Q1. Evaluate:
(i) $7x^2 + 18x^2 + 3x^2 - 5x^2$
(ii) $b^2 - 9b^2y + 2b^2y - 5b^2y$
(iii) $abx - 15 abx - 10abx + 32abx$
(iv) $7x - 9y + 3 - 3x - 5y + 8$
(v) $3x^2 + 5xy - 4y^2 + x^2 - 8xy - 5y^2$
Show Answer
(i) $7x^2 + 18x^2 + 3x^2 - 5x^2$
$\Rightarrow (-7 + 18 + 3 - 5)x^2$
$\Rightarrow (21 - 13)x^2 = 8x^2 $ Ans
(ii) $b^2 - 9b^2y + 2b^2y - 5b^2y$
$\Rightarrow (1 - 9 + 2 - 5)b^2y$
$\Rightarrow (3 - 14)b^2y = -11b^2y $ Ans
(iv) $7x - 9y + 3 - 3x - 5y + 8$
$\Rightarrow (7 - 3)x + (-9 - 5)y + (3 + 8)$
$\Rightarrow 4x + (-14y) + 11$
$\Rightarrow 4x - 14y + 11 $ Ans
(v) $3x^2 + 5xy - 4y^2 + x^2 - 8xy - 5y^2$
$\Rightarrow (3 + 1)x^2 + (-4 - 5)y^2 + (5 - 8)xy$
$\Rightarrow 4x^2 + (-9)y^2 + (-3)xy$
$\Rightarrow 4x^2 - 9y^2 - 3xy $ Ans
$\Rightarrow (-7 + 18 + 3 - 5)x^2$
$\Rightarrow (21 - 13)x^2 = 8x^2 $ Ans
(ii) $b^2 - 9b^2y + 2b^2y - 5b^2y$
$\Rightarrow (1 - 9 + 2 - 5)b^2y$
$\Rightarrow (3 - 14)b^2y = -11b^2y $ Ans
(iv) $7x - 9y + 3 - 3x - 5y + 8$
$\Rightarrow (7 - 3)x + (-9 - 5)y + (3 + 8)$
$\Rightarrow 4x + (-14y) + 11$
$\Rightarrow 4x - 14y + 11 $ Ans
(v) $3x^2 + 5xy - 4y^2 + x^2 - 8xy - 5y^2$
$\Rightarrow (3 + 1)x^2 + (-4 - 5)y^2 + (5 - 8)xy$
$\Rightarrow 4x^2 + (-9)y^2 + (-3)xy$
$\Rightarrow 4x^2 - 9y^2 - 3xy $ Ans
Q2. Add:
(i) $5a + 3b, a - 2b, 3a + 5b$
(ii) $8x - 3y + 7z, -4x + 5y - 4z, -x - y - 2z$
(iii) $3b - 7c + 10, 5c - 2b - 15, 15 + 12c + b$
(iv) $a - 3b + 3, 2a + 5 - 3c, 6c - 15 + 6b$
(v) $13ab - 9cd - xy, 5xy, 15cd - 7ab, 6xy - 3cd$
(vi) $x^3 - x^2y + 5xy^2 + y^3, -x^3 - 9xy^2 + y^3, 3x^2y + 9xy^2$
Show Answer
(i) $5a + 3b, a - 2b, 3a + 5b$
$\Rightarrow 5a + 3b + a - 2b + 3a + 5b$
$\Rightarrow (5 + 1 + 3)a + (3 - 2 + 5)b$
$\Rightarrow 9a + 6b $ Ans
(ii) $8x - 3y + 7z, -4x + 5y - 4z, -x - y - 2z$
$\Rightarrow 8x - 3y + 7z + (-4x + 5y - 4z) + (-x - y - 2z)$
$\Rightarrow 8x - 3y + 7z - 4x + 5y - 4z - x - y - 2z$
$\Rightarrow (8 - 4 - 1)x + (-3 + 5 - 1)y + (7 - 4 - 2)z$
$\Rightarrow (3)x + (1)y + (1)z$
$\Rightarrow 3x + y + z $ Ans
(iii) $3b - 7c + 10, 5c - 2b - 15, 15 + 12c + b$
$\Rightarrow 3b - 7c + 10 + 5c - 2b - 15 + 15 + 12c + b$
$\Rightarrow (3 - 2 + 1)b + (-7 + 5 + 12)c + (10 - 15 + 15)$
$\Rightarrow (2)b + (10)c + (10)$
$\Rightarrow 2b + 10c + 10 $ Ans
(iv) $a - 3b + 3, 2a + 5 - 3c, 6c - 15 + 6b$
$\Rightarrow a - 3b + 3 + 2a + 5 - 3c + 6c - 15 + 6b$
$\Rightarrow (1 + 2)a + (-3 + 6)b + (-3 + 6)c + (3 + 5 - 15)$
$\Rightarrow (3)a + (3)b + (3)c + (-7)$
$\Rightarrow 3a + 3b + 3c - 7 $ Ans
(v) $13ab - 9cd - xy, 5xy, 15cd - 7ab, 6xy - 3cd$
$\Rightarrow 13ab - 9cd - xy + 5xy + 15cd - 7ab + 6xy - 3cd$
$\Rightarrow (13 - 7)ab + (-9 + 15 - 3)cd + (-1 + 5 + 6)xy$
$\Rightarrow (6)ab + (3)cd + (10)xy$
$\Rightarrow 6ab + 3cd + 10xy$ Ans
(vi) $x^3 - x^2y + 5xy^2 + y^3, -x^3 - 9xy^2 + y^3, 3x^2y + 9xy^2$
$\Rightarrow (x^3 - x^2y + 5xy^2 + y^3) + (-x^3 - 9xy^2 + y^3) + (3x^2y + 9xy^2)$
$\Rightarrow (1 - 1)x^3 + (-1 + 3)x^2y + (5 - 9 + 9)xy^2 + (1 + 1)y^3$
$\Rightarrow 0 + (2)x^2y + (5)xy^2 + (2)y^3$
$\Rightarrow 2x^2y + 5xy^2 + 2y^3 $ Ans
$\Rightarrow 5a + 3b + a - 2b + 3a + 5b$
$\Rightarrow (5 + 1 + 3)a + (3 - 2 + 5)b$
$\Rightarrow 9a + 6b $ Ans
(ii) $8x - 3y + 7z, -4x + 5y - 4z, -x - y - 2z$
$\Rightarrow 8x - 3y + 7z + (-4x + 5y - 4z) + (-x - y - 2z)$
$\Rightarrow 8x - 3y + 7z - 4x + 5y - 4z - x - y - 2z$
$\Rightarrow (8 - 4 - 1)x + (-3 + 5 - 1)y + (7 - 4 - 2)z$
$\Rightarrow (3)x + (1)y + (1)z$
$\Rightarrow 3x + y + z $ Ans
(iii) $3b - 7c + 10, 5c - 2b - 15, 15 + 12c + b$
$\Rightarrow 3b - 7c + 10 + 5c - 2b - 15 + 15 + 12c + b$
$\Rightarrow (3 - 2 + 1)b + (-7 + 5 + 12)c + (10 - 15 + 15)$
$\Rightarrow (2)b + (10)c + (10)$
$\Rightarrow 2b + 10c + 10 $ Ans
(iv) $a - 3b + 3, 2a + 5 - 3c, 6c - 15 + 6b$
$\Rightarrow a - 3b + 3 + 2a + 5 - 3c + 6c - 15 + 6b$
$\Rightarrow (1 + 2)a + (-3 + 6)b + (-3 + 6)c + (3 + 5 - 15)$
$\Rightarrow (3)a + (3)b + (3)c + (-7)$
$\Rightarrow 3a + 3b + 3c - 7 $ Ans
(v) $13ab - 9cd - xy, 5xy, 15cd - 7ab, 6xy - 3cd$
$\Rightarrow 13ab - 9cd - xy + 5xy + 15cd - 7ab + 6xy - 3cd$
$\Rightarrow (13 - 7)ab + (-9 + 15 - 3)cd + (-1 + 5 + 6)xy$
$\Rightarrow (6)ab + (3)cd + (10)xy$
$\Rightarrow 6ab + 3cd + 10xy$ Ans
(vi) $x^3 - x^2y + 5xy^2 + y^3, -x^3 - 9xy^2 + y^3, 3x^2y + 9xy^2$
$\Rightarrow (x^3 - x^2y + 5xy^2 + y^3) + (-x^3 - 9xy^2 + y^3) + (3x^2y + 9xy^2)$
$\Rightarrow (1 - 1)x^3 + (-1 + 3)x^2y + (5 - 9 + 9)xy^2 + (1 + 1)y^3$
$\Rightarrow 0 + (2)x^2y + (5)xy^2 + (2)y^3$
$\Rightarrow 2x^2y + 5xy^2 + 2y^3 $ Ans
Q3. Find total savings of a boy who saves Rs.$(4x - 6y)$, Rs.$(6x + 2y)$, Rs.$(4y - x)$, Rs.$(y - 2x)$ in 4 weeks.
Show Answer
$\Rightarrow (4x - 6y) + (6x + 2y) + (4y - x) + (y - 2x)$
$\Rightarrow (4 + 6 - 1 - 2)x + (-6 + 2 + 4 + 1)y$
$\Rightarrow (7)x + (1)y$
$\Rightarrow$ Rs.$(7x + y) $ Ans
$\Rightarrow (4 + 6 - 1 - 2)x + (-6 + 2 + 4 + 1)y$
$\Rightarrow (7)x + (1)y$
$\Rightarrow$ Rs.$(7x + y) $ Ans
Q4. Subtract:
(i) $3xy^2 - 4xy^2$
(ii) $8x^2y - (-2x^2y + 3xy^2)$
(iii) $7a - 3b + c - 2d - (3a - 5b + c + 2d)$
(iv) $3x^3 - x^2 + 6 - (x^3 - 4x - 1)$
(v) $a^3 - 3a^2 + 4a + 1 - (6a + 3)$
(vi) $3abc + 5bcd - cda - (cab - 4cad - cbd)$
Show Answer
(i) $3xy^2 - 4xy^2 = -xy^2 $ Ans
(ii) $8x^2y - (-2x^2y + 3xy^2)$
$\Rightarrow 8x^2y + 2x^2y - 3xy^2$
$\Rightarrow 10x^2y - 3xy^2 $ Ans
(iii) $7a - 3b + c - 2d - (3a - 5b + c + 2d)$
$\Rightarrow 7a - 3b + c - 2d - 3a + 5b - c - 2d$
$\Rightarrow (7 - 3)a + (-3 + 5)b + (1 - 1)c + (-2 - 2)d$
$\Rightarrow (4)a + (2)b + (0)c + (-4)d$
$\Rightarrow 4a + 2b - 4d $ Ans
(iv) $3x^3 - x^2 + 6 - (x^3 - 4x - 1)$
$\Rightarrow 3x^3 - x^2 + 6 - x^3 + 4x + 1$
$\Rightarrow (3 - 1)x^3 - x^2 + 4x + (6 + 1)$
$\Rightarrow 2x^3 - x^2 + 4x + 7 $ Ans
(v) $a^3 - 3a^2 + 4a + 1 - (6a + 3)$
$\Rightarrow a^3 - 3a^2 + (4 - 6)a + (1 - 3)$
$\Rightarrow a^3 - 3a^2 + (-2)a + (-2)$
$\Rightarrow a^3 - 3a^2 - 2a - 2 $ Ans
(vi) $3abc + 5bcd - cda - (cab - 4cad - cbd)$
$\Rightarrow 3abc + 5bcd - cda - cab + 4cad + cbd$
$\Rightarrow (3abc - cab) + (-cda + 4cad) + (5bcd + cbd)$
$\Rightarrow (3 - 1)abc + (-1 + 4)cad + (5 + 1)bcd$ (Higer values retain their values)
$\Rightarrow 2abc + 3cad + 6bcd $ Ans
(ii) $8x^2y - (-2x^2y + 3xy^2)$
$\Rightarrow 8x^2y + 2x^2y - 3xy^2$
$\Rightarrow 10x^2y - 3xy^2 $ Ans
(iii) $7a - 3b + c - 2d - (3a - 5b + c + 2d)$
$\Rightarrow 7a - 3b + c - 2d - 3a + 5b - c - 2d$
$\Rightarrow (7 - 3)a + (-3 + 5)b + (1 - 1)c + (-2 - 2)d$
$\Rightarrow (4)a + (2)b + (0)c + (-4)d$
$\Rightarrow 4a + 2b - 4d $ Ans
(iv) $3x^3 - x^2 + 6 - (x^3 - 4x - 1)$
$\Rightarrow 3x^3 - x^2 + 6 - x^3 + 4x + 1$
$\Rightarrow (3 - 1)x^3 - x^2 + 4x + (6 + 1)$
$\Rightarrow 2x^3 - x^2 + 4x + 7 $ Ans
(v) $a^3 - 3a^2 + 4a + 1 - (6a + 3)$
$\Rightarrow a^3 - 3a^2 + (4 - 6)a + (1 - 3)$
$\Rightarrow a^3 - 3a^2 + (-2)a + (-2)$
$\Rightarrow a^3 - 3a^2 - 2a - 2 $ Ans
(vi) $3abc + 5bcd - cda - (cab - 4cad - cbd)$
$\Rightarrow 3abc + 5bcd - cda - cab + 4cad + cbd$
$\Rightarrow (3abc - cab) + (-cda + 4cad) + (5bcd + cbd)$
$\Rightarrow (3 - 1)abc + (-1 + 4)cad + (5 + 1)bcd$ (Higer values retain their values)
$\Rightarrow 2abc + 3cad + 6bcd $ Ans
Q5. Subtract:
(i) Take away $(-3x^3 + 4x^2 - 5x + 6)$ from $(3x^3 - 4x^2 + 5x - 6)$
(ii) Take away $(m^2 + m + 4)$ from $(-m^2 + 3m + 6)$ and the result from $(m^2 + m + 1)$
Show Answer
(i) $3x^3 - 4x^2 + 5x - 6 - (-3x^3 + 4x^2 - 5x + 6)$
$\Rightarrow 3x^3 - 4x^2 + 5x - 6 + 3x^3 - 4x^2 + 5x - 6$
$\Rightarrow (3 + 3)x^3 + (-4 - 4)x^2 + (5 + 5)x + (-6 - 6)$
$\Rightarrow (6)x^3 + (-8)x^2 + (10)x + (-12)$
$\Rightarrow 6x^3 - 8x^2 + 10x - 12 $ Ans
(ii) Case 1:
$\Rightarrow -m^2 + 3m + 6 - (m^2 + m + 4)$
$\Rightarrow -m^2 + 3m + 6 - m^2 - m - 4$
$\Rightarrow (-1 - 1)m^2 + (3 - 1)m + (6 - 4)$
$\Rightarrow (-2)m^2 + (2)m + (2)$
$\Rightarrow - 2m^2 + 2m + 2$
Case 2:
$\Rightarrow m^2 + m + 1 - (-2m^2 + 2m + 2)$
$\Rightarrow m^2 + m + 1 + 2m^2 - 2m - 2)$
$\Rightarrow (1 + 2)m^2 + (1 - 2)m + (1 - 2)$
$\Rightarrow (3)m^2 + (-1)m + (-1)$
$\Rightarrow 3m^2 - m - 1 $ Ans
$\Rightarrow 3x^3 - 4x^2 + 5x - 6 + 3x^3 - 4x^2 + 5x - 6$
$\Rightarrow (3 + 3)x^3 + (-4 - 4)x^2 + (5 + 5)x + (-6 - 6)$
$\Rightarrow (6)x^3 + (-8)x^2 + (10)x + (-12)$
$\Rightarrow 6x^3 - 8x^2 + 10x - 12 $ Ans
(ii) Case 1:
$\Rightarrow -m^2 + 3m + 6 - (m^2 + m + 4)$
$\Rightarrow -m^2 + 3m + 6 - m^2 - m - 4$
$\Rightarrow (-1 - 1)m^2 + (3 - 1)m + (6 - 4)$
$\Rightarrow (-2)m^2 + (2)m + (2)$
$\Rightarrow - 2m^2 + 2m + 2$
Case 2:
$\Rightarrow m^2 + m + 1 - (-2m^2 + 2m + 2)$
$\Rightarrow m^2 + m + 1 + 2m^2 - 2m - 2)$
$\Rightarrow (1 + 2)m^2 + (1 - 2)m + (1 - 2)$
$\Rightarrow (3)m^2 + (-1)m + (-1)$
$\Rightarrow 3m^2 - m - 1 $ Ans
Q6. Subtract the sum of $5y^2 + y - 3$ and $y^2 - 3y + 7$ from $6y^2 + y - 2$
Show Answer
$\Rightarrow 5y^2 + y - 3 + y^2 - 3y + 7$
$\Rightarrow (5 + 1)y^2 + (1 - 3)y + (-3 + 7)$
$\Rightarrow (6)y^2 + (-2)y + (4)$
$\Rightarrow 6y^2 - 2y + 4$
For Subtraction:
$\Rightarrow 6y^2 + y - 2 - (6y^2 - 2y + 4)$
$\Rightarrow 6y^2 + y - 2 - 6y^2 + 2y - 4$
$\Rightarrow (6 - 6)y^2 + (1 + 2)y + (-2 - 4)$
$\Rightarrow (0)y^2 + (3)y + (-6)$
$\Rightarrow 3y - 6 $ Ans
$\Rightarrow (5 + 1)y^2 + (1 - 3)y + (-3 + 7)$
$\Rightarrow (6)y^2 + (-2)y + (4)$
$\Rightarrow 6y^2 - 2y + 4$
For Subtraction:
$\Rightarrow 6y^2 + y - 2 - (6y^2 - 2y + 4)$
$\Rightarrow 6y^2 + y - 2 - 6y^2 + 2y - 4$
$\Rightarrow (6 - 6)y^2 + (1 + 2)y + (-2 - 4)$
$\Rightarrow (0)y^2 + (3)y + (-6)$
$\Rightarrow 3y - 6 $ Ans
Q7. What must be added to $x^4 - x^3 + x^2 + x + 3$ to obtain $x^4 + x^2 - 1$
Show Answer
$\Rightarrow x^4 - x^3 + x^2 + x + 3 + \text{Number} = x^4 + x^2 - 1$
$\Rightarrow x^4 - x^3 + x^2 + x + 3 - (x^4 + x^2 - 1) = \text{Number}$
$\Rightarrow \text{Number} = x^4 - x^3 + x^2 + x + 3 - (x^4 + x^2 - 1)$
$\Rightarrow \text{Number} = x^4 - x^3 + x^2 + x + 3 - x^4 - x^2 + 1$
$\Rightarrow \text{Number} = (1 - 1)x^4 - x^3 + (1 - 1)x^2 + x + (3 + 1)$
$\Rightarrow \text{Number} = (0)x^4 - x^3 + (0)x^2 + x + (4)$
$\Rightarrow \text{Number} = - x^3 + x + 4$
$\because$ The required number is to be subtracted:
$\therefore -(- x^3 + x + 4) = x^3 - x - 4 $ Ans
$\Rightarrow x^4 - x^3 + x^2 + x + 3 - (x^4 + x^2 - 1) = \text{Number}$
$\Rightarrow \text{Number} = x^4 - x^3 + x^2 + x + 3 - (x^4 + x^2 - 1)$
$\Rightarrow \text{Number} = x^4 - x^3 + x^2 + x + 3 - x^4 - x^2 + 1$
$\Rightarrow \text{Number} = (1 - 1)x^4 - x^3 + (1 - 1)x^2 + x + (3 + 1)$
$\Rightarrow \text{Number} = (0)x^4 - x^3 + (0)x^2 + x + (4)$
$\Rightarrow \text{Number} = - x^3 + x + 4$
$\because$ The required number is to be subtracted:
$\therefore -(- x^3 + x + 4) = x^3 - x - 4 $ Ans
Q8. (i) How much more than $2x^2 + 4xy + 2y^2$ is $5x^2 + 10xy - y^2$?
(ii) How much less than $2a^2 + 1$ is $3a^2 - 6$?
Show Answer
(i) $5x^2 + 10xy - y^2 - (2x^2 + 4xy + 2y^2)$
$\Rightarrow 5x^2 + 10xy - y^2 - 2x^2 - 4xy - 2y^2$
$\Rightarrow (5 - 2)x^2 + (10 - 4)xy + (-1 - 2)y^2$
$\Rightarrow (3)x^2 + (6)xy + (-3)y^2$
$\Rightarrow 3x^2 + 6xy - 3y^2 $ Ans
(ii) $3a^2 - 6 - (2a^2 + 1)$
$\Rightarrow 3a^2 - 6 - 2a^2 + 1$
$\Rightarrow (3 - 2)a^2 - (6 + 1)$
$\Rightarrow (1)a^2 - (7)$
$\Rightarrow a^2 - 7 $ Ans
$\Rightarrow 5x^2 + 10xy - y^2 - 2x^2 - 4xy - 2y^2$
$\Rightarrow (5 - 2)x^2 + (10 - 4)xy + (-1 - 2)y^2$
$\Rightarrow (3)x^2 + (6)xy + (-3)y^2$
$\Rightarrow 3x^2 + 6xy - 3y^2 $ Ans
(ii) $3a^2 - 6 - (2a^2 + 1)$
$\Rightarrow 3a^2 - 6 - 2a^2 + 1$
$\Rightarrow (3 - 2)a^2 - (6 + 1)$
$\Rightarrow (1)a^2 - (7)$
$\Rightarrow a^2 - 7 $ Ans
Q9. If $x = 6a + 8b + 9x$; $y = 2b - 3a - 6c$ and $z = c - b + 3a$. Find:
If $x = 6a + 8b + 9x$; $y = 2b - 3a - 6c$ and $z = c - b + 3a$. Find:
(i) $x + y + z$
(ii) $x - y + z$
(iii) $2x - y - 3z$
(iv) $3y - 2z - 5x$
Show Answer
(i) $x + y + z$
$\Rightarrow (6a + 8b + 9c) + (2b - 3a - 6c) + (c - b + 3a)$
$\Rightarrow 6a + 8b + 9c + 2b - 3a - 6c + c - b + 3a$
$\Rightarrow (6 - 3 + 3)a + (8 + 2 - 1)b + (9 - 6 + 1)c$
$\Rightarrow (6)a + (9)b + (4)c$
$\Rightarrow 6a + 9b + 4c $ Ans
(ii) $x - y + z$
$\Rightarrow (6a + 8b + 9c) - (2b - 3a - 6c) + (c - b + 3a)$
$\Rightarrow 6a + 8b + 9c - 2b + 3a + 6c + c - b + 3a$
$\Rightarrow (6 + 3 + 3)a + (8 - 2 - 1)b + (9 + 6 + 1)c$
$\Rightarrow (12)a + (5)b + (16)c$
$\Rightarrow 12a + 5b + 16c $ Ans
(iii) $2x - y - 3z$
$\Rightarrow 2(6a + 8b + 9c) - (2b - 3a - 6c) - 3(c - b + 3a)$
$\Rightarrow 12a + 16b + 18c - 2b + 3a + 6c - 3c + 3b - 9a$
$\Rightarrow (12 + 3 - 9)a + (16 - 2 + 3)b + (18 + 6 - 3)c$
$\Rightarrow (6)a + (17)b + (21)c$
$\Rightarrow 6a + 17b + 21c $ Ans
(iv) $3y - 2z - 5x$
$\Rightarrow 3(2b - 3a - 6c) - 2(c - b + 3a) - 5(6a + 8b + 9c)$
$\Rightarrow 6b - 9a - 18c - 2c + 2b - 6a - 30a - 40b - 45c)$
$\Rightarrow (-9 - 6 - 30)a + (6 + 2 - 40)b + (-18 - 2 - 45)c$
$\Rightarrow (-45)a + (-32)b + (-65)c$
$\Rightarrow -45a - 32b - 65c $ Ans
$\Rightarrow (6a + 8b + 9c) + (2b - 3a - 6c) + (c - b + 3a)$
$\Rightarrow 6a + 8b + 9c + 2b - 3a - 6c + c - b + 3a$
$\Rightarrow (6 - 3 + 3)a + (8 + 2 - 1)b + (9 - 6 + 1)c$
$\Rightarrow (6)a + (9)b + (4)c$
$\Rightarrow 6a + 9b + 4c $ Ans
(ii) $x - y + z$
$\Rightarrow (6a + 8b + 9c) - (2b - 3a - 6c) + (c - b + 3a)$
$\Rightarrow 6a + 8b + 9c - 2b + 3a + 6c + c - b + 3a$
$\Rightarrow (6 + 3 + 3)a + (8 - 2 - 1)b + (9 + 6 + 1)c$
$\Rightarrow (12)a + (5)b + (16)c$
$\Rightarrow 12a + 5b + 16c $ Ans
(iii) $2x - y - 3z$
$\Rightarrow 2(6a + 8b + 9c) - (2b - 3a - 6c) - 3(c - b + 3a)$
$\Rightarrow 12a + 16b + 18c - 2b + 3a + 6c - 3c + 3b - 9a$
$\Rightarrow (12 + 3 - 9)a + (16 - 2 + 3)b + (18 + 6 - 3)c$
$\Rightarrow (6)a + (17)b + (21)c$
$\Rightarrow 6a + 17b + 21c $ Ans
(iv) $3y - 2z - 5x$
$\Rightarrow 3(2b - 3a - 6c) - 2(c - b + 3a) - 5(6a + 8b + 9c)$
$\Rightarrow 6b - 9a - 18c - 2c + 2b - 6a - 30a - 40b - 45c)$
$\Rightarrow (-9 - 6 - 30)a + (6 + 2 - 40)b + (-18 - 2 - 45)c$
$\Rightarrow (-45)a + (-32)b + (-65)c$
$\Rightarrow -45a - 32b - 65c $ Ans
Q10. The sides of a triangle are $(x^2 - 3xy + 8), (4x^2 + 5xy - 3)$ and $(6 - 3x^2 + 4xy)$. Find its perimeter.
Show Answer
Solution 10:
$\Rightarrow$ Perimeter of a triangle = sum length of its 3 sides
$\therefore (x^2 - 3xy + 8) + (4x^2 + 5xy - 3) + (6 - 3x^2 + 4xy)$
$\Rightarrow (1 + 4 - 3)x^2 + (-3 + 5 + 4)xy + (8 - 3 + 6)$
$\Rightarrow (2)x^2 + (6)xy + (11)$
$\Rightarrow 2x^2 + 6xy + 11 $ Ans
$\Rightarrow$ Perimeter of a triangle = sum length of its 3 sides
$\therefore (x^2 - 3xy + 8) + (4x^2 + 5xy - 3) + (6 - 3x^2 + 4xy)$
$\Rightarrow (1 + 4 - 3)x^2 + (-3 + 5 + 4)xy + (8 - 3 + 6)$
$\Rightarrow (2)x^2 + (6)xy + (11)$
$\Rightarrow 2x^2 + 6xy + 11 $ Ans
Q11. The perimeter of a triangle is $(8y^2 - 9y + 4)$ and its two sides are $(3y^2 - 5y)$ and $(4y^2 + 12)$. Find its third side.
Show Answer
$\Rightarrow$ Perimeter of a triangle = sum length of its 3 sides
$\therefore (3y^2 - 5y) + (4y^2 + 12) + $ third side = $(8y^2 - 9y + 4)$
$\Rightarrow (3 + 4)y^2 + (-5)y + (12) + $ third side = $(8y^2 - 9y + 4)$
$\Rightarrow 7y^2 - 5y + 12 + $ third side = $(8y^2 - 9y + 4)$
$\Rightarrow$ third side = $(8y^2 - 9y + 4) - (7y^2 - 5y + 12)$
$\Rightarrow$ third side = $(8y^2 - 9y + 4) - 7y^2 + 5y - 12$
$\Rightarrow$ third side = $(8 - 7)y^2 + (-9 + 5)y + (4 - 12)$
$\Rightarrow$ third side = $(1)y^2 + (-4)y + (-8)$
$\Rightarrow$ third side = $y^2 - 4y - 8 $ Ans
$\therefore (3y^2 - 5y) + (4y^2 + 12) + $ third side = $(8y^2 - 9y + 4)$
$\Rightarrow (3 + 4)y^2 + (-5)y + (12) + $ third side = $(8y^2 - 9y + 4)$
$\Rightarrow 7y^2 - 5y + 12 + $ third side = $(8y^2 - 9y + 4)$
$\Rightarrow$ third side = $(8y^2 - 9y + 4) - (7y^2 - 5y + 12)$
$\Rightarrow$ third side = $(8y^2 - 9y + 4) - 7y^2 + 5y - 12$
$\Rightarrow$ third side = $(8 - 7)y^2 + (-9 + 5)y + (4 - 12)$
$\Rightarrow$ third side = $(1)y^2 + (-4)y + (-8)$
$\Rightarrow$ third side = $y^2 - 4y - 8 $ Ans
Q12. The two adjacent sides of a rectangle are $2x^2 - 5xy + 3z^2$ and $(4xy - x^2 - z^2)$. Find its perimeter
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$\Rightarrow$ Perimeter of a rectangle = 2(L $+$ B)
$\therefore 2((2x^2 - 5xy + 3z^2) + (4xy - x^2 - z^2))$
$\Rightarrow (4x^2 - 10xy + 6z^2) + (8xy - 2x^2 - 2z^2)$
$\Rightarrow (4 - 2)x^2 + (-10 + 8)xy + (6 - 2)z^2$
$\Rightarrow (2)x^2 + (-2)xy + (4)z^2$
$\Rightarrow 2x^2 - 2xy + 4z^2 $ Ans
$\therefore 2((2x^2 - 5xy + 3z^2) + (4xy - x^2 - z^2))$
$\Rightarrow (4x^2 - 10xy + 6z^2) + (8xy - 2x^2 - 2z^2)$
$\Rightarrow (4 - 2)x^2 + (-10 + 8)xy + (6 - 2)z^2$
$\Rightarrow (2)x^2 + (-2)xy + (4)z^2$
$\Rightarrow 2x^2 - 2xy + 4z^2 $ Ans
Q13. What must be subtracted from $(19x^4 + 2x^3 + 30x - 37)$ to get $(8x^4 + 22x^3 - 7x - 60)$?
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$\Rightarrow$ Let the required no. be = x
$\therefore (19x^4 + 2x^3 + 30x - 37) - x = (8x^4 + 22x^3 - 7x - 60)$
$\Rightarrow (19x^4 + 2x^3 + 30x - 37) - (8x^4 + 22x^3 - 7x - 60) = x$
$\Rightarrow (19x^4 + 2x^3 + 30x - 37) - 8x^4 - 22x^3 + 7x + 60 = x$
$\Rightarrow x = (19 - 8)x^4 + (2 - 22)x^3 + (30 + 7)x + (-37 + 60)$
$\Rightarrow x = (11)x^4 + (-20)x^3 + (37)x + (23)$
$\Rightarrow x = 11x^4 - 20x^3 + 37x + 23 $ Ans
$\therefore (19x^4 + 2x^3 + 30x - 37) - x = (8x^4 + 22x^3 - 7x - 60)$
$\Rightarrow (19x^4 + 2x^3 + 30x - 37) - (8x^4 + 22x^3 - 7x - 60) = x$
$\Rightarrow (19x^4 + 2x^3 + 30x - 37) - 8x^4 - 22x^3 + 7x + 60 = x$
$\Rightarrow x = (19 - 8)x^4 + (2 - 22)x^3 + (30 + 7)x + (-37 + 60)$
$\Rightarrow x = (11)x^4 + (-20)x^3 + (37)x + (23)$
$\Rightarrow x = 11x^4 - 20x^3 + 37x + 23 $ Ans
Q14. How much smaller is $(15x - 18y + 19z)$ than $(22x - 20y - 13z + 26)$?
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$\Rightarrow (22x - 20y - 13z + 26) - (15x - 18y + 19z)$
$\Rightarrow (22x - 20y - 13z + 26) - 15x + 18y - 19z$
$\Rightarrow (22 - 15)x + (-20 + 18)y + (-13 - 19)z + 26$
$\Rightarrow (7)x + (-2)y + (-32)z + 26$
$\Rightarrow 7x - 2y - 32z + 26 $ Ans
$\Rightarrow (22x - 20y - 13z + 26) - 15x + 18y - 19z$
$\Rightarrow (22 - 15)x + (-20 + 18)y + (-13 - 19)z + 26$
$\Rightarrow (7)x + (-2)y + (-32)z + 26$
$\Rightarrow 7x - 2y - 32z + 26 $ Ans
Q15. How much bigger is $(5x^2y^2 - 18xy^2 - 10x^2y)$ than $(-5x^2 + 6x^2y - 7xy)$?
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$\Rightarrow (5x^2y^2 - 18xy^2 - 10x^2y) - (-5x^2 + 6x^2y - 7xy)$
$\Rightarrow (5x^2y^2 - 18xy^2 - 10x^2y) + 5x^2 - 6x^2y + 7xy$
$\Rightarrow 5x^2y^2 - 18xy^2 + (-10 - 6)x^2y + 5x^2 + 7xy$
$\Rightarrow 5x^2y^2 - 18xy^2 + (-16)x^2y + 5x^2 + 7xy$
$\Rightarrow 5x^2y^2 - 18xy^2 - 16x^2y + 5x^2 + 7xy $ Ans
$\Rightarrow (5x^2y^2 - 18xy^2 - 10x^2y) + 5x^2 - 6x^2y + 7xy$
$\Rightarrow 5x^2y^2 - 18xy^2 + (-10 - 6)x^2y + 5x^2 + 7xy$
$\Rightarrow 5x^2y^2 - 18xy^2 + (-16)x^2y + 5x^2 + 7xy$
$\Rightarrow 5x^2y^2 - 18xy^2 - 16x^2y + 5x^2 + 7xy $ Ans