Category : 7th Class
Abatement of equality which holds, for all values of the variable is called algebraic identities.
Now we recall some important algebraic identities:
If \[(a+b+c)=0\]\[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc\]
Simplify: \[{{(2p+3q-+4r)}^{2}}+{{(2p-3q-4r)}^{2}}\]
(a) \[2(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-16rp)\]
(b) \[-2(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-16rp)\]
(c) \[2(-4{{p}^{2}}+9{{q}^{2}}-16{{r}^{2}}+16rp)\]
(d)\[~2(5{{p}^{2}}+9{{q}^{2}}-16{{r}^{2}}-98rp)\]
(e) None of these
Answer: (a)
Explanation
Let us first solve,
\[{{\left[ 2p+3q+(-4r) \right]}^{2}}={{(2p)}^{2}}+{{(3q)}^{2}}\]\[+{{(-4r)}^{2}}+2(2p)(3q)+2(3q)(-4r)\] \[+2(-4r)(2p)=4{{p}^{2}}+4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}\]\[+12pq-24qr-16rp..........(i)\]
Now solve, \[~{{(2p-3q-4r)}^{2}}={{\left[ 2p+(-3q)+(-4r) \right]}^{2}}\]
\[={{(2p)}^{2}}+{{(-3q)}^{2}}+{{(-4r)}^{2}}+2(2p)(-3q)+2(-3q)\]\[(-4r)+2(-4r)(2p)\]
\[=4{{p}^{2}}+9{{q}^{2}}+16r2-12pq+24qr-16rp\text{ }.......\left( ii \right)\]
Adding, (i) & (ii) we get,
\[{{(2p+3a-4r)}^{2}}+{{(2p-3q-4r)}^{2}}\]
\[=4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}+12pq-24qr-16rp\]+ \[(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-12pq+24qr-16rp)\]
\[=4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}+12pq-24qr-16rp+4{{p}^{2}}+9{{q}^{2}}\] \[+16{{r}^{2}}-12pq+24qr-16rp\]
\[=8{{p}^{2}}+18{{q}^{2}}+32{{r}^{2}}-32rp\]
\[=2(4{{p}^{2}}+9{{q}^{2+}}16r2-16rp)\]
The expanded form of \[{{(2x+3y-5z)}^{2}}\] is:
(a) \[4{{x}^{2}}+9{{y}^{2}}+25{{z}^{2}}+12xy-30yz-20zx\]
(b) \[5{{x}^{2}}-6{{y}^{3}}+15{{z}^{3}}+12xy-36{{y}^{6}}+21x{{y}^{2}}\]
(c) \[8{{a}^{2}}+{{a}^{3}}-{{c}^{2}}+7{{x}^{2}}+2{{c}^{2}}+9{{c}^{2}}y\]
(d) \[1{{z}^{2}}-{{z}^{4}}-{{8}^{c}}-{{75}^{2}}+2{{c}^{2}}-98{{c}^{2}}\]
(e) None of these
Answer: (a)
Explanation
\[{{(2x+3y-5z)}^{2}}={{(2x)}^{2}}+{{(3y)}^{2}}+{{(-5z)}^{2}}\]\[+2(2x)(3y)+2(3y)(-5z)+2(5z)(2x)\text{ }\]
Expand: \[{{(3a-b+4c)}^{2}}\]
(a) \[~5{{x}^{2}}-6{{y}^{3}}+15{{z}^{3}}+12xy-36y+21xy\]
(b) \[9{{a}^{2}}+{{b}^{2}}+16{{c}^{2}}-6ab-8bc+24ac\]
(c) \[2{{x}^{2}}+7{{y}^{2}}+2{{z}^{2}}+xy+3yz+2zx\]
(d) \[15a2+22{{b}^{2}}+2{{c}^{4}}+ab+4bc+102ca\]
(e) \[5{{x}^{2}}-6{{y}^{3}}+15{{z}^{3}}+12xy-36y+21xy\]
Answer: (b)
Explanation
\[{{(3a-b+4c)}^{2}}={{(3a(-b)+4c)}^{2}}={{(3a)}^{2}}\]\[+{{(-b)}^{2}}+{{(4c)}^{2}}+2(3a)(-b)+2(-b)\] \[(4c)+2(4c)(3a)=9{{a}^{2}}+{{b}^{2}}+16{{c}^{2}}-6ab-8bc+24ac\]
Find the value of \[{{(3a+5b)}^{3}}.\]
(a) \[27{{a}^{3}}+125{{b}^{3}}+135{{a}^{2}}b+225a{{b}^{2}}\]
(b) \[27{{a}^{3}}-125{{b}^{3}}-135{{a}^{2}}b+2225a{{b}^{2}}\]
(c) \[27{{a}^{2}}+155{{b}^{2}}-135{{a}^{2}}b-225{{a}^{2}}b\]
(d) \[29{{a}^{2}}-156{{b}^{2}}-156{{a}^{2}}b-225{{a}^{4}}c\]
(e) None of these
Answer: (a)
Find the Cube of \[x-2y.\]
(a)\[~{{x}^{3}}+8{{y}^{2}}+6{{x}^{2}}y-12x{{y}^{3}}\]
(b) \[{{x}^{3}}-8{{y}^{3}}-6{{x}^{2}}y+12x{{y}^{2}}\]v
(c)\[{{x}^{2}}+87y+7xy-7xy\]
(d) \[7x{{y}^{3}}-6x-74{{x}^{2}}y+2xy\]
(e) None of these
Answer: (b)
Evaluate: \[{{\left( -6p+\frac{1}{3}q-r \right)}^{2}}\]
(a) \[36{{p}^{2}}\frac{9}{1}{{p}^{2}}+{{5}^{2}}-4pq-\frac{2}{3}gr+12rp\]
(b) \[36{{p}^{3}}\frac{9}{1}{{q}^{2}}+{{p}^{2}}+4qp-\frac{3}{2}qr+13rp\]
(c)\[~36{{p}^{2}}+\frac{1}{9}{{q}^{2}}+{{r}^{2}}-4qp-\frac{2}{3}qr+12rp\]
(d) \[36{{x}^{2}}-\frac{99}{8}{{s}^{2}}-{{s}^{2}}-4pq-\frac{2}{3}qr+14rp\]
(e) None of these
Answer: (c)
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