4. Chapter 4. Special Production and Factorization NIOS Class 10 Mathematics Textbook Solutions

Chapter 4. Special Production and Factorization NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation

Chapter 4. Special Production and Factorization

CHECK YOUR PROGRESS 4.1 SPECIAL PRODUCTS AND FACTORIZATION

(i)     (ii)   (iii)  (iv)  (v) (vi)(vii)   (viii)    (ix)   (x)    (xi)   (xii)  

Solution: (i) We have,   

    

 (ii) We have, 

 

(iii) We have, 

  

 (iv) We have,  

  

(v) We have,   

(vi) We have,

(vii) We have,   

(viii) We have,    

(ix) We have,     

(x) We have,      

(xi) We have,    

(xii) We have,

2. Simplify :    (i)       (ii)      (iii)     (iv)²

Solution: (i) We have,  

 

   

(ii) We have,

  

(iii) We have,  

 

(iv) We have,  

 

3. Using special products, calculate each of the following:

(i) 102 × 102 (ii) 108 × 108 (iii) 69 × 69   (iv) 998 × 998 (v) 84 × 76 (vi) 157 × 143   (vii) 306 × 294 (viii) 508 × 492 (ix) 105 × 109    (x) 77 × 73 (xi) 94 × 95 (xii) 993 × 996

Solution: (i) We have,   

(ii) We have,   

(iii) We have, 

(iv) We have,    

(v)    

(vi)  

(vii)  

(viii)   

(ix)   

(x)   

(xi)    

(xii)  

    

CHECK YOUR PROGRESS 4.2

1. Write the expansion of each of the following:

(i)  (ii)     (iii)        (iv)     (v)  (vi)  

Solution:  (i) We have,  

  

(ii)

 

(iii) 

(iv)  

(v) 

 

(vi) 

 

  

2. Using special products, find the cube of each of the following:

(i) 8 (ii) 12 (iii) 18 (iv) 23  (v) 53 (vi) 48 (vii) 71 (viii) 69 (ix) 97 (x) 99

Solution: (i)  

 

(ii)

  

(iii)  

 

  

(iv)

    

(v)  

 

(vi)

    

(vii)

 

(viii)   

 

   

(ix)  

  

(x)

 

 

 

3. Without actual multiplication, find each of the following products:

(i)       (ii)    (iii)      (iv)    (v)    (vi)  

Solution: (i) We have,   

  

(ii)  

(iii)  

(iv)    

(v)    

(vi) 

   

4. Find the value of:

 (i) if   

[Hint: ]

(ii)  when and 

Solution:  (i) We have,  

  

(ii)  We have,   

 

 

 

5. Find the value of  if (i)   

(ii)  

Solution: (i)  

 

 

 

 

 

  

 (ii)

 

 

 

 

 

  

 

6. Simplify:

(i)  

(ii)  

[Hint put 7x + 5y = a and 7x – 5y = b so that a – b = 10y]

(iii)

(iv)   

Solution:  (i)

 

 

 

 

(ii) We have,  

Let,   and   

 

Now,   

(iii)

 

   

(iv) We have,  

 

   

7. Simplify:

(i)  

(ii)

Solution:  (i) We have,   

Here,  

Now,       

(ii) We have, 

Here,

Now,    

CHECK YOUR PROGRESS 4.3

Factorise:

1.      2.        3.      4.         5.  

6.        7.    8.   9.   10.      11.        12.

13.     14.      15.     16.      17.     18.      19.       20. 

Solution: 1. We have

2. We have,  

3. We have,  

4. We have, 

5.We have,  

 

 

6. We have,   

 

 

 

7. We have,     

8.  

 

 

9. We have,

   

10.

 

 

 

11.

 

 

 

12.  

 

 

13.  

14.

15.  

16.  

17.   

 

 

 

18.   

 

 

 

19. 

 

 

 

20.

 

 

21. Find the value of n if

(i)         (ii)   

Solution: (i) We have,  

 

 (ii) We have,   

 

  

CHECK YOUR PROGRESS 4.4

Factorise:   1.     2.      3.     4.   5.       6.     7.         8.          9.       10.       11.      12.   

Solution: 1.  

 

 

2.  

3.  

 

 

4.

 

5.  

 

 

6.  

 

7.

 

  

8. 

 

9.   

 

  

10.  

 

 

 

 

11.  

 

 

12.   

 

 

CHECK YOUR PROGRESS 4.5

Factorise:

1.      2.     3.      4.    5.       6.

7.      8.       9.      10.     11. [Hint:  ]

12. [Hint: Put  ]

Solution: 1.

2.

 

3.

4.

5.  

 

6.

 

7.  

  

8.  

 

   

9.

10. 

 

 

 

 

 

   

11.

Let,   

 

   

12. We have,  

Let  and  

 

 

 

  

CHECK YOUR PROGRESS 4.6

1. Find the HCF of the following polynomials:

(i)    (ii)   (iii)   

(iv) and   (v)  (vi)

(vii)  (viii)   (ix)    (x)  

Solution:  (i)    

      

Hence, the HCF of polynomials

(ii)   

     

Hence, the HCF of polynomials 

(iii)  

 

Hence, the HCF of polynomials  .

(iv)and   

  and    

Hence, the HCF of polynomials  .

(v)   

 

  

Hence, the HCF of polynomials .

(vi)

 

Hence, the HCF of polynomials .

(vii)   

 

Hence, the HCF of polynomials .

(viii)   

 

Hence, the HCF of polynomials .

(ix)

   

Hence, the HCF of polynomials .

(x)  

  

   

Hence, the HCF of polynomials .

2. Find the LCM of the following polynomials:

(i)  (ii)    (iii)     (iv)

(v)    (vi)

(vii)  (viii)

(ix)   (x)  

Solution:  (i)  

Hence, the LCM of given polynomials 

(ii)  

   

Hence, the LCM of given polynomials

(iii)    

     

Hence, the LCM of given polynomials

(iv)  

  

Hence, the LCM of given polynomials

(v)   

    

Hence, the LCM of given polynomials

(vi)  

 

 

Hence, the LCM of given polynomials

(vii)   

  

Hence, the LCM of given polynomials 

(viii)   

   

Hence, the LCM of given polynomials

(ix)   

   

Hence, the LCM of given polynomials 

(x)  

 

 

 

Hence, the LCM of given polynomials  

CHECK YOUR PROGRESS 4.7

1. Which of the following algebraic expressions are rational expressions?

(i)     (ii)     (iii)     (iv)     (v)   (vi)     (vii)    (viii)

Solution: (i)  

Numerator: 2x−3 is a polynomial (degree 1).

Denominator: 4x−1 is a polynomial (degree 1).

Therefore, this is a rational expression.

(ii)   

Numerator:  8 is a constant polynomial.

Denominator: x²+y² is a polynomial in two variables.

Therefore, this is also a rational expression.

(iii)   

Numerator:  ​ is a polynomial (coefficients can be irrational numbers).

Denominator:  ​ is a constant (nonzero), which is allowed.
Thus, this is a rational expression.

(iv)      

Numerator:   . The term  , which is not a non-negative integer power of x, so the numerator is not a polynomial.
Thus, the whole expression is not a rational expression.

(v)  

Both 200 and  are constants, so the expression is a constant polynomial. Therefore, it is a rational expression.

(vi)

The numerator: ​ is not a polynomial because it has b in the denominator.
Also, the denominator contains , which is not a polynomial term.

This is not a rational expression.               

(vii)  is a polynomial.

This is a rational expression.

(viii)  

The numerator : 5 is a polynomial .

 The denominator : (a+3b) is a polynomial . So it is a rational expression.

2. For each of the following, cite two examples:

(i) A rational expression is one variable

(ii) A rational expression is two variables

(iii) A rational expression whose numerator is a binomial and whose denominator is trinomial

(iv) A rational expression whose numerator is a constant and whose denominator is a quadratic polynomial

(v) A rational expression in two variables whose numerator is a polynomial of degree 3 and whose denominator is a polynomial of degree 5 .

(vi) An algebraic expression which is not a rational expression.

Solution: (i) A rational expression is one variable

Example:   

(ii) A rational expression is two variables.

Example : 

(iii) A rational expression whose numerator is a binomial and whose denominator is trinomial.

Example:   

(iv) A rational expression whose numerator is a constant and whose denominator is a quadratic polynomial.

Example :    

(v) A rational expression in two variables whose numerator is a polynomial of degree 3 and whose denominator is a polynomial of degree 5 .

Example :     

(vi) An algebraic expression which is not a rational expression.

Example:    

CHECK YOUR PROGRESS 4.8

1. Find the sum of rational expressions:

(i)     (ii)     (iii)    (iv)    (v)  

(vi)      (vii)     (viii)   

Solution: (i) We have,  

(ii) We have,

 

 

(iii) We have,

 

 

(iv) We have,

 

  

(v) We have,  

(vi) We have,  

 

(vii) We have,  

(viii) We have,

2. Subtract

(i)    (ii)      (iii)   (iv)   (v)   (vi)   (vii)    (viii)   

Solution: (i) We have,

(ii) We have,

 

(iii) We have, 

(iv) We have,  

(v) We have, 

(vi) We have, 

(vii)  

 

 

(viii)   

 

3. Find the value of :

(i)  when       (ii)    (iii)      (iv)   (v)       (vi)     (vii)      (viii)     (ix)         (x)   

Solution: (i) when

We have,

 

 (ii) 

We have, 

 

 

 

 

(iii)   

We have,   

 

 

 

 

(iv) 

We have,  

 

 

 

 

(v)  

We have,  

 

 

 

  

(vi)  

We have,  

 

 

 

 (vii)  

We have,

 

 

  

(viii)

We have,  

 

 

  

 

Again,   

 

(ix) 

We have,   

 

(x)   

We have,   

 

 

 

 

Again,   

 

 

 

TERMINAL EXERCISE

1. Mark a tick against the correct alternative:

(i) If  , then p is equal to

(A) 16 (B) 140 (C) 560 (D) 14000

Answer: (C) 560

[ We have,

    ]

(ii) 2a² + 3²-2a²-3² is equal to

(A)      (B)         (C)        (D)

Answer: (A)

[We have,

   ]

(iii)  is equal to

(A)       (B)       (C)        (D)

(iv) If   , then is equal to

(A) 0    (B)    (C)      (D)  

Answer:  (A) 0

[ We have,     

 

 

   ]

(v)   is equal to

(A) 650       (B) 327       (C) 323        (D) 4

Solution:  (D)  4

We have,  ]

(vi)   is equal to:

(A) (2m – n)(4m² – 2mn + n²) (B) (2m – n)(4m² + 2mn + n²)

(C) (2m – n)(4m² – 4mn + n²) (D) (2m – n)(4m² + 4mn + n²)

Solution:   (B) 

[ We have,   

   ]

(vii)    is equal to

(A) 66 (B) 198 (C) 1000 (D) 3000

Answer: (C) 1000

[ We have,   

Let,  

 ]

(viii) The HCF of  and   is

(A)  (B)  (C)   (D)  

Answer: (B)  

[ We have, The HCF of  

and   

The HCF of  and  is  ]

(ix) The LCM of x² – 1 and x² – x – 2 is

(A) (x² – 1) (x – 2)     (B) (x² – 1) (x + 2)       (C) (x – 1)² (x + 2)     (D) (x + 1)² (x – 2)

Answer: (A)   

[ We have,

 

The LCM of x² – 1 and x² – x – 2 is  ]

(x) Which of the following is not a rational expression?

(A)  (B)     (C)   (D)  

Answer: (C)    

2. Find each of the following products:

(i)   (ii) (x + y + 2)(x – y + 2)     (iii) (2x + 3y) (2x + 3y)     (iv) (3a – 5b)(3a – 5b)     (v) (5x + 2y) ( 25x² – 10xy + 4y²)     (vi) (2x – 5y) (4x² + 10xy + 25y²)

(vii)       (viii) (2z² + 3)(2z² – 5)     (ix) 99 × 99 × 99     (x) 103 × 103 × 103     (xi) (a + b – 5) (a + b – 6)    (xii) (2x + 7z)(2x + 5z)

Solution: (i) We have,    

(ii) We have,  

 

  

(iii) We have,  

  

(iv) We have,   

  

(v) We have,   

    

(vi) We have,  

   

(vii) We have,

  

(viii) We have,  

 

  

(ix) We have,  

  

(x) We have,   

 

(xi) We have,   

 

 

  

(xii) We have

  

3. If x = a – b and y = b – c, show that (a – c) (a + c – 2b) = x² – y²

Solution:  We have,

 

and  

LHS:  RHS  Proved.

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

Solution: We have,  

 

 

  

 

5. Factorise :   (i)   (ii)   (iii)    (iv)  (v)  

(vi)  (vii)   (viii)   (ix)  (x)  (xi)  (xii)   

Solution: (i) We have,     

(ii) We have,   

 

 

 

(iii) We have,   

 

(iv) We have,  

(v) We have,   

 

(vi) We have,  

(vii) We have,   

 

 

(viii) We have,  

 

 

(ix) We have,

(x) We have,

(xi) We have, 

 

 

(xii)

 

6. Find the HCF of :   (i) (ii) 

Solution:  (i)  

We have,

and   

The HCF of  is  .

(ii)

We have, 

 

And  

Therefore, the HCF of is 

7. Find the LCM of :  (i)        (ii)

Solution: (i)   

We have,  

and 

The LCM of  is  

(ii)

We have,  

 

 

The LCM of   is

8. Perform the indicated operation:

(i)        (ii)     (iii)    (iv)  

Solution:  (i) We have 

 

(ii) We have,

 

 

 

(iii) We have,

   

(iv) We have,  

 

 

 

9. Simpify:   

[ Hint:   ; now combine next term and so on]

Solution: We have, 

 

10. If   , find .

Solution: We have,

 

And   

 

 

 

 

 

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