y×3y)


Taking x common from 1st and 2nd term and y from 3rd and 4th,


2x + 3xy + 2y + 3y2 = x(2 + 3y) + (y)(2 + 3y)


⇒ 2x + 3xy + 2y + 3y2= (x + y) (2 + 3y)



Question 19.

Factorize:

15b2 – 3bx2 – 5b + x2


Answer:

Given 15b2 – 3bx2 – 5b + x2 = (3b×5b – 3b×x2 + ( – 1)× 5b + x2 )


Taking 3b common from 1st and 2nd term and ( – 1) from 3rd and 4th,


15b2 – 3bx2 – 5b + x2= 3b(5b – x2) + ( – 1)(5b – x2)


⇒ 15b2 – 3bx2 – 5b + x2= (5b – x2) (3b – 1)



Question 20.

Factorize:

a2x2 + axy + abx + by


Answer:

Given a2x2 + axy + abx + by = (ax × ax + ax × y + b × ax + b × y )


Taking ax common from 1st and 2nd term and b from 3rd and 4th,


a2x2 + axy + abx + by = ax (ax + y) + b (ax + y)


⇒ a2x2 + axy + abx + by = (ax + y) (ax + b)



Question 21.

Factorize:

a2x + abx + ac + aby + b2y + bc


Answer:

Given a2x + abx + ac + aby + b2y + bc = (ax × a + ax × b + by × a + by × b + c × a + c × b)


Taking ax common from 1st and 2nd term and by from 3rd and 4th and c from 5th and 6th,


a2x + abx + ac + aby + b2y + bc = ax (a + b) + by (a + b) + c (a + b)


⇒ a2x + abx + ac + aby + b2y + bc = (ax + by + c) (a + b)



Question 22.

Factorize:

ax3 – bx2 + ax – b


Answer:

Given ax3 – bx2 + ax – b = (x2 × ax – x2 × b + ax – b)


Taking x2 common from 1st and 2nd term and 1 from 3rd and 4th,


ax3 – bx2 + ax – b = (x2 × ax – x2 × b + ax – b)


⇒ ax3 – bx2 + ax – b = x2 (ax – b) + (ax – b)


⇒ ax3 – bx2 + ax – b = (x2 + 1)(ax – b)



Question 23.

Factorize:

mx – my – nx + ny


Answer:

Given mx – my – nx + ny =(m × x – m × y + ( – n) × x + n × y)


Taking m common from 1st and 2nd term and ( – n) from 3rd and 4th,


⇒ mx – my – nx + ny = m (x – y) + ( – n)(x – y)


⇒ mx – my – nx + ny = (m – n)(x – y)



Question 24.

Factorize:

2m3 + 3m – 2m2 – 3


Answer:

Given 2m3 + 3m – 2m2 – 3=(m × 2m2 + m × 3 + ( – 1) × 2m2 + ( – 1) × 3)


Taking m common from 1st and 2nd term and ( – 1) from 3rd and 4th,


⇒ 2m3 + 3m – 2m2 – 3= m (2m2 + 3) + ( – 1)(2m2 + 3)


⇒ 2m3 + 3m – 2m2 – 3= (2m2 + 3)(m – 1)



Question 25.

Factorize:

a2 + 11b + 11ab + a


Answer:

Given a2 + 11b + 11ab + a =(a × a + a × 1 + (11b) × 1 + (11b) × a)


Taking a common from 1st and 2nd term and (11b) from 3rd and 4th,


⇒ a2 + 11b + 11ab + a = a (a + 1) + (11b)(1 + a)


⇒ a2 + 11b + 11ab + a = (a + 1)(a + 11b)



Question 26.

Factorize :

a2 + 14a + 49


Answer:

Given a2 + 14a + 49 = a2 + 2× 7× a + (7)2


Comparing with (a+ b)2 = a2 + b2 + 2a×b


a = a and b = 7


So, a2 + 14a + 49 = (a + 7)(a + 7)



Question 27.

Factorize :

x2 – 12x + 36


Answer:

Given x2 – 12x + 36= x2 – 2× 6× x + (6)2


Comparing with (a– b)2 = a2 + b2 – 2a×b


a = x and b = 6


So, x2 – 12x + 36= (x – 6)(x – 6)



Question 28.

Factorize :

4p2 – 25q2


Answer:

Given 4p2 – 25q2= (2p)2 – (5q)2


Comparing with a2 – b2 = (a + b) (a – b)


a = 2p and b = 5q


So, 4p2 – 25q2 = (2p + 5q)(2p – 5q)



Question 29.

Factorize :

25x2 – 20xy + 4y2


Answer:

Given 25x2 – 20xy + 4y2 = (5x)2 – 2× 5x ×2y x + (2y)2


Comparing with (a– b)2 = a2 + b2 – 2a×b


a = 5x and b = 2y


So, 25x2 – 20xy + 4y2 = (5x – 2y)(5x – 2y)



Question 30.

Factorize :

169m2 – 625n2


Answer:

Given 169m2 – 625n2 = (13m)2 – (25n)2


Comparing with a2 – b2 = (a + b) (a – b)


a = 13m and b = 25n


So, 169m2 – 625n2 = (13m + 25n)(13m – 25m)



Question 31.

Factorize :



Answer:

Given 


Comparing with (a+ b)2 = a2 + b2 + 2a×b


a =x and 


So, 



Question 32.

Factorize :

121a2 + 154ab + 49b2


Answer:

Given 121a2 + 154ab + 49b2 = (11a)2 + 2× (11a)× (7b) + (7b)2


Comparing with (a+ b)2 = a2 + b2 + 2a×b


a = 11a and b = 7b


So, 121a2 + 154ab + 49b2 = (11a + 7b)(11a + 7b)



Question 33.

Factorize :

3x3 – 75x


Answer:

Given 3x3 – 75x = 3x × x2 – 3x × (5)2


Taking 3x common,


3x3 – 75x = 3x (x2 – (5)2)


Comparing with a2 – b2 = (a + b) (a – b)


a = x and b = 5


So, 3x3 – 75x = 3x(x + 5)(x – 5)



Question 34.

Factorize :

36 – 49x2


Answer:

Given 36 – 49x2= (6)2 – (7x)2


Comparing with a2 – b2 = (a + b) (a – b)


a = 6 and b = 7x


So, 36 – 49x2= (6 + 7x)(6 – 7x)



Question 35.

Factorize :

1 – 6x + 9x2


Answer:

Given 1 – 6x + 9x2= (1)2 – 2× 1 ×3x + (3x)2


Comparing with (a– b)2 = a2 + b2 – 2a×b


a = 1 and b = 3x


So, 1 – 6x + 9x2 = (1 – 3x)(1 – 3x)



Question 36.

Factorize :

x2 + 7x + 12


Answer:

Given x2 + 7x + 12


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ x2 + 7x + 12= x2 + (4 + 3)x + (4)×3


On comparing with (I),


a = 4 and b = 3


So, x2 + 7x + 12= (x + 4)(x + 3)



Question 37.

Factorize :

p2 – 6p + 8


Answer:

Given p2 – 6p + 8


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ p2 – 6p + 8= p2 + ( – 4 – 2)p + ( – 4)×( – 2)


On comparing with (I),


a = – 4 and b = – 2


So, p2 – 6p + 8= (p – 4)(p – 2)



Question 38.

Factorize :

m2 – 4m – 21


Answer:

Given m2 – 4m – 21


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ m2 – 4m – 21= m2 + ( – 7 + 3)m + ( – 7)×(3)


On comparing with (I),


a = – 7 and b = 3


So, m2 – 4m – 21= (m – 7)(m + 3)



Question 39.

Factorize :

x2 – 14x + 45


Answer:

Given x2 – 14x + 45


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ x2 – 14x + 45= x2 + ( – 9 – 5)x + ( – 9)×( – 5)


On comparing with (I),


a = – 9 and b = – 5


So, x2 – 14x + 45= (x – 9)(x – 5)



Question 40.

Factorize :

x2 – 24x + 108


Answer:

Given x2 – 24x + 108


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ x2 – 24x + 108= x2 + ( – 18 – 6)x + ( – 18)×( – 6)


On comparing with (I),


a = – 18 and b = – 6


So, x2 – 24x + 108= (x – 18)(x – 6)



Question 41.

Factorize :

a2 + 13a + 12


Answer:

Given a2 + 13a + 12


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ a2 + 13a + 12= a2 + (12 + 1)a + (12)×(1)


On comparing with (I),


a = 12 and b = 1


So, a2 + 13a + 12= (a + 12)(a + 1)



Question 42.

Factorize :

x2 – 5x + 6


Answer:

Given x2 – 5x + 6


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ x2 – 5x + 6= x2 + ( – 2 – 3)x + ( – 2)×( – 3)


On comparing with (I),


a = – 2 and b = – 3


So, x2 – 5x + 6= (x – 2)(x – 3)



Question 43.

Factorize :

x2 – 14xy + 24y2


Answer:

Given x2 – 14xy + 24y2


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ x2 – 14xy + 24y2 = x2 + ( – 12y – 2y)x + ( – 12y)×( – 2y)


On comparing with (I),


a = – 12 and b = – 2


So, x2 – 14xy + 24y2 = (x – 12y)(x – 2y)



Question 44.

Factorize :

m2 – 21m – 72


Answer:

Given m2 – 21m – 72


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ m2 – 21m – 72 = m2 + ( – 24 + 3)m + ( – 24)×(3)


On comparing with (I),


a = – 24 and b = 3


So, m2 – 21m – 72 = (m – 24)(m + 3)



Question 45.

Factorize :

x2 – 28x + 132


Answer:

Given x2 – 28x + 132


Using (x + a)(x + b) = x2 + (a + b)x + ab … (I)


⇒ x2 – 28x + 132= x2 + ( – 22 – 6)x + ( – 22)×( – 6)


On comparing with (I),


a = – 22 and b = – 6


So, x2 – 28x + 132 = (x – 22)(x – 6)




Exercise 1.5
Question 1.

Simplify:

16x4 ÷ 32x


Answer:






Question 2.

Simplify:

-42y3 ÷ 7y2


Answer:




= -6y



Question 3.

Simplify:

30a3b3c3 ÷ 45abc


Answer:






Question 4.

Simplify:

(7m2 - 6m) ÷ m


Answer:





= 7m – 6



Question 5.

Simplify:

25x3y2 ÷ 15x2y


Answer:






Question 6.

Simplify:

(-72l4 m5 n8) ÷ (-8l2 m2 n3)


Answer:




= 9l2m3n5



Question 7.

Work out the following divisions:

5y3 – 4y2 + 3y ÷ y


Answer:





= 5y2 – 4y + 3



Question 8.

Work out the following divisions:

(9x5 – 15x4 – 21 x2) ÷ (3x2)


Answer:





= 3x3 – 5x2 – 7



Question 9.

Work out the following divisions:

(5x3 – 4x2 +3x) ÷ (2x)


Answer:







Question 10.

Work out the following divisions:

4x2y – 28xy + 4xy2 ÷ (4xy)


Answer:





= x – 7 + y



Question 11.

Work out the following divisions:

(8x4yz – 4xy3z +3x2yz4) ÷ (xyz)


Answer:





= 8x3 – 4y2 + 3xz3



Question 12.

Simplify the following expressions:

(x2 +7x + 10) ÷ (x + 2)


Answer:

Factorize the numerator,


x2 +7x + 10 = x2 + 5x + 2x + 10


= x(x + 5) + 2(x + 5)


= (x + 2)(x + 5)


Now,




= x + 5



Question 13.

Simplify the following expressions:

(a2 + 24a + 144) ÷ (a + 12)


Answer:

Factorize the numerator,


a2 + 24a + 144 = a2 + 12a + 12a + 144


= a(a + 12) + 12(a + 12)


= (a + 12)(a + 12)


Now,




= a + 12



Question 14.

Simplify the following expressions:

(m2 + 5m – 14) ÷ (m + 7)


Answer:

Factorize the numerator,


m2 + 5m – 14 = m2 + 7m – 2m – 14


= m(m + 7) – 2(m + 7)


= (m – 2)( m + 7)


Now,




= m – 2



Question 15.

Simplify the following expressions:

(25m2 – 4n2)÷ (5m + 2n)


Answer:

Factorize the numerator,


25m2 – 4n2 = (5m)2 – (2n)2


= (5m + 2n)(5m – 2n) [∵ a2 – b2 = (a + b)(a – b)]


Now,




= 5m – 2n



Question 16.

Simplify the following expressions:

(4a2 – 4ab – 15b2) ÷ (2a – 5b)


Answer:

Factorize the numerator,


4a2 – 4ab – 15b2 = 4a2 + 6ab – 10ab – 15b2


= 2a(2a + 3b) – 5b(2a + 3b)


= (2a + 3b)(2a – 5b)


Now,




= (2a + 3b)



Question 17.

Simplify the following expressions:

(a4 – b4) ÷ (a – b)


Answer:

Factorize the numerator,


a4 – b4 = (a2)2 – (b2)2


= (a2 + b2)(a2 – b2) [∵ a2 – b2 = (a + b)(a – b)]


= (a2 + b2)(a + b)(a – b) [∵ a2 – b2 = (a + b)(a – b)]


Now,




= (a2 + b2)(a + b)




Exercise 1.6
Question 1.

Solve the following equations:

3x + 5 = 23


Answer:

Subtracting 5 from both sides,


⇒ 3x + 5 – 5 = 23 – 5


⇒ 3x = 18


Dividing both sides by 3,



⇒ x = 6



Question 2.

Solve the following equations:

17 = 10 - y


Answer:

Subtracting 10 from both sides,


⇒ 17 - 10 = 10 – y – 10


⇒ 7 = -y


Dividing both sides by -1,



⇒ y = -7



Question 3.

Solve the following equations:

2y - 7 = 1


Answer:

Adding 7 to both sides,


⇒ 2y – 7 + 7 = 1 + 7


⇒ 2y = 8


Dividing both sides by 2,



⇒ y = 4



Question 4.

Solve the following equations:

6x = 72


Answer:

Dividing both sides by 6,



⇒ x = 12



Question 5.

Solve the following equations:



Answer:

Multiplying both sides by 11,



⇒ y = -77



Question 6.

Solve the following equations:

3(3x - 7) = 5(2x - 3)


Answer:

9x – 21 = 10x – 15


⇒ 10x – 15 – 9x + 21 = 0


⇒ x + 6 = 0


⇒ x = -6



Question 7.

Solve the following equations:

4(2x - 3) + 5(3x - 4) = 14


Answer:

⇒ 8x – 12 + 15x – 20 = 14


⇒ 8x + 15x = 14 + 12 + 20


⇒ 23x = 46


Dividing both sides by 23,



⇒ x = 2



Question 8.

Solve the following equations:



Answer:

⇒ 7(x – 7) = 5(x – 5)


⇒ 7x – 49 = 5x – 25


⇒ 7x – 5x = 49 – 25


⇒ 2x = 24


Dividing both sides by 2,



⇒ x = 12



Question 9.

Solve the following equations:



Answer:

⇒ 5(2x + 3) = 3(3x + 7)


⇒ 10x + 15 = 9x + 21


⇒ 10x – 9x = 21 – 15


⇒ x = 6



Question 10.

Solve the following equations:



Answer:



⇒ 7m × 2 = 12


⇒ 14m = 12


Dividing both sides by 2,





Question 11.

Frame and solve the equations for the following statements:

Half of a certain number added to its one third gives 15. Find the number.


Answer:

Let the number be x


Then according to question,







⇒ x = 18


Hence, the number is 18.



Question 12.

Frame and solve the equations for the following statements:

Sum of three consecutive numbers is 90. Find the numbers.


Answer:

Let the numbers be x, x+1 and x+2


Then according to the question,


x + (x + 1) + (x + 2) = 90


⇒ x + x + 1 + x + 2 = 90


⇒ 3x + 3 = 90


⇒ 3x = 90 – 3


⇒ 3x = 87



⇒ x = 29


⇒ x + 1 = 29 + 1 = 30


⇒ x + 2 = 29 + 2 = 31


Hence, the numbers are 29, 30 and 31.



Question 13.

Frame and solve the equations for the following statements:

The breadth of a rectangle is 8 cm less than its length. If the perimeter is 60 cm, find its length and breadth.


Answer:

Let breadth of rectangle = x


Then length of rectangle = x + 8


Perimeter = 60 cm


We know that,


Perimeter of rectangle = 2 (length of rectangle + breadth of rectangle)


⇒ 60 = 2(x + (x + 8))


⇒ 60 = 2(x + x + 8)


⇒ 60 = 2(2x + 8)


⇒ 60 = 4x + 16


⇒ 4x = 60 – 16


⇒ 4x = 44


⇒ x = 11


⇒ x + 8 = 11 + 8 = 19


Hence, breadth of rectangle = 11 cm


length of rectangle = 19 cm



Question 14.

Frame and solve the equations for the following statements:

Sum of two numbers is 60. The bigger number is 4 times the smaller one. Find the numbers.


Answer:

Let the smaller number be x


Then bigger number = 4x


Then according to the question,


x + 4x = 60


⇒ 5x = 60



⇒ x = 12


⇒ 4x = 4 × 12 = 48


Hence, the numbers are 12 and 48.



Question 15.

Frame and solve the equations for the following statements:

The sum of the two numbers is 21 and their difference is 3. Find the numbers. (Hint: Let the bigger number be x and smaller number be x – 3)


Answer:

Let the bigger number be x


Then the smaller number = x – 3 [∵ the difference is 3]


Then according to the question,


x + (x – 3) = 21


⇒ 2x – 3 = 21


⇒ 2x = 21 + 3



⇒ x = 12


⇒ x – 3 = 12 – 3 = 9


Hence, the numbers are 12 and 9.



Question 16.

Frame and solve the equations for the following statements:

Two numbers are in the ratio 5 : 3. If they differ by 18, what are the numbers?


Answer:

Let the numbers be 5x and 3x


Then according to the question,


5x – 3x = 18


⇒ 2x = 18



⇒ x = 9


⇒ 5x = 5 × 9 = 45


⇒ 3x = 3 × 9 = 27


Hence, the numbers are 45 and 27.



Question 17.

Frame and solve the equations for the following statements:

A number decreased by 5% of it is 3800. What is the number?


Answer:

Let the number be x


Then according to the question,







⇒ x = 4000


Hence, the number is 4000.



Question 18.

Frame and solve the equations for the following statements:

The denominator of a fraction is 2 more than its numerator. If one is added to both the numerator and their denominator the fraction becomes  Find the fraction.


Answer:

Let the numerator be x


Then the denominator = x + 2


Then according to the question,




⇒ 3(x + 1) = 2(x + 3)


⇒ 3x + 3 = 2x + 6


⇒ 3x – 2x = 6 – 3


⇒ x = 3 (Numerator)


⇒ x + 2 = 3 + 2 = 5 (Denominator)


Hence, the fraction is 



Question 19.

Frame and solve the equations for the following statements:

Mary is 3 times older than Nandhini. After 10 years the sum of their ages will be 80. Find their present ages.


Answer:

At present,


Let age of Nandini = x


Then age of Mary = 3x


After 10 years,


Age of Nandini = x + 10


Age of Mary = 3x + 10


Then according to the question,


[x + 10] + [3x + 10] = 80


⇒ x + 10 + 3x + 10 = 80


⇒ 4x + 20 = 80


⇒ 4x = 80 – 20


⇒ 4x = 60


⇒ x = 15


⇒ 3x = 3 × 15 = 45


Hence, Present age of Nandini = 15 years


Present age of Mary = 45 years



Question 20.

Frame and solve the equations for the following statements:

Murali gives half of his savings to his wife, two third of the remainder to his son and the remaining ` 50,000 to his daughter. Find the shares of his wife and son.


Answer:

Let the savings of Murali be x












According to the question,



⇒ x = 50000 × 6


⇒ x = 300000




Hence, Share of his wife = Rs 1,50,000


Share of his son = Rs 1,00,000


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