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#### Question 1:

(i) 3x and 7x
(ii) −5xy and 9xy

We have
(i) 3x + 7x = (3 + 7)x = 10x
(ii) -5xy + 9xy = ( -5 + 9)xy = 4xy

#### Question 2:

Simplify each of the following:
(i) 7x3y + 9yx3
(ii) 12a2b + 3ba2

Simplifying the given expressions, we have
(i) 7x3y + 9yx3 = (7 + 9)x3y = 16x3y
(ii) 12a2b + 3ba2 = (12 + 3)a2b = 15a2b

#### Question 3:

(i) 7abc, −5abc, 9abc, −8abc
(ii) 2x2y, − 4x2y, 6x2y, −5x2y

Adding the given terms, we have
(i) 7abc + (- 5abc) + (9 abc) + (- 8abc)
= 7abc - 5abc + 9abc - 8abc
= (7 - 5 + 9 - 8)abc
= (16 - 13)abc
= 3abc

(ii) 2x2y + (- 4x2y) + 6x2y + (- 5x2y)
= 2x2y - 4x2y + 6x2y - 5x2y
= (2 - 4 + 6 - 5)x2y
= (8 - 9)x2y
= -x2y

#### Question 4:

(i)
(ii) ${a}^{4}-2{a}^{3}b+3a{b}^{3}+4{a}^{2}{b}^{2}+3{b}^{4},-2{a}^{4}-5a{b}^{3}+7{a}^{3}b-6{a}^{2}{b}^{2}+{b}^{4}$

Adding the given expressions, we have
(i) x3- 2x2y + 3xy2- y3+ 2x3- 5xy2 + 3x2y- 4y3
Collecting positive and negative like terms together, we get
x3+ 2x3- 2x2y + 3x2y + 3xy2- 5xy2 - y3 - 4y3
= 3x3 + x2y - 2xy2 - 5y3

(ii) (a4- 2a3b + 3ab3 + 4a2b2 + 3b4) + (-2a4- 5ab3 + 7a3b - 6a2b2 + b4)
a4- 2a3b + 3ab3 + 4a2b2 + 3b4 - 2a4- 5ab3 + 7a3b - 6a2b2 + b4
Collecting positive and negative like terms together, we get
a4 - 2a4 - 2a3b + 7a3b + 3ab3 - 5ab3 + 4a2b2 - 6a2b2 + 3b4 + b4
= - a4 + 5a3b - 2ab3 -  2a2b2 + 4b4

#### Question 5:

(i)
(ii)

(i) Required expression = (8a - 6ab + 5b) + (- 6a - ab - 8b) + ( - 4a + 2ab + 3b)
Collecting positive and negative like terms together, we get
8a - 6a - 4a - 6ab - ab + 2ab + 5b - 8b + 3b
= 8a - 10a - 7ab + 2ab + 8b - 8b
= - 2a - 5ab

(ii) Required expression = (5x3 + 7 + 6x - 5x2) + (2x2 - 8 - 9x) + (4x - 2x2 + 3x3) + (3x3- 9x - x2) + ( x - x2 - x3- 4)
Collecting positive and negative like terms together, we get
5x3+ 3x3 + 3x3- x3- 5x2 + 2x2 - 2x2 - x2- x2 + 6x - 9x + 4x - 9x + x + 7 - 8 - 4
= 11x3 - x3 - 7x2 + 11x - 18x + 7 - 12
= 10x3 - 7x2 - 7x - 5

#### Question 6:

(i) $x-3y-2z\phantom{\rule{0ex}{0ex}}5x+7y-8z\phantom{\rule{0ex}{0ex}}3x-2y+5z$
(ii) $4ab-5bc+7ca\phantom{\rule{0ex}{0ex}}-3ab+2bc-3ca\phantom{\rule{0ex}{0ex}}5ab-3bc+4ca$

(i)  Required expression = (x - 3y - 2z) + (5x +7y - 8z) +(3x - 2y + 5z)
Collecting positive and negative like terms together, we get
x + 5x + 3x - 3y + 7y - 2y - 2z - 8z + 5z
= 9x - 5y + 7y - 10z + 5z
= 9x + 2y - 5z

(ii) Required expression = (4ab - 5bc + 7ca) + (- 3ab + 2bc - 3ca ) + (5ab - 3bc + 4ca)
Collecting positive and negative like terms together, we get
4ab - 3ab + 5ab - 5bc + 2bc - 3bc + 7ca - 3ca + 4ca
= 9ab - 3ab - 8bc + 2bc + 11 ca  - 3ca
= 6ab - 6bc + 8ca

#### Question 7:

Add 2x2 − 3x + 1 to the sum of 3x2 − 2x and 3x + 7.

Sum of 3x2 - 2x and 3x + 7
= (3x2 - 2x) + ( 3x +7)
= 3x2 - 2x + 3x + 7
= (3x2 + x  + 7)
Now, required expression = (2x2 - 3x + 1) + (3x2 + x  + 7)
= 2x2 + 3x2 - 3x + x + 1 + 7
= 5x2 - 2x + 8

#### Question 8:

Add x2 + 2xy + y2 to the sum of x2 − 3y2 and 2x2 − y2+ 9.

Sum of x2 - 3y2 and 2x2 - y2 + 9
= (x2 - 3y2) + (2x2 - y2 + 9)
= x2 + 2x2 - 3y2 - y2+ 9
= 3x2 - 4y2 + 9

Now, required expression = (x2 + 2xy + y2) + (3x2 - 4y2 + 9)
= x2 + 3x2 + 2xy + y2 - 4y2 + 9
= 4x2 + 2xy  - 3y2 + 9

#### Question 9:

Add a3 + b3 − 3 to the sum of 2a3 − 3b3 − 3ab + 7 and −a3 + b3 + 3ab −9.

First, we need to find the sum of 2a3 - 3b3 - 3ab + 7 and - a3 + b3 + 3ab - 9.
= (2a3 - 3b3 - 3ab + 7) + (- a3 + b3 + 3ab - 9)
Collecting positive and negative like terms together, we get
= 2a3 - a3- 3b3 + b3 - 3ab + 3ab + 7 - 9
= a3 - 2b3 - 2

Now, the required expression = (a3 + b3 - 3) + (a3 - 2b3 - 2).
= a3 + a3 + b3- 2b3 - 3 - 2
= 2a3 - b3 - 5

#### Question 10:

Subtract:
(i) 7a2b from 3a2b
(ii) 4xy from −3xy

(i) Required expression  =  3a2b - 7a2b
= (3 - 7)a2b
= - 4a2b

(ii) Required expression  = - 3 xy - 4xy
= - 7xy

#### Question 11:

Subtract:
(i) −4x from 3y
(ii) −2x from −5y

(i) Required expression = (3y) - (-4x)
= 3y + 4x

(ii) Required expression = (-5y) - (-2x)
= -5y + 2x

#### Question 12:

Subtract:
(i)
(ii)
(iii)

(i) Required expression = (4 - 5x + 6x2 - 8x3) - (6x3 - 7x2 + 5x - 3)
= 4 - 5x + 6x2 - 8x3 - 6x3 + 7x2 - 5x + 3
= - 8x3- 6x3 + 7x2 + 6x2- 5x - 5x + 3 + 4
= - 14x3 + 13x2 - 10x +7

(ii) Required expression  = (5x2 - y + z + 7) - (- x2 - 3z)
= 5x2 - y + z + 7 + x2 + 3z
= 5x2+ x2 - y + z + 3z + 7
= 6x2 - y + 4z + 7

(iii) Required expression = (y3 - 3xy2 - 4x2y) - (x3 + 2x2y + 6xy2 - y3)
= y3 - 3xy2 - 4x2y - x3 - 2x2y - 6xy2 + y3
= y3 + y3 - 3xy2- 6xy2 - 4x2y - 2x2y - x3
= 2y3- 9xy2 - 6x2y - x3

#### Question 13:

From
(i) p3 − 4 + 3p2, take away 5p2 − 3p3 + p − 6
(ii) 7 + xx2, take away 9 + x + 3x2 + 7x3
(iii) 1 − 5y2, take away y3 + 7y2 + y + 1
(iv) x3 − 5x2 + 3x + 1, take away 6x2 − 4x3 + 5 + 3x

(i) Required expression = (p3 - 4 + 3p2) - (5p2 - 3p3 + p - 6)
= p3 - 4 + 3p2 - 5p2 + 3p3 - p + 6
= p3 + 3p3 + 3p2 - 5p2- p - 4+ 6
= 4p3 - 2p2 - p + 2

(ii) Required expression = (7 + x - x2) - (9 + x + 3x2 + 7x3)
= 7 + x - x2 - 9 - x - 3x2 - 7x3
= - 7x3- x2 - 3x2 + 7 - 9
= - 7x3 - 4x2 - 2

(iii) Required expression = (1 - 5y2) - (y3+ 7y2 + y + 1)
= 1 - 5y2 - y3 - 7y2 - y - 1
= - y3- 5y2 - 7y2 - y
= - y3- 12y2 - y

(iv) Required expression = (x3 - 5x2 + 3x + 1) - (6x2 - 4x3 + 5 +3x)
= x3 - 5x2 + 3x + 1 - 6x2 + 4x3 - 5 - 3x
= x3 + 4x3 - 5x2 - 6x2 + 1 - 5
= 5x3 - 11x2 - 4

#### Question 14:

From the sum of 3x2 − 5x + 2 and −5x2 − 8x + 9 subtract 4x2 − 7x + 9.

Required expression = {(3x2 - 5x + 2) + (- 5x2 - 8x + 9)} - (4x2 - 7x + 9)
= {3x2 - 5x + 2 - 5x2 - 8x + 9} -  (4x2 - 7x + 9)
= {3x2 - 5x2 - 5x - 8x + 2 + 9} -  (4x2 - 7x + 9)
= {- 2x2 - 13x +11} - (4x2 - 7x + 9)
= - 2x2 - 13x + 11 - 4x2 + 7x - 9
= - 2x2 - 4x2 - 13x + 7x + 11 - 9
= - 6x2 - 6x + 2

#### Question 15:

Subtract the sum of 13x − 4y + 7z and −6z + 6x + 3y from the sum of 6x − 4y − 4z and 2x + 4y − 7.

Sum of (13x - 4y + 7z) and ( - 6z + 6x + 3y)
= {(13x - 4y + 7z) + (- 6z + 6x + 3y)
={13x - 4y + 7z - 6z + 6x + 3y}
= {13x + 6x - 4y + 3y + 7z - 6z}
= 19x - y + z

Sum of (6x − 4y − 4z) and (2x + 4y − 7)
= (6x − 4y − 4z) + (2x + 4y − 7)
= 6x − 4y − 4z + 2x + 4y − 7
= 8x - 4z - 7

Now, required expression = {(8x - 4z - 7) - (19x - y + z)}
= 8x - 4z - 7 - 19x + y - z
= 8x - 19x + y - 4z - z - 7
= - 11x + y - 5z - 7

#### Question 16:

From the sum of x2 + 3y2 − 6xy, 2x2y2 + 8xy, y2 + 8 and x2 − 3xy subtract −3x2 + 4y2xy + xy + 3.

Sum of (x2 + 3y2 - 6xy), (2x2 - y2 + 8xy), (y2 + 8) and (x2 - 3xy)
={(x2 + 3y2 - 6xy) + (2x2 - y2 + 8xy) + ( y2 + 8) + (x2 - 3xy)}
={x2 + 3y2 - 6xy + 2x2 - y2 + 8xy + y2 + 8 + x2 - 3xy}
= {x2+ 2x2+ x2 + 3y2- y2 + y2- 6xy + 8xy - 3xy + 8}
= 4x2 + 3y2 - xy + 8

Now, required expression = (4x2 + 3y2 - xy + 8) - (- 3x2 + 4y2 - xy + x - y + 3)
= 4x2 + 3y2 - xy + 8 + 3x2 - 4y2 + xy - x + y - 3
= 4x2 + 3x2+ 3y2- 4y2- x + y - 3 + 8
= 7x2 - y2- x + y + 5

#### Question 17:

What should be added to xy − 3yz + 4zx to get 4xy − 3zx + 4yz + 7?

The required expression can be got by subtracting xy - 3yz + 4zx from 4xy - 3zx + 4yz + 7.
Therefore, required expression = (4xy - 3zx + 4yz + 7) - (xy - 3yz + 4zx)
= 4xy - 3zx + 4yz + 7 - xy + 3yz - 4zx
= 4xy - xy - 3zx - 4zx + 4yz + 3yz + 7
= 3xy - 7zx + 7yz + 7

#### Question 18:

What should be subtracted from x2 xy + y2x + y + 3 to obtain −x2 + 3y2 − 4xy + 1?

Let 'M' be the required expression. Then, we have
x2 - xy + y2 - x + y + 3 - M = - x2 + 3y2 - 4xy + 1
Therefore,
M = (x2 - xy + y2 - x + y + 3) - (- x2 + 3y2 - 4xy + 1)
= x2 - xy + y2 - x + y + 3 + x2 - 3y2 + 4xy - 1
Collecting positive and negative like terms together, we get
x2 + x2- xy + 4xy + y2- 3y2 - x + y + 3 - 1
= 2x2 + 3xy- 2y2- x + y + 2

#### Question 19:

How much is x − 2y + 3z greater than 3x + 5y − 7?

Required expression  = (x - 2y + 3z) - (3x + 5y - 7)
=  x - 2y + 3z - 3x - 5y + 7
Collecting positive and negative like terms together, we get
x - 3x - 2y - 5y + 3z + 7
= - 2x - 7y + 3z + 7

#### Question 20:

How much is x2 − 2xy + 3y2 less than 2x2 − 3y2 + xy?

Required expression = (2x2 - 3y2 + xy) - (x2 - 2xy + 3y2)
= 2x2 - 3y2 + xy - x2 + 2xy - 3y2
Collecting positive and negative like terms together, we get
2x2 - x2 - 3y2 - 3y2 + xy + 2xy
= x2 - 6y2 + 3xy

#### Question 21:

How much does a2 − 3ab + 2b2 exceed 2a2 − 7ab + 9b2?

Required expression = (a2 - 3ab + 2b2) - (2a2 - 7ab + 9b2)
= a2 - 3ab + 2b2 - 2a2 + 7ab - 9b2
Collecting positive and negative like terms together, we get
=  a2 - 2a2  - 3ab + 7ab  + 2b2 -  9b2
= - a2 + 4ab - 7b2

#### Question 22:

What must be added to 12x3 − 4x2 + 3x − 7 to make the sum x3 + 2x2 − 3x + 2?

Let 'M' be the required expression. Thus, we have
12x3 - 4x2 + 3x - 7 + M = x3 + 2x2 - 3x + 2
Therefore,
M = (x3 + 2x2 - 3x + 2) - (12x3 - 4x2 + 3x - 7)
=  x3 + 2x2 - 3x + 2 - 12x3 + 4x2 - 3x + 7
Collecting positive and negative like terms together, we get
x3- 12x3 + 2x2 + 4x2 - 3x - 3x + 2 + 7
= - 11x3 + 6x2 - 6x + 9

#### Question 23:

If P = 7x2 + 5xy − 9y2, Q = 4y2 − 3x2 − 6xy and R = −4x2 + xy + 5y2, show that P + Q + R = 0.

We have
P + Q + R = (7x2 + 5xy - 9y2) + (4y2 - 3x2 - 6xy) + (- 4x2 + xy + 5y2)
= 7x2 + 5xy - 9y2 + 4y2 - 3x2 - 6xy - 4x2 + xy + 5y2
Collecting positive and negative like terms together, we get
7x2- 3x2 - 4x2 + 5xy - 6xy + xy - 9y2 + 4y2 + 5y2
= 7x2- 7x+ 6xy - 6xy  - 9y2 + 9y2
= 0

#### Question 24:

If P = a2 b2 + 2ab, Q = a2 + 4b2 − 6ab, R = b2 + b, S = a2 − 4ab and T = −2a2 + b2ab + a. Find P + Q + R + S − T.

We have
P + Q + R + S - T = {(a2 - b2 + 2ab) + (a2 + 4b2 - 6ab) + (b2 + b) + (a2 - 4ab)} - (-2a2 + b2 - ab + a)
= {a2 - b2 + 2ab + a2 + 4b2 - 6ab + b2 + b + a2 - 4ab}- (- 2a2 + b2 - ab + a)
= {3a2 + 4b2 - 8ab + b } - (-2a2 + b2 - ab + a)
= 3a2+ 4b2 - 8ab + b + 2a2 - b2 + ab - a
Collecting positive and negative like terms together, we get
3a2 + 2a2 + 4b2 - b2 - 8ab + ab - a + b
= 5a2 + 3b2- 7ab - a + b

#### Question 1:

Place the last two terms of the following expressions in parentheses preceded by a minus sign:
(i) x + y − 3z + y
(ii) 3x 2y − 5z − 4
(iii) 3a 2b + 4c − 5
(iv) 7a + 3b + 2c + 4
(v) 2a2 b2 − 3ab + 6
(vi) a2 + b2 c2 + ab − 3ac

We have
(i) x + y − 3z + y = x  + y − (3z - y )
(ii) 3x − 2y − 5z − 4 = 3x - 2y - (5z + 4)
(iii) 3a − 2b + 4c − 5 = 3a - 2b - (- 4c + 5)
(iv) 7a + 3b + 2c + 4 = 7a + 3b - (- 2c - 4)
(v) 2a2 − b2 − 3ab + 6 = 2a2 − b2 − (3ab - 6)
(vi) a2 + b2 − c2 + ab − 3ac = a2 + b2 − c2 - (- ab + 3ac)

#### Question 2:

Write each of the following statements by using appropriate grouping symbols:
(i) The sum of a b and 3a − 2b + 5 is subtracted from 4a + 2b − 7.
(ii) Three times the sum of 2x + y − {5 − (x − 3y)} and 7x − 4y + 3 is subtracted from 3x − 4y + 7.
(iii) The subtraction of x2y2 + 4xy from 2x2 + y2 − 3xy is added to 9x2 − 3y2xy.

(i) The sum of a − b and 3a − 2b + 5 = {(a - b) + (3a − 2b + 5)}.
This is subtracted from 4a + 2b - 7.
Thus, the required expression is {4a + 2b - 7) - {(a - b) + (3a − 2b + 5)}.

(ii) Three times the sum of 2x + y − {5 − (x − 3y)} and 7x − 4y + 3 = 3[(2x + y) − {5 − (x − 3y)} + (7x − 4y + 3)].
This is subtracted from 3x - 4y +7.
Thus, the required expression is (3x - 4y +7) - 3[(2x + y) − {5 − (x − 3y)} + (7x − 4y + 3)].

(iii) The product of subtraction of x2 − y2 + 4xy from 2x2 + y2 − 3xy is given by {(2x2 + y2 − 3xy) - (x2 − y2 + 4xy)}.
When the above equation is added to 9x2 − 3y2 − xy, we get
{(2x2 + y2 − 3xy) - (x2 − y2 + 4xy)} + (9x2 − 3y2 − xy)

#### Question 1:

Simplify each of the following algebraic expressions by removing grouping symbols.
2x + (5x − 3y)

We have
2x + (5x − 3y)
Since the '+' sign precedes the parentheses, we have to retain the sign of each term in the parentheses when we remove them.
= 2x + 5x - 3y
= 7x - 3y

#### Question 2:

Simplify each of the following algebraic expressions by removing grouping symbols.
3x − (y − 2x)

We have
3x − (y − 2x)
Since the '-' sign precedes the parentheses, we have to change the sign of each term in the parentheses when we remove them. Therefore, we have
3x − y + 2x
= 5x - y

#### Question 3:

Simplify each of the following algebraic expressions by removing grouping symbols.
5a − (3b − 2a + 4c)

We have
5a − (3b − 2a + 4c)
Since the '-' sign precedes the parentheses, we have to change the sign of each term in the parentheses when we remove them.
= 5a - 3b + 2a - 4c
= 7a - 3b - 4c

#### Question 4:

Simplify each of the following algebraic expressions by removing grouping symbols.
−2 (x2y2 + xy) − 3(x2 + y2xy)

We have
− 2(x2 − y2 + xy) − 3(x2 + y2 − xy)
Since the '-' sign precedes the parentheses, we have to change the sign of each term in the parentheses when we remove them.
= - 2x2 + 2y2 - 2xy - 3x2 - 3y2 + 3xy
= - 2x2 - 3x2 + 2y2- 3y2 - 2xy + 3xy
= - 5x2 - y2 + xy

#### Question 5:

Simplify each of the following algebraic expressions by removing grouping symbols.
3x + 2y − {x − (2y − 3)}

We have
3x + 2y − {x − (2y − 3)}
First, we have to remove the small brackets (or parentheses): ( ). Then, we have to remove the curly brackets (or braces): { }.
Therefore,
= 3x + 2y − {x − 2y + 3}
= 3x + 2y − x + 2y - 3
= 2x + 4y - 3

#### Question 6:

Simplify each of the following algebraic expressions by removing grouping symbols.
5a − {3a − (2 − a) + 4}

We have
5a − {3a − (2 − a) + 4}
First, we have to remove the small brackets (or parentheses): ( ). Then, we have to remove the curly brackets (or braces): { }.
Therefore,
= 5a − {3a − 2 + a + 4}
= 5a − 3a + 2 - a - 4
= 5a - 4a - 2
= a - 2

#### Question 7:

Simplify each of the following algebraic expressions by removing grouping symbols.
a − [b − {a − (b − 1) + 3a}]

First we have to remove the parentheses, or small brackets, ( ), then the curly brackets, { }, and then the square brackets [ ].
Therefore, we have
a - [b - {a - (b - 1) + 3a}]
= a - [b - {a - b + 1 + 3a}]
= a - [b - {4a - b + 1}]
= a - [b - 4a + b - 1]
= a - [2b - 4a - 1]
= a - 2b + 4a + 1
= 5a - 2b + 1

#### Question 8:

Simplify each of the following algebraic expressions by removing grouping symbols.
a − [2b − {3a − (2b − 3c)}]

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
a - [2b - {3a - (2b - 3c)}]
= a - [2b - {3a - 2b + 3c}]
= a - [2b - 3a + 2b - 3c]
= a - [4b - 3a - 3c]
= a - 4b + 3a + 3c
= 4a - 4b + 3c

#### Question 9:

Simplify each of the following algebraic expressions by removing grouping symbols.
x + [5y − {2x − (3y − 5x)}]

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
- x + [5y - {2x - (3y - 5x)}]
= - x + [5y - {2x - 3y + 5x}]
= - x + [5y - {7x - 3y}]
= - x + [5y - 7x + 3y]
= - x + [8y - 7x]
= - x + 8y - 7x
= - 8x + 8y

#### Question 10:

Simplify each of the following algebraic expressions by removing grouping symbols.
2a − [4b − {4a − 3(2ab)}]

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
2a - [4b - {4a - 3(2a - b)}]
= 2a - [4b - {4a - 6a + 3b}]
= 2a - [4b - {- 2a + 3b}]
= 2a - [4b + 2a - 3b]
= 2a - [b + 2a]
= 2a - b - 2a
= - b

#### Question 11:

Simplify each of the following algebraic expressions by removing grouping symbols.
a − [a + {a + b − 2a − (a − 2b)} − b]

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets,{ }, and then the square brackets, [ ].
Therefore, we have
- a - [a + {a + b - 2a - (a - 2b)} - b]
= - a - [a + {a + b - 2a - a + 2b} - b]
= - a - [a + {- 2a + 3b} - b]
= - a - [a - 2a + 3b - b]
= - a - [- a + 2b]
= - a + a - 2b
= - 2b

#### Question 12:

Simplify each of the following algebraic expressions by removing grouping symbols.
2x − 3y − [3x − 2y − {x − z − (x − 2y)}]

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
2x - 3y - [3x - 2y - {x - z - (x - 2y)}]
= 2x - 3y - [3x - 2y - {x - z - x + 2y}]
= 2x - 3y - [3x - 2y - {- z + 2y}]
= 2x - 3y - [3x - 2y + z - 2y]
= 2x - 3y - [3x - 4y + z]
= 2x - 3y - 3x + 4y - z
= - x + y - z

#### Question 13:

Simplify each of the following algebraic expressions by removing grouping symbols.
5 + [x − {2y − (6x + y − 4) + 2x} − {x − (y − 2)}]

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
5 + [x - {2y - (6x + y - 4) + 2x} - {x - (y - 2)}]
= 5 + [x - {2y - 6x - y + 4 + 2x} - {x - y + 2}]
= 5 + [x - {y - 4x + 4} - {x - y + 2}]
= 5 + [x - y + 4x - 4 - x + y - 2]
= 5 + [4x - 6]
= 5 + 4x - 6
= 4x - 1

#### Question 14:

Simplify each of the following algebraic expressions by removing grouping symbols.
x2 − [3x + {2x − (x2 − 1) + 2}]

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
x2 - [3x + {2x - (x2 - 1)} + 2]
= x2 - [3x + {2x - x2 + 1} + 2]
= x2 - [3x + 2x - x2 + 1+ 2]
= x2 - [5x - x2 + 3]
= x2 - 5x + x2 - 3
= 2x2 - 5x - 3

#### Question 15:

Simplify each of the following algebraic expressions by removing grouping symbols.
20 − [5xy + 3{x2 − (xyy) − (xy)}]

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
20 - [5xy + 3{x2 - (xy - y) - (x - y)}]
= 20 - [5xy + 3{x2 - xy + y - x + y}]
= 20 - [5xy + 3{x2 - xy + 2y - x}]
= 20 - [5xy + 3x2 - 3xy + 6y - 3x]
= 20 - [2xy + 3x2 + 6y - 3x]
= 20 - 2xy - 3x2 - 6y + 3x
= - 3x2 - 2xy - 6y + 3x + 20

#### Question 16:

Simplify each of the following algebraic expressions by removing grouping symbols.
85 − [12x − 7(8x − 3) − 2 {10x − 5(2 − 4x)}]

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
85 - [12x - 7(8x - 3) - 2{10x - 5(2 - 4x)}]
= 85 - [12x - 56x + 21 - 2{10x - 10 + 20x}]
= 85 - [12x - 56x + 21 - 2{30x - 10}]
= 85 - [12x - 56x + 21 - 60x + 20]
= 85 - [12x - 116x + 41]
= 85 - [- 104x + 41]
= 85 + 104x - 41
= 44 + 104x

#### Question 17:

Simplify each of the following algebraic expressions by removing grouping symbols.
xy [yzzx − {yx − (3yxz) − (xyzy)}]

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
xy - [yz - zx - {yx - (3y - xz) - (xy - zy)}]
= xy - [yz - zx - {yx - 3y + xz - xy + zy}]
= xy - [yz - zx - {- 3y + xz + zy}]
= xy - [yz - zx + 3y - xz - zy]
= xy - [- zx + 3y - xz]
= xy - [- 2zx + 3y]
= xy + 2xz - 3y

#### Question 1:

Mark the correct alternative in the following question:

Which of the following pairs is/are like terms?
(1) x           (2) x2           (3) 3x3           (4) 4x3

(a) 1, 2                               (b) 2, 3                               (c) 3, 4                              (d) None of these

Since, 3x3 and 4x3 is the pair of like terms.

Hence, the correct option is (c).

#### Question 2:

Mark the correct alternative in the following question:

Which of the following is not a monomial?

(a) 2x2 + 1                           (b) 3x4                           (c) ab                           (d) x2y

Since, 2x2 + 1 has two terms 2x2 and 1.

So, 2x2 + 1 is a binomial.

Hence, the correct alternative is option (a).

#### Question 3:

Mark the correct alternative in the following question:

The sum of the coefficients in the monomials 3a2b and $-$2ab2 is

(a) 5                           (b) $-$1                           (c) 1                           (d) 6

Since, the coefficient in the monomial 3a2b is 3 and the coefficient in the monomial $-$2ab2 is $-$2.

So, the sum of the coefficients in the monomials 3a2b and $-$2ab2 = 3 + ($-$2) = 3 $-$ 2 = 1

Hence, the correct alternative is option (c).

#### Question 4:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (b).

#### Question 5:

Mark the correct alternative in the following question:

If a, b and c are respectively the coefficients of x2 in $-$x2, 2x2 + x and 2x $-$ x2, respectively, then a + b + c =

(a) 0                                         (b) $-$2                                         (c) 2                                         (d) $-$1

As, the coefficient x2 in $-$x= $-$1, the coefficient x2 in 2x2 + x = 2 and the coefficient x2 in 2x $-$ x= $-$1.

Now, a + b + c = ($-$1) + 2 + ($-$1) = $-$2 + 2 = 0

Hence, the correct alternative is option (a).

#### Question 6:

Mark the correct alternative in the following question:

The sum of the coefficients in the terms of 2x2y $-$ 3xy2 + 4xy is

(a) $-$3                                     (b) 3                                     (c) 9                                     (d) 5

As, the coefficient in the term 2x2y = 2, the coefficient in the term $-$3xy2 = $-$3 and the coefficient in the term 4xy = 4.

So, the sum of the coefficients in the terms of 2x2y $-$ 3xy2 + 4xy = 2 + ($-$3) + 4 = $-$3 + 6 = 3

Hence, the correct alternative is option (b).

#### Question 7:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (c).

#### Question 8:

Mark the correct alternative in the following question:

If a and b are respectively the sum and product of coefficients of terms in the expression x2 + y2 + z2 $-$ xy $-$ yz $-$ zx, then a + 2b =

(a) 0                                             (b) 2                                             (c) $-$2                                             (d) $-$1

We have,
The expression x2 + y2 + z2 $-$ xy $-$ yz $-$ zx,

 Terms Coefficients x2 1 y2 1 z2 1 $-$xy $-$1 $-$yz $-$1 $-$zx $-$1 Sum, a 0 Product, b $-$1

So, a + 2b = 0 + 2($-$1) = $-$2

Hence, the correct alternative is option (c).

#### Question 9:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (d).

#### Question 10:

Mark the correct alternative in the following question:

The sum of the values of the expression 2x2 + 2x + 2 when x = $-$1 and x = 1 is

(a) 6                                         (b) 8                                         (c) 4                                         (d) 2

Since, when x = $-$1, the value of the expression 2x2 + 2x + 2 = 2($-$1)2 + 2($-$1) + 2 = 2 $-$ 2 + 2 = 2

And, when x = 1, the value of the expression 2x2 + 2x + 2 = 2(1)2 + 2(1) + 2 = 2 + 2 + 2 = 6

So, the sum of the values of the expression 2x2 + 2x + 2 when x = $-$1 and x = 1 = 2 + 6 = 8

Hence, the correct alternative is option (b).

#### Question 11:

Mark the correct alternative in the following question:

What should be added to 3x2 + 4 to get 9x2 $-$ 7?

(a) 6x2 $-$ 11                               (b) 6x2 + 11                               (c) 12x2 $-$ 11                               (d) 12x2 + 11

Since, (9x2 $-$ 7) $-$ (3x2 + 4) = 9x2 $-$ 7 $-$ ​3x2 $-$ 4 = 6x2 $-$ 11

So, 6x2 $-$ 11 should added to 3x2 + 4 to get 9x2 $-$ 7.

Hence, the correct alternative is option (a).

#### Question 12:

Mark the correct alternative in the following question:

How much is a2 $-$ 3a greater than 2a2 + 4a?

(a) a2 $-$ 7a                                (b) a2 + 7a                                (c) $-$a2 $-$ 7a ​                               (d) $-$a2 + 7a

Since, (a2 $-$ 3a$-$ (2a2 + 4a) = a2 $-$ 3a $-$ 2a2 $-$ 4a$-$ a2 $-$​7a

So, a2 $-$ 3a is greater than 2a2 + 4a by $-$ a2 $-$​7a.

Hence, the correct alternative is option (c).

#### Question 13:

Mark the correct alternative in the following question:

How much is $-$2x2 + x + 1 less than x2 + 2x $-$ 3?

(a) $-$x2 + 3x $-$ 2                      (b) 3x2 + x $-$ 4                      (c) $-$3x2 $-$ x + 4                       (d) 3x2 + 3x $-$ 4

Since, (x2 + 2x $-$ 3) $-$ ($-$2x2 + x + 1) = x2 + 2x $-$ 3 + 2x2 $-$ x $-$ 1 = 3x2 + x $-$ 4

So, $-$2x2 + x + 1 is less than x2 + 2x $-$ 3 by 3x2 + x $-$ 4.

Hence, the correct alternative is option (b).

#### Question 14:

Mark the correct alternative in the following question:

​What should be added to xy + yz + zx to get $-$xy $-$ yz $-$ zx?

(a) $-$2xy $-$ 2yz $-$ 2zx                       (b) $-$3xy $-$ yz $-$ zx                       (c) $-$3xy $-$ 3yz $-$ 3zx ​                      (d) 2xy + 2yz + 2zx

Since, ($-$xy $-$ yz $-$ zx$-$ (xy + yz + zx) = $-$xy $-$ yz $-$ zx $-$ xy $-$ yz $-$ zx$-$2xy $-$ 2yz $-$ 2zx

So, $-$2xy $-$ 2yz $-$ 2zx should be added to xy + yz + zx to get $-$xy $-$ yz $-$ zx.

Hence, the correct alternative is option (a).

#### Question 1:

Identify the monomials, binomials, trinomials and quadrinomials from the following expressions:
(i) a2
(ii) a2 b2
(iii) x3 + y3 + z3
(iv) x3 + y3 + z3 + 3xyz
(v) 7 + 5
(vi) abc + 1
(vii) 3x − 2 + 5
(viii) 2x − 3x + 4
(ix) xy + yz + zx
(x) ax3 + bc2 + cx + d

The monomials, binomials, trinomials and quadrinomials are as follows.
(i) a2 is a monomial expression as it contains one term only.
(ii) a2 - b2 is a binomial expression as it contains two terms.
(iii) x3 + y3 + z3 is a trinomial expression as it contains three terms.
(iv) x3 + y3 + z3 + xyz is a quadrinomial expression as it contains four terms.
(v) 7 + 5 = 12 is a monomial expression as it contains one term only.
(vi) abc +1 is a binomial expression as it contains two terms.
(vii) 3x - 2 + 5 = 3x + 3 is a binomial expression as it contains two terms.
(viii) 2x - 3y + 4 is a trinomial expression as it contains three terms.
(ix) xy + yz + zx is a trinomial expression as it contains three terms.
(x) ax3 +bx2 +cx + d is a quadrinomial expression as it contains four terms.

#### Question 2:

Write all the terms of each of the following algebraic expressions:
(i) 3x
(ii) 2x 3
(iii) 2x2 − 7
(iv) 2x2 + y2 − 3xy + 4

The terms of each of the given algebraic expressions are as follows.
(i) 3x is the only term of the given algebraic expression.
(ii) 2x and -3 are the terms of the given algebraic expression.
(iii) 2x2 and -7 are the terms of the given algebraic expression.
(iv) 2x2, y2, -3xy  and 4 are the terms of the given algebraic expression.

#### Question 3:

Identify the like terms and also mention the numerical coefficients of those terms:
(i) 4xy, 5x2y, −3yx, 2xy2
(ii)

Like terms                                          Numerical coefficients
(i)   4xy, -3yx                                                4, -3
(ii) {7a2bc ,-3ca2b, $-\frac{4}{3}cb{a}^{2}$}                      { 7, -3,$-\frac{4}{3}$}
{$-\frac{5}{2}ab{c}^{2},\frac{3}{2}ab{c}^{2}$ }                                 {$-\frac{5}{2},\frac{3}{2}$}

#### Question 4:

Identify the like terms in the following algebraic expressions:
(i) ${a}^{2}+{b}^{2}-2{a}^{2}+{c}^{2}+4a$
(ii) $3x+4xy-2yz+\frac{5}{2}zy$
(iii) $abc+a{b}^{2}c+2ac{b}^{2}+3{c}^{2}ab+{b}^{2}ac-2{a}^{2}bc+3ca{b}^{2}$

The like terms in the given algebraic expressions are as follows.
(i) The like terms in the given algebraic expression are a2  and -2a2.

(ii) The like terms in the given algebraic expression are .
(iii) The like terms in the given algebraic expression are ab2c, 2acb2, b2ac and 3cab2.

#### Question 5:

Write the coefficient of x in the following:
(i) −12x
(ii) −7xy
(iii) xyz
(iv) −7ax

The coefficients of x are as follows.
(i) The numerical coefficient of x is -12.
(ii) The numerical coefficient of x is -7y.
(iii) The numerical coefficient of x is yz.
(iv) The numerical coefficient of x is -7a.

#### Question 6:

Write the coefficient of x2 in the following:
(i) −3x2
(ii) 5x2yx
(iii) $\frac{5}{7}{x}^{2}z$
(iv) $-\frac{3}{2}a{x}^{2}+yx$

The coefficients of x2 are as follows.
(i) The numerical coefficient of x2 is -3.
(ii) The numerical coefficient of x2 is 5yz.

(iii) The numerical coefficient of x2 is $\frac{5}{7}z$.
(iv) The numerical coefficient of x2 is $-\frac{3}{2}a$.

#### Question 7:

Write the coefficient of:
(i) y in −3y
(ii) a in 2ab
(iii) z in −7xyz
(iv) p in −3pqr
(v) y2 in 9xy2z
(vi) x3 in x3 + 1
(vii) x2 in −x2

The coefficients are as follows.

(i) The coefficient of  y is -3.
(ii) The coefficient of  a is 2b.
(iii) The coefficient of  z is -7xy.
(iv) The coefficient of  p is -3qr.
(v) The coefficient of  y2 is 9xz.
(vi) The coefficient of  x3 is 1.
(vii) The coefficient of  -x2 is - 1.

#### Question 8:

Write the numerical coefficient of each of the following:
(i) xy
(ii) −6yz
(iii) 7abc
(iv) −2x3y2z

The numerical coefficient of each of the given terms is as follows.
(i) The numerical coefficient in the term xy is 1.
(ii) The numerical coefficient in the term -6yz is - 6.
(iii) The numerical coefficient in the term 7abc is 7.
(iv) The numerical coefficient in the term -2x3y2z is - 2.

#### Question 9:

Write the numerical coefficient of each term in the following algebraic expressions:
(i) $4{x}^{2}y-\frac{3}{2}xy+\frac{5}{2}x{y}^{2}$
(ii) $-\frac{5}{3}{x}^{2}y+\frac{7}{4}xyz+3$

The numerical coefficient of each term in the given algebraic expressions is as follows.
Term                         Coefficient

(i)       4x2y                                   4

$-\frac{3}{2}xy$                              $-\frac{3}{2}$
$\frac{5}{2}x{y}^{2}$                                 $\frac{5}{2}$

(ii)    $-\frac{5}{3}{x}^{2}y$                            $-\frac{5}{3}$
$\frac{7}{4}xyz$                                 $\frac{7}{4}$
3                                        3

#### Question 10:

Write the constant term of each of the following algebraic expressions:
(i) x2y xy2 + 7xy − 3
(ii) a3 − 3a2 + 7a + 5

The constant term of each of the given algebraic expressions is as follows.
(i) The constant term in the given algebraic expression is -3.
(ii) The constant term in the given algebraic expression is 5.

#### Question 11:

Evaluate each of the following expressions for x = − 2, y = −1, z = 3:
(i) $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$
(ii) ${x}^{2}+{y}^{2}+{z}^{2}-xy-yz-zx$

We have x = −2, y = −1 and z = 3.
Thus,
(i)

(ii) x2 + y2 + z2- xy - yz - zx
= (-2)2 + (-1)2 + (3)2 - (-2)(-1) - (-1)(3) - (3)(-2)
= 4 + 1 + 9 - 2 + 3 + 6
= (4 + 1 + 9 + 3 + 6) - 2
= 23 - 2 = 21

#### Question 12:

Evaluate each of the following algebraic expressions for x = 1, y = −1, z = 2, a = −2, b = 1, c = −2:
(i) ax + by + cz
(ii) ax2 + by2cz2
(iii) axy + byz + cxy

We have x = 1, y = −1, z = 2, a = −2, b = 1 and c = −2.
Thus,
(i) ax + by + cz
= (-2)(1) + (1)(-1) + (-2)(2)
= - 2 + (- 1) + (- 4)
= - 2 -1 - 4 = -7

(ii) ax2 + by2 - cz2
= (-2)(1)2 + (1)(-1)2 - (-2)(2)2
= -2 x 1 + 1 - (-2 x 4)
= -2 + 1 - (-8)
= -2 + 1 + 8
= -2 + 9
= 7

(iii) axy + byz + cxy
= (-2)(1)(-1) + (1)(-1)(2) + (-2)(1)(-1)
= 2 + (-2) + 2
= 2 - 2 + 2
= 4 - 2
= 2

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