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Identify the terms, their coefficients for each of the following expressions: (i) \[5xy{{z}^{2}}-3zy\] (ii) \[1+x+{{x}^{2}}\] (iii) \[4{{x}^{2}}{{y}^{2}}-4{{x}^{2}}{{y}^{2}}{{z}^{2}}+{{z}^{2}}\] (iv) \[3-pq+qr-rp\] (v) \[\frac{x}{2}+\frac{y}{2}-xy\] (vi) \[0.3a-0.6ab+0.5b\]
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Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories? \[x+y,\,1000,\,x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}},7+y+5x,\,2y\] \[-3{{y}^{2}},\,2y-3{{y}^{2}}+4{{y}^{3}},\,5x-4y+3xy,\,4z-15{{z}^{2}},\] \[ab+bc+cd+da,\,pqr,\,{{p}^{2}}q\,+p{{q}^{2}},\,2p\,+2q\]
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Add the following: (i) \[ab-bc,\,bc-ca,\,ca-ab\] (ii) \[a-b+ab,\,b-c+bc,\,c-a+ac\] (iii) \[2{{p}^{2}}{{q}^{2}}-3pq+4,\,5+7pq-3{{p}^{2}}{{q}^{2}}\] (iv) \[{{l}^{2}}+{{m}^{2}},\,{{m}^{2}}+{{n}^{2}},\,{{n}^{2}}+{{l}^{2}},\,2lm\,+2mn+2nl\].
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(a) Subtract \[4a-7ab+3b+12\] from \[12a-9ab+5b-3\] (b) Subtract \[3xy+5yz-7zx\] from \[5xy-2yz-2zx+10xyz\] (c) Subtract \[4{{p}^{2}}q-3pq+5p{{q}^{2}}-8p+7p-10\] from \[18-3p-11q+5pq+2p{{q}^{2}}+5{{p}^{2}}q\].
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Find the product of the following pairs of monomials: (i) 4, 7p (ii) ? 4p, 7p (iii) -4p, 7pq (iv) \[4{{p}^{3}}-3p\]
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Find the areas of rectangles with the following Paris of monomials as their lengths and breaths respectively. (p, q); (10m, 5n); \[(20{{x}^{2}},\,5{{y}^{2}});\,(4x,\,3{{x}^{2}});\,(3mn,\,4np)\].
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Complete the table of products.
\[\frac{\text{First}\,\text{monomial}\,\to }{\text{Second}\,\text{monomial}\,\downarrow }\] | \[2x\] | \[-5y\] | \[3{{x}^{2}}\] | \[-4xy\] | \[7{{x}^{2}}y\] | \[-9{{x}^{2}}{{y}^{2}}\] |
\[2x\] | \[4{{x}^{2}}\] | - | - | - | - | - |
\[-5y\] | - | - | \[-15{{x}^{2}}y\] | - | - | - |
\[3{{x}^{2}}\] | - | - | - | - | - | - |
\[-4xy\] | - | - | - | - | - | - |
\[7{{x}^{2}}y\] | - | - | - | - | - | - |
\[-9{{x}^{2}}{{y}^{2}}\] | - | - | - | - | - | - |
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Obtain the following of rectangular boxes with the following length, breadth and height respectively: (i)\[5a,\,3{{a}^{2}},\,7{{a}^{4}}\] (ii) \[2p,\,4q,\,8r\] (iii) \[xy,\,2{{x}^{2}}y,\,2x{{y}^{2}}\] (iv) \[a,\,2b,\,3c\].
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Obtain the product of (i) \[xy,\,yz,\,zx\] (ii) \[a,\,-{{a}^{2}},\,{{a}^{3}}\] (iii) \[2,\,4y,\,8{{y}^{2}},\,16{{y}^{3}}\] (iv) \[a,\,2b,\,3c,\,6abc\] (v) \[m,\,-mn,\,mnp\].
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Carry out the multiplication of the expressions in each of the following pairs: (i) \[4p+q+r\] (ii) \[ab,\,a-b\] (iii) \[a+b,\,7{{a}^{2}}{{b}^{2}}\] (iv) \[{{a}^{2}}-9,\,4a\] (v) \[pq+qr+2p,\,0\]
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Complete the table:
| First expression | Second expression | Product |
(i) | \[a\] | \[b+c+d\] | - |
(ii) | \[x+y-5\] | \[5xy\] | - |
(iii) | \[p\] | \[6{{p}^{2}}-7p+5\] | - |
(iv) | \[4{{p}^{2}}{{q}^{2}}\] | \[{{p}^{2}}-{{q}^{2}}\] | - |
(v) | \[a+b+c\] | \[abc\] | - |
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Find the product: (i) \[({{a}^{2}})\times (2{{a}^{22}})\times (4{{a}^{26}})\] (ii) \[\left( \frac{2}{3}xy \right)\times \left( \frac{-9}{10}{{x}^{2}}{{y}^{2}} \right)\] (iii) \[\left( -\frac{10}{3}\,p{{q}^{3}} \right)\,\times \left( \frac{6}{5}\,{{p}^{3}}q \right)\] (iv) \[x\times {{x}^{2}}\times {{x}^{3}}\times {{x}^{4}}\].
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(a) Simplify \[3x(4x-5)\,+3\] and find its values for (i) \[x=3,\] (ii) \[x=\frac{1}{2}\]. (b) Simplify: \[a({{a}^{2}}+a+1)\,+5\] and find its value for (i) \[a=0,\] (ii) a = 1 and ( (iii) a = - 1.
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(a) Add: \[p(p-q),\,q(q-r)\] and \[r(r-p)\] (b) Add: \[2x(z-x-y)\] and \[2y(z-y-x)\] (c) Subtract: \[3l(l-4m+5n)\] from \[(10n-3m+2l)\] (d) Subtract: \[3a(a+b+c)\,-2b(a-b+c)\] from \[4c(-a+b+c)\].
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Multiply the binomials: (i) (2x + 5) and (4x - 3) (ii) (y - 8) and (3y - 4) (iii) (2.51 - 0.5 m) and (2.51 + 0.5m) (iv) (a + 3b) and (x + 5) (v) (2pq + 3q2) and (3pq - 2q2) (vi) \[\left( \frac{3}{4}{{a}^{2}}+3{{b}^{2}} \right)\,\] and \[4\left( {{a}^{2}}-\frac{2}{3}{{b}^{2}} \right)\]
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Find the product: (i) \[(5-2x)\,(3+x)\,\] (ii) \[(x+7y)\,(7x-y)\] (iii) \[({{a}^{2}}+b)(a+{{b}^{2}})\] (iv) \[({{p}^{2}}-{{q}^{2}})\,(2p+q)\]
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Simplify: (i) \[({{x}^{2}}-5)\,(x+5)\,+25\] (ii) \[({{a}^{2}}+5)\,({{b}^{3}}+3)+5\] (iii) \[(t+{{s}^{2}})\,({{t}^{2}}-s)\] (iv) \[(a+b)\,(c-d)\,+(a-b)\,(c+d)\]\[+2\,(ac+bd)\] (v) \[(x+y)\,(2x+y)\,+(x+2y)(x-y)\] (vi) \[(x+y)\,({{x}^{2}}-xy+{{y}^{2}})\] (vii) \[(1.5x-4y)\,(1.5x+4y+3)\,-4.5x+12y\] (viii) \[(a+b+c)\,(a+b-c)\].
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Use a suitable identity to get each of the following products: (i) \[(x+3)(x-3)\] (ii) \[(2y+5)(2y+5)\] (iii) \[(2a-7)(2a-7)\] (iv) \[\left( 3a-\frac{1}{2} \right)\,\left( 3a-\frac{1}{2} \right)\] (v) \[(1.1\,m\,-\,0.4)\,(1.1\,m\,+\,0.4)\] (vi) \[({{a}^{2}}+{{b}^{2}})\,(-{{a}^{2}}+{{b}^{2}})\] (vii) \[(6x-7)\,(6x+7)\] (viii) \[(-a+c)\,(-a+c)\] (ix) \[\left( \frac{x}{2}+\frac{3y}{4} \right)\,\left( \frac{x}{2}\,+\frac{3y}{4} \right)\] (x) \[(7a-9b)\,(7a-9b)\].
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Use the identity \[(x+a)\,(x+b)\,={{x}^{2}}\]\[+(a+b)\,x+ab\] to find the following products: (i) \[(x+3)\,(x+7)\] (ii) \[(4x+5)\,(4x+1)\] (iii) \[(4x-5)\,(4x-1)\] (iv) \[(4x+5)\,(4x-1)\] (v) \[(2x+5y)\,(2x+3y)\] (vi) \[(2{{a}^{2}}+9)\,(2{{a}^{2}}+5)\] (vii) \[(xyz-4)\,(xyz\,-2)\]
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Find the following squares by using the identities. (i) \[{{(b-7)}^{2}}\] (ii) \[{{(xy+3z)}^{2}}\] (iii) \[{{(6{{x}^{2}}-5y)}^{2}}\] (iv) \[{{\left( \frac{2}{3}\,m+\frac{3}{2}n \right)}^{3}}\] (v) \[{{(0.4\,p-0.5\,q)}^{2}}\] (vi) \[{{(2xy\,+5y)}^{2}}\].
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Simplify: (i) \[{{({{a}^{2}}-{{b}^{2}})}^{2}}\] (ii) \[{{(2x+5)}^{2}}-{{(2x-5)}^{2}}\] (iii) \[{{(7m-8n)}^{2}}+{{(7m+8n)}^{2}}\] (iv) \[{{(4m+5n)}^{2}}+{{(5m+4n)}^{2}}\] (v) \[{{(2.5p-1.5q)}^{2}}-{{(1.5p-2.5q)}^{2}}\] (vi) \[{{(ab+bc)}^{2}}-2a{{b}^{2}}c\] (vii) \[{{({{m}^{2}}-{{n}^{2}}m)}^{2}}\,+2{{m}^{3}}{{n}^{2}}\]S
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Show that (i) \[{{(3x+7)}^{2}}-84x={{(3x-7)}^{2}}\] (ii) \[{{(9p-5q)}^{2}}+180pq={{(9p+5q)}^{2}}\] (iii) \[{{\left( \frac{4}{3}m-\frac{3}{4}n \right)}^{2}}+2mn=\frac{16}{9}\,{{m}^{2}}+\frac{9}{16}{{n}^{2}}\] (iv) \[{{(4pq+3q)}^{2}}-{{(4pq-3q)}^{2}}=48p{{q}^{2}}\] (v) \[(a-b)\,(a+b)\,+(b-c)\,(b+c)\,+(c-a)\]\[(c+a)=0\]
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Using identities, evaluate: (i) \[{{71}^{2}}\] (ii) \[{{99}^{2}}\] (iii) \[{{102}^{2}}\] (iv) \[{{998}^{2}}\] (v) \[{{5.2}^{2}}\] (vi) \[297\times 303\] (vii) \[78\times 82\] (viii) \[{{8.9}^{2}}\] (ix) \[1.05\times 9.5\].
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Using \[{{a}^{2}}-{{b}^{2}}=(a+b)\,(a-b),\] find (i) \[{{51}^{2}}-{{49}^{2}}\] (ii) \[{{(1.02)}^{2}}-{{(0.98)}^{2}}\] (iii) \[{{153}^{2}}-{{147}^{2}}\] (iv) \[{{12.1}^{2}}-{{7.9}^{2}}\].
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Using \[(x+a)(x+b)\,={{x}^{2}}+(a+b)\]\[x+ab,\] find (i) \[103\times 104\] (ii) \[5.1\times 5.2\] (iii) \[103\times 98\] (iv) \[9.7\times 9.8\]
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