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Ncert Solutions Maths class 12th Maths Class 12th Application of Derivatives

65. Using differentials, find the approximate value of each of the following up to 3 places of decimal.

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Vishal Baghel | Contributor-Level 10

4 months ago

(i) Let y= ?x : Let x = 25 and x = 0. 3.

Then, ?y = ?x+?x $?x$

$=?25.3 ?25$

$=?25.3 ? 5$

$?25.3 = 5 + ? y ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? y ~ d y .$

= 5 + dy

= $5 + (\frac{d y}{d x}) A x .$

$= 5 + \frac{1}{2?x} ? x .$

$= 5 + \frac{1}{2?25} ? 0.3$

$= 5 + \frac{0.3}{2 * 5}$

= 5 + 0.03

$?25.3 = 5.030$

(ii) ?49.5

A.(ii)

Let y = ?x $.$ Let x = 49 and x = 0.5.

Then, $? y =?x+?x ?x .$

$=?49.5 ?49$

$?49.5 = 7 + ? y = 7 + {(\frac{d y}{d x})}_{}^{? x}$

$= 7 + \frac{1}{2?x} ? x$

$= 7 + \frac{1}{2?49} * (0.5)$

$= 7 + \frac{0.5}{14}$

= 7 + 0.0357.

$?49.5 = 7 ? 035$

(iii) ?0.6

A.(iii)

Let y = ?x $.$ Let x = 0.64 and ?x = 0.04.

Then, $? y =?x+Ax ?x$

$=?0.64-0.04 ?0.64 .$

$=?0.6 ? 0.8$

$?0.6= 0.8 + ? y = 0.8 + (\frac{d y}{d x}) ? x$

$= 0.8 + \frac{1}{2?x} ? x$

$= 0.8 + \frac{1}{2?0.64} * (? ? 0.04)$

$= 0.8 \frac{? 0.04}{2 * 0.8}$

= 0.8 - 0.025.

= 0.775.

(iv) ${(0.009)}^{\frac{1}{3}}$

A.(iv)

Let $y = x^{\frac{1}{3}}$ Let x = 0.008 and ?x = 0.00 1.

Then, ?y = ${[x + ? x]}^{\frac{1}{3}} ? {[x]}^{\frac{1}{3}}$

$= {(0.009)}^{\frac{1}{3}} ? {(0.008)}^{\frac{1}{3}}$

${(0.009)}^{\frac{1}{3}} = 0.2 + ? y = 0.2 + \frac{d y}{d x} ? ? x$

$= 0.2 + \frac{1}{3 {(x^{\frac{1}{3}})}^{2}} ? x$

$= 0.2 + \frac{1}{3 {[{(0.008)}^{\frac{1}{3}}]}^{2}} * (0.001)$

$= 0.2 + \frac{0.001}{3 * {(0.0)}^{2}} = 0.2 + \frac{0.001}{0.12}$

= 0.2 + 0.0083.

= 0.208.

(v) ${(0.999)}^{\frac{1}{10}}$

A.(v)

Let $y = x^{\frac{1}{10}} .$ Let x = 1 and ?x = -0.001

Then, $? y = {(x + ? x)}^{\frac{1}{10}} ? x^{\frac{1}{10}}$

$= {(0.999)}^{\frac{1}{10}} ? {(1)}^{\frac{1}{10}}$

$? 1 {(0.999)}^{\frac{1}{10}} = 1 + A y$

$= 1 + \frac{d y}{d x} ? x .$

$= 1 + \frac{1}{10 x^{\frac{9}{10}}} * ? x$

$= 1 ? \frac{0.001}{10 (. 1^{\frac{9}{10}})} = 1 ? \frac{0.001}{10} = 1 ? 0.0001$

= 0.999.

(vi) ${(15)}^{\frac{1}{4}}$

A.(vi)

Let $y = x^{\frac{1}{4}} .$ Then, x = 16 and ?x = 1.

Then, $? y = {(x + A x)}^{\frac{1}{4}} ? x^{\frac{1}{4}} .$

$= {(15)}^{\frac{1}{4}} ? {(16)}^{\frac{1}{4}}$

$? {(15)}^{\frac{1}{4}} = 2 + A y = 2 + \frac{d y}{d x} A x$

$= 2 + \frac{1}{4 x^{\frac{3}{4}}} ? (? 1)$

$= 4 ? \frac{1}{4 {(16)}^{\frac{3}{4}}}$

$= 2 ? \frac{1}{4 * 8}$

$= 4 ? \frac{1}{32}$

$= \frac{64 ? 1}{32} = \frac{63}{32} = 1.968$

(vii) ${(26)}^{\frac{1}{3}}$

A.(vii)

Let $y = x^{\frac{1}{3}} .$ Let x = 27 and ?x = 1.

Then, $? y = {(x + ? x)}^{\frac{1}{3}} ? x^{\frac{1}{3}}$

$= {(26)}^{\frac{1}{3}} ? {(27)}^{\frac{1}{3}}$

$? {(26)}^{\frac{1}{3}} = 3 + 4 y = 3 + \frac{d y}{d x} A x = 3 + \frac{1}{3 x^{\frac{2}{3}}} A x,$

$= 3 ? \frac{1}{3 {(2 F)}^{\frac{2}{3}}}$

$= 3 ? \frac{1}{27}$

$= \frac{81 ? 1}{27} = \frac{80}{27} = 2.962 .$

(viii) ${(255)}^{\frac{1}{4}}$

A.(viii)

Let $y = x^{\frac{1}{4}} .$ Let x = 256 and ?x = 1.

Then, $? y = {(x + ? x)}^{\frac{1}{4}} ? x^{\frac{1}{4}} .$

$= {(255)}^{\frac{1}{4}} ? {(256)}^{\frac{1}{4}}$

$? {(255)}^{\frac{1}{4}} = {(256)}^{\frac{1}{4}} + 4 y = 4 + \frac{d y}{d x} A x .$

$= 4 + \frac{1}{4 x^{\frac{3}{4}}} ? 4 x$

$= 4 \frac{? 1}{4 {(256)}^{\frac{3}{4}}}$

$= 4 ? \frac{1}{256}$

$= \frac{1024 ? 1}{256}$

$= \frac{1023}{256} = 3.996$

(ix) ${(82)}^{\frac{1}{4}}$

A.(ix)

...more

(i) Let y= ?x : Let x = 25 and x = 0. 3.

Then, ?y = ?x+?x $?x$

$=?25.3 ?25$

$=?25.3 ? 5$

$?25.3 = 5 + ? y ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? y ~ d y .$

= 5 + dy

= $5 + (\frac{d y}{d x}) A x .$

$= 5 + \frac{1}{2?x} ? x .$

$= 5 + \frac{1}{2?25} ? 0.3$

$= 5 + \frac{0.3}{2 * 5}$

= 5 + 0.03

$?25.3 = 5.030$

(ii) ?49.5

A.(ii)

Let y = ?x $.$ Let x = 49 and x = 0.5.

Then, $? y =?x+?x ?x .$

$=?49.5 ?49$

$?49.5 = 7 + ? y = 7 + {(\frac{d y}{d x})}_{}^{? x}$

$= 7 + \frac{1}{2?x} ? x$

$= 7 + \frac{1}{2?49} * (0.5)$

$= 7 + \frac{0.5}{14}$

= 7 + 0.0357.

$?49.5 = 7 ? 035$

(iii) ?0.6

A.(iii)

Let y = ?x $.$ Let x = 0.64 and ?x = 0.04.

Then, $? y =?x+Ax ?x$

$=?0.64-0.04 ?0.64 .$

$=?0.6 ? 0.8$

$?0.6= 0.8 + ? y = 0.8 + (\frac{d y}{d x}) ? x$

$= 0.8 + \frac{1}{2?x} ? x$

$= 0.8 + \frac{1}{2?0.64} * (? ? 0.04)$

$= 0.8 \frac{? 0.04}{2 * 0.8}$

= 0.8 - 0.025.

= 0.775.

(iv) ${(0.009)}^{\frac{1}{3}}$

A.(iv)

Let $y = x^{\frac{1}{3}}$ Let x = 0.008 and ?x = 0.00 1.

Then, ?y = ${[x + ? x]}^{\frac{1}{3}} ? {[x]}^{\frac{1}{3}}$

$= {(0.009)}^{\frac{1}{3}} ? {(0.008)}^{\frac{1}{3}}$

${(0.009)}^{\frac{1}{3}} = 0.2 + ? y = 0.2 + \frac{d y}{d x} ? ? x$

$= 0.2 + \frac{1}{3 {(x^{\frac{1}{3}})}^{2}} ? x$

$= 0.2 + \frac{1}{3 {[{(0.008)}^{\frac{1}{3}}]}^{2}} * (0.001)$

$= 0.2 + \frac{0.001}{3 * {(0.0)}^{2}} = 0.2 + \frac{0.001}{0.12}$

= 0.2 + 0.0083.

= 0.208.

(v) ${(0.999)}^{\frac{1}{10}}$

A.(v)

Let $y = x^{\frac{1}{10}} .$ Let x = 1 and ?x = -0.001

Then, $? y = {(x + ? x)}^{\frac{1}{10}} ? x^{\frac{1}{10}}$

$= {(0.999)}^{\frac{1}{10}} ? {(1)}^{\frac{1}{10}}$

$? 1 {(0.999)}^{\frac{1}{10}} = 1 + A y$

$= 1 + \frac{d y}{d x} ? x .$

$= 1 + \frac{1}{10 x^{\frac{9}{10}}} * ? x$

$= 1 ? \frac{0.001}{10 (. 1^{\frac{9}{10}})} = 1 ? \frac{0.001}{10} = 1 ? 0.0001$

= 0.999.

(vi) ${(15)}^{\frac{1}{4}}$

A.(vi)

Let $y = x^{\frac{1}{4}} .$ Then, x = 16 and ?x = 1.

Then, $? y = {(x + A x)}^{\frac{1}{4}} ? x^{\frac{1}{4}} .$

$= {(15)}^{\frac{1}{4}} ? {(16)}^{\frac{1}{4}}$

$? {(15)}^{\frac{1}{4}} = 2 + A y = 2 + \frac{d y}{d x} A x$

$= 2 + \frac{1}{4 x^{\frac{3}{4}}} ? (? 1)$

$= 4 ? \frac{1}{4 {(16)}^{\frac{3}{4}}}$

$= 2 ? \frac{1}{4 * 8}$

$= 4 ? \frac{1}{32}$

$= \frac{64 ? 1}{32} = \frac{63}{32} = 1.968$

(vii) ${(26)}^{\frac{1}{3}}$

A.(vii)

Let $y = x^{\frac{1}{3}} .$ Let x = 27 and ?x = 1.

Then, $? y = {(x + ? x)}^{\frac{1}{3}} ? x^{\frac{1}{3}}$

$= {(26)}^{\frac{1}{3}} ? {(27)}^{\frac{1}{3}}$

$? {(26)}^{\frac{1}{3}} = 3 + 4 y = 3 + \frac{d y}{d x} A x = 3 + \frac{1}{3 x^{\frac{2}{3}}} A x,$

$= 3 ? \frac{1}{3 {(2 F)}^{\frac{2}{3}}}$

$= 3 ? \frac{1}{27}$

$= \frac{81 ? 1}{27} = \frac{80}{27} = 2.962 .$

(viii) ${(255)}^{\frac{1}{4}}$

A.(viii)

Let $y = x^{\frac{1}{4}} .$ Let x = 256 and ?x = 1.

Then, $? y = {(x + ? x)}^{\frac{1}{4}} ? x^{\frac{1}{4}} .$

$= {(255)}^{\frac{1}{4}} ? {(256)}^{\frac{1}{4}}$

$? {(255)}^{\frac{1}{4}} = {(256)}^{\frac{1}{4}} + 4 y = 4 + \frac{d y}{d x} A x .$

$= 4 + \frac{1}{4 x^{\frac{3}{4}}} ? 4 x$

$= 4 \frac{? 1}{4 {(256)}^{\frac{3}{4}}}$

$= 4 ? \frac{1}{256}$

$= \frac{1024 ? 1}{256}$

$= \frac{1023}{256} = 3.996$

(ix) ${(82)}^{\frac{1}{4}}$

A.(ix)

Let $y = x^{\frac{1}{4}} .$ let x = 81 and ?x = 1.

Then, $? y = {(x + 4 x)}^{\frac{1}{4}} ? x^{\frac{1}{4}}$

$= {(82)}^{\frac{1}{4}} ? {(81)}^{\frac{1}{4}}$

$7 {(82)}^{\frac{1}{4}} = 3 + 4 y = 3 + \frac{d y}{d x} ? x$

$= 3 + \frac{1}{4 x^{\frac{3}{4}}} ? ? x$

$= 3 + \frac{1}{4 {(81)}^{\frac{3}{4}}}$

$= 3 + \frac{1}{4 * 3^{3}}$

$= 3 + \frac{1}{108} = \frac{324 + 1}{108} = \frac{325}{108} = 3.001 .$

(x) ${(401)}^{\frac{1}{2}}$

A.(x)

Let y = $x^{\frac{1}{2}}$ = ?x $.$ Let x = 400 and ?x = 1.

Then, $? y = {(x + ? x)}^{\frac{1}{2}} ? x^{\frac{1}{2}} .$

$? ? ? y = {(401)}^{\frac{1}{2}} ? {(400)}^{\frac{1}{2}}$

$? ? {(401)}^{\frac{1}{2}} = 20 ? + ? y = 20 + \frac{d y}{d x} ? x$

$= 20 + \frac{1}{2?x} ? x$

$= 20 + \frac{1}{2?400}$

$= 20 + \frac{1}{2 * 20}$

$= 20 + \frac{1}{40}$

$= \frac{800 + 1}{40} = \frac{801}{40} = 20.025 .$

(xi) ${(0.0037)}^{\frac{1}{2}}$

A.(xi)

Let $y = x^{\frac{1}{2}}$ Let x = 0.0036 and ?x = 0.0001

Then, $? y =?x+?x ?x$

$=?0.0037 ?0.0036$

$?0.0037 = 0.06 + D y = 0.06 + \frac{d y}{d x} ? x = 0.06 + \frac{x}{2?x}$

$= 0.06 + \frac{0.001}{2?0.0036}$

$= 0.06 + \frac{0.001}{2 * 0.06}$

$= 0.06 + \frac{0.001}{0.12 .}$

= 0.06 + 0.0008.

= 0.0608.

(xii) ${(26.57)}^{\frac{1}{3}}$

A.(xii)

Let $y = x^{\frac{1}{3}} .$ Let x = 27 and ?x = 0.43.

Then, $? y = {(x + A x)}^{\frac{1}{3}} ? {(x)}^{\frac{1}{3}}$

$= {(26.57)}^{\frac{1}{3}} ? {(27)}^{\frac{1}{3}}$

$? {(26.57)}^{\frac{1}{3}} = 3 + 4 y = 3 + \frac{d y}{d x} ? x = 3 + \frac{1}{3 x^{\frac{2}{3}}} ? ? x .$

$= 3 + \frac{1}{3 {(27)}^{\frac{2}{3}}} ? less(i) Let y= ?x<math><mrow></mrow></math> : Let x = 25 and x = 0. 3.Then, ?y = ?x+?x<math><mrow><mo>?x</mo></mrow></math><math><mrow><mo>=?25.3</mo><mo>?25</mo></mrow></math><math><mrow><mo>=?25.3</mo><mo>?</mo><mn>5</mn></mrow></math><math><mrow><mo>?25.3</mo><mo>=</mo><mn>5</mn><mo>+</mo><mi>?</mi><mi>y</mi><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mtext>?</mtext><mi>?</mi><mi>y</mi><mo>~</mo><mi>d</mi><mi>y</mi><mo>.</mo></mrow></math>= 5 + dy= <math><mrow><mn>5</mn><mo>+</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mi>A</mi><mi>x</mi><mo>.</mo></mrow></math><math><mrow><mo>=</mo><mn>5</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2?x</mn></mrow></mfrac><mi>?</mi><mi>x</mi><mo>.</mo></mrow></math><math><mrow><mo>=</mo><mn>5</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2?25</mn></mrow></mfrac><mo>?</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></math><math><mrow><mo>=</mo><mn>5</mn><mo>+</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>3</mn></mrow><mrow><mn>2</mn><mo>\times</mo><mn>5</mn></mrow></mfrac></mrow></math>= 5 + 0.03<math><mrow><mo>?25.3</mo><mo>=</mo><mn>5</mn><mo>.</mo><mn>0</mn><mn>3</mn><mn>0</mn></mrow></math>(ii) ?49.5<math><mrow></mrow></math>A.(ii)Let y = ?x<math><mrow><mo>.</mo></mrow></math> Let x = 49 and x = 0.5.Then, <math><mrow><mi>?</mi><mi>y</mi><mo>=?x+?x</mo><mo>?x</mo><mo>.</mo></mrow></math><math><mrow><mo>=?49.5</mo><mo>?49</mo></mrow></math><math><mrow><mo>?49.5</mo><mo>=</mo><mn>7</mn><mo>+</mo><mi>?</mi><mi>y</mi><mo>=</mo><mn>7</mn><mo>+</mo><msubsup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow></mrow><mrow><mi>?</mi><mi>x</mi></mrow></msubsup></mrow></math><math><mrow><mo>=</mo><mn>7</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2?x</mn></mrow></mfrac><mi>?</mi><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>7</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2?49</mn></mrow></mfrac><mo>\times</mo><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></math><math><mrow><mo>=</mo><mn>7</mn><mo>+</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow><mrow><mn>1</mn><mn>4</mn></mrow></mfrac></mrow></math>= 7 + 0.0357.<math><mrow><mo>?49.5</mo><mo>=</mo><mn>7</mn><mo>?</mo><mn>0</mn><mn>3</mn><mn>5</mn></mrow></math>(iii) ?0.6<math><mrow></mrow></math>A.(iii)Let y = ?x<math><mrow><mo>.</mo></mrow></math> Let x = 0.64 and ?x = 0.04.Then, <math><mrow><mi>?</mi><mi>y</mi><mo>=?x+Ax</mo><mo>?x</mo></mrow></math><math><mrow><mo>=?0.64-0.04</mo><mo>?0.64</mo><mo>.</mo></mrow></math><math><mrow><mo>=?0.6</mo><mo>?</mo><mn>0</mn><mo>.</mo><mn>8</mn></mrow></math><math><mrow><mo>?0.6=</mo><mn>0</mn><mo>.</mo><mn>8</mn><mo>+</mo><mi>?</mi><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>8</mn><mo>+</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mi>?</mi><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>8</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2?x</mn></mrow></mfrac><mi>?</mi><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>8</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2?0.64</mn></mrow></mfrac><mo>\times</mo><mo stretchy="false">(</mo><mo>?</mo><mtext>?</mtext><mn>0</mn><mo>.</mo><mn>0</mn><mn>4</mn><mo stretchy="false">)</mo></mrow></math><math><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>8</mn><mfrac><mrow><mo>?</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>4</mn></mrow><mrow><mn>2</mn><mo>\times</mo><mn>0</mn><mo>.</mo><mn>8</mn></mrow></mfrac></mrow></math>= 0.8 - 0.025.= 0.775.(iv) <math><mrow><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>9</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math>A.(iv)Let <math><mrow><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math> Let x = 0.008 and ?x = 0.00 1.Then, ?y = <math><mrow><msup><mrow><mo stretchy="false">[</mo><mi>x</mi><mo>+</mo><mi>?</mi><mi>x</mi><mo stretchy="false">]</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>9</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>8</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>9</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo>+</mo><mi>?</mi><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>?</mo><mi>?</mi><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mi>?</mi><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><msup><mrow><mrow><mo>[</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>8</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow><mo>]</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>\times</mo><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>1</mn><mo stretchy="false">)</mo></mrow></math><math><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>1</mn></mrow><mrow><mn>3</mn><mo>\times</mo><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>1</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>1</mn><mn>2</mn></mrow></mfrac></mrow></math>= 0.2 + 0.0083.= 0.208.(v) <math><mrow><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>9</mn><mn>9</mn><mn>9</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mrow></msup></mrow></math>A.(v)Let <math><mrow><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mrow></msup><mo>.</mo></mrow></math> Let x = 1 and ?x = -0.001Then, <math><mrow><mi>?</mi><mi>y</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>?</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>9</mn><mn>9</mn><mn>9</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>?</mo><mn>1</mn><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>9</mn><mn>9</mn><mn>9</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mrow></msup><mo>=</mo><mn>1</mn><mo>+</mo><mi>A</mi><mi>y</mi></mrow></math><math><mrow><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>?</mi><mi>x</mi><mo>.</mo></mrow></math><math><mrow><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>9</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>\times</mo><mi>?</mi><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>1</mn><mo>?</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn><mo stretchy="false">(</mo><mo>.</mo><msup><mrow><mn>1</mn></mrow><mrow><mfrac bevelled="true"><mrow><mn>9</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mrow></msup><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mn>1</mn><mo>?</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac><mo>=</mo><mn>1</mn><mo>?</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>0</mn><mn>1</mn></mrow></math>= 0.999.(vi) <math><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></math>A.(vi)Let <math><mrow><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>.</mo></mrow></math> Then, x = 16 and ?x = 1.Then, <math><mrow><mi>?</mi><mi>y</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>A</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>.</mo></mrow></math><math><mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mn>6</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>=</mo><mn>2</mn><mo>+</mo><mi>A</mi><mi>y</mi><mo>=</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>A</mi><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>?</mo><mo stretchy="false">(</mo><mo>?</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math><math><mrow><mo>=</mo><mn>4</mn><mo>?</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mn>6</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>2</mn><mo>?</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><mo>\times</mo><mn>8</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>4</mn><mo>?</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mn>2</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mfrac><mrow><mn>6</mn><mn>4</mn><mo>?</mo><mn>1</mn></mrow><mrow><mn>3</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn><mn>3</mn></mrow><mrow><mn>3</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mn>1</mn><mo>.</mo><mn>9</mn><mn>6</mn><mn>8</mn></mrow></math>(vii) <math><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>6</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math>A.(vii)Let <math><mrow><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>.</mo></mrow></math> Let x = 27 and ?x = 1.Then, <math><mrow><mi>?</mi><mi>y</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>?</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>6</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>7</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>6</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>=</mo><mn>3</mn><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>A</mi><mi>x</mi><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mi>A</mi><mi>x</mi><mo>,</mo></mrow></math><math><mrow><mo>=</mo><mn>3</mn><mo>?</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">F)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>3</mn><mo>?</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mfrac><mrow><mn>8</mn><mn>1</mn><mo>?</mo><mn>1</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>8</mn><mn>0</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mo>.</mo><mn>9</mn><mn>6</mn><mn>2</mn><mo>.</mo></mrow></math>(viii) <math><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>5</mn><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></math>A.(viii)Let <math><mrow><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>.</mo></mrow></math> Let x = 256 and ?x = 1.Then, <math><mrow><mi>?</mi><mi>y</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>?</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>.</mo></mrow></math><math><mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>5</mn><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>5</mn><mn>6</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>5</mn><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>5</mn><mn>6</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>4</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>A</mi><mi>x</mi><mo>.</mo></mrow></math><math><mrow><mo>=</mo><mn>4</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>?</mo><mn>4</mn><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>4</mn><mfrac><mrow><mo>?</mo><mn>1</mn></mrow><mrow><mn>4</mn><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>5</mn><mn>6</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>4</mn><mo>?</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mn>5</mn><mn>6</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mn>0</mn><mn>2</mn><mn>4</mn><mo>?</mo><mn>1</mn></mrow><mrow><mn>2</mn><mn>5</mn><mn>6</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mn>0</mn><mn>2</mn><mn>3</mn></mrow><mrow><mn>2</mn><mn>5</mn><mn>6</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mo>.</mo><mn>9</mn><mn>9</mn><mn>6</mn></mrow></math>(ix) <math><mrow><msup><mrow><mo stretchy="false">(</mo><mn>8</mn><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></math>A.(ix)Let <math><mrow><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>.</mo></mrow></math> let x = 81 and ?x = 1.Then, <math><mrow><mi>?</mi><mi>y</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn>8</mn><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>8</mn><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mn>7</mn><msup><mrow><mo stretchy="false">(</mo><mn>8</mn><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>=</mo><mn>3</mn><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>?</mi><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>?</mo><mi>?</mi><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><msup><mrow><mo stretchy="false">(</mo><mn>8</mn><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><mo>\times</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>8</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mn>2</mn><mn>4</mn><mo>+</mo><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>8</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mn>2</mn><mn>5</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>8</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>1</mn><mo>.</mo></mrow></math>(x) <math><mrow><msup><mrow><mo stretchy="false">(</mo><mn>4</mn><mn>0</mn><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math>A.(x)Let y = <math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math> = ?x<math><mrow><mo>.</mo></mrow></math> Let x = 400 and ?x = 1.Then, <math><mrow><mi>?</mi><mi>y</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>?</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>.</mo></mrow></math><math><mrow><mo>?</mo><mtext>?</mtext><mi>?</mi><mi>y</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn>4</mn><mn>0</mn><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>4</mn><mn>0</mn><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>?</mo><mtext>?</mtext><msup><mrow><mo stretchy="false">(</mo><mn>4</mn><mn>0</mn><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>=</mo><mn>2</mn><mn>0</mn><mtext>?</mtext><mo>+</mo><mi>?</mi><mi>y</mi><mo>=</mo><mn>2</mn><mn>0</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>?</mo><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>2</mn><mn>0</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2?x</mn></mrow></mfrac><mi>?</mi><mi>x</mi></mrow></math><math><mrow><mo>=</mo><mn>2</mn><mn>0</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2?400</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>2</mn><mn>0</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mo>\times</mo><mn>2</mn><mn>0</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>2</mn><mn>0</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><mn>0</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mfrac><mrow><mn>8</mn><mn>0</mn><mn>0</mn><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><mn>0</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>8</mn><mn>0</mn><mn>1</mn></mrow><mrow><mn>4</mn><mn>0</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mn>0</mn><mo>.</mo><mn>0</mn><mn>2</mn><mn>5</mn><mo>.</mo></mrow></math>(xi) <math><mrow><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>3</mn><mn>7</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math>A.(xi)Let <math><mrow><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math> Let x = 0.0036 and ?x = 0.0001Then, <math><mrow><mi>?</mi><mi>y</mi><mo>=?x+?x</mo><mo>?x</mo></mrow></math><math><mrow><mo>=?0.0037</mo><mo>?0.0036</mo></mrow></math><math><mrow><mo>?0.0037</mo><mo>=</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>6</mn><mo>+</mo><mi>D</mi><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>6</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>?</mi><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>6</mn><mo>+</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2?x</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>6</mn><mo>+</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>1</mn></mrow><mrow><mn>2?0.0036</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>6</mn><mo>+</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>1</mn></mrow><mrow><mn>2</mn><mo>\times</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>6</mn></mrow></mfrac></mrow></math><math><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>6</mn><mo>+</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0</mn><mn>0</mn><mn>1</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>1</mn><mn>2</mn><mo>.</mo></mrow></mfrac></mrow></math>= 0.06 + 0.0008.= 0.0608.(xii) <math><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>6</mn><mo>.</mo><mn>5</mn><mn>7</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math>A.(xii)Let <math><mrow><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>.</mo></mrow></math> Let x = 27 and ?x = 0.43.Then, <math><mrow><mi>?</mi><mi>y</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>A</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>6</mn><mo>.</mo><mn>5</mn><mn>7</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>7</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></math><math><mrow><mo>?</mo><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>6</mn><mo>.</mo><mn>5</mn><mn>7</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>=</mo><mn>3</mn><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>?</mi><mi>x</mi><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><msup><mrow><mi>x</mi></mrow><mrow><mfrac bevelled="true"><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>?</mo><mi>?</mi><mi>x</mi><mo>.</mo></mrow></math><math><mrow><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mn>7</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac bevelled="true"><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>?</mo><mo st0 Upvotes 0 Upvotes 0 Downvotes Report Abuse CommentCharacters 0 /2500 Disclaimer: Please note that external links or URLs are not allowed. Non adherence will lead to removal of the comment Cancel PostpaginationHtml = `<ul class="pagniatn-ul"></ul>`;Similar Questions for you If the normal to the curve y(x) = \int(from 2 to x) (2t² - 15t + 10)dt at a point (a, b) is parallel to the line x + 3y = -5, a > 1, then the value of |a + 6b| is equal to.......... R Raj Pandey y (x) = \int? (2t² - 15t + 10)dtdy/dx = 2x² - 15x + 10.For tangent at (a, b), slope is m = dx/dy = 1 / (dy/dx) = 1 / (2a² - 15a + 10).Given slope is -1/3.2a² - 15a + 10 = -32a² - 15a + 13 = 0 (The provided solution has 2a²-15a+7=0, suggesting a different problem or a typo)Following the image: 2a² - 15a + 7 = 0 (2a - 1) (a - 7) = 0a = 1/2 or a = 7.a = 1/2 Rejected as a > 1. So a = 7.b = \int? (2t² - 15t + 10)dt = [2t³/3 - 15t²/2 + 10t] from 0 to 7.6b = [4t³ - 45t² + 60t] from 0 to 7 = 4 (7)³ - 45 (7)² + 60 (7) = 1372 - 2205 + 420 = -413.|a + 6b| = |7 - 413| = |-406| ...more y (x) = \int? (2t² - 15t + 10)dtdy/dx = 2x² - 15x + 10.For tangent at (a, b), slope is m = dx/dy = 1 / (dy/dx) = 1 / (2a² - 15a + 10).Given slope is -1/3.2a² - 15a + 10 = -32a² - 15a + 13 = 0 (The provided solution has 2a²-15a+7=0, suggesting a different problem or a typo)Following the image: 2a² - 15a + 7 = 0 (2a - 1) (a - 7) = 0a = 1/2 or a = 7.a = 1/2 Rejected as a > 1. So a = 7.b = \int? (2t² - 15t + 10)dt = [2t³/3 - 15t²/2 + 10t] from 0 to 7.6b = [4t³ - 45t² + 60t] from 0 to 7 = 4 (7)³ - 45 (7)² + 60 (7) = 1372 - 2205 + 420 = -413.|a + 6b| = |7 - 413| = |-406| = 406. less If the tangent to the curve, y = f(x) = xlog? x, (x > 0) at a point (c, f(c)) is parallel to the line - segment joining the points (1,0) and (e, e) then c is equal to: R Raj Pandey f' (c) = 1 + lnc = e/ (e-1)lnc = e/ (e-1) - 1 = (e - (e-1)/ (e-1) = 1/ (e-1)c = e^ (1/ (e-1) The lengths of the sides of a triangle are 10 + x2, 10 + x2 and 20 - 2x2. If for x = k, the area of the triangle is maximum, then 3k2 is equal to: R Raj Pandey C D = \sqrt[]{{(10 + x^{2})}^{2} - {(10 - x^{2})}^{2}} = 2 \sqrt[]{10} | x | Area = \frac{1}{2} \times C D \times A B = \frac{1}{2} \times 2 \sqrt[]{10} | x | (20 - 2 x^{2}) \Rightarrow 10 - x^{2} = 2 x 3x 2 = 10 x = k 3k 2 = 10 Let F 1 (A,B,C) = (A \land \sim B) \lor [\sim C \land (A \lor B)] \lor \sim A a n d F_{2} (A, B) = (A \lor B) \lor (B \to \sim A) be two logical expressions. Then: V Vishal Baghel By truth table So F 1 (A, B, C) is not a tautology Now again by truth table So F 2 (A, B) be a tautology. The triangle of maximum area that can be inscribed in a given circle of radius ‘r’ is: V Vishal Baghel From option let it be isosceles where AB = AC then x = \sqrt[]{r^{2} - {(h - r)}^{2}} = \sqrt[]{r^{2} - h^{2} - r^{2} + 2 r h} x = \sqrt[]{2 h r - h^{2}} . . . . . . . . (i) Now ar (Δ A B C) = Δ = \frac{1}{2} B C \times A L Δ = \frac{1}{2} \times 2 \sqrt[]{2 h r - a^{2}} \times h then x = \sqrt[]{2 \times \frac{3 r}{2} \times r - \frac{9 r^{2}}{4}} = \frac{\sqrt[]{3}}{2} r f r o m (i) \Rightarrow B C = \sqrt[]{3} r So A B = \sqrt[]{h^{2} + x^{2}} = \sqrt[]{\frac{9 r^{2}}{4} + \frac{3 r^{2}}{4}} = \sqrt[]{3} r . Hence Δ be equilateral having each side of length \sqrt[]{3} r .var postBodyTracking = 'null';Taking an Exam? Selecting a College? Get authentic answers from experts, students and alumni that you won't find anywhere else Sign Up on Shiksha On Shiksha, get access to 65k Colleges 1.2k Exams 688k Reviews 1800k Answers Learn more about... Maths Ncert Solutions class 12th 2023 Exam View Exam Details 2.5k Answered Questions Most viewed information Download Guide .write-a-review-widget h2.tbSec2 {
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