# RD Sharma Solutions Class 7 Algebraic Expressions Exercise 7.2

## RD Sharma Solutions Class 7 Chapter 7 Exercise 7.2

### RD Sharma Class 7 Solutions Chapter 7 Ex 7.2 PDF Free Download

#### Exercise 7.2

(i) 3x and 7x

(ii) -5xy and 9xy

Solution:

We have

(i) 3x + 7x = (3 + 7)x = 10x

(ii) -5xy + 9xy = (-5 + 9)xy = 4xy

Q2) Simplify each of the following:

$(i) 7x^{3}y+9yx^{3}$

$(ii) 12a^{2}b+3ba^{2}$

Solution:

Simplifying the given expressions, we have

$(i) 7x^{3}y+9yx^{3}=(7+9)x^{3}y=16x^{3}y$

$(ii) 12a^{2}b+3ba^{2}=(12+3)a^{2}b=15a^{2}b$

(i) 7abc, -5abc, 9abc, -8abc

$(ii) 2x^{2}y,\;-4x^{2}y,\;6x^{2}y,\;-5x^{2}y$

Solution:

Adding the given terms, we have

(i) 7abc + (-5abc) + (9abc) + (-8abc)

= 7abc – 5abc + 9abc – 8abc

= (7 – 5 + 9 – 8)abc

= (16 – 13)abc

= 3abc

$(ii) 2x^{2}y+(-4x^{2}y)+(6x^{2}y)+(-5x^{2}y)$

$= 2x^{2}y-4x^{2}y+6x^{2}y-5x^{2}y$

$= (2-4+6-5)x^{2}y$

$= (8-9)x^{2}y$

$= -x^{2}y$

$(i) x^{3}-2x^{2}y+3xy^{2}-y^{3},\;2x^{3}-5xy^{2}+3x^{2}y-4y^{3}$

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

Solution:

Adding the given expressions, we have

$(i) x^{3}-2x^{2}y+3xy^{2}-y^{3}+2x^{3}-5xy^{2}+3x^{2}y-4y^{3}$

Collecting positive and negative like terms together, we get

$x^{3}+2x^{3}-2x^{2}y+3x^{2}y +3xy^{2}-5xy^{2}-y^{3}-4y^{3}$

$= 3x^{3}+x^{2}y-2xy^{2}-5y^{3}$

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

$a^{4}-2a^{3}b+3ab^{3}+4a^{2}b^{2}+3b^{4}-2a^{4}-5ab^{3}+7a^{3}b-6a^{2}b^{2}+b^{4}$

Collecting positive and negative like terms together, we get

$a^{4}-2a^{4}-2a^{3}b+7a^{3}b+3ab^{3}-5ab^{3}+4a^{2}b^{2}-6a^{2}b^{2}+3b^{4}+b^{4}$

$= -a^{4}+5a^{3}b-2ab^{3}-2a^{2}b^{2}+4b^{4}$

(i) 8a – 6ab + 5b, –6a – ab – 8b and –4a + 2ab + 3b

$(ii) 5x^{3}+7+6x-5x^{2},\;2x^{2}-8-9x,\;4x-2x^{2}+3x^{3},\;3x^{3}-9x-x^{2}\;and\;x-x^{2}-x^{3}-4$

Solution:

(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 = $(5x^{3}+7+6x-5x^{2})+(2x^{2}-8-9x)+(4x-2x^{2}+3x^{3})+(3x^{3}-9x-x^{2})+(x-x^{2}-x^{3}-4)$

Collecting positive and negative like terms together, we get

$5x^{3}+3x^{3}+3x^{3}-x^{3}-5x^{2}+2x^{2}-2x^{2}-x^{2}-x^{2}+6x-9x+4x-9x+x+7-8-4$

$= 10x^{3}-7x^{2}-7x-5$

(i) x – 3y – 2z

5x + 7y – 8z

3x – 2y + 5z

(ii) 4ab – 5bc + 7ca

–3ab + 2bc – 3ca

5ab – 3bc + 4ca

Solution:

(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 + 11ca – 3ca

= 6ab – 6bc + 8ca

Q7) Add $2x^{2}-3x+1$ to the sum of $3x^{2}-2x$ and 3x + 7.

Solution:

Sum of $3x^{2}-2x$ and 3x + 7

$= (3x^{2}-2x) + (3x + 7)$

$= 3x^{2}-2x+3x+7$

$= (3x^{2}+x+7)$

Now, required expression = $2x^{2}-3x+1$ + $(3x^{2}+x+7)$

$= 2x^{2}+3x^{2}-3x+x+1+7$

$= 5x^{2}-2x+8$

Q8) Add $x^{2}+2xy+y^{2}$ to the sum of $x^{2}-3y^{2}$ and $2x^{2}-y^{2}+9$.

Solution:

Sum of $x^{2}-3y^{2}$ and $2x^{2}-y^{2}+9$

$= (x^{2}-3y^{2})+(2x^{2}-y^{2}+9)$

$= x^{2}+2x^{2}-3y^{2}-y^{2}+9$

$= 3x^{2}-4y^{2}+9$

Now, required expression = $(x^{2}+2xy+y^{2})$ + $3x^{2}-4y^{2}+9$

$= x^{2}+3x^{2}+2xy+y^{2}-4y^{2}+9$

$= 4x^{2}+2xy-3y^{2}+9$

Q9) Add $a^{3}+b^{3}-3$ to the sum of $2a^{3}-3b^{3}-3ab+7$ and $-a^{3}+b^{3}+3ab-9$.

Solution:

First, we need to find the sum of $2a^{3}-3b^{3}-3ab+7$ and $-a^{3}+b^{3}+3ab-9$

$= (2a^{3}-3b^{3}-3ab+7)+(-a^{3}+b^{3}+3ab-9)$

Collecting positive and negative like terms together, we get

$= 2a^{3}-a^{3}-3b^{3}+b^{3}-3ab+3ab+7-9$

$= a^{3}-2b^{3}-2$

Now, the required expression = $(a^{3}+b^{3}-3)$ + $(a^{3}-2b^{3}-2)$

$= a^{3}+a^{3}+b^{3}-2b^{3}-3-2$

$= 2a^{3}-b^{3}-5$

Q10) Subtract:

$(i) 7a^{2}b$ from $3a^{2}b$

(ii) 4xy from -3xy

Solution:

(i) Required expression = $3a^{2}b-7a^{2}b$

$= (3-7)a^{2}b$

$= -4a^{2}b$

(ii) Required expression = –3xy – 4xy

= –7xy

Q11) Subtract:

(i) -4x from 3y

(ii) -2x from -5y

Solution:

(i) Required expression = (3y) – (–4x)

= 3y + 4x

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

= –5y + 2x

Q12) Subtract:

$(i) 6x^{3}-7x^{2}+5x-3$ from $4-5x+6x^{2}-8x^{3}$

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

$(iii) x^{3}+2x^{2}y+6xy^{2}-y^{3}$ from $y^{3}-3xy^{2}-4x^{2}y$

Solution:

(i) Required expression = $(4-5x+6x^{2}-8x^{3})-(6x^{3}-7x^{2}+5x-3)$

$= 4-5x+6x^{2}-8x^{3}-6x^{3}+7x^{2}-5x+3$

$= -8x^{3}-6x^{3}+7x^{2}+6x^{2}-5x-5x+3+4$

$= -14x^{3}+13x^{2}-10x+7$

(ii) Required expression = $(5x^{2}-y+z+7)-(-x^{2}-3z)$

$= 5x^{2}-y+z+7+x^{2}+3z$

$= 5x^{2}+x^{2}-y+z+3z +7$

$= 6x^{2}-y+4z +7$

(iii) Required expression = $(y^{3}-3xy^{2}-4x^{2}y)-(x^{3}+2x^{2}y+6xy^{2}-y^{3})$

$= y^{3}-3xy^{2}-4x^{2}y-x^{3}-2x^{2}y-6xy^{2}+y^{3}$

$y^{3}+y^{3}-3xy^{2}-6xy^{2}-4x^{2}y-2x^{2}y-x^{3}$

$= 2y^{3}-9xy^{2}-6x^{2}y-x^{3}$

Q13) From

(i) $p^{3}-4+3p^{2}$, take away $5p^{2}-3p^{3}+p-6$

(ii) $7+x-x^{2}$, take away $9+x+3x^{2}+7x^{3}$

(iii) $1-5y^{2}$, take away $y^{3}+7y^{2}+y+1$

(iv) $x^{3}-5x^{2}+3x+1$, take away $6x^{2}-4x^{3}+5+3x$

Solution:

(i) Required expression = $(p^{3}-4+3p^{2})-(5p^{2}-3p^{3}+p-6)$

$= p^{3}-4+3p^{2}-5p^{2}+3p^{3}-p+6$

$= p^{3}+3p^{3}+3p^{2}-5p^{2}-p-4+6$

$= 4p^{3}-2p^{2}-p+2$

(ii) Required expression = $(7+x-x^{2})-(9+x+3x^{2}+7x^{3})$

$= 7+x-x^{2}-9-x-3x^{2}-7x^{3}$

$= -7x^{3}-x^{2}-3x^{2}+7-9$

$= -7x^{3}-4x^{2}-2$

(iii) Required expression = $(1-5y^{2})-(y^{3}+7y^{2}+y+1)$

$= 1-5y^{2}-y^{3}-7y^{2}-y-1$

$= -y^{3}-5y^{2}-7y^{2}-y$

$= -y^{3}-12y^{2}-y$

(iv) Required expression = $(x^{3}-5x^{2}+3x+1)-(6x^{2}-4x^{3}+5+3x)$

$= x^{3}-5x^{2}+3x+1-6x^{2}+4x^{3}-5-3x$

$= x^{3}+4x^{3}-5x^{2}-6x^{2}+1-5$

$= 5x^{3}-11x^{2}-4$

Q14) From the sum of $3x^{2}-5x+2$ and $-5x^{2}-8x+9$ subtract $4x^{2}-7x+9$.

Solution:

Required expression = $[(3x^{2}-5x+2)+(-5x^{2}-8x+9)]-(4x^{2}-7x+9)$

$= [3x^{2}-5x+2-5x^{2}-8x+9]-(4x^{2}-7x+9)$

$= [3x^{2}-5x^{2}-5x-8x+2+9]-(4x^{2}-7x+9)$

$= [-2x^{2}-13x+11]-(4x^{2}-7x+9)$

$= -2x^{2}-13x+11-4x^{2}+7x-9$

$= -2x^{2}-4x^{2}-13x+7x+11-9$

$= -6x^{2}-6x+2$

Q15) Subtract the sum of 13x – 4y + 7z and –6z + 6x + 3y from the sum of 6x – 4y – 4z and   2x + 4y – 7.

Solution:

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)

= (6x + 2x – 4z – 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

Q16) From the sum of $x^{2}+3y^{2}-6xy,\;2x^{2}-y^{2}+8xy,\;y^{2}+8\;and\;x^{2}-3xy$ subtract $-3x^{2}+4y^{2}-xy+x-y+3$.

Solution:

Sum of $(x^{2}+3y^{2}-6xy),\;(2x^{2}-y^{2}+8xy),\;(y^{2}+8)\;and\;(x^{2}-3xy)$

$= (x^{2}+3y^{2}-6xy)+(2x^{2}-y^{2}+8xy)+(y^{2}+8)+(x^{2}-3xy)$

$= (x^{2}+3y^{2}-6xy+2x^{2}-y^{2}+8xy+y^{2}+8+x^{2}-3xy)$

$= (x^{2}+2x^{2}+x^{2}+3y^{2}-y^{2}+y^{2}-6xy+8xy-3xy+8)$

$= (4x^{2}+3y^{2}-xy+8)$

Now, required expression = $(4x^{2}+3y^{2}-xy+8)-(-3x^{2}+4y^{2}-xy+x-y+3)$

$= 4x^{2}+3y^{2}-xy+8+3x^{2}-4y^{2}+xy-x+y-3$

$= 4x^{2}+3x^{2}+3y^{2}-4y^{2}-xy+xy-x+y-3+8$

$= 7x^{2}-y^{2}-x+y+5$

Q17) What should be added to xy – 3yz + 4zx to get 4xy – 3zx + 4yz + 7?

Solution:

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

Q18) What should be subtracted from $x^{2}-xy+y^{2}-x+y+3$ to obtain $-x^{2}+3y^{2}-4xy+1$?

Solution:

Let ‘M’ be the required expression. Then, we have

$x^{2}-xy+y^{2}-x+y+3-M = -x^{2}+3y^{2}-4xy+1$

Therefore,

$M = (x^{2}-xy+y^{2}-x+y+3)-(-x^{2}+3y^{2}-4xy+1)$

$= x^{2}-xy+y^{2}-x+y+3+x^{2}-3y^{2}+4xy-1$

Collecting positive and negative like terms together, we get

$x^{2}+x^{2}-xy+4xy+y^{2}-3y^{2}-x+y+3-1$

$= 2x^{2}+3xy-2y^{2}-x+y+2$

Q19) How much is x – 2y + 3z greater than 3x + 5y – 7?

Solution:

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

Q20) How much is $x^{2}-2xy+3y^{2}$ less than $2x^{2}-3y^{2}+xy$?

Solution:

Required expression = $(2x^{2}-3y^{2}+xy)-(x^{2}-2xy+3y^{2})$

$= 2x^{2}-3y^{2}+xy-x^{2}+2xy-3y^{2}$

Collecting positive and negative like terms together, we get

$2x^{2}-x^{2}-3y^{2}-3y^{2}+xy+2xy$

$x^{2}-6y^{2}+3xy$

Q21) How much does $a^{2}-3ab+2b^{2}$ exceed $2a^{2}-7ab+9b^{2}$?

Solution:

Required expression = $(a^{2}-3ab+2b^{2})-(2a^{2}-7ab+9b^{2})$

$= a^{2}-3ab+2b^{2}-2a^{2}+7ab-9b^{2}$

Collecting positive and negative like terms together, we get

$= a^{2}-2a^{2}-3ab+7ab+2b^{2}-9b^{2}$

$= -a^{2}+4ab-7b^{2}$

Q22) What must be added to $12x^{3}-4x^{2}+3x-7$ to make the sum $x^{3}+2x^{2}-3x+2$?

Solution:

Let ‘M’ be the required expression. Thus, we have

$12x^{3}-4x^{2}+3x-7+M=x^{3}+2x^{2}-3x+2$

Therefore,

$M = (x^{3}+2x^{2}-3x+2) – (12x^{3}-4x^{2}+3x-7)$

$M = x^{3}+2x^{2}-3x+2-12x^{3}+4x^{2}-3x+7$

Collecting positive and negative like terms together, we get

$M = x^{3}-12x^{3}+2x^{2}+4x^{2}-3x-3x+7+2$

$x^{3}-12x^{3}+2x^{2}+4x^{2}-3x-3x+7+2$

$= -11x^{3}+6x^{2}-6x+9$

Q23) If P = $7x^{2}+5xy-9y^{2}$, Q = $4y^{2}-3x^{2}-6xy$ and R = $-4x^{2}+xy+5y^{2}$, show that P + Q + R = 0.

Solution:

We have

P + Q + R = $(7x^{2}+5xy-9y^{2})$ + $(4y^{2}-3x^{2}-6xy)$ + $(-4x^{2}+xy+5y^{2})$

$= 7x^{2}+5xy-9y^{2}+4y^{2}-3x^{2}-6xy-4x^{2}+xy+5y^{2}$

Collecting positive and negative like terms together, we get

$7x^{2}-3x^{2}-4x^{2}+5xy-6xy+xy-9y^{2}+4y^{2} +5y^{2}$

$= 7x^{2}-7x^{2}+6xy-6xy-9y^{2}+9y^{2}$

= 0

Q24) If P = $a^{2}-b^{2}+2ab$, Q = $a^{2}+4b^{2}-6ab$, R = $b^{2}+b$, S = $a^{2}-4ab$ and T = $-2a^{2}+b^{2}-ab+a$. Find P + Q + R + S – T.

Solution:

We have

$P + Q + R + S – T = [(a^{2}-b^{2}+2ab)+(a^{2}+4b^{2}-6ab)+(b^{2}+b)+(a^{2}-4ab)]-(-2a^{2}+b^{2}-ab+a)$

$= [a^{2}-b^{2}+2ab+a^{2}+4b^{2}-6ab+b^{2}+b+a^{2}-4ab]-(-2a^{2}+b^{2}-ab+a)$

$= [3a^{2}+4b^{2}-8ab+b]-(-2a^{2}+b^{2}-ab+a)$

$= 3a^{2}+4b^{2}-8ab+b+2a^{2}-b^{2}+ab-a$

Collecting positive and negative like terms together, we get

$3a^{2}+2a^{2}+4b^{2}-b^{2}-8ab+ab-a+b$

$= 5a^{2}+3b^{2}-7ab-a+b$

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