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Get the algebraic expressions in the following cases using variables, constants and arithmetic operations. (i) Subtraction of \[z\] from\[y\]. (ii) One-half of the sum of numbers \[x\] and\[y\]. (iii) The number \[z\] multiplied by itself. (iv) One-fourth of the product of numbers p and q. (v) Numbers \[x\] and \[y\] both squared and added. (vi) Number 5 added to three times the product of numbers m and n. (vii) Product of numbers \[y\] and \[z\] subtracted from 10. (viii) Sum of numbers a and b subtracted from their product.
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(i) Identify the terms and `their factors in the following expressions. Show the terms and factors by tree diagrams: (a) \[x-3\] (b) \[1+x+{{x}^{2}}\] (c) \[y-{{y}^{3}}\] (d) \[5x{{y}^{2}}+7{{x}^{2}}y\] (e) \[-ab+2{{b}^{2}}-3{{a}^{2}}\] (ii) Identity terms and factors in the expressions given below: (a) \[-4x+5\] (b) \[-4x+5y\] (c) \[5y+3{{y}^{2}}\] (d) \[xy+2{{x}^{2}}{{y}^{2}}\] (e) \[pq+q\] (f) \[1.2ab-2.4b+3.6a\] (g) \[\frac{3}{4}x+\frac{1}{4}\] (h) \[0.1{{p}^{2}}+0.2{{q}^{2}}\]
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Identify the numerical coefficients of terms (other than constants) in the following expressions: (i) \[5-3{{t}^{2}}\] (ii) \[1+t+{{t}^{2}}+{{t}^{3}}\] (iii) \[x+2xy+3y\] (iv) \[100m+1000n\] (v) \[-{{p}^{2}}{{q}^{2}}+7pq\] (vi) \[1.2a+0.8b\] (vii) \[3.14{{r}^{2}}\] (viii) \[2(l+b)\] (ix) \[0.1y+0.01{{y}^{2}}\]
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(a) Identify terms which contain x and give that coefficient of \[x\] (i) \[{{y}^{2}}x+y\] (ii) \[13{{y}^{2}}-8yx\] (iii) \[x+y+2\] (iv) \[5+z+zx\] (v) \[1+x+xy\] (vi) \[12x{{y}^{2}}+25\] (vii) \[7x+x{{y}^{2}}\] (b) identify terms which contain \[{{y}^{2}}\]and give the coefficient of \[{{y}^{2}}\]. (i) \[8-x{{y}^{2}}\] (ii) \[5{{y}^{2}}+7x\] (iii) \[2{{x}^{2}}y-15x{{y}^{2}}+7{{y}^{2}}\]
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Classify into monomials, binomials and trinomials. (i) \[4y-7z\] (ii) \[{{y}^{2}}\] (iii) \[x+y-xy\] (iv) 100 (v) \[ab-a-b\] (vi) \[5-3t\] (vii) \[4{{p}^{2}}q-4p{{q}^{2}}\] (viii) \[7mn\] (ix) \[{{z}^{2}}-3z+8\] (x) \[{{a}^{2}}+{{b}^{2}}\] (xi) \[{{z}^{2}}+z\] (xii) \[1+x+{{x}^{2}}\]
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State whether a given pair of terms is of like or unlike terms: (i) \[1,100\] (ii) \[-7x,\frac{5}{2}x\] (iii) \[-29x,-29y\] (iv) \[14xy,42yx\] (v) \[4{{m}^{2}}p,4m{{p}^{2}}\] (vi) \[12xz,12{{x}^{2}}{{z}^{2}}\]
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Identify like terms in the following: (a) \[-x{{y}^{2}},-4y{{x}^{2}},8{{x}^{2}},2x{{y}^{2}},7y,-11{{x}^{2}},-100x,\]\[-11yx,20{{x}^{2}}y,-6{{x}^{2}},y,2xy,3x\]. (b) \[10pq,7p,8q,-{{p}^{2}}{{q}^{2}},-7qp,-100q,-23,\]\[12{{q}^{2}}{{p}^{2}},\] \[-5{{p}^{2}},41,2405p,78qp,13{{p}^{2}}q,\]\[q{{p}^{2}},\] \[701{{p}^{2}}\]
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Simplify combining like terms: (i) \[21b-32+7b-20b\] (ii) \[-{{z}^{2}}+13{{z}^{2}}-5z+7{{z}^{3}}-15z\] (iii) \[p-(p-q)-q-(q-p)\] (iv) \[3a-2b-ab-(a-b+ab)+3ab+b-a\] (v) \[5{{x}^{2}}y-5{{x}^{2}}+3y{{x}^{2}}-3{{y}^{2}}+{{x}^{2}}-{{y}^{2}}\]\[+8x{{y}^{2}}-3{{y}^{2}}\] (vi) \[(3{{y}^{2}}+5y-4)-(8y-{{y}^{2}}-4)\]
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Add: (i) \[3mn,-5mn,8mn,-4mn\] (ii) \[t-8tz,3tz-z,z-t\] (iii) \[-7mn+5,12mn+2,9mn-8,-2mn-3\] (iv) \[a+b-3,b-a+3,a-b+3\] (v) \[14x+10y-12xy-13,18-7x-10y\]\[+8xy,\]\[4xy\] (vi) \[5m-7n,3n-4m+2,2m-3mn-5\] (vii) \[4{{x}^{2}}y,-3x{{y}^{2}}-5x{{y}^{2}},5{{x}^{2}}y\] (viii) \[3{{p}^{2}}{{q}^{2}}-4pq+5,-10{{p}^{2}}{{q}^{2}},15+9pq\]\[+7{{p}^{2}}{{q}^{2}}\] (ix) \[ab-4a,4b-ab;4a-4b\] (x) \[{{x}^{2}}-{{y}^{2}}-1,{{y}^{2}}-1-{{x}^{2}},1-{{x}^{2}}-{{y}^{2}}\]
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Subtract: (i) \[-5{{y}^{2}}\] from \[{{y}^{2}}\] (ii) \[6xy\] from \[-12xy\] (iii) \[(a-b)\] from \[(a+b)\] (iv) \[a(b-5)\] from \[b(5-a)\] (v) \[-{{m}^{2}}+5mn\] from \[4{{m}^{2}}-3mn+8\] (vi) \[-{{x}^{2}}+10x-5\] from \[5x-10\] (vii) \[5{{a}^{2}}-7ab+5{{b}^{2}}\] from \[3ab-2{{a}^{2}}-2{{b}^{2}}\] (viii) \[4pq-5{{q}^{2}}-3{{p}^{2}}\] from \[5{{p}^{2}}+3{{q}^{2}}-pq\]
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(a) What should be added to \[{{x}^{2}}+xy+{{y}^{2}}\] to obtain \[2{{x}^{2}}+3xy\]? (b) What should be subtracted from \[2a+8b+10\] to get \[-3a+7b+16\]?
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(a) From the sum of \[3x-y+11\] and \[-y-11,\] subtract \[3x-y-11\]. (b) From the sum of \[4+3x\] and \[5-4x+2{{x}^{2}},\] subtract the sum of \[3{{x}^{2}}-5x\] and \[-{{x}^{2}}+2x+5\].
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If \[m=2,\] find the value of: (i) \[m-2\] (ii) \[3m-5\] (iii) \[9-5m\] (iv) \[3{{m}^{2}}-2m-7\] (v) \[\frac{5m}{2}-4\]
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If \[p=-2,\] find the value of : (i) \[4p+7\] (ii) \[-3{{p}^{2}}+4p+7\] (iii) \[-2{{p}^{3}}-3{{p}^{2}}+4p+7\]
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Find the value of the following expressions, when \[x=-1:\] (i) \[2x-7\] (ii) \[-x+2\] (iii) \[{{x}^{2}}+2x+1\] (iv) \[2{{x}^{2}}-x-2\]
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If \[a=2,b=-2\], find the value of : (i) \[{{a}^{2}}+{{b}^{2}}\] (ii) \[{{a}^{2}}+ab+{{b}^{2}}\] (iii) \[{{a}^{2}}-{{b}^{2}}\].
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When \[a=0,b=-1,\] find the value of the given expressions: (i) \[2a+2b\] (ii) \[2{{a}^{2}}+{{b}^{2}}+1\] (iii) \[2{{a}^{2}}b+2a{{b}^{2}}+ab\] (iv) \[{{a}^{2}}+ab+2\]
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Simplify the expressions and find the value if \[x\]is equal to 2. (i) \[x+7+4(x-5)\] (ii) \[3(x+2)+5x-7\] (iii) \[6x+5(x-2)\] (iv) \[4(2x-1)+3x+11\]
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Simplify these expressions and find their value if \[x=3,a=-1,b=-2\]. (i) \[3x-5-x+9\] (ii) \[2-8x+4x+4\] (iii) \[3a+5-8a+1\] (iv) \[10-3b-4-5b\] (v) \[2a-2b-4-5+a.\]
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(a) If \[z=10,\]find the value of \[{{z}^{3}}-3(z-10)\]. (b) If \[p=-10,\]find the value of \[{{p}^{2}}-2p-100.\]
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What should be the value of if the value of \[2{{x}^{2}}+x-a\] equals to 5, when \[x=0?\]
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Simplify the expression and find its value when \[a=5\] and \[b=-3\] \[2({{a}^{2}}+ab+)+3-ab\].
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Observe the patterns of digits made from line segments of equal length. You will find such segmented digits on the display of electronic watches or calculators. (a)
(b)
(c)
If the number of digits formed is taken to be n, the number of segments required to form n digits is given by the algebraic expression appearing on the right of each pattern. How many segments are required to form 5, 10, 100 digit of the kind
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Use the given algebraic expression to complete the table of number patterns.
S. No. | Expression | Terms |
1st | 2nd | 3rd | 4th | 5th | ? | 10th | ? | 100th | ? |
(i) | \[2n-2\] | 1 | 3 | 5 | 7 | 9 | ? | 19 | ? | ? | ? |
(ii) | \[3n+2\] | 5 | 8 | 11 | 14 | ? | ? | ? | ? | ? | ? |
(iii) | \[4n+1\] | 5 | 9 | 13 | 17 | ? | ? | ? | ? | ? | ? |
(iv) | \[7n+20\] | 27 | 34 | 41 | 48 | ? | ? | ? | ? | ? | ? |
(v) | \[{{n}^{2}}+1\] | 2 | 5 | 10 | 17 | ? | ? | ? | ? | 10,001 | ? |
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