## Selina Concise Mathematics class 7 ICSE Solutions – Fundamental Concepts (Including Fundamental Operations)

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**POINTS TO REMEMBER**

**Constants and Variables :**The numbers which has fixed value is called constant and same at English alphabet which can be assigned any value according to the requirement is called variables.**Term :**A term is a number, (constant), a variable or a combination of numbers and variables.**Algebraic Expression :**An algebraic expression is a collection of one or more terms, which are separated from each other by addition (+) or subtraction (-) signs.**Types of algebraic expressions :**

(i) Monomial : It has only one term

(ii) Binomial : It has two terms

(iii) Trinomial : It has three terms

(iv) Multinomial : It has more than three terms

(v) Polynomial : It has two or more than two terms.

**Note**: An expression of the type does not form a monomial unless JC is not equal to zero.**Product:**When two or more quantities are multiplied together, the result is called their product.**Factors :**Each of the quantities (numbers or variables) multiplied together to form a term is called a factor of the given term.**Co-efficient:**In a monomial, any factor or group of factors of a term is called the co-efficient of the remaining part of the monomial.**Degree of a monomial:**The degree of a monomial is the exponent of its variable or the sum of the exponents of its variables.**Degree of a polynomial:**The degree of a polynomial is the degree of its highest degree term.**Like and unlike terms**: Terms having the same literal co-efficients or alphabetic letters are called like terms ; whereas the terms with different literal co-efficients are called unlike terms.**Addition and subtraction**: Addition and subtraction of only like terms is possible by adding or subtracting the numerical co-efficients.**Multiplication and division**:

**(A) Multiplication :**

(i) Multiplications of monomials.

(a) Multiply the numerical co-efficient together

(ii) Multiply the literal co-efficients separately together.

(iii) Combine the like terms.

**(B) Division :**

(i) Dividing a polynomial by a monomial Divide each term of the polynomial by monomial and simplify each fractions.

(ii) While dividing one polynomial by another polynomial ; arrange the terms of both the dividend and the divisior both in descending or in ascending order of their powers and then divide.

**SOME IMPORTANT POINTS**

**TYPES OF BRACKETS:**

The name of different types of brackets and the order in which they are removed is shown below:

(a) ____ ; Bar (Vinculum) bracket

(b) ( ); Circular bracket .

(c) { } ; Curly bracket and then

(d) [ ]; square bracket

**EXERCISE 11 (A)**

**Question 1.**

**Separate constant terms and variable terms from tile following :**

**Solution:**

Constant is only 8 others are variables

**Question 2.**

**Constant is only 8 others are variables**

** (i) 2x ÷ 15**

** (ii) ax+ 9**

** (iii) 3x ^{2} × 5x**

**(iv) 5 + 2a-3b**

**(v) 2y – z÷x**

**(vi) 3p x q ÷ z**

**(vii) 12z ÷ 5x + 4**

**(viii) 12 – 5z – 4**

**(ix) a**

^{3}– 3ab^{2}x c**Answer:**

**Question 3.**

**Write the coefficient of:**

** (i) xy in – 3axy**

** (ii) z ^{2} in p^{2}yz^{2}**

**(iii) mn in -mn**

**(iv) 15 in – 15p**

^{2}**Solution:**

**(i)** Co-efficient of xy in – 3 axy = – 3a

**(ii)** Co-efficient of z^{2} in p^{2}yz^{2} = p^{2}y

**(iii)** Co-efficient of mn in – mn = – 1

**(iv)** Co-efficient of 15 in – 15p^{2} is -p^{2}

**Question 4.**

**For each of the following monomials, write its degree :**

** (i) 7y**

** (ii) – x ^{2}y**

**(iii) xy**

^{2}z**(iv) – 9y**

^{2}z^{3}**(v) 3 m**

^{3}n^{4}**(vi) – 2p**

^{2}q^{3}r^{4}**Solution:**

**(i)** Degree of 7y = 1

**(ii)** Degree of – x^{2}y = 2+1=3

**(iii)** Degree of xy^{2}z = 1 + 2 + 1 = 4

**(iv)** Degree of – 9y^{2}z^{3} = 2 + 3 = 5

**(v)** Degree of 3m^{3}n^{4} = 3 + 4 = 7

**(vi)** Degree of – 2p^{2}q^{3}r^{4} = 2 + 3 + 4 = 9

**Question 5.**

**Write the degree of each of the following polynomials :**

** (i) 3y ^{3}-x^{2}y^{2} + 4x**

**(ii) p**

^{3}q^{2}– 6p^{2}q^{5}+ p^{4}q^{4}**(iii) – 8mn**

^{6}+ 5m^{3}n**(iv) 7 – 3x**

^{2}y + y^{2}**(v) 3x – 15**

**(vi) 2y**

^{2}z + 9yz^{3}**Solution:**

**(i)** The degree of 3y^{3} – x^{2}y^{2}+ 4x is 4 as x^{2}

y^{2} is the term which has highest degree.

**(ii)** The degree of p^{3}q^{2} – 6p^{2}q^{5}-p^{4}q^{4} is 8 as p^{4} q^{4} is the term which has highest degree.

**(iii)** The degree of- 8mn^{6} + 5m^{3}n is 7 as – 8mx^{6} is the term which has the highest degree.

**(iv)** The degree of 7 – 3x^{2} y + y^{2} is 3 as – 3x^{2}y is the term which has the highest degree.

**(v)** The degree of 3x – 15 is 1 as 3x is the term which is highest degree.

**(vi)** The degree of 2y^{2} z + 9y z^{3} is 4 as 9yz^{3} has the highest degree.

**Question 6.**

**Group the like term together :**

** (i) 9x ^{2}, xy, – 3x^{2}, x^{2} and – 2xy**

**(ii) ab, – a**

^{2}b, – 3ab, 5a^{2}b and – 8a^{2}b**(iii) 7p, 8pq, – 5pq – 2p and 3p**

**Solution:**

**(i)** 9x^{2}, – 3x^{2} and x^{2} are like terms

xy and – 2xy are like terms

**(ii)** ab, – 3ab, are like terms,

– a^{2}b, 5a^{2}b, – 8a^{2}b are like terms

**(iii)** 7p, – 2p and 3p are like terms,

8pq, – 5pq are like terms.

**Question 7.**

**Write numerical co-efficient of each of the followings :**

** (i) y**

** (ii) -y**

** (iii) 2x ^{2}y**

**(iv) – 8xy**

^{3}**(v) 3py**

^{2}**(vi) – 9a**

^{2}b^{3}**Solution:**

**(i)** Co-efficient of y = 1

**(ii)** Co-efficient of-y = – 1

**(iii)** Co-efficient of 2x2y is = 2

**(iv)** Co-efficient of – 8xy3 is = – 8

**(v)** Co-efficient of Ipy2 is = 3

**(vi)** Co-efficient of – 9a2b3 is = – 9

**Question 8.**

**In -5x ^{3}y^{2}z^{4}; write the coefficient of:**

**(i) z**

^{2}**(ii) y**

^{2}**(iii) yz**

^{2}**(iv) x**

^{3}y**(v) -xy**

^{2}**(vi) -5xy**

^{2}z**Also, write the degree of the given algebraic expression.**

**Solution:**

-5x^{3}y^{2}z^{4}

**(i)** Co-efficient of z2 is -5x^{3}y^{2}z^{2}

**(ii)** Co-efficient of y2 is -5x^{3}z^{4}

**(iii)** Co-efficient of yz^{2} is -5x^{3}yz^{2}

**(iv)** Co-efficient of x^{3}y is -5yz^{4}

**(v)** Co-efficient of -xy^{2} is 5x^{2}z^{4}

**(vi)** Co-efficient of -5xy^{2}z is x^{2}z^{3}

Degree of the given expression is 3 + 2 + 4 = 9

**EXERCISE 11 (B)**

**Question 1.**

**Fill in the blanks :**

** (i) 8x + 5x = ………**

** (ii) 8x – 5x =……..**

** (iii) 6xy ^{2} + 9xy^{2} =……..**

**(iv) 6xy**

^{2}– 9xy^{2}= ………**(v) The sum of 8a, 6a and 5b = ……..**

**(vi) The addition of 5, 7xy, 6 and 3xy = …………**

**(vii) 4a + 3b – 7a + 4b = ……….**

**(viii) – 15x + 13x + 8 = ………**

**(ix) 6x**

^{2}y + 13xy^{2}– 4x^{2}y + 2xy^{2}= ……..**(x) 16x**

^{2}– 9x^{2}= and 25xy^{2}– 17xy^{2}=………**Solution :**

**Question 2.**

**Add :**

** (i)- 9x, 3x and 4x**

** (ii) 23y ^{2}, 8y^{2} and – 12y^{2}**

**(iii) 18pq – 15pq and 3pq**

**Solution:**

**Question 3.**

**Simplify :**

** (i) 3m + 12m – 5m**

** (ii) 7n ^{2} – 9n^{2} + 3n^{2}**

**(iii) 25zy—8zy—6zy**

**(iv) -5ax**

^{2}+ 7ax^{2}– 12ax^{2}**(v) – 16am + 4mx + 4am – 15mx + 5am**

**Solution:**

**Question 4.**

**Add : **

** (i) a + i and 2a + 3b**

** (ii) 2x + y and 3x – 4y**

** (iii)- 3a + 2b and 3a + b**

** (iv) 4 + x, 5 – 2x and 6x**

**Solution:**

**Question 5.**

**Find the sum of:**

** (i) 3x + 8y + 7z, 6y + 4z- 2x and 3y – 4x + 6z**

** (ii) 3a + 5b + 2c, 2a + 3b-c and a + b + c.**

** (iii) 4x ^{2}+ 8xy – 2y^{2} and 8xy – 5y^{2} + x^{2}**

**(iv) 9x**

^{2}– 6x + 7, 5 – 4x and 6 – 3x^{2}**(v) 5x**

^{2}– 2xy + 3y^{2}and – 2x^{2}+ 5xy + 9y^{2}**and 3x**

^{2}-xy- 4y^{2}**(vi) a**

^{2}+ b^{2}+ 2ab, 2b^{2}+ c^{2}+ 2bc**and 4c**

^{2}-a^{2}+ 2ac**(vii) 9ax – 6bx + 8, 4ax + 8bx – 7**

**and – 6ax – 46x – 3**

**(viii) abc + 2 ba + 3 ac, 4ca – 4ab + 2 bca**

**and 2ab – 3abc – 6ac**

**(ix) 4a**

^{2}+ 5b^{2}– 6ab, 3ab, 6a^{2}– 2b^{2}**and 4b**

^{2}– 5 ab**(x) x**

^{2}+ x – 2, 2x – 3x^{2}+ 5 and 2x^{2}– 5x + 7**(xi) 4x**

^{3}+ 2x^{2}– x + 1, 2x^{3}– 5x^{2}– 3x + 6, x^{2}+ 8 and 5x^{3}– 7x**Solution:**

**Question 6.**

**Find the sum of:**

** (i) x and 3y**

** (ii) -2a and +5**

** (iii) – 4x ^{2 }and +7x**

**(iv) +4a and -7b**

**(v) x**

^{3}+3x^{2}y and 2y^{2}**(vi) 11 and -by**

**Solution:**

**Question 7.**

**The sides of a triangle are 2x + 3y, x + 5y and 7x – 2y, find its perimeter.**

**Solution:**

**Question 8.**

**The two adjacent sides of a rectangle are 6a + 96 and 8a – 46. Find its, perimeter.**

**Solution**

**Question 9.**

**Subtract the second expression from the first:**

**Solution:**

**Question 10.**

**Subtract:**

**Solution:**

**Question 11.**

**Subtract – 5a ^{2} – 3a + 1 from the sum of 4a^{2} + 3 – 8a and 9a – 7.**

**Solution:**

**Question 12.**

**By how much does 8x ^{3} – 6x^{2} + 9x – 10 exceed 4x^{3} + 2x^{2} + 7x -3 ?**

**Solution:**

**Question 13.**

**What must be added to 2a ^{3} + 5a – a^{2} – 6 to get a^{2} – a – a^{3} + 1 ?**

**Solution:**

**Question 14.**

**What must be subtracted from a ^{2} + b^{2} + lab to get – 4ab + 2b^{2} ?**

**Solution:**

**Question 15.**

**Find the excess of 4m ^{2} + 4n^{2} + 4p^{2 }over m^{2}+ 3n^{2} – 5p^{2}**

**Solution:**

**Question 16.**

**By how much is 3x ^{3} – 2x^{2}y + xy^{2} -y^{3} less than 4x^{3} – 3x^{2}y – 7xy^{2} +2y^{3}**

**Solution:**

**Question 17.**

**Subtract the sum of 3a ^{2} – 2a + 5 and a^{2} – 5a – 7 from the sum of 5a^{2} -9a + 3 and 2a – a^{2} – 1**

**Solution:**

**Question 18.**

**The perimeter of a rectangle is 28x ^{3}+ 16x^{2} + 8x + 4. One of its sides is 8x^{2} + 4x. Find the other side**

**Solution:**

**Question 19.**

**The perimeter of a triangle is 14a ^{2} + 20a + 13. Two of its sides are 3a^{2} + 5a + 1 and a^{2} + 10a – 6. Find its third side.**

**Solution:**

**Question 20.**

**Solution:**

**Question 21.**

**Solution:**

**Question 22.**

**Simplify:**

**Solution:**

**EXERCISE 11 (C)**

**Question 1.**

**Multiply:**

**Solution:**

**Question 2.**

**Copy and complete the following multi-plications :**

**Solution:**

**Question 3.**

**Evaluate :**

**Solution:**

**Question 4.**

**Evaluate:**

**Solution:**

**Question 5.**

E**valuate :**

**Solution:**

**Question 6.**

**Multiply:**

**Solution:**

**Question 7.**

**Multiply:**

**Solution:**

**EXERCISE 11 (D)**

**Question 1.**

**Divide:**

**Solution:**

**Question 2.**

**Divide :**

**Solution:**

**Question 3.**

**The area of a rectangle is 6x ^{2}– 4xy – 10y^{2} square unit and its length is 2x + 2y unit. Find its breadth**

**Solution:**

**Question 4.**

**The area of a rectangular field is 25x ^{2} + 20xy + 3y^{2} square unit. If its length is 5x + 3y unit, find its breadth, Hence find its perimeter.**

**Solution:**

**Question 5.**

**Divide:**

**Solution:**

**EXERCISE 11 (E)**

**Simplify**

**Question 1.**

**Solution:**

**Question 2.**

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

**Solution:**

**Question 10.**

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

**Question 14.**

**Solution:**

**Question 15.**

**Solution:**

**Question 16.**

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

**Solution:**

**Question 19.**

**Solution:**

**Question 20.**

**Solution:**

**Question 21.**

**Solution:**

**Question 22.**

**Solution:**

**Question 23.**

**Solution:**

**Question 24.**

**Solution:**

**Question 25.**

**Solution:**

**Question 26.**

**Solution:**

**EXERCISE 11 (F)**

**Enclose the given terms in brackets as required :**

**Question 1. **

**x – y – z = x-{…….)**

**Solution: **x – y – z = x – (y + z)

**Question 2. **

**x**

^{2}– xy^{2}– 2xy – y^{2}= x^{2}– (…….. )**Solution: **x

^{2 }– xy

^{2 }– 2xy – y

^{2 }= x

^{2}– (xy

^{2}+ 2xy + y

^{2})

**Question 3. **

**4a – 9 + 2b – 6 = 4a – (…….. )**

**Solution: **4a – 9 + 2b – 6

= 4a – (9 – 2b + 6)

**Question 4. **

**x**

^{2}-y^{2}+ z^{2}+ 3x – 2y = x^{2}– (…….. )**Solution: **x

^{2}– y

^{2}+ z

^{2}+ 3x – 2y

= x

^{2}– (y

^{2}– z

^{2}– 3x + 2y)

**Question 5. **

**– 2a**

^{2}+ 4ab – 6a^{2}b^{2}+ 8ab^{2}= – 2a (……… )**Solution: ** – 2a

^{2}+ 4ab – 6a

^{2}b

^{2}+ 8ab

^{2 }= – 2a (a – 2b + 3ab

^{2}– 4b

^{2})

**Simplify :**

**Question 6. **

**2x – (x + 2y- z)**

**Solution: **2x-(x + 2y-z) = 2x – x – 2y + z

= x – 2y + z

**Question 7. **

**p + q – (p – q) + (2p – 3q)**

**Solution: **p + q – (p – q) + (2p- 3q)

= p + q – p + q + 2p – 3q = 2p – q

**Question 8. **

**9x – (-4x + 5)**

**Solution: **9x – (-4x + 5) = 9x + 4x – 5

= 13x- 5

**Question 9. **

**6a – (- 5a – 8b) + (3a + b)**

**Solution:**

6a – (- 5a – 8b) + (3a + b)

= 6a + 5a + 8b + 3a + b

= 6a + 5a + 3a + 8b + b

= 14a + 9b

** ****Question 10. **

**(p – 2q) – (3q – r)**

**Solution: **(p-2q) –

*(3*q – r) =p – 2q – 3q

*+*r =p – 5q + r

**Question 11. **

**9a (2b – 3a + 7c)**

**Solution: **9a (2b – 3a + 7c)

= 18ab – 27a

^{2}+ 63ca

**Question 12. **

**-5m (-2m**

*+ 3*n – 7p)**Solution: **-5m (-2m + 3n- 7p)

= – 5m x (-2m) + (-5m) (3n) – (-5m) (7p)

= 10m

^{2}– 15mn + 35 mp.

**Question 13. **

**-2x (x + y) + x**

^{2 }**Solution: **– 2x (x + y) + x

^{2 }= -2x x x + (-2x)y + x

^{2 }= – 2x

^{2}– 2xy + x

^{2}

= – 2x

^{2}+ x

^{2}– 2xy = – x

^{2}– 2xy

**Question 14.**

**Solution:**

**Question 15. **

**8 (2a + 3b – c) – 10 (a + 2b + 3c)**

**Solution: **8 (2a + 3b -c)- 10 (a + 2b + 3c)

= 16a + 24b – 8c – 10a – 20b- 30c

= 16a – 10a + 24b – 20b – 8c – 30c

= 6a + 4b – 38c

**Question 16.**

**Solution:**

**Question 17. **

**5 x (2x + 3y) – 2x (x – 9y)**

**Solution: **5x (2x + 3y) – 2x (x – 9y)

= 10x

^{2}+ 15xy – 2x

^{2}+ 18xy

= 10x

^{2 }– 2x

^{2}+ 15xy+ 18xy

= 8x

^{2}+ 33 xy

**Question 18. **

**a + (b + c – d)**

**Solution: **a + (b + c – d) = a + (b + c – d)

= a + b + c – d

**Question 19. **

**5 – 8x – 6 – x**

**Solution: **5 – 8x – 6 – x

= 5 – 6 – 8x – x

= -1 -7x

**Question 20. **

**2a + (6- )**

**Solution: **2a + (6 – )

= 2a + (b – a + b)

= 2a + b – a + b

*=*a

*+*2b

**Question 21. **

**3x + [4x – (6x – 3)]**

**Solution: **3x + [4x – (6x – 3)]

= 3x + [4x – 6x + 3]

= 3x + 4x – 6x + 3

= 3x + 4x – 6x + 3

= 7x – 6x + 3= x + 3

**Question 22. **

**5b – {6a + (8 – b – a)}**

**Solution: **5b- {6a + 8- 6-a}

= 5b – 6a – 8 + b + a

= -6a + a + 5b +b – 8

= -5a + 6b-8

**Question 23. **

**2x-[5y- (3x -y) + x]**

**Solution: **2x – [5y- (3x – y) + x]

= 2x – {5y – 3x +y + x}

= 2x – 5y + 3x -y – x

= 2x + 3x – x – 5y – y

= 4x – 6y

**Question 24. **

**6a – 3 (a + b – 2)**

**Solution: **6a –

*3*(a

*+*b –

*2)*

=6a – 3a – 3b + 6

=

= 3a -3b + 6

**Question 25. **

**8 [m + 2n-p – 7 (2m -n + 3p)]**

**Solution: **8 [m + 2n-p -1 (2m – n + 3p)]

8 [m + 2n-p- 14m + 7n-21p]

= 8m+ 16n -8p- 112m + 56n – 168p

= 8m – 112m + 16n + 56n -8p – 168p

= -104m + 72n – 176p

**Question 26. **

**{9 – (4p – 6q)} – {3q – (5p – 10)}**

**Solution: **{9 – {4p – 6q)} – {3q – (5p – 10)}

{9 – 4p + 6q} – {3q -5p+ 10}

*= 9 –*4p + 6q – 3q + 5p

*– 10*

= 9 – 4p +5p + 6q – 3q

= 9 – 4p +

*– 10*

= p + 3q – 1

**Question 27. **

**2 [a – 3 {a + 5 {a – 2) + 7}]**

**Solution: **2 [a – 3 {a + 5 {a – 2) + 7}]

= 2 [a- 3 {a + 5a- 10 + 7}]

= 2 [a -3a- 15a + 30 -21]

= 2a-6a- 30a + 60-42

= 2a- 36a + 60-42

= -34a + 18

**Question 28. **

**5a – [6a – {9a – (10a – )}]**

**Solution: **5a – [6a – {9a – (10a – 4a + 3a)}]

= 5a – [6a – {9a – (10a – 4a + 3a)}]

= 5a – [6a – {9a – 10a + 4a – 3a}]

= 5a- [6a – 9a + 10a – 4a + 3a]

= 5a – 6a + 9a – 10a + 4a – 3a

= 5a + 9a + 4a – 6a – 10a – 3a

= 18a – 19a = – a

**Question 29. **

**9x + 5 – [4x – {3x – 2 (4x – 3)}]**

**Solution: **9x + 5 – [4x – {3x – 2 (4x – 3)}]

= 9x + 5 – [4x – {3x – 8x + 6}]

= 9x + 5 – [4x – 3x + 8x – 6]

= 9x + 5-4x + 3x-8x + 6

= 9x + 3x-4x-8x + 5 + 6

= 12x- 12x+ 11 = 11

**Question 30. **

**(x + y – z)x + (z + x – y)y – (x + y – z)z**

**Solution: **(x + y – z)x + (z + x -y )y – (x + y -z)z

= x

^{2 }+ xy – zx + yz + xy -y

^{2 }– zx – yz + z

^{2 }

*= x*-y

^{2}^{2}

*+*z

^{2}

*+ 2*xy

*– 2*zx

**Question 31. **

**-1 [a-3 {b -4 (a-b-8) + 4a} + 10]**

**Solution: **– 1 [a – 3 {b – 4(a – b – 8) + 4a} + 10]

= -1 [a-3 {b-4{a-b-8) + 4a} + 10]

= -1[a-3 {b-4a + Ab +32 + 4a} + 10]

= -1 [a-3b+ 12a- 126-96- 12a + 10]

= -a + 3b – 12a + 12b + 96 + 12a – 10

= -a-12a + 12a+ 3b+ 12b-96-10

= – a + 15b – 106

**Question 32. **

**Solution: **

**Question 33. **

**10 – {4a – (7 – ) – (5a – )}**

**Solution: **

10 – {4a – (7 – ) – (5a – )}

= 10 – {4a – (7 – a + 5) – (5a – 1 – a)}

= 10- {4a -(12 -a) -(4a- 1)}

= 10 – {4a – 12 + a- 4a + 1}

= 10 – 4a + 12 – a + 4a- 1

= 10 + 12 – 1 – 4a – a + 4a

= 21 -a

**Question 34. **

**7a- [8a- (11a-(12a- )}]**

**Solution: **7a – [8a – {1 la – (12a

**–**)}]

= 7a-[8a-{11a-(12a-6a + 5a)}]

= 7a -[8a -{11a -(17a -6a)}]

= 7a- [8a- {11a-(11a)}]

= 7a- [8a- {11a- 11a}]

= 7a – 8a = -a

**Question 35. **

**Solution: **

**Question 36. **

**x-(3y- +2z- )**

**Solution: **x-(3y- +2z- )

= x – (3y – 4z + 3x + 2z -5y + 7x)

= x-(-2y-2z+10x)

= x + 2y + 2z- 10x

= -9x + 2y + 2z

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