\begin{array}{l}
\text {Q1. Add the following: } \\
\text {(i) } \quad 3 x \text { and } 7 x \quad
\text {(ii) } \quad -5 xy \text { and } 9 x y
\end{array}\begin{array}{l}
\text {Sol. } \\\text {(i) } \quad 3x+7x=(3+7)x=10 x \\ \\\text {(ii) } \quad -5xy+9xy=(-5+9)xy = 4xy \\
\end{array}\begin{array}{l}
\text {Q2. Simplify each of the following: } \\
\text {(i) } \quad 7 x^{3} y + 9 y x^{3} \quad
\text {(ii) } \quad 12 a^{2} b + 3 b a^{2}
\end{array}\begin{array}{l}
\text {Sol. } \\\text {(i) } \quad 7 x^{3} y + 9 y x^{3} =(7+9) x^{3}y=16 x^{3}y \\ \\\text {(ii) } \quad 12 a^{2}b + 3ba^{2}=(12+3)a^{2}b=15 a^{2}b \\
\end{array}\begin{array}{l}
\text {Q3. Add the following: } \\
\text {(ii) } \quad 2 x^{2} y,-4 x^{2} y, 6 x^{2} y,-5 x^{2} y \quad
\text {(i) } \quad 7abc, -5abc, 9abc,-8abc
\end{array}\begin{array}{l}
\text {Sol. } \\
\text {(i) } \quad 7abc+ (-5abc) + 9abc + (-8abc) \\
=7abc -5abc+ 9abc -8abc \\
=(7-5+9-8)abc
=3abc \\ \\\text {(ii) } \quad 2 x^{2} y + (-4 x^{2}y) + 6 x^{2}y + (-5x^{2}y) \\
=2 x^{2}y – 4 x^{2}y + 6 x^{2}y – 5x^{2}y \\
=(2-4+6-5)x^{2}y \\
=-x^{2}y \\\end{array}\begin{array}{l}
\text {Q4. Add the following expressions: } \\
\text {(i) } \quad 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} \\
\text {(ii) } \quad 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}
\end{array}\begin{array}{l}
\text {Sol. } \\\text {(i) } \quad 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} \\
\text {Collecting positive and negative like terms together, we get } \\
=x^{3}+2 x^{3}-2 x^{2} y+3 x^{2} y+3 x y^{2}-5 x y^{2}-y^{3}-4 y^{3} \\
=3 x^{3}+x^{2} y-2 x y^{2}-5 y^{3} \\ \\\text {(ii) } \quad (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}) \\=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} \\
\text {Collecting positive and negative like terms together, we get } \\
=a^{4}-2 a^{4}-2 a^{3} b+7 a^{3} b+3 a b^{3}-5 a b^{3}+4a^{2} b^{2}-6 a^{2} b^{2}+3 b^{4}+b^{4} \\
=-a^{4}+5 a^{3}b – 2ab^{3} – 2a^{2}b^{2} + 4b^{4} \\\end{array}\begin{array}{l}
\text {Q5. Add the following expressions: } \\
\text {(i) } \quad 8a – 6ab + 5b,-6a – ab – 8b { and } -4a + 2ab + 3b \\
\text {(ii) } \quad 5 x^{3} + 7 + 6x – 5 x^{2}, 2 x^{2}-8-9 x, 4x – 2x^{2} + 3x^{3}, 3x^{3} – 9x – x^{2} { and } x -x^{2} – x^{3} – 4
\end{array}\begin{array}{l}
\text {Sol. } \\\text {(i) } \quad (8 a-6 a b+5 b)+(-6 a-a b-8 b)+(-4 a+2 a b+3 b) \\
\text {Collecting positive and negative like terms together, we get } \\
=8 a-6 a-4 a-6 a b-a b+2 a b+5 b-8 b+3 b \\
=8 a-10 a-7 a b+2 a b+8 b-8 b \\
= -2a -5ab \\ \\\text {(ii) } \quad (5 \times 3+7+6 x-5 x^{2})+(2 \times 2-8-9 x)+(4 x-2 x^{2}+3 x 3)+(3 x 3-9 x-x^{2})+(x-x^{2}-x^{3}-4) \\
\text {Collecting positive and negative like terms together, we get } \\
= 5 x^{3}+3 x^{3}+3 x^{3}-x^{3}-5 x^{2}+2 x^{2}-2 x^{2}-x^{2}-x^{2}+6 x-9 x+4 x-9 x+x+7-8-4 \\
=10x^{3} – 7x^{2} – 7x -5 \\\end{array}\begin{array}{l}
\text {Q 6. Add the following: } \\
\text {(i) } \quad \quad x-3 y-2 z, 5 x+7 y-8 z, 3 x-2 y+5 z \\
\text {(ii) } \quad 4ab – 5bc + 7ca, -3ab + 2bc – 3ca, 5ab – 3bc + 4ca
\end{array}\begin{array}{l}
\text {Sol. } \\\text {(i) } \quad (x-3 y-2 z)+(5 x+7 y-8 z)+(3 x-2 y+5 z) \\
\text {Collecting positive and negative like terms together, we get } \\
=x+5 x+3 x-3 y+7 y-2 y-2 z-8 z+5 z \\
=9 x-5 y+7 y-10 z+5 z \\
=9 x+2 y-5 z \\ \\\text {(ii) } \quad (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) \\
\text {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 \\\end{array}\begin{array}{l}
\text {Q 7. Add } 2x^{2}- 3x+1 \text { to the sum of } 3 x^{2}-2 x \text { and } 3x+7
\end{array}\begin{array}{l}
\text {Sol. The required sum is} \\(3x^{2}-2x)+(3 x+7) + (2x^{2} – 3x+1) \\
=3x^{2} + x+7 + 2x^{2} – 3x+1 \\
=5 x^{2}-2 x+8 \\\end{array}\begin{array}{l}
\text {Q 8. Add } x^{2}+2xy+y^{2} \text { to the sum of } x^{2}-3 y^{2} \text { and } 2 x^{2}-y^{2}+9
\end{array}\begin{array}{l}
\text {Sol. The required sum is} \\x^{2}+2xy+y^{2} + (x^{2}-3 y^{2} + 2 x^{2}-y^{2}+9 ) \\
=4x^{2}+ 2xy -3y^{2}+9 \\\end{array}\begin{array}{l}
\text {Q9. Add } a^{3}+b^{3}-3 \text { to the sum of } 2 a^{3}-3 b^{3}-3ab+7 \text { and } -a^{3}+b^{3} + 3ab – 9
\end{array}\begin{array}{l}
\text {Sol. The required sum is} \\a^{3}+b^{3}-3 + (2 a^{3}-3 b^{3}-3ab+7 -a^{3}+b^{3} + 3ab – 9 ) \\
\text {Collecting positive and negative like terms together, we get} \\
=a^{3} + 2 a^{3} -a^{3} +b^{3}-3 b^{3}+b^{3}-3ab+ 3ab-3+7- 9 \\
=2 a^{3}-b^{3}-5 \\\end{array}\begin{array}{l}
\text {Q10. Subtract: } \\
\text {(i) } \quad 7a^{2}b \text { from } 3a^{2}b \quad
\text {(ii) } \quad 4xy \text { from } -3xy\end{array}\begin{array}{l}
\text {Sol. } \\\text {(i) } \quad 3 a^{2} b-7 a^{2} b=(3-7) a^{2}b=-4 a^{2} b \\ \\\text {(ii) } \quad -3xy -4xy =-7xy \\\end{array}\begin{array}{l}
\text {Q11. Subtract: } \\
\text {(i) } \quad -4x \text { from } 3y \quad
\text {(ii) } \quad -2x \text { from } -5y
\end{array}\begin{array}{l}
\text {Sol. } \\\text {(i) } \quad (3y) – (-4x) =3y+4x \\ \\\text {(ii) } \quad (-5y) – (-2x)=-5y+2x \\\end{array}\begin{array}{l}
\text {Q12. Subtract: } \\
\text {(i) } \quad 6x^{3} – 7x^{2} + 5x-3 \text { from } 4-5x+6x^{2}-8x^{3} \\
\text {(ii) } \quad -x^{2}-3z \text { from } 5 x^{2}-y+z+7 \\
\text {(ii) } \quad x^{3}+2 x^{2} y+6 x y^{2}-y^{3} \text { from } y^{3} – 3xy^{2} – 4x^{2}y
\end{array}\begin{array}{l}
\text {Sol. } \\\text {(i) } \quad (4-5x+6 x^{2}-8 x^{3})-(6 x^{3}-7 x^{2}+5 x-3) \\
=4-5 x+6 x^{2}-8 x^{3}-6 x^{3}+7 x^{2}-5 x+3 \\
=-8 x^{3}-6 x^{3}+7 x^{2}+6 x^{2}-5 x-5 x+3+4 \\
=-14 x^{3}+13 x^{2}-10 x+7 \\ \\\text {(ii) } \quad (5 x^{2}-y+z+7)-(-x^{2}-3 z) \\
=5 x^{2}-y+z+7+x^{2}+3 z \\
=5 x^{2}+x^{2}-y+z+3 z+7 \\
=6 x^{2}-y+4 z+7 \\ \\\text {(iii) } \quad (y^{3}-3 x y^{2}-4 x^{2} y)-(x^{3}+2 x^{2} y+6 x y^{2}-y^{3}) \\
=y^{3}-3 x y^{2}-4 x^{2} y-x^{3}-2 x^{2} y-6 x y^{2}+y^{3} \\
=y^{3}+y^{3}-3 x y^{2}-6 x y^{2}-4 x^{2} y-2 x^{2} y-x^{3} \\
=2 y^{3}-9 x y^{2}-6 x^{2} y-x^{3} \\\end{array}\begin{array}{l}
\text {Q13. From } \\
\text {(i) } \quad p^{3}-4+3 p^{2} \text { , take away } 5 p^{2}-3 p^{3}+p-6 \\
\text {(ii) } \quad 7+x-x^{2} \text { , take away } 9+x+3 x^{2}+7 x^{3} \\
\text {(iii) } \quad 1-5 y^{2} \text { , take away } y^{3}+7 y^{2}+y+1 \\
\text {(iv) } \quad x^{3}-5 x^{2}+3 x+1 \text { , take away } 6 x^{2}-4 x^{3}+5+3x
\end{array}\begin{array}{l}
\text {Sol. } \\\text {(i) } \quad (p^{3}-4+3 p^{2})-(5 p^{2}-3 p^{3}+p-6) \\=p^{3}-4+3 p^{2}-5 p^{2}+3 p^{3}-p+6 \\
=p^{3}+3 p^{3}+3 p^{2}-5 p^{2}-p-4+6 \\
=4 p^{3}-2 p^{2}-p+2 \\ \\\text {(ii) } \quad (7+x-x^{2})-(9+x+3 x^{2}+7 x^{3}) \\
=7+x-x^{2}-9-x-3 x^{2}-7 x^{3} \\
=-7 x^{3}-x^{2}-3 x^{2}+7-9 \\
=-7 x^{3}-4 x^{2}-2 \\ \\\text {(iii) } \quad (1-5 y^{2})-(y^{3}+7 y^{2}+y+1) \\
=1-5 y^{2}-y^{3}-7 y^{2}-y-1 \\
=-y^{3}-5 y^{2}-7 y^{2}-y \\
=-y^{3}-12 y^{2}-y \\ \\\text {(iv) } \quad (x^{3}-5 x^{2}+3 x+1)-(6 x^{2}-4 x^{3}+5+3 x) \\
=x^{3}-5 x^{2}+3 x+1-6 x^{2}+4 x^{3}-5-3 x \\
=x^{3}+4 x^{3}-5 x^{2}-6 x^{2}+1-5 \\
=5 x^{3}-11 x^{2}-4 \\
\end{array}\begin{array}{l}
\text {Q14. From the sum of } 3x^{2}-5 x+2 \text { and } -5 x^{2}-8 x+9 \text { subtract } 4 x^{2}-7 x+9
\end{array}\begin{array}{l}
\text {Sol. } \\[(3 x^{2}-5 x+2)+(-5 x^{2}-8 x+9)]-(4 x^{2}-7 x+9) \\
=[3 x^{2}-5 x+2-5 x^{2}-8 x+9]-(4 x^{2}-7 x+9) \\
=[3 x^{2}-5 x^{2}-5 x-8 x+2+9]-(4 x^{2}-7 x+9) \\
=[-2 x^{2}-13 x+11]-(4 x^{2}-7 x+9) \\
=-2 x^{2}-13 x+11-4 x^{2}+7 x-9 \\
=-2 x^{2}-4 x^{2}-13 x+7 x+11-9 \\
=-6 x^{2}-6 x+2 \\\end{array}\begin{array}{l}
\text {Q15. Subtract the sum of } 13 x-4 y+7 z \text { and } -6 z+6 x+3 y \text { from the sum of } \\
6 x-4 y-4 z \text { and } 2 x+4 y-7
\end{array}\begin{array}{l}
\text {Sol. } \\\text {Sum of } \quad (13 x-4 y+7 z) \text { and } (-6 z+6 x+3 y) \\
=(13 x-4 y+7 z)+(-6 z+6 x+3 y) \\
=13 x-4 y+7 z-6 z+6 x+3 y \\
=13 x+6 x-4 y+3 y+7 z-6 z \\
=19 x-y+z \\ \\\text {Sum of } \quad (6 x-4 y-4 z) \text { and } (2 x+4 y-7) \\
=(6 x-4 y-4 z)+(2 x+4 y-7) \\
=6 x-4 y-4 z+2 x+4 y-7 \\
=6 x+2 x-4 z-7 \\
=8 x-4 z-7 \\
\text {Hence, the required expression } =(8 x-4 z-7)-(19 x-y+z) \\
=8 x-4 z-7-19 x+y-z \\
=8 x-19 x+y-4 z-z-7 \\
=-11 x+y-5 z-7 \\\end{array}\begin{array}{l}
\text {Q16. From the sum of } x^{2}+3 y^{2}-6 x y, 2 x^{2}-y^{2}+8 x y, y^{2}+8 \text { and } x^{2}-3 x y \text { subtract } \\
-3 x^{2}+4 y^{2}-x y+x-y+3
\end{array}\begin{array}{l}
\text {Sol. } \\\text {Sum of } \quad (x^{2}+3 y^{2}-6 x y),(2 x^{2}-y^{2}+8 x y),(y^{2}+8) \text { and } (x^{2}-3 x y) \\
=(x^{2}+3 y^{2}-6 x y)+(2 x^{2}-y^{2}+8 x y)+(y^{2}+8)+(x^{2}-3 x y) \\
=(x^{2}+3 y^{2}-6 x y+2 x^{2}-y^{2}+8 x y+y^{2}+8+x^{2}-3 x y) \\
=(x^{2}+2 x^{2}+x^{2}+3 y^{2}-y^{2}+y^{2}-6 x y+8 x y-3 x y+8) \\
=(4 x^{2}+3 y^{2}-x y+8) \\ \\\text {Hence, the required expression } =(4 x^{2}+3 y^{2}-x y+8)-(-3 x^{2}+4 y^{2}-x y+x-y+3) \\
=4 x^{2}+3 y^{2}-x y+8+3 x^{2}-4 y^{2}+x y-x+y-3 \\
=4 x^{2}+3 x^{2}+3 y^{2}-4 y^{2}-x y+x y-x+y-3+8 \\
=7 x^{2}-y^{2}-x+y+5 \\\end{array}\begin{array}{l}
\text {Q17. What should be added to } xy-3yz+4zx \text { to get } 4xy-3zx + 4yz+7 ?
\end{array}\begin{array}{l}
\text {Sol. } \\\text {We will get the required expression by subtracting } xy-3yz+4zx \text { from } 4xy – 3zx + 4yz+7 \\
\therefore \text {The required expression } =(4xy-3zx+4 yz+7)-(xy-3yz+4zx) \\
=4xy-3zx+4yz+7-xy+3yz-4zx \\
=4xy-xy-3zx-4zx+4yz+3yz+7 \\
=3xy – 7zx + 7yz+7 \\\end{array}\begin{array}{l}
\text {Q18. What should be subtracted from } x^{2}-x y+y^{2}-x+y+3 \text { to obtain } -x^{2}+3y^{2}-4xy+1 ?
\end{array}\begin{array}{l}
\text {Sol. } \\\text {Let’s assume ‘A’ be the required expression. } \\
\text {According to question, we have } \\
x^{2}-x y+y^{2}-x+y+3 – A = -x^{2}+3 y^{2}-4 x y+1 \\
\Rightarrow \quad A=(x^{2}-x y+y^{2}-x+y+3)-(-x^{2}+3 y^{2}-4 x y+1) \\
\Rightarrow \quad A=x^{2}-x y+y^{2}-x+y+3+x^{2}-3 y^{2}+4 x y-1 \\
\Rightarrow \quad A = x^{2}+x^{2}-x y+4 x y+y^{2}-3 y^{2}-x+y+3-1 \\
\Rightarrow \quad A=2 x^{2}+3xy-2y^{2}-x+y+2 \\\end{array}\begin{array}{l}
\text {Q19. How much is } x-2y+3z \text { greater than } 3 x+5 y-7 ?
\end{array}\begin{array}{l}
\text {Sol. The Required expression } =(x-2 y+3 z)-(3 x+5 y-7) \\
=x-2y+3z-3x-5y+7 \\
\text {Collecting positive and negative like terms together, we get } \\
=x-3x-2y+5y+3z+7 \\
=-2x-7y+3z+7 \\\end{array}\begin{array}{l}
\text {Q20. How much is } x^{2} – 2xy + 3y^{2} \text { less than } 2 x^{2}-3 y^{2}+x y ?
\end{array}\begin{array}{l}
\text {Sol. The Required expression }=(2 x^{2}-3 y^{2}+x y)-(x^{2}-2 x y+3 y^{2}) \\
=2 x^{2}-3 y^{2}+x y-x^{2}+2 x y-3 y^{2} \\\text {Collecting positive and negative like terms together, we get } \\
=2 x^{2}-x^{2}-3y^{2}-3y^{2}+xy+2xy \\
=x^{2}-6 y^{2}+3xy \\
\end{array}\begin{array}{l}
\text {Q21. How much does } a^{2}-3 a b+2 b^{2} \text { exceed } 2 a^{2}-7 a b+9 b^{2} ?
\end{array}\begin{array}{l}
\text {Sol. The Required expression } =(a^{2}-3 a b+2 b^{2})-(2 a^{2}-7 a b+9 b^{2}) \\
=a^{2}-3 a b+2 b^{2}-2 a^{2}+7 a b-9 b^{2} \\
\text {Collecting positive and negative like terms together, we get } \\
=a^{2}-2 a^{2}-3 a b+7 a b+2 b^{2}-9 b^{2} \\
=-a^{2}+4 a b-7 b^{2} \\
\end{array}\begin{array}{l}
\text {Q22 . What must be added to } 12 x^{3}-4 x^{2}+3 x-7 \text { to make the sum } x^{3}+2 x^{2}-3 x+2 ?
\end{array}\begin{array}{l}
\text {Sol. Let’s assume ‘A’ be the required expression.} \\
\text {According to question, we have } \\
12 x^{3}-4 x^{2}+3 x-7+M=x^{3}+2 x^{2}-3 x+2 \\
\Rightarrow \quad A=(x^{3}+2 x^{2}-3 x+2)-(12 x^{3}-4 x^{2}+3 x-7) \\
\Rightarrow \quad A=x^{3}+2 x^{2}-3 x+2-12 x^{3}+4 x^{2}-3 x+7 \\
\text {Collecting positive and negative like terms together, we get } \\
\Rightarrow \quad A =x^{3}-12 x^{3}+2 x^{2}+4 x^{2}-3 x-3 x+7+2 \\
\Rightarrow \quad A=-11 x^{3}+6 x^{2}-6 x+9 \\\end{array}\begin{array}{l}
\text {Q23. If } P=7 x^{2}+5 x y-9 y^{2}, Q=4 y^{2}-3 x^{2}-6 x y \text { and } R=-4 x^{2}+x y+5 y^{2} \\
\text {, show that } P+Q+R=0
\end{array}\begin{array}{l}
\text {Sol. } \\P+Q+R=(7 x^{2}+5 x y-9 y^{2})+(4y^{2}-3x^{2}-6xy)+(-4 x^{2}+xy+5y^{2}) \\
=7x^{2}+5xy-9y^{2}+4y^{2}-3x^{2}-6xy-4x^{2}+xy+5y^{2} \\
\text {Collecting positive and negative like terms together, we get } \\
=7x^{2}-3x^{2}-4x^{2}+5xy-6xy+xy-9y^{2}+4y^{2}+5y^{2} \\
=7 x^{2} – 7 x^{2}+6 xy -6 xy – 9 y^{2}+9 y^{2} \\
=0\end{array}\begin{array}{l}
\text {Q24. If } P=a^{2}-b^{2}+2ab, Q=a^{2}+4 b^{2}-6ab, R=b^{2}+b, S=a^{2}-4ab \text { and } T=-2 a^{2}+b^{2}-a b \\
\text {Find } P+Q+R+S-T\end{array}\begin{array}{l}
\text {Sol. } \\
P+Q+R+S-T=\{(a^{2}-b^{2}+2ab)+(a^{2}+4 b^{2}-6 ab)+(b^{2}+b)+(a^{2}-4 ab)\}-1-(-2 a^{2}+b^{2}-ab+a) \\
=\{a^{2}-b^{2}+2 a b+a^{2}+4 b^{2}-6 a b+b^{2}+b+a^{2}-4 a b\}-(-2 a^{2}+b^{2}-a b+a) \\
=\{3 a^{2}+4 b^{2}-8 a b+b\}-(-2 a^{2}+b^{2}-a b+a) \\
=3 a^{2}+4 b^{2}-8 a b+b+2 a^{2}-b^{2}+a b-a \\
\text {Collecting positive and negative like terms together, we get } \\
=3 a^{2}+2 a^{2}+4 b^{2}-b^{2}-8 a b+a b-a+b \\
=5 a^{2}+3 b^{2}-7 ab-a+b \\\end{array}
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