**NCERT Exemplar Problems Class 8 Mathematics Chapter 7 Algebraic Expressions, Identities and Factorisation**

**Question. 1 The product of a monomial and a binomial is a**

** (a) monomial (b) binomial**

** (c) trinomial (d) None of these**

** Solution.** (b) Monomial consists of only single term and binomial contains two terms. So, the multiplication of a binomial by a monomial will always produce a binomial, whose first term is the product of monomial and the binomial’s first term and second term is the product of monomial and the binomial’s second term.

**Question. 2 In a polynomial, the exponents of the variables are always (a)’integers (b) positive integers ****(c) non-negative integers (d) non-positive integers**

** Solution.** (c) In a polynomial, the exponents of the variables are either positive integers or 0. Constant term C can be written as C x°. We do not consider the expressions as a polynomial which consist of the variables having negative/fractional exponent.

**Question. 3 Which of the following is correct?**

** (a) (b) **

** (c) (d) **

** Solution.**

**Question. 4 The sum of -7pq and 2pq is**

** (a) -9pq (b) 9pq**

** (c) 5pq (d) -5pq**

** Solution.**

**Question. 5 If we subtract from , then we get**

** **

** Solution.**

**Question. 6 Like term as is**

** (a) (b) **

** (c) (d) **

** Solution.** (b) We know that, the like terms contain the same literal factor. So, the like term as , is , as it contains the same literal factor .

**Question. 7 Which of the following is a binomial?**

** **

** Solution.**

**Question. 8 Sum of a – b + ab, b + c – bc and c – a – ac is**

** **

** Solution.**

**Question. 9 Product of the monomials 4p, -7, -7pq is**

** **

** Solution.**

**Question. 10 Area of a rectangle with length 4ab and breadth 6 is**

** **

** Solution.**

**Question. 11 Volume of a rectangular box (cuboid) with length = 2ab, breadth = 3ac and height = 2ac is**

** **

** Solution.**

**Question. 12 Product of 6 -7b + 5ab and 2ab is**

** **

** Solution.**

**Question. 13 Square of 3x – 4y is**

** **

** Solution.**

**Question. 14 Which of the following are like terms?**

** **

** Solution.**

**Question. 15 Coefficient of y in the term of is**

** (a)-1 (b)-3 (c) (d)**

** Solution.**

**Question. 16 is equal to**

** **

** Solution.**

**Question. 17 Common factor Of 17abc, 34a, 51b is**

** (a)17abc (b)17ab (c)17ac (d)17c**

** Solution.**

**Question. 18 Square of 9x – 7xy is**

** **

** Solution.**

**Question. 19 Factorised form of 23xy – 46x + 54y -108 is**

** **

** Solution.**

**Question. 20 Factorised form of -10r + 21 is**

** (a)(r-1)(r-4) (b)(r-7)(r-3) (c)(r-7)(r+3) (d)(r+7)(r+3)**

** Solution.**

**Question. 21 Factorised form of – 17p – 38 is**

** (a) (p -19)(p + 2) (b) (p -19) (p – 2) (c) (p +19) (p + 2) (d) (p + 19) (p – 2)**

** Solution.**

**Question. 22 On dividing 57 qr by 114pq, we get**

** **

** Solution.**

**Question. 23 On dividing p(4 – 16) by 4p (p – 2), we get**

** (a) 2p + 4 (b) 2p – 4 (c) p + 2 (d) p – 2**

** Solution.**

**Question. 24 The common factor of 3ab and 2cd is**

** (a) 1 (b) -1 (c) a (d) c**

** Solution.** (a) We have, monomials 3ab and 2cd Now, 3ab = 3xaxb 2cd =2 x c x d

Observing the monomials, we see that, there is no common factor (neither numerical nor literal) between them except 1.

**Question. 25 An irreducible factor of24 is**

** (a) (b) (c)x (d)24x**

** Solution.** (c) A factor is said to be irreducible, if it cannot be factorised further.

We have, 24 =2 x 2 x 2 x 3 x x x x x y x y Hence, an irreducible factor of 24 is x.

**Question. 26 Number of factors of is**

** (a) 4 (b) 3 (c) 2 (d) 1**

** Solution.** (c) We can write as, (a + b) (a + b) and this cannot be factorised further.

Hence, number of factors of is 2.

**Question. 27 The factorised form of 3x – 24 is**

** (a) 3x x 24 (b)3 (x – 8) (c)24(x – 3) (d)3(x-12)**

** Solution.** (b) We have,

3x – 24 = 3 x x – 3 x 8= 3 (x – 8) [taking 3 as common]

**Question. 28 The factors of – 4 are**

** (a) (x – 2), (x – 2) (b) (x + 2), (x – 2)**

** (c) (x + 2), (x + 2) (d) (x – 4), (x – 4)**

** Solution.**

**Question. 29 The value of is**

** (a)3xy (b)-3xy (c)-3x (d)3x**

** Solution.**

**Question. 30 The value of is**

** **

** Solution.**

**Question. 31 The value of is**

** **

** Solution.**

**Question. 32 The value of is**

** **

** Solution.**

**Question. 33 The value of is**

** **

** Solution.**

**Fill in the Blanks**

**In questions 34 to 58, fill in the blanks to make the statements true.**

** Question. 34 The product of two terms with like signs is a term.**

** Solution.** Positive

If both the like terms are either positive or negative, then the resultant term will always be positive.

**Question. 35 The product of two terms with unlike signs is a term.**

** Solution.** Negative

As the product of a positive term and a negative term is always negative.

**Question. 36 a (b + c) = a x ——– + a x ———-**

** Solution.** b,c

we have , a(b+c)=a x b + a x c [using left distributive law]

**Question. 37 (a-b) ————- =**

** Solution.**

**Question. 38 =(a+b)—————-**

** Solution.**

**Question. 39 +—————-=**

** Solution.**

**Question. 40 -2ab=————- + ———–.**

** Solution.**

**Question. 41 (x+a)(x+b)= + (a+b) x + ———–.**

** Solution.**

**Question. 42 The product of two polynomials is a ————–.**

** Solution.** Polynomial

As the product of two polynomials is again a polynomial.

**Question. 43 Common factor of ax2 + bx is——————.**

** Solution.**

**Question. 44 Factorised form of 18mn + 10mnp is —————–.**

** Solution.**

**Question. 45 Factorised form of 4 – 12y + 9 is———– .**

** Solution.**

**Question. 46 is equal to———–.**

** Solution.**

**Question. 47 Volume of a rectangular box with length 2x, breadth 3y and height 4z is ——.**

** Solution.** 24 xyz

We know that, the volume of a rectangular box,

V = Length x Breadth x Height = 2x x 3y x 4z = (2 x 3 x 4) xyz = 24 xyz

**Question. 48 =(67 -37) x ———–=————.**

** Solution.**

**Question. 49 =————- x (103-102)=————–.**

** Solution.**

**Question. 50 Area of a rectangular plot with sides 4 and 3 is————–.**

** Solution.**

**Question. 51 Volume of a rectangular box with l = b = h = 2x is ———-.**

** Solution.**

**Question. 52 The numerical coefficient in -37abc is————–.**

** Solution.** -37

The constant term (with their sign) involved in term of an algebraic expression is called the numerical coefficient of that term.

**Question. 53 Number of terms in the expression and + bc x d is –.**

** Solution.**

**Question. 54 The sum of areas of two squares with sides 4o and 4b is————-.**

** Solution.**

**Question. 55 The common factor method of factorisation for a polynomial is based on————-property.**

** Solution.**Distributive

In this method, we regroup the terms in such a way, so that each term in the group contains a common literal or number or both.

**Question. 56 The side of the square of area 9 is————.**

** Solution.**

**Question. 57 On simplification, =————.**

** Solution.**

**Question. 58 The factorisation of 2x + 4y is————-.**

** Solution.** 2 (x + 2y)

We have, 2x + 4y = 2x + 2 x 2y = 2 (x + 2y)

**True/False**

**In questions 59 to 80, state whether the statements are True or False**

** Question. 59 .**

** Solution.**

**Question. 60 .**

** Solution.**

**Question. 61 (a+b) (a-b)=**

** Solution.**

**Question. 62 The product of two negative terms is a negative term.**

** Solution.**False

Since, the product of two negative terms is always a positive term, i.e. (-) x (-) = (+).

**Question. 63 The product of one negative and one positive term is a negative term.**

** Solution.**True

When we multiply a negative term by a positive term, the resultant will be a negative term, i-e. (-) x (+) = (-).

**Question. 64 The numerical coefficient of the term -6 is -6.**

** Solution.** True

Since, the constant term (i.e. a number) present in the expression -6 is -6.

**Question. 65 q+r+q is a binomial.**

** Solution.** False

Since, the given expression contains three unlike terms, so it is a trinomial.

**Question. 66 The factors of – 2ab + are (a + b) and (a + b).**

** Solution.**

**Question. 67 h is a factor of .**

** Solution.**

**Question. 68 Some of the factors of are and (n+1).**

** Solution.**

**Question. 69 An equation is true for all values of its variables.**

** Solution.** False

As equation is true only for some values of its variables, e.g. 2x – 4= 0 is true, only for x =2.

**Question. 70 + (a+b)x +ab =(a+b)(x +ab)**

** Solution.**

**Question. 71 Common factors of is **

** Solution.**

**Question. 72 Common factors of 12 +4a -32 is 4.**

** Solution.**

**Question. 73 Factorisation of -3+3ab+3ac is 3a (-a-b-c).**

** Solution.**

**Question. 74 Factorised form of +30p+216 is (p+18) (p-12).**

** Solution.**

**Question. 75 The difference of the squares of two consecutive numbers is their sum.**

** Solution.**

**Question. 76 abc + bca + cab is a monomial.**

** Solution. **True

The given expression seems to be a trinomial but it is not as it contains three like terms which can be added to form a monomial, i.e. abc + abc + abc = 3abc

**Question. 77 On dividing by ,the quotient is 9**

** Solution.**

**Question. 78 The value of p for 5-4=100 p is 2.**

** Solution.**

**Question. 79 is x-51.**

** Solution.**

**Question. 80 The value of (a+1) (a-1)( +1) is -1.**

** Solution.**

**Question. 81 Add:**

** **

** Solution.**

**Question. 82 Subtract**

** **

** Solution.**

**Question. 83 Multiply the following:**

** Solution.**

**Question. 84 Simplify**

**Solution.**

**Question. 85 Expand the following, using suitable identities.**

**Solution.**

**Question. 86 Using suitable identities, evaluate the following:**

**Solution.**

**Question. 87 Write the greatest common factor in each of the following terms.**

**Solution.**

**Question. 88 Factorise the following expressions.**

**Solution.**

**Question. 89Factorise the following, using the identity,**

**Solution.**

**Question. 90 Factorise the following, using the identity,**

**Solution.**

**Question. 91 Factorise the following**

**Solution.**

**Question. 92 Factorise the following using the identity ,=(a+b)(a-b).**

** Solution.**

**Question. 93 The following expressions are the areas of rectangles. Find the possible lengths and breadths of these rectangles.**

** Solution.**

**Question. 94 Carry out the following divisions:**

** Solution.**

**Question. 95 Perform the following divisions:**

** Solution.**

**Question. 96 Factorise the expressions and divide them as directed.**

** Solution.**

**Question. 97 The area of a square is given by 4+ 12xy + 9. Find the side of the square.**

** Solution.**

**Question. 98 The area of a square is 9 + 24xy + 16. Find the side of the square.**

** Solution.**

**Question. 99 The area of a rectangle is + 7x + 12. If its breadth is (x + 3), then find its length.**

** Solution.**

**Question. 100 The curved surface area of a cylinder is and its radius is (y – 3). Find the height of the cylinder (CSA of cylinder = )**

** Solution.**

**Question. 101 The area of a circle is given by the expression . Find the radius of the circle.**

** Solution.**

**Question.102 The sum of first n natural numbers is given by the expression Factorise this expression.**

** Solution.**

**Question.103 The sum of (x + 5) observations is – 625. Find the mean of the observations.**

** Solution.**

**Question.104 The height of a triangle is + and its base is 14xy. Find the area of the triangle.**

** Solution.**

**Question.105 The cost of a chocolate is Rs (x + 4) and Rohit bought (x + 4) chocolates. Find the total amount paid by him in terms of x. If x = 10, find the amount paid by him.**

** Solution.**

**Question.106 The base of a parallelogram is (2x + 3) units and the corresponding height is (2x – 3) units. Find the area of the parallelogram in terms of x. What will be the area of a parallelogram of x = 30 units?**

** Solution.**

**Question.107 The radius of a circle is 7ab – 7be – 14ac . Find the circumference of the circle,**

** Solution.**

**Question.108 If p + q = 12 and pq = 22, then find + .**

** Solution.**

**Question.109 If a + b = 25 and + then find ab.**

** Solution.**

**Question.110 If x – y = 13 and xy = 28, then find + .**

** Solution.**

**Question.111 If m – n = 16 and + = 400, then find mn.**

** Solution.**

**Question.112 If + = 74 and ab = 35, then find a + b ?**

** Solution.**

**Question.113 Verify the following:**

**Solution.**

**Question.114 Find the value of a, if**

**Solution.**

**Question.115 What should be added to 4c (-a + b + c) to obtain 3a(a + b + c) – 2b (a – b + c)?**

** Solution.**

**Question.116 Subtract b( + b – 7) + 5 from 3 – 8 and find the value of expression obtained for b = – 3.**

** Solution.**

**Question.117 If x – = 1, then find the value of .**

** Solution.**

**Question.118 Factorise .**

** Solution.**

**Question.119 Factorise .**

** Solution.**

**Question.120 Find the value of**

**Solution.**

**Question.121 The product of two expressions is + + x . If one of them is + x + 1, find the other.**

** Solution.**

**Question.122 Find the length of the side of the given square, if area of the square is 625sq units and then find the value of x.**

**Solution.**

**Question.123 Take suitable number of cards given in the adjoining diagram [G(x x x) representing , R (x x 1) representing x and Y (1 x 1) representing 1] to factorise the following expressions, by arranging to cards in the form of rectangles: (i) 2 + 6x + 4 (ii) + 4x + 4. Factorise 2 + 6x + 4 by using the figure.**

** **

** Calculate the area of figure.**

** Solution.** The given information is incomplete for solution of this question.

**Question.124 The figure shows the dimensions of a wall having a window and a door of a room. Write an algebraic expression for the area of the wall to be painted.**

** **

** Solution.**

**Question.125 Match the expressions of column I with that of column II**

** **

** Solution.**