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

**Multiple Choice Questions**

**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) \({{\left( a-b \right)}^{2}}={{a}^{2}}+2ab-{{b}^{2}}\) (b) \({{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\)**

**(c) \({{\left( a-b \right)}^{2}}={{a}^{2}}-{{b}^{2}}\) (d) \({{\left( a+b \right)}^{2}}={{a}^{2}}+2ab-{{b}^{2}}\)**

**Solution.**

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

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

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

**Solution.**

**Question. 5 If we subtract \(-3{ x }^{ 2 }{ y }^{ 2 }\) from \({ x }^{ 2 }{ y }^{ 2 }\), then we get**

**Solution.**

**Question. 6 Like term as \(4{ m }^{ 3 }{ n }^{ 2 }\) is**

**(a)\(4{ m }^{ 2 }{ n }^{ 2 }\) (b) \(-6{ m }^{ 3 }{ n }^{ 2 }\)**

**(c) \(6p{ m }^{ 3 }{ n }^{ 2 }\) (d) \(4{ m }^{ 3 }{ n }\)**

**Solution.** (b) We know that, the like terms contain the same literal factor. So, the like term as \(4{ m }^{ 3 }{ n }^{ 2 }\) , is \(-6{ m }^{ 3 }{ n }^{ 2 }\), as it contains the same literal factor \({ m }^{ 3 }{ n }^{ 2 }\).

**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\({ q }^{ 3 }\), -7pq is**

**Solution.**

**Question. 10 Area of a rectangle with length 4ab and breadth 6\({ b }^{ 2 }\) 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\({ a }^{ 2 }\) -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 \({ -y }^{ 3 }\) is**

**(a)-1 (b)-3 (c)\({ -1 }^{ 3 }\) (d)\({ 1 }^{ 3 }\)**

**Solution.**

**Question. 16 \({ a }^{ 2 }-{ b }^{ 2 }\) is equal to**

**Solution.**

**Question. 17 Common factor Of 17abc, 34a\({ b }^{ 2 }\), 51\({ a }^{ 2 }\)b is**

**(a)17abc (b)17ab (c)17ac (d)17\({ a }^{ 2 }\)\({ b }^{ 2 }\)c**

**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 \({ r }^{ 2 }\)-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 \({ p }^{ 2 }\) – 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 \({ p }^{ 2 }\) qr by 114pq, we get**

**Solution.**

**Question. 23 On dividing p(4\({ p }^{ 2 }\) – 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\({ x }^{ 2 }\)\({ y }^{ 2 }\) is**

**(a)\({ a }^{ 2 }\) (b)\({ y }^{ 2 }\) (c)x (d)24x**

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

We have, 24\({ x }^{ 2 }\)\({ y }^{ 2 }\) =2 x 2 x 2 x 3 x x x x x y x y Hence, an irreducible factor of 24\({ x }^{ 2 }\)\({ y }^{ 2 }\) is x.

**Question. 26 Number of factors of \({{\left( a+b \right)}^{2}}\) is**

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

**Solution.** (c) We can write \({{\left( a+b \right)}^{2}}\) as, (a + b) (a + b) and this cannot be factorised further.

Hence, number of factors of \({{\left( a+b \right)}^{2}}\) 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 \({ x }^{ 2 }\) – 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 \((-27{ x }^{ 2 }y)\div (-9xy)\) is**

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

**Solution.**

**Question. 30 The value of \((2{ x }^{ 2 }+4)\div (2)\) is**

**Solution.**

**Question. 31 The value of \((3{ x }^{ 3 }+9{ x }^{ 2 }+27x)\div 3x\) is**

**Solution.**

**Question. 32 The value of \({{\left( a+b \right)}^{2}}+{{(a-b)}^{2}}\) is**

**Solution.**

**Question. 33 The value of \({{\left( a+b \right)}^{2}}-{{(a-b)}^{2}}\) 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) ————- =\( { a }^{ 2 }-2ab+{ b }^{ 2 }\)**

**Solution.**

**Question. 38 \({ a }^{ 2 }-{ b }^{ 2 }\)=(a+b)—————-**

**Solution.**

**Question. 39 \({{(a-b)}^{2}}\)+—————-=\({ a }^{ 2 }-{ b }^{ 2 }\)**

**Solution.**

**Question. 40 \({{(a+b)}^{2}}\)-2ab=————- + ———–.**

**Solution.**

**Question. 41 (x+a)(x+b)=\({ x }^{ 2 }\) + (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\({ y }^{ 2 }\) – 12y + 9 is———– .**

**Solution.**

**Question. 46 \(38{ x }^{ 2 }{ y }^{ 2 }z\div 19x{ y }^{ 2 }\) 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 \( 6{ 7 }^{ 2 }-3{ 7 }^{ 2 }\) =(67 -37) x ———–=————.**

**Solution.**

**Question. 49 \( { 103 }^{ 2 }-{ 102 }^{ 2 }\)=————- x (103-102)=————–.**

**Solution.**

**Question. 50 Area of a rectangular plot with sides 4\({ y }^{ 2 }\) and 3\({ y }^{ 2 }\) 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 \({ a }^{ 2 }\) 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\({ y }^{ 2 }\) is————.**

**Solution.**

**Question. 57 On simplification, \(\frac { 3x+3 }{ 3 }\) =————.**

**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 \({{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}\).**

**Solution.**

**Question. 60 \({{(a-b)}^{2}}={{a}^{2}}-{{b}^{2}}\).**

**Solution.**

**Question. 61 (a+b) (a-b)=\({{a}^{2}}-{{b}^{2}}\)**

**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\({ x }^{ 2 }{ y }^{ 2 }\) is -6.**

**Solution.** True

Since, the constant term (i.e. a number) present in the expression -6\({ x }^{ 2 }{ y }^{ 2 }\) is -6.

**Question. 65 \({ p }^{ 2 }\)q+\({ q }^{ 2 }\)r+\({ r }^{ 2 }\)q is a binomial.**

**Solution.** False

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

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

**Solution.**

**Question. 67 h is a factor of \(2\pi (h+r)\).**

**Solution.**

**Question. 68 Some of the factors of \(\frac { { n }^{ 2 } }{ 2 } +\frac { n }{ 2 }\) are \(\frac { 1 }{ 2 } n\) 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 \({ x }^{ 2 }\) + (a+b)x +ab =(a+b)(x +ab)**

**Solution.**

**Question. 71 Common factors of \(11p{ q }^{ 2 },121{ p }^{ 2 }{ q }^{ 3 },1331{ p }^{ 2 }q\) is \(11{ p }^{ 2 }{ q }^{ 2 }\)**

**Solution.**

**Question. 72 Common factors of 12 \(11{ a }^{ 2 }{ b }^{ 2 }\) +4a\({ b }^{ 2 }\) -32 is 4.**

**Solution.**

**Question. 73 Factorisation of -3\({ a }^{ 2 }\)+3ab+3ac is 3a (-a-b-c).**

**Solution.**

**Question. 74 Factorised form of \({ p }^{ 2 }\)+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 \(\frac { p }{ 3 }\) by \(\frac { 3 }{ p }\) ,the quotient is 9**

**Solution.**

**Question. 78 The value of p for 5\({ 1 }^{ 2 }\)-4\({ 9 }^{ 2 }\)=100 p is 2.**

**Solution.**

**Question. 79 \((9x-51)\div 9\) is x-51.**

**Solution.**

**Question. 80 The value of (a+1) (a-1)(\({ a }^{ 2 }\) +1) is \({ a }^{ 4 }\)-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,\({{a}^{2}}+2ab+{{b}^{2}}={{(a+b)}^{2}}\)**

**Solution.**

**Question. 90 Factorise the following, using the identity,\({{a}^{2}}-2ab+{{b}^{2}}={{(a-b)}^{2}}\)**

**Solution.**

**Question. 91 Factorise the following**

**Solution.**

**Question. 92 Factorise the following using the identity ,\({{a}^{2}}-{{b}^{2}}\)=(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\({{x}^{2}}\)+ 12xy + 9\({{y}^{2}}\). Find the side of the square.**

**Solution.**

**Question. 98 The area of a square is 9\({{x}^{2}}\) + 24xy + 16\({{y}^{2}}\). Find the side of the square.**

**Solution.**

**Question. 99 The area of a rectangle is \({{x}^{2}}\) + 7x + 12. If its breadth is (x + 3), then find its length.**

**Solution.**

**Question. 100 The curved surface area of a cylinder is \(2\pi ({ y }^{ 2 }-7y+12)\) and its radius is (y – 3). Find the height of the cylinder (CSA of cylinder = \(2\pirh\))**

**Solution.**

**Question. 101 The area of a circle is given by the expression \( \pi { x }^{ 2 }+6\pi x+9\pi \). Find the radius of the circle.**

**Solution.**

**Question.102 The sum of first n natural numbers is given by the expression \(\frac { { n }^{ 2 } }{ 2 } +\frac { n }{ 2 }\) Factorise this expression.**

**Solution.**

**Question.103 The sum of (x + 5) observations is \({ x }^{ 4 }\) – 625. Find the mean of the observations.**

**Solution.**

**Question.104 The height of a triangle is \({ x }^{ 4 }\) + \({ y }^{ 4 }\) 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,\( (\pi =\frac { 22 }{ 7 } )\)**

**Solution.**

**Question.108 If p + q = 12 and pq = 22, then find \({ p }^{ 2 }\) + \({ q }^{ 2 }\) .**

**Solution.**

**Question.109 If a + b = 25 and \({ a }^{ 2 }\) + \({ b }^{ 2 }\) then find ab.**

**Solution.**

**Question.110 If x – y = 13 and xy = 28, then find \({ x }^{ 2 }\) + \({ y }^{ 2 }\).**

**Solution.**

**Question.111 If m – n = 16 and \({ m }^{ 2 }\) + \({ n }^{ 2 }\) = 400, then find mn.**

**Solution.**

**Question.112 If \({ a }^{ 2 }\) + \({ b }^{ 2 }\) = 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(\({ b }^{ 2 }\) + b – 7) + 5 from 3\({ b }^{ 2 }\) – 8 and find the value of expression obtained for b = – 3.**

**Solution.**

**Question.117 If x – \(\frac { 1 }{ x }\) = 1, then find the value of \({ x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } }\) .**

**Solution.**

**Question.118 Factorise \({ x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } +2-3x-\frac { 3 }{ x }\).**

**Solution.**

**Question.119 Factorise \({ p }^{ 4 }+{ q }^{ 4 }+{ p }^{ 2 }{ q }^{ 2 }\).**

**Solution.**

**Question.120 Find the value of**

**Solution.**

**Question.121 The product of two expressions is \({ x }^{ 5 }\) + \({ x }^{ 3 }\)+ x . If one of them is \({ x }^{ 2 }\) + 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 \({ x }^{ 2 }\), 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\({ x }^{ 2 }\) + 6x + 4 (ii) \({ x }^{ 2 }\) + 4x + 4. Factorise 2\({ x }^{ 2 }\) + 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.**

**NCERT Exemplar Class 8 Maths Solutions**

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