Rd Sharma 2020 2021 Solutions for Class 7 MATHS Chapter 7

Question 1:

Add the following:
(i) 3x and 7x
(ii) −5xy and 9xy

Answer:

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

Answer:

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

Question 3:

Add the following:
(i) 7abc, −5abc, 9abc, −8abc
(ii) 2x²y, − 4x²y, 6x²y, −5x²y

Answer:

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:

Add the following expressions:
(i) $x^{3} - 2 x^{2} y + 3 x y^{2} - y^{3}, 2 x^{3} - 5 x y^{2} + 3 x^{2} y - 4 y^{3}$
(ii) $a^{4} - 2 a^{3} b + 3 a b^{3} + 4 a^{2} b^{2} + 3 b^{4}, - 2 a^{4} - 5 a b^{3} + 7 a^{3} b - 6 a^{2} b^{2} + b^{4}$

Answer:

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:

Add the following expressions:
(i) $8 a - 6 a b + 5 b, - 6 a - a b - 8 b and - 4 a + 2 a b + 3 b$
(ii) $5 x^{3} + 7 + 6 x - 5 x^{2}, 2 x^{2} - 8 - 9 x, 4 x - 2 x^{2} + 3 x^{3}, 3 x^{3} - 9 x - x^{2} and x - x^{2} - x^{3} - 4$

Answer:

(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:

Add the following:
(i) $x - 3 y - 2 z 5 x + 7 y - 8 z 3 x - 2 y + 5 z$
(ii) $4 a b - 5 b c + 7 c a - 3 a b + 2 b c - 3 c a 5 a b - 3 b c + 4 c a$

Answer:

(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.

Answer:

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.

Answer:

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.

Answer:

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

Answer:

(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

Answer:

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

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

Question 12:

Subtract:
(i) $6 x^{3} - 7 x^{2} + 5 x - 3 from 4 - 5 x + 6 x^{2} - 8 x^{3}$
(ii) $- x^{2} - 3 z from 5 x^{2} - y + z + 7$
(iii) $x^{3} + 2 x^{2} y + 6 x y^{2} - y^{3} from y^{3} - 3 x y^{2} - 4 x^{2} y$

Answer:

(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 + x − x², 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

Answer:

(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.

Answer:

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.

Answer:

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, 2x² − y² + 8xy, y² + 8 and x² − 3xy subtract −3x² + 4y² − xy + x − y + 3.

Answer:

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?

Answer:

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 + y² −x + y + 3 to obtain −x² + 3y² − 4xy + 1?

Answer:

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?

Answer:

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?

Answer:

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?

Answer:

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?

Answer:

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.

Answer:

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- 7x²+ 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 + b² − ab + a. Find P + Q + R + S − T.

Answer:

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

Answer:

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 x² − y² + 4xy from 2x² + y² − 3xy is added to 9x² − 3y² − xy.

Answer:

(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)

Answer:

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)

Answer:

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)

Answer:

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 (x² − y² + xy) − 3(x² + y² − xy)

Answer:

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)}

Answer:

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}

Answer:

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}]

Answer:

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)}]

Answer:

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)}]

Answer:

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(2a − b)}]

Answer:

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]

Answer:

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)}]

Answer:

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)}]

Answer:

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}]

Answer:

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 − (xy − y) − (x − y)}]

Answer:

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)}]

Answer:

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 [yz − zx − {yx − (3y − xz) − (xy − zy)}]

Answer:

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

Answer:

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

Answer:

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

Answer:

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:

$The coefficient of x^{2} in - \frac{5}{3} x^{2} y is equal to (a) - \frac{5}{3} (b) - \frac{5}{3} y (c) \frac{5}{3} (d) \frac{5}{3} y$

Answer:

$Since, the coefficient of x^{2} in - \frac{5}{3} x^{2} y is equal to - \frac{5}{3} y .$

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

Answer:

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

Answer:

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:

$The product of the coefficients of terms in - \frac{4}{3} a b^{2} + \frac{1}{4} b c^{2} + 3 c a^{2} is (a) 1 (b) \frac{1}{2} (c) - 1 (d) 3$

Answer:

$As, the coefficient of the term - \frac{4}{3} a b^{2} is - \frac{4}{3}, the coefficient of the term \frac{1}{4} b c^{2} is \frac{1}{4} and the coefficient of the term 3 c a^{2} is 3 . So, the product of the coefficients of the terms = - \frac{4}{3} \times \frac{1}{4} \times 3 = - 1$

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

Answer:

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:

$If P = 3 x^{3} + 3 x^{2} + 3 x + 3 and Q = 3 x^{2} - 3 x + 3, then P - Q = (a) 3 x^{3} (b) 3 x^{3} + 6 x^{2} + 6 x + 6 (c) 6 x^{2} + 6 x + 6 (d) 3 x^{3} + 6 x$

Answer:

$We have, P = 3 x^{3} + 3 x^{2} + 3 x + 3 and Q = 3 x^{2} - 3 x + 3 Now, P - Q = (3 x^{3} + 3 x^{2} + 3 x + 3) - (3 x^{2} - 3 x + 3) = 3 x^{3} + 3 x^{2} + 3 x + 3 - 3 x^{2} + 3 x - 3 = 3 x^{3} + 6 x$

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
â€‹
(a) 6 (b) 8 (c) 4 (d) 2

Answer:

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?
â€‹
(a) 6x² $-$ 11 (b) 6x² + 11 (c) 12x² $-$ 11 (d) 12x² + 11

Answer:

Since, (9x² $-$ 7) $-$ (3x² + 4) = 9x² $-$ 7 $-$ â€‹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 â€‹ (d) $-$ a² + 7a

Answer:

Since, (a² $-$ 3a) $-$ (2a² + 4a) = a² $-$ 3a $-$ 2a² $-$ 4a = $-$ a² $-$ â€‹7a

So, a² $-$ 3a is greater than 2a² + 4a by $-$ a² $-$ â€‹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

Answer:

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:

â€‹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 â€‹ (d) 2xy + 2yz + 2zx

Answer:

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

Answer:

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

Answer:

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) $7 a^{2} b c, - 3 c a^{2} b, - \frac{5}{2} a b c^{2}, \frac{3}{2} a b c^{2}, - \frac{4}{3} c b a^{2}$

Answer:

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

Question 4:

Identify the like terms in the following algebraic expressions:
(i) $a^{2} + b^{2} - 2 a^{2} + c^{2} + 4 a$
(ii) $3 x + 4 x y - 2 y z + \frac{5}{2} z y$
(iii) $a b c + a b^{2} c + 2 a c b^{2} + 3 c^{2} a b + b^{2} a c - 2 a^{2} b c + 3 c a b^{2}$

Answer:

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 $- 2 yz and \frac{5}{2} zy$ .
(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

Answer:

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) $- \frac{3}{2} a x^{2} + y x$

Answer:

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

Answer:

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

Answer:

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

Answer:

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

(i)       4x²y                                   4

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

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

Answer:

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) $x^{2} + y^{2} + z^{2} - x y - y z - z x$

Answer:

We have x = −2, y = −1 and z = 3.
Thus,
(i)
$\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = (\frac{- 2}{- 1}) + (\frac{- 1}{3}) + (\frac{3}{- 2}) = 2 - \frac{1}{3} - \frac{3}{2} = \frac{12 - 2 - 9}{6} = \frac{12 - 11}{6} = \frac{1}{6}$

(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 + by² − cz²
(iii) axy + byz + cxy

Answer:

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

RD Sharma 2020 2021 Solutions for Class 7 Maths Chapter 7 - Algebraic Expressions

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