Question 1: Expand:

i)  {(x+3)}^2       ii)  { ( 2a+7 ) }^2       iii)  {(8+3p)}^2       iv)  {(\sqrt{3}x+2)}^2 

v)  {(4+\sqrt{5}y)}^2       vi)  {(6x+11y)}^2       vii)  \Big( \frac{x}{2} + \frac{y}{3} \Big)^2       viii)  \Big( \frac{3a}{5} + \frac{5b}{3} \Big)^2 

Answer:

i)  {(x+3)}^2 = x^2+6x+9

ii)  { ( 2a+7 ) }^2 = {4a}^2+28a+49

iii)  {(8+3p)}^2 = {64+48p+9p}^2

iv)  {(\sqrt{3}x+2)}^2 = {3x}^2+4\sqrt{3}x+4

v)  {(4+\sqrt{5}y)}^2 = {16+8\sqrt{5}y+5y}^2

vi)  {(6x+11y)}^2 = {36x^2+132xy+121y}^2

vii)  \Big( \frac{x}{2} + \frac{y}{3} \Big)^2 = \frac{x^2}{4} + \frac{1}{3} xy+ \frac{y^2}{9}

viii)  \Big( \frac{3a}{5} + \frac{5b}{3} \Big)^2 = \frac{9a^2}{25} +2ab + \frac{25b^2}{9}

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Question 2: Expand:

i)  {(x-9)}^2       ii)  {(6-y)}^2      iii)  {(3a-2)}^2       iv)  {(8y-5z)}^2 

v)  \Big( \frac{x}{2} - \frac{y}{2} \Big)^2       vi)  \Big( 2a- \frac{5}{2} \Big)^2       vii)  \Big( \frac{2}{a} - \frac{3}{b} \Big) 

Answer:

i)  {(x-9)}^2 = x^2-18x+81

ii)  {(6-y)}^2 = 36-127+y^2

iii)  {(3a-2)}^2 = 9a^2-12a+4

iv)  {(8y-5z)}^2 = {64y}^2-80yz+25z^2

v)  \Big( \frac{x}{2} - \frac{y}{2} \Big)^2 = \frac{x^2}{4} - \frac{1}{2} xy+ \frac{y^2}{4}

vi)  \Big( 2a- \frac{5}{2} \Big)^2 = {4a}^2-2a+ \frac{25}{4}

vii)  \Big( \frac{2}{a} - \frac{3}{b} \Big)^2 = \frac{4}{a^2} - \frac{12}{ab} + \frac{9}{b^2}

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Question 3: Using special expressions to find the value ofː

i)  ({53)}^2       ii)  ({84)}^2       iii)  ({1011)}^2       iv)  ({988)}^2 

Answer:

i)  ({53)}^2 = ({50+3)}^2 = 2500+300+9 = 2809

ii)  ({84)}^2 = ({100-16)}^2 = 1000-3200+256 = 7056

iii)  ({1011)}^2 = ({1000+11)}^2 = 1000000+22000+121 = 1022121

iv)  ({988)}^2 = ({1000-12)}^2 = 1000000-24000+144 = 976144

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Question 4: Using special expressions find the value ofː

i)  ({67)}^2       ii)  ({795)}^2       iii)  ({10.9)}^2       iv)  ({9.2)}^2 

Answer:

i)  ({67)}^2 = ({70-3)}^2 = 4900-420+9 = 4489

ii)  ({795)}^2 = ({800-5)}^2 = 640000-8000+25 = 632025

iii)  ({10.9)}^2 = ({11-0.1)}^2 = 121-2.2+0.01 = 118.81

iv)  ({9.2)}^2 = ({10-0.8)}^2 = 100-16+0.64 = 84.64

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Question 5: If \Big( x+ \frac{1}{x} \Big) = 4 , find the value of:

i)  x- \frac{1}{x}          ii)  \Big( x^2+ \frac{1}{x^2} \Big)           iii)  \Big( x^4+ \frac{1}{x^4} \Big)

Answer:

Given \Big( x+ \frac{1}{x} \Big) = 4

\Rightarrow \Big( x+ \frac{1}{x} \Big)^2 = 16

\Rightarrow x^2+ \frac{1}{x^2} +2 = 16

\Rightarrow x^2+ \frac{1}{x^2} = 14

Now  \Big( x- \frac{1}{x} \Big)^2 = x^2+ \frac{1}{x^2} -2

\Rightarrow \Big( x- \frac{1}{x} \Big)^2 = 14-2 = 12

Therefore \Big( x- \frac{1}{x} \Big) = \sqrt{12} = \pm 2\sqrt{3}

\Rightarrow \Big( x^4+ \frac{1}{x^4} \Big) = \Big( x^2+ \frac{1}{x^2} \Big)^2-2 = {14}^2-2 = 194

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Question 6: If \Big( z- \frac{1}{z} \Big) = 6 find the value of

i)  \Big( z+ \frac{1}{z} \Big)           ii)  \Big( z^2+ \frac{1}{z^2} \Big)           iii)  \Big( z^4+ \frac{1}{z^4} \Big)

Answer:

Given z- \frac{1}{z} = 6

\Rightarrow z^2+ \frac{1}{z^2} -2 = 36

\Rightarrow z^2+ \frac{1}{z^2} = 38

\Rightarrow  \Big( z+ \frac{1}{z} \Big)^2 = z^2+ \frac{1}{z^2} +2 = 38+2 = 40

\Rightarrow z+ \frac{1}{z} = \pm{}2\sqrt{10}

z^4+ \frac{1}{z^4} = \Big( z^2+ \frac{1}{z^2} \Big)^2-2 = {38}^2-2 = 1442

\Rightarrow \Big( z+ \frac{1}{z} \Big) = \pm 2\sqrt{10}

\Big( z^2+ \frac{1}{z^2} \Big) = 38

\Big( z^4+ \frac{1}{z^4} \Big) = 1442

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Question 7: If \Big( a^2+ \frac{1}{a^2} \Big) = 23 , find the value of \Big( a+ \frac{1}{a} \Big)

Answer:

\Big( a+ \frac{1}{a} \Big)^2 = a^2+ \frac{1}{a^2} +2   = 23+2 = 25

\Rightarrow \Big( a+ \frac{1}{a} \Big) = \pm 5

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Question 8: If \Big( x^2+ \frac{1}{x^2} \Big) = 102 , find the value of \Big( x- \frac{1}{x} \Big)

Answer:

\Big( x- \frac{1}{x} \Big)^2 = x^2+ \frac{1}{x^2} -2   = 102-2 = 100

\Rightarrow \Big( x- \frac{1}{x} \Big) = \pm 10

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Question 9: If \Big( 2p+ \frac{1}{2p} \Big) = 5 , find the value of \Big( 4p^2+ \frac{1}{4p^2} \Big)

Answer:

\Big( 2p+ \frac{1}{2p} \Big)^2 = 4p^2+ \frac{1}{{4p}^2} +2

\Rightarrow 4p^2+ \frac{1}{4p^2} = 25-2 = 23

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Question 10: If \Big( 3c- \frac{1}{3c} \Big) = 8 , find the value of \Big( {9c}^2+ \frac{1}{{9c}^2} \Big)

Answer:

\Big( 3c - \frac{1}{3c} \Big)^2 = 9c^2+ \frac{1}{9c^2} -2

\Rightarrow 9c^2+ \frac{1}{9c^2} = 8^2+2 = 64+2 = 66

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Question 11: If ( a+b ) = 8 , and ab = 15 , find the value of a^2+b^2

Answer:

( a+{b)}^{2\ } = {(a}^2+b^2+2ab )

\Rightarrow a^2+b^2 = 8^2-2\times{}15 = 64-30 = 34

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Question 12: If a+b = 11 , and a^2+b^2 = 61 , find the value of ab

Answer:

( a+{b)}^{2\ } = {(a}^2+b^2+2ab )

\Rightarrow 2ab = {11}^2-61 = 121-61 = 60

\Rightarrow ab = 30

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Question 13: If \ a^2+b^2 = 13 , and ab = 6 , find the value of ( a+b )

Answer:

( a+{b)}^{2\ } = (a^2+b^2+2ab ) = 13+12 = 25

\Rightarrow ( a+b ) = \pm 5

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Question 14: If a+b = 15 , and \ ab = 56 , find ( a^2+b^2 )

Answer:

( a+{b)}^{2\ } = {(a}^2+b^2+2ab )

\Rightarrow a^2+b^2 = {15}^2+2\times 56 = 225-112 = 133

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Question 15: If a-b = 1   and ab = 12 , find ( a^2+b^2 )

Answer:

( a-{b)}^{2\ } = a^2+b^2-2ab 

\Rightarrow a^2+b^2 = 1^2+2\times{}12 = 25

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Question 16: If a-b = 5\ \&\ a^2+b^2 = 52\ find\ the\ value\ of\ ab

Answer:

( a-{b)}^{2\ } = {(a}^2+b^2-ab 

\Rightarrow 2ab = a^2+b^2-(a-{b)}^2 = 53-25 = 28

\Rightarrow ab = 14

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Question 17: If a^2+b^2 = 52\ \&\ ab = 24 Find (a-b)

Answer:

{ ( a-b ) }^2 = a^2+b^2-2ab = 52-48 = 4

\Rightarrow ( a-b ) = \ \pm 2

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Question 18: Find the value of: 

(i) 36x^2+49y^2+84xy Given \ x = 3\ \&\ y = 6

(ii) 25x^2+16y^2-40xy Given \ x = 6\ \&\ y = 7

Answer:

(i) 36x^2+49y^2+84xy   Given x = 3\ \&\ y = 6

\Rightarrow 36x^2+49y^2+84xy = (6x+7y)^2 = (18+42)^2 = 3600

(ii) 25x^2+16y^2-40xy Given x = 6\ \&\ y = 7

\Rightarrow 25x^2+16y^2-40xy = (5x-4y)^2 = (30-28)^2 = 4