Algebra Class 8th Mathematics Term 2 Tamilnadu Board Solution

Class 8^th 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…

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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.

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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

=  y^{2 + 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 ( x^{2 + 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) (a^{1 + 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( m^{1 + 1 + 2 + 3})

= -72 m⁷

v. xyz, y²z, yx²

= (x x²) 

= (x^{1 + 2})  ( )  (  )

= (x³)  (y⁴)  ()

= x³y⁴

vi. lm², mn², ln²

= ( l

= 

= 

vii. -2p, -3q, -5p²

= (-2 p²)

= (-30)  p^{1 + 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 a^{3 + 5 + 15}

⇒ 8 a²³ (∵ a^{n + m} = aⁿ × a^m)

ii. 5 × (4 + x) -2x × (4 + x)

⇒ (5× 4 + 5x) – (2x × 4 + 2x × x)

⇒ (20 + 5x) – (8x + 2x^{1 + 1} ) (∵ a^{n + m} = aⁿ × a^m)

⇒ (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))

⇒ (3x^{1 + 1} - xy) + ((3× 3)xy – 3y^{1 + 1} ) (∵ a^{n + m} = aⁿ × a^m)

⇒ (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)x^{1 + 1} -9x) + (8x -6) (∵ a^{n + m} = aⁿ × a^m)

⇒ 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)

⇒( 2a^{2 + 1} – 5a^{1 + 1}b + 3ab²) + (2a²b – 5ab² + 3b^{2 + 1})

(∵ a^{n + m} = aⁿ × a^m)

⇒( 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)

⇒ (2x^{2 + 1} – 2x^{1 + 1}y + 2xy²) + (3yx² -3xy^{1 + 1} + 3y^{2 + 1})

(∵ a^{n + m} = aⁿ × a^m)

⇒ (2x^{2 + 1} – 2x^{1 + 1}y + 2xy²) + (3yx² -3xy^{1 + 1} + 3y^{2 + 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)

⇒ (x^{1 + 1} + xy -xz ) + (yx + y^{1 + 1} -yz) + (zx + zy -z^{1 + 1})

(∵ a^{n + m} = aⁿ × a^m)

⇒ 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)

⇒ (a^{2 + 1} + 2a^{1 + 1}b + ab²) + (ba² + 2ab^{1 + 1} + b^{2 + 1})

(∵ a^{n + m} = aⁿ × a^m)

⇒ (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)

⇒ (m^{2 + 1} + m^{1 + 1}n + mn²) –( nm² + mn^{1 + 1} + n^{2 + 1})

(∵ a^{n + m} = aⁿ × a^m)

⇒ (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

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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 1^st and 2^nd 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 1^st and 2^nd term and – 3 from 3^rd and 4^th,

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 1^st and 2^nd term and y from 3^rd and 4^th,

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 1^st and 2^nd term and ( – 1) from 3^rd and 4^th,

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 1^st and 2^nd term and b from 3^rd and 4^th,

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

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 1^st and 2^nd term and 1 from 3^rd and 4^th,

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

⇒ 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 1^st and 2^nd term and ( – 1) from 3^rd and 4^th,

⇒ 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 1^st and 2^nd term and (11b) from 3^rd and 4^th,

⇒ 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)

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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)

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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