Q 1 . \(a^{3}+8b^{3}+64c^{3}-24abc\)
SOLUTION :
= \(\left ( a \right )^{3}+\left ( 2b \right )^{3}+\left ( 4c \right )^{3}-3\times a\times 2b\times 4c\)
= \(\left ( a+2b+4c \right )\left ( a^{2}+\left ( 2b \right )^{2}+\left ( 4c \right )^{2}-a\times 2b-2b\times 4c-4c\times a \right )\)
= \(\left ( a+2b+4c \right )\left ( a^{2}+ 4b^{2}+ 16c^{2}-2ab-8bc -4ac \right )\)
\(∴\)
Q 2 . \(x^{3}-8y^{3}+27z^{3}+18xyz\)
SOLUTION :
= \(x^{3}-\left ( 2y \right )^{3}+\left ( 3z \right )^{3}-3\times x\times \left ( -2y \right )\left ( 3z\right )\)
= \(\left ( x+\left ( -2y \right )+3z \right )\left ( x^{2}+\left (-2y \right )^{2}+\left ( 3z \right )^{2}-x\left (- 2y \right )-\left ( -2y \right )\left ( 3z \right )-3z\left ( x \right ) \right )\)
=\(\left ( x+\left ( -2y \right )+3z \right )\left ( x^{2}+4y^{2}+9z^{2}+2xy+6yz -3z x \right )\)
\(∴\)
Q 3 . \(\frac{1}{27}x^{3}-y^{3}+125z^{3}+5xyz\)
SOLUTION :
= \(\left ( \frac{x}{3} \right )^{3}+\left ( -y \right )^{3}+\left ( 5z \right )^{3}-3\times \frac{x}{3}\left ( -y \right )\left ( 5z \right )\)
= \(\left ( \frac{x}{3}+\left ( -y\right ) +5z\right )\left ( \left ( \frac{x}{3} \right )^{2}+\left ( -y \right )^{2}+\left ( 5z \right )^{2}-\frac{x}{3}\left ( -y \right )-\left ( -y \right )5z-5z\left ( \frac{x}{3} \right ) \right )\)
= \(\left ( \frac{x}{3} -y +5z\right )\left ( \frac{x^{2}}{9} +y^{2}+25z^{2}+\frac{xy}{3} +5yz – \frac{5zx}{3} \right )\)
\(∴\)
Q 4 . \(8x^{3}+27y^{3}-216z^{3}+108xyz\)
SOLUTION :
= \(\left ( 2x \right )^{3}+\left ( 3y \right )^{3}+\left ( -6y \right )^{3}-3\left ( 2x \right )\left (3y \right )\left ( -6z \right )\)
= \(\left ( 2x+3y+\left ( -6z \right ) \right )\left ( \left ( 2x \right )^{2}+\left ( 3y \right )^{2}+\left ( -6z \right )^{2}-2x\times 3y-3y\left ( -6z \right )-\left ( -6z \right )2x \right )\)
= \(\left ( 2x+3y+\left ( -6z \right ) \right )\left (4x^{2}+9y^{2}+36z ^{2}-6xy+18yz+12zx \right )\)
\(∴\)
Q 5 . \(125+8x^{3}-27y^{3}+90xy\)
SOLUTION :
= \(\left ( 5 \right )^{3}+\left ( 2x \right )^{3}+\left ( -3y \right )^{3}-3\times 5\times 2x\times \left ( -3y \right )\)
= \(\left ( 5+2x+\left ( -3y \right ) \right )\left ( 5^{2}+\left ( 2x \right )^{2}+\left ( -3y \right )^{2}-5\left ( 2x \right )-2x\left ( -3y \right )-\left ( -3y \right )5 \right )\)
= \(\left ( 5+2x-3y \right )\left ( 25+ 4x^{2}+9y^{2}-10x+6xy +15y \right )\)
\(∴\)
Q 6 . \(\left ( 3x-2y \right )^{3}+\left ( 2y-4z \right )^{3}+\left ( 4z-3x \right )^{3}\)
SOLUTION :
Let \(\left ( 3x-2y \right )\)
\(∴ a+b+c=3x-2y+2y-4z+4z-3x=0\)
\(∵ a+b+c=0∴ a^{3}+b^{3}+c^{3}=3abc\)
= \(3\left ( 3x-2y \right )\left ( 2y-4z \right )\left ( 4z-3x \right )\)
\(∴\)
Q 7 . \(\left ( 2x-3y \right )^{3}+\left (4z-2x \right )^{3}+\left ( 3y-4z \right )^{3}\)
SOLUTION :
Let 2x – 3y = a , 4z – 2x = b , 3y – 4z = c
\(∴ a+b+c=2x-3y+4z-2x+3y-4z=0\)
\(∵ a+b+c=0∴ a^{3}+b^{3}+c^{3}=3abc\)
= \(3\left ( 2x-3y \right )\left ( 4z-2x \right )\left ( 3y-4z \right )\)
\(∴\)
Q 8 . \(\left [ \frac{x}{2}+y+\frac{z}{3} \right ]^{3}+\left [ \frac{x}{3}-\frac{2y}{3}+z \right ]^{3}+\left [ -\frac{5x}{6}-\frac{y}{3}-\frac{4z}{3} \right ]^{3}\)
SOLUTION :
\(Let\;\left [ \frac{x}{2}+y+\frac{z}{3} \right ]=a , \left [ \frac{x}{3}-\frac{2y}{3}+z \right ]=b,\left [ -\frac{5x}{6}-\frac{y}{3}-\frac{4z}{3} \right ]=c\)
\(a+b+c=\frac{x}{2}+y+\frac{z}{3}+\frac{x}{3}-\frac{2y}{3}+z-\frac{5x}{6}-\frac{y}{3}-\frac{4z}{3}\)
\(a+b+c=\left ( \frac{x}{2}+\frac{x}{3}-\frac{5x}{6} \right )+\left (y-\frac{2y}{3} -\frac{y}{3} \right )+\left ( \frac{z}{3}+z-\frac{4z}{3} \right )\)
\(a+b+c=\frac{3x}{6}+\frac{2x}{6}-\frac{5x}{6}+\frac{3y}{3}-\frac{2y}{3}-\frac{y}{3}+\frac{z}{3}+\frac{3z}{3}-\frac{4z}{3}\)
\(a+b+c=\frac{5x-5x}{6}+\frac{3y-3y}{3}+\frac{4z-4z}{3}\)
\(a+b+c=0\)
\(∵ a+b+c=0\;\;\;\;\;\;\;\;\;\;∴ a^{3}+b^{3}+c^{3}=3abc\)
=\(3\left ( \frac{x}{2}+y+\frac{z}{3} \right )\left ( \frac{x}{3}-\frac{2y}{3}+z\right )\left (-\frac{5x}{6}-\frac{y}{3}-\frac{4z}{3} \right )\)
\(∴\)
Q 9 . \(\left (a-3b \right )^{3}+\left (3b-c \right )^{3}+\left (c-a \right )^{3}\)
SOLUTION :
Let a – 3b = x , 3b – c = y , c – a = z
\(x+y+z=a-3b+3b-c+c-a=0\)
\((∵ x+y+z=0)\;\;\;\;\;\;\;\;∴ x^{3}+y^{3}+z^{3}=3xyz\)
= \(3\left ( a-3b \right )\left ( 3b-c \right )\left ( c-a \right )\)
\(∴\)
Q 10 . \(2\sqrt{2}a^{3}+3\sqrt{3}b^{3}+c^{3}-3\sqrt{6abc}\)
SOLUTION :
= \(\left ( \sqrt{2}a \right )^{3}+\left ( \sqrt{3}b \right )^{3}+c^{3}-3\times \sqrt{2a}\times \sqrt{3b}\times c\)
= \(\left ( \sqrt{2a} +\sqrt{3b}+c\right )\left ( \left ( \sqrt{2a} \right )^{2}+\left ( \sqrt{3b} \right )^{2}+c^{2}-\left ( \sqrt{2a} \right )\left ( \sqrt{3b} \right )-\left ( \sqrt{3b} \right )c-\left ( \sqrt{2a}c \right ) \right )\)
= \(\left ( \sqrt{2a} +\sqrt{3b}+c\right )\left ( 2a^{2}+3b^{2}+c^{2}-\sqrt{6}ab-\sqrt{3}bc-\sqrt{2}ac \right )\)
\(∴\)
Q 11 . \(3\sqrt{3}a^{3}-b^{3}-5\sqrt{5}c^{3}-3\sqrt{15}abc\)
SOLUTION :
= \(\left ( \sqrt{3}a \right )^{3}+\left ( -b \right )^{3}-\left ( \sqrt{5}c \right )^{3}-3\left ( \sqrt{3a} \right )\left ( -b \right )\left ( -\sqrt{5}c \right )\)
= \(\left ( \sqrt{3}a +\left ( -b \right )+\left ( -\sqrt{5}c \right )\right )\left ( \left ( \sqrt{3}a \right )^{2}+\left ( -b \right )^{2} +\left ( -\sqrt{5}c \right )^{2}-\sqrt{3}a\left ( -b \right )-\left ( -b \right )\left ( -\sqrt{5}c \right )-\left ( -\sqrt{5}c\right )\sqrt{3}a\right )\)
=\(\left ( \sqrt{3}a -b -\sqrt{5}c\right )\left ( 3a ^{2}+b^{2} +5c^{2}+\sqrt{3}ab -\sqrt{5}bc +\sqrt{15}ac\right )\)
\(∴\)
Q 12 . \(8x^{3}-125y^{3}+216+180xy\)
SOLUTION :
= \(\left ( 2x \right )^{3}+\left ( -5y \right )^{3}+6^{3}-3\times \left ( 2x \right )\left ( -5y \right )\left ( 6 \right )\)
= \(\left ( 2x+\left ( -5y \right ) +6\right )\left ( \left ( 2x \right )^{2}+\left ( -5y \right ) ^{2}+6^{2}-2x\times \left ( -5y \right )-\left ( -5y \right )6-6\left ( 2x \right )\right )\)
= \(\left ( 2x-5y +6\right )\left ( 4x^{2}+25y ^{2}+36+10xy+30y-12x\right)\)
\(∴\)
Q 13 . \(2\sqrt{2}a^{3}+16\sqrt{2}b^{3}+c^{3}-12abc\)
SOLUTION :
= \(\left ( \sqrt{2}a \right )^{3}+\left ( 2\sqrt{2}b \right )^{3}+c^{3}-3\times \sqrt{2}a\times 2\sqrt{2}b\times c\)
= \(\left ( \sqrt{2}a + 2\sqrt{2}b+c \right )\left ( \left ( \sqrt{2}a \right )^{2} +\left (2\sqrt{2}b \right )^{2}+c^{2}-\left ( \sqrt{2}a \right )\left (2\sqrt{2}b \right )-\left (2\sqrt{2}b \right )c-\left ( \sqrt{2}a \right )c\right )\)
= \(\left ( \sqrt{2}a + 2\sqrt{2}b+c \right )\left ( 2a^{2}+8b^{2}+c^{2}-4ab-2\sqrt{2}bc-\sqrt{2}ac \right )\)
\(∴\)
Q 14 . Find the value of \(x^{3}+y^{3}-12xy+64\)
SOLUTION :
= \(x^{3}+y^{3}+64-12xy\)
= \(x^{3}+y^{3}+4^{3}-3\left ( x \right )\left ( y \right )\left ( 4 \right )\)
= \(\left ( x+y+4 \right )\left ( x^{2}+y^{2}+4^{2} -xy-y\times 4-4\times x\right )\)
= \(\left ( -4+4 \right )\left ( x^{2}+y^{2}+16 -xy- 4y-4x\right )\)
=0
\(∴\)
Q 15 . MULTIPLY :
(i) . \(x^{2}+y^{2}+z^{2}-xy+xz+yz\;\;by\;\;x+y-z\)
SOLUTION :
= \(\left (x^{2}+y^{2}+z^{2}-xy+xz+yz \right )\left (x+y-z \right )\)
= \(x^{3}+y^{3}+z^{3}-3xyz\)
(ii) . \(x^{2}+4y^{2}+z^{2}+2xy+xz-2yz\;\;by\;\;x-2y-z\)
SOLUTION :
\(x^{2}+\left (-2y \right )^{2}+\left (-z \right )^{2}-\left ( -2y \right )\left ( -z \right )-\left ( -z \right )\left ( x \right )\)
\(\Rightarrow x^{2}+4y^{2}+ z^{2}+2xy-2yz+zx = x^{3}-8y^{3}-z^{3}-6xyz\)
(iii) . \(x^{2}+4y^{2}+2xy-3x+6y+9\)
SOLUTION :
\(\left ( x \right )^{2}+\left ( -2y \right )^{2}+\left ( 3 \right )^{2}-\left ( x \right )\left ( -2y \right )-\left ( -2y \right )\left ( 3 \right )-3\left ( x \right )\)
\(\Rightarrow x^{2}+4y^{2}+9+2xy+6y-3x\)
(iv) . \(9x^{2}+25y^{2}+15xy+12x-20y+16\;\;by\;\;3x-5y+4\)
SOLUTION :
\(\left (3x \right )^{2}+\left (5y \right )^{2}+4^{2}-\left ( -3x \right )\left ( 5y \right )-\left ( 5y \right )\left ( 4 \right )-\left ( 4 \right )\left ( -3x \right )=\left ( -3x \right )^{3}+\left ( 5y \right )^{3}+4^{3}-3\left ( -3x \right )\left ( 5y \right )\left ( 4 \right )\)
\(\Rightarrow 9x^{2}+25y^{2}+16+15xy-20y+12x=-27x^{3}+125y^{3}+64+180xy\)
