Turito Academy

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<img src="data:image/png;base64,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" class="Wirisformula" alt="L t subscript left parenthesis x rightwards arrow infinity right parenthesis invisible function application left square bracket left parenthesis a to the power of left parenthesis 1 divided by x right parenthesis plus b to the power of left parenthesis 1 divided by x right parenthesis plus c to the power of left parenthesis 1 divided by x right parenthesis right parenthesis divided by 3 right square bracket to the power of x" width="311" height="23" style="vertical-align:-6px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»L«/mi»«msub»«mi»t«/mi»«mo»(«/mo»«/msub»«mi»x«/mi»«mo»§#8594;«/mo»«mo»§#8734;«/mo»«mo»)«/mo»«mo»§#8289;«/mo»«mo»[«/mo»«mo»(«/mo»«msup»«mi»a«/mi»«mo»(«/mo»«/msup»«mn»1«/mn»«mo»/«/mo»«mi»x«/mi»«mo»)«/mo»«mo»+«/mo»«msup»«mi»b«/mi»«mo»(«/mo»«/msup»«mn»1«/mn»«mo»/«/mo»«mi»x«/mi»«mo»)«/mo»«mo»+«/mo»«msup»«mi»c«/mi»«mo»(«/mo»«/msup»«mn»1«/mn»«mo»/«/mo»«mi»x«/mi»«mo»)«/mo»«mo»)«/mo»«mo»/«/mo»«mn»3«/mn»«msup»«mo»]«/mo»«mi»x«/mi»«/msup»«/math»'/> where a,b,c are real and non-zero=

Solution for the question - (t^(x)+oo)[(a^((1//x))+b^((1//x))+c^((1//x)))//3]^(x):} where a,b,c arereal and non-zero= [theta '/> (abc)^((1//x)) '/> (abc)(1//3) '/>

(t^(x)+oo)[(a^((1//x))+b^((1//x))+c^((1//x)))//3]^(x):} where-Turito

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Solution for the question - {:delta_(()x rarr oo)((x^(2)+5x+3)//(x^(2)+x+2))^(x)= e^(t) '/> e^(2) '/> theta^(3) '/> epsi^(4) '/>

{:delta_(()x rarr oo)((x^(2)+5x+3)//(x^(2)+x+2))^(x)= -Turito

Solution for the question - (4(x rarr oo)((x+6)//(x+1))(x+4)= theta^(3) '/>eepsi^(4) '/> a^(6) '/>

(4(x rarr oo)((x+6)//(x+1))(x+4)= -Turito

Solution for the question - (u_(()x rarr oo)((x+5)//(x+2))(x+3)= e-e^(3) '/> theta^(3) '/> e^(2) '/>

(u_(()x rarr oo)((x+5)//(x+2))(x+3)= -Turito

Solution for the question - {:u(x rarr oo)((x+a)//(x+b))^(()x+b)= e^(q_(t)) '/> e(a+b) '/> e(e-b) '/>

{:u(x rarr oo)((x+a)//(x+b))^(()x+b)= -Turito

Solution for the question - (t^(x)rarr0)(e^(x)-e^(()(nx))//(2(x-sin x))= 1//2 '/>-121

(t^(x)rarr0)(e^(x)-e^(()(nx))//(2(x-sin x))= -Turito

In this question we have to find limit of  <img src="data:image/png;base64,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" class="Wirisformula" alt="limit as x minus greater than 0 of fraction numerator left parenthesis 1 minus e to the power of x right parenthesis sin open parentheses x close parentheses over denominator x squared plus x cubed end fraction" width="150" height="40" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mo»(«/mo»«mn»1«/mn»«mo»-«/mo»«msup»«mi»e«/mi»«mi»x«/mi»«/msup»«mo»)«/mo»«mi»sin«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«/math»'/> Step1: Rearranging the expression <img src="data:image/png;base64,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" class="Wirisformula" alt="limit as x minus greater than 0 of fraction numerator left parenthesis 1 minus e to the power of x right parenthesis over denominator x end fraction cross times open parentheses fraction numerator sin open parentheses x close parentheses over denominator x end fraction close parentheses open parentheses limit as x minus greater than 0 of fraction numerator 1 over denominator 1 plus x end fraction close parentheses" width="292" height="43" style="vertical-align:-17px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mo»(«/mo»«mn»1«/mn»«mo»-«/mo»«msup»«mi»e«/mi»«mi»x«/mi»«/msup»«mo»)«/mo»«/mrow»«mi»x«/mi»«/mfrac»«mo»§#215;«/mo»«mfenced»«mfrac»«mrow»«mi»sin«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mi»x«/mi»«/mfrac»«/mfenced»«mfenced»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mn»1«/mn»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»x«/mi»«/mrow»«/mfrac»«/mrow»«/mfenced»«/math»'/> Step2: Using Standard Limits We know that <img src="data:image/png;base64,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" class="Wirisformula" alt="limit as x minus 0 of fraction numerator left parenthesis e to the power of x minus 1 right parenthesis over denominator x end fraction equals 1" width="119" height="40" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mo»(«/mo»«msup»«mi»e«/mi»«mi»x«/mi»«/msup»«mo»-«/mo»«mn»1«/mn»«mo»)«/mo»«/mrow»«mi»x«/mi»«/mfrac»«mo»=«/mo»«mn»1«/mn»«/math»'/> and <img src="data:image/png;base64,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" class="Wirisformula" alt="limit as x minus greater than 0 of fraction numerator sin open parentheses x close parentheses over denominator x end fraction equals 1" width="122" height="40" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mi»sin«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mi»x«/mi»«/mfrac»«mo»=«/mo»«mn»1«/mn»«/math»'/> =&gt; <img src="data:image/png;base64,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" class="Wirisformula" alt="negative 1 cross times open parentheses limit as x minus greater than 0 of fraction numerator 1 over denominator 1 plus x end fraction close parentheses" width="143" height="43" style="vertical-align:-17px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mn»1«/mn»«mo»§#215;«/mo»«mfenced»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mn»1«/mn»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»x«/mi»«/mrow»«/mfrac»«/mrow»«/mfenced»«/math»'/> =&gt; <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAANCAYAAABCZ/VdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAMyZLetQAAADBJREFUeNpjYKAcqAFxLRBfYKABWAzEaUD8n4GGYNRw0g3/TwQeDfMRYDipEU8dAADUJhpljLE7wQAAAFN0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+LTwvbW8+PG1uPjE8L21uPjwvbWF0aD5mjCijAAAAAElFTkSuQmCC" class="Wirisformula" alt="negative 1" width="23" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mn»1«/mn»«/math»'/>. So, the value of the limit is <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAANCAYAAABCZ/VdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAMyZLetQAAADBJREFUeNpjYKAcqAFxLRBfYKABWAzEaUD8n4GGYNRw0g3/TwQeDfMRYDipEU8dAADUJhpljLE7wQAAAFN0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+LTwvbW8+PG1uPjE8L21uPjwvbWF0aD5mjCijAAAAAElFTkSuQmCC" class="Wirisformula" alt="negative 1" width="23" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mn»1«/mn»«/math»'/>

Solution for the question - (t(x rarr0)((1-e^(x))sin x)//(x^(2)+x^(3))=:} e^(2) '/>-101

(t(x rarr0)((1-e^(x))sin x)//(x^(2)+x^(3))=:} -Turito

If <img src="data:image/png;base64,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" class="Wirisformula" alt="a greater than 0 text  and  end text Lt invisible function application left parenthesis x not stretchy rightwards arrow a right parenthesis open parentheses a to the power of x minus x to the power of 4 close parentheses divided by open parentheses x to the power of x minus a cubed close parentheses equals negative 1 text , then  end text a equals" width="397" height="19" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mo»§#62;«/mo»«mn»0«/mn»«mtext»§#160;and§#160;«/mtext»«mi»Lt«/mi»«mo»§#8289;«/mo»«mo»(«/mo»«mi»x«/mi»«mo stretchy=¨false¨»§#8594;«/mo»«mi»a«/mi»«mo»)«/mo»«mfenced separators=¨|¨»«mrow»«msup»«mi»a«/mi»«mi»x«/mi»«/msup»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«/mrow»«/mfenced»«mo»/«/mo»«mfenced separators=¨|¨»«mrow»«msup»«mi»x«/mi»«mi»x«/mi»«/msup»«mo»§#8722;«/mo»«msup»«mi»a«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«mtext», then§#160;«/mtext»«mi»a«/mi»«mo»=«/mo»«/math»'/>

Solution for the question - if a > 0 and (t(x rarr a)(a^^x-x^(^^)4)//(x^(^^)^^x-a^^^(^^))=-1:}, thena= [theta '/>-110

If a > 0 and (t(x rarr a)(a^^x-x^(^^)4)//(x^(^^)^^x-a^^^(^^))-Turito

If <img src="data:image/png;base64,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" class="Wirisformula" alt="stack a with minus on top equals left parenthesis 3 comma negative 1 , 5 right parenthesis comma stack b with minus on top equals left parenthesis 1 , 2 comma negative 3 right parenthesis" width="195" height="23" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»a«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mn»3«/mn»«mo»,«/mo»«mo»-«/mo»«mn»1,5«/mn»«mo»)«/mo»«mo»,«/mo»«mover accent=¨true¨»«mrow»«mi»b«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mn»1,2«/mn»«mo»,«/mo»«mo»-«/mo»«mn»3«/mn»«mo»)«/mo»«/math»'/> are given vectors. A vector <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAARCAYAAAAL4VbbAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAGxJREFUeNpjYECA/3jwKCAK8ABxFxA/BeJfQLwOiAVxKTwHxK1AzAfEbEBcAsQ+2BR3APEkYqxnAuLXuKxEB6ZAfJhYj3lBPUMUMAbiG6QE2wVoSLBAQ6MAiK1xKZYF4oNA/AeILwFxMtmxBQBxjBaVFcpzEAAAAHB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+YzwvbWk+PG1vPi08L21vPjwvbW92ZXI+PC9tYXRoPrxG3ZMAAAAASUVORK5CYII=" class="Wirisformula" alt="stack c with minus on top" width="11" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»c«/mi»«/mrow»«mo»-«/mo»«/mover»«/math»'/> which is perpendicular to <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAKCAYAAACNMs+9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAAExJREFUeNpjYECAeiD+jwWXMxAAkkB8EIhF8SkSBOLdQCyNTxEHEG8CYh1CVi4HYltCiqYBsQ8hRZVAHE9IUSQxwQACH3CEIRsDqQAA8N4SDfBGJwkAAABJdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1pPno8L21pPjwvbWF0aD6hUlFuAAAAAElFTkSuQmCC" class="Wirisformula" alt="z" width="10" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mi»z«/mi»«/math»'/>-axis satisfying <img src="data:image/png;base64,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" class="Wirisformula" alt="stack c with minus on top times stack a with minus on top equals 9" width="61" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»c«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»⋅«/mo»«mover accent=¨true¨»«mrow»«mi»a«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»=«/mo»«mn»9«/mn»«/math»'/> and <img src="data:image/png;base64,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" class="Wirisformula" alt="stack c with minus on top times stack b with minus on top equals negative 4" width="78" height="21" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»c«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»⋅«/mo»«mover accent=¨true¨»«mrow»«mi»b«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»=«/mo»«mo»-«/mo»«mn»4«/mn»«/math»'/> then

Solution for the question - if bar(a)=(3,-1,5),b=(1,2,-3) are given vectors. a vector e which isperpendicular to -axis satisfying bar(c)- bar(a)=9 and bar(c)* b

If bar(a)=(3,-1,5),b=(1,2,-3) are given vectors. a vector e whi-Turito

<img src="data:image/png;base64,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" class="Wirisformula" alt="a with minus on top equals left parenthesis c o s invisible function application theta right parenthesis times i with minus on top minus left parenthesis s i n invisible function application theta right parenthesis j with minus on top comma b with minus on top equals left parenthesis s i n invisible function application theta right parenthesis times i with minus on top plus left parenthesis c o s invisible function application theta right parenthesis j with minus on top comma c with minus on top equals k with minus on top comma r with minus on top equals 17 i with minus on top plus 7 j with minus on top plus 15 k with minus on top" width="543" height="23" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover accent=¨true¨»«mi»a«/mi»«mo»-«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#8289;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mo»§#8901;«/mo»«mover accent=¨true¨»«mi»i«/mi»«mo»-«/mo»«/mover»«mo»-«/mo»«mo»(«/mo»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#8289;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mover accent=¨true¨»«mi»j«/mi»«mo»-«/mo»«/mover»«mo»,«/mo»«mover accent=¨true¨»«mi»b«/mi»«mo»-«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#8289;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mo»§#8901;«/mo»«mover accent=¨true¨»«mi»i«/mi»«mo»-«/mo»«/mover»«mo»+«/mo»«mo»(«/mo»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#8289;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mover accent=¨true¨»«mi»j«/mi»«mo»-«/mo»«/mover»«mo»,«/mo»«mover accent=¨true¨»«mi»c«/mi»«mo»-«/mo»«/mover»«mo»=«/mo»«mover accent=¨true¨»«mi»k«/mi»«mo»-«/mo»«/mover»«mo»,«/mo»«mover accent=¨true¨»«mi»r«/mi»«mo»-«/mo»«/mover»«mo»=«/mo»«mn»17«/mn»«mover accent=¨true¨»«mi»i«/mi»«mo»-«/mo»«/mover»«mo»+«/mo»«mn»7«/mn»«mover accent=¨true¨»«mi»j«/mi»«mo»-«/mo»«/mover»«mo»+«/mo»«mn»15«/mn»«mover accent=¨true¨»«mi»k«/mi»«mo»-«/mo»«/mover»«/math»'/> and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHcAAAAfCAYAAADdnlK9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAc1CXO0QAAArJJREFUeNrtmUFkXFEUhp9RI6JKRVRFhdFVVTdVFVXddBFRNbqpWUREN12OKlUxoqpEVRZRpYuqqCpVs6iYTVVEFtnEiOiiQpdRI1RVVVVI/8Ofeq5z572Xefc1xvn4ybv3zXt3/nPvuedmoshIy55HhmEYhmF7icXOBm0YhmEYhmEYhmEYhmEYhvH/+QWVzIb+4zS0aTYcDvL+oaEKvTlkY0rNLHQPug2tQ795nSe3oDml/SH74kj6ewB1oD/QBjQawMgX0E2lvQ49cvyZ4Zi+0p9l6GSg4M5G+i9jWkyOQo+hbXrVhI7Hb3jLzidQJWGASepGGzoRu56Gnir3LUELHGSJ46sGMHKKxsQZhLagIcefLzR3f0zzGVdzLytXJtEKNKwEts2JeAwqQ3eha/Gb5MvcKCBDjNMU4Sr0UbmnRjPjLEITAYycUN4lq6bhtIk/Z5w2MfNHAcGVyfQBGlH65rgIvJRYCRaFDPQKU+2Q0i/pbsxpW0lIywfNKPL+T7HrYa7QgRT+DCQEt9cst/+O99BZT9x23BTscpGGpp19vQ64zr3hXMojh/z9M2DxEg/cPMeXxp8xT+bJc+VK2r/s6bsArSY9QNLgy4JW7SXoHU2see7ZUT7TDhjcVdYZFa7aUkp/7ijpO8/gPnP3TmVLaSY9ZEGpVkNQYYoZZCGw7knL205abHBChAqu7OfXodfQpMefaaetzHNvqGr5Pou9bpyHPic9qJmxWDkIspe1nGDKrHzlORo95woapemtgMGt80i02cWfLRaAEQubphLwvMZUy3AM3WClfIQFXp2Z7h/fnJWSN2WaMeLZU8aV9hlWsQ2OrZPxKJT1HxR7XL0aHfZJgHdpaDWgX989tUxZufcUi81dTs4iMnCQlR8KCdxaZPQlLeXoZfQBcn5c6scv9hcRO84owsuTSAAAAVt0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+cjwvbWk+PG1pPiYjeDIwM0U7PC9taT48L21vdmVyPjxtbz49PC9tbz48bWk+eDwvbWk+PG1vdmVyIGFjY2VudD0idHJ1ZSI+PG1pPmE8L21pPjxtaT4mI3gyMDNFOzwvbWk+PC9tb3Zlcj48bW8+KzwvbW8+PG1pPnk8L21pPjxtb3ZlciBhY2NlbnQ9InRydWUiPjxtaT5iPC9taT48bWk+JiN4MjAzRTs8L21pPjwvbW92ZXI+PG1vPis8L21vPjxtaT56PC9taT48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+YzwvbWk+PG1pPiYjeDIwM0U7PC9taT48L21vdmVyPjwvbWF0aD582ruHAAAAAElFTkSuQmCC" class="Wirisformula" alt="stack r with ‾ on top equals x stack a with ‾ on top plus y stack b with ‾ on top plus z stack c with ‾ on top" width="119" height="31" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»r«/mi»«/mrow»«mi»‾«/mi»«/mover»«mo»=«/mo»«mi»x«/mi»«mover accent=¨true¨»«mrow»«mi»a«/mi»«/mrow»«mi»‾«/mi»«/mover»«mo»+«/mo»«mi»y«/mi»«mover accent=¨true¨»«mrow»«mi»b«/mi»«/mrow»«mi»‾«/mi»«/mover»«mo»+«/mo»«mi»z«/mi»«mover accent=¨true¨»«mrow»«mi»c«/mi»«/mrow»«mi»‾«/mi»«/mover»«/math»'/>. The range of <img src="data:image/png;base64,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" class="Wirisformula" alt="x plus y plus z" width="64" height="13" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«mo»+«/mo»«mi»z«/mi»«/math»'/> is

Solution for the question - bar(a)=(cos theta)*i-(sin theta)j,b=(sin theta)*i+(cos theta)j, bar(c)=bar(k), bar(r)=17 hat(i)+7j+15 k and r=xa+yb+zc . the range o

Bar(a)=(cos theta)*i-(sin theta)j,b=(sin theta)*i+(cos theta)j,-Turito

<img src="data:image/png;base64,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" class="Wirisformula" alt="stack a with minus on top comma stack b with minus on top comma stack c with minus on top" width="43" height="23" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»a«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»,«/mo»«mover accent=¨true¨»«mrow»«mi»b«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»,«/mo»«mover accent=¨true¨»«mrow»«mi»c«/mi»«/mrow»«mo»-«/mo»«/mover»«/math»'/> are non-coplanar vectors and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEMAAAAVCAYAAAAdHVOZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAU2v5G4wAAAXtJREFUeNpjYBg48B8PHgWjYBSMgoEGJJVRowXaKBgFgxJ8A2Km0WBgYFAB4kvDvcQmFgQA8XI62kcSACXXRiB+CcS/gPgCEMvT0HH1QFwNtfM5EP8A4v1ALEnjwOAB4i4gfgr15zogFkRXtAWIJ0ElQAGzChp7tAoMkPl3gbgcyc4+ElPLfzIC4hwQtwIxHxCzAXEJEPsgK4qEOg4ZLARiLxoGxi0g1kITAznwEw0DowMa4XgBKHlaookdJCKb/CcC48qS37CIcxAIDHLtg9n5GluWIFTFgdhfaFigmUMjAB2AImQvjVKGKRAfJkbhazS+NTRv0SowQNlyPhbxIiCupVFgeEELS4LgKTSJwgDIQWtoGBigfJuIJsYGbXfQqjYxBuIbxChsBuKZ0OwBKieWAvE2GgbGOmgB6gLlS0PFEmnczrgArUlYoIV1ATQXYIBqaI1SC00lL8moWokFILP9oAHyB+pIWtmFDGShFcMfaCpMJlaj6EjrHwAA+vt2iHyLWmEAAADmdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vdmVyIGFjY2VudD0idHJ1ZSI+PG1pPmE8L21pPjxtbz4tPC9tbz48L21vdmVyPjxtbz4rPC9tbz48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+YjwvbWk+PG1vPi08L21vPjwvbW92ZXI+PG1vPis8L21vPjxtb3ZlciBhY2NlbnQ9InRydWUiPjxtaT5jPC9taT48bW8+LTwvbW8+PC9tb3Zlcj48L21hdGg+PiHXDQAAAABJRU5ErkJggg==" class="Wirisformula" alt="stack a with minus on top plus stack b with minus on top plus stack c with minus on top" width="67" height="21" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»a«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»+«/mo»«mover accent=¨true¨»«mrow»«mi»b«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»+«/mo»«mover accent=¨true¨»«mrow»«mi»c«/mi»«/mrow»«mo»-«/mo»«/mover»«/math»'/> and <img src="data:image/png;base64,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" class="Wirisformula" alt="stack b with minus on top plus stack c with minus on top plus stack d with minus on top" width="67" height="21" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»b«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»+«/mo»«mover accent=¨true¨»«mrow»«mi»c«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»+«/mo»«mover accent=¨true¨»«mrow»«mi»d«/mi»«/mrow»«mo»-«/mo»«/mover»«/math»'/> are collinear with <img src="data:image/png;base64,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" class="Wirisformula" alt="stack d with minus on top" width="11" height="21" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»d«/mi»«/mrow»«mo»-«/mo»«/mover»«/math»'/> and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAQCAYAAADAvYV+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAPUJuPDwAAAH1JREFUeNpjYECA/3jwyANMQNwIxC+B+BcQXwBieVyKtwDxJCAWhGpcBcQB2BRGQiWRwUIg9sKmeD8QW6KJHcTljG9Qq5Hd/wWXe1+j8a2B+BwuxU+BmAOJXwvEa3ApbgbimVDrQe5cCsTb8IVzNTREaqG2vMQVdNiAKLoAAHqFG/BdleSaAAAAcHRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtb3ZlciBhY2NlbnQ9InRydWUiPjxtaT5hPC9taT48bW8+LTwvbW8+PC9tb3Zlcj48L21hdGg+oYkMMQAAAABJRU5ErkJggg==" class="Wirisformula" alt="stack a with minus on top" width="11" height="16" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»a«/mi»«/mrow»«mo»-«/mo»«/mover»«/math»'/> respectively then <img src="data:image/png;base64,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" class="Wirisformula" alt="stack a with minus on top plus stack b with minus on top plus stack c with minus on top plus stack d with minus on top" width="95" height="21" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»a«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»+«/mo»«mover accent=¨true¨»«mrow»«mi»b«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»+«/mo»«mover accent=¨true¨»«mrow»«mi»c«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»+«/mo»«mover accent=¨true¨»«mrow»«mi»d«/mi»«/mrow»«mo»-«/mo»«/mover»«/math»'/> is

Solution for the question - bar(a),b,c are non-coplanar vectors and bar(a)+ bar(b)+ bar(c) and theta+bar(c)+theta are collinear with phi and respectively then b

Bar(a),b,c are non-coplanar vectors and bar(a)+ bar(b)+ bar(c) -Turito

If <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAOCAYAAAAi2ky3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAAJhJREFUeNpjYEAAJiBuBOKXQPwLiC8AsTwDGWALEE8CYkGooauAOIBUQyKhGpHBQiD2ItWg/UBsiSZ2kByvfYN6Bzm8vpATPq/R+NZAfA6HWjUgroVGBgZ4CsQcSHyQwjU4DFoMxGlA/B+bZDMQz4R6CRQuS4F4GwFf/MclUQ2NuVqo614SiP7/xIabKLkuIhUMQ4P+Y8EMAE95I2eLUPoaAAAAYHRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtc3ViPjxtaT5hPC9taT48bW4+MTwvbW4+PC9tc3ViPjwvbWF0aD7pKfJKAAAAAElFTkSuQmCC" class="Wirisformula" alt="a subscript 1 end subscript" width="18" height="14" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«msub»«mrow»«mi»a«/mi»«/mrow»«mrow»«mn»1«/mn»«/mrow»«/msub»«/math»'/> and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAOCAYAAAAi2ky3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAALFJREFUeNpjYEAAJiBuBOKXQPwLiC8AsTwDGWALEE8CYkGooauAOIBUQyKhGpHBQiD2ItWg/UBsiSZ2kByvfYN6Bzm8vpATPq/R+NZAfA6Luv9QDIqMo0Csgq7gKRBzIPFrgXgNHotBLs7CZlkzEM+EKgCFy1Ig3kaET35gE6yGxlwt1HUvCUS/PTRCCAJRPHKS0DBSYqAAqEETryQlhkhCw46PgUIAMkSDgQrgPxYMBgBL1SSzSaTjmQAAAGB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bXN1Yj48bWk+YTwvbWk+PG1uPjI8L21uPjwvbXN1Yj48L21hdGg+RoC/gAAAAABJRU5ErkJggg==" class="Wirisformula" alt="a subscript 2 end subscript" width="18" height="14" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«msub»«mrow»«mi»a«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msub»«/math»'/> are two values of a for which the unit vector <img src="data:image/png;base64,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" class="Wirisformula" alt="stack a i with minus on top plus b j with minus on top plus 1 half k with minus on top" width="100" height="35" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover accent=¨true¨»«mrow»«mi»a«/mi»«mi»i«/mi»«/mrow»«mo»-«/mo»«/mover»«mo»+«/mo»«mi»b«/mi»«mover accent=¨true¨»«mi»j«/mi»«mo»-«/mo»«/mover»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mover accent=¨true¨»«mi»k«/mi»«mo»-«/mo»«/mover»«/math»'/> is linearly dependent with <img src="data:image/png;base64,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" class="Wirisformula" alt="i with minus on top plus 2 j with minus on top" width="49" height="23" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover accent=¨true¨»«mi»i«/mi»«mo»-«/mo»«/mover»«mo»+«/mo»«mn»2«/mn»«mover accent=¨true¨»«mi»j«/mi»«mo»-«/mo»«/mover»«/math»'/> and <img src="data:image/png;base64,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" class="Wirisformula" alt="j with minus on top minus 2 k with minus on top" width="49" height="23" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover accent=¨true¨»«mi»j«/mi»«mo»-«/mo»«/mover»«mo»-«/mo»«mn»2«/mn»«mover accent=¨true¨»«mi»k«/mi»«mo»-«/mo»«/mover»«/math»'/> then <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator 1 over denominator a subscript 1 end subscript end fraction plus fraction numerator 1 over denominator a subscript 2 end subscript end fraction" width="69" height="39" style="vertical-align:-16px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mn»1«/mn»«/mrow»«mrow»«msub»«mrow»«mi»a«/mi»«/mrow»«mrow»«mn»1«/mn»«/mrow»«/msub»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«mn»1«/mn»«/mrow»«mrow»«msub»«mrow»«mi»a«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msub»«/mrow»«/mfrac»«/math»'/> is equal to

Solution for the question - if 41 and 42 are two values of a for which the unit vector bar(ai)+bbar(j)+(1)/(2) bar(k) is linearly dependent with i+2j and j-2k t

If 41 and 42 are two values of a for which the unit vector bar(-Turito

<img src="data:image/png;base64,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" class="Wirisformula" alt="a with rightwards arrow on top equals left parenthesis c o s invisible function application theta right parenthesis i with rightwards arrow on top minus left parenthesis s i n invisible function application theta right parenthesis j with rightwards arrow on top comma b with rightwards arrow on top equals left parenthesis s i n invisible function application theta right parenthesis i with rightwards arrow on top plus left parenthesis c o s invisible function application theta right parenthesis j with rightwards arrow on top comma c with rightwards arrow on top equals k with rightwards arrow on top comma r with rightwards arrow on top equals 7 i with rightwards arrow on top plus j with rightwards arrow on top plus 10 k with rightwards arrow on top" width="511" height="22" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover accent=¨true¨»«mi»a«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#8289;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mover accent=¨true¨»«mi»i«/mi»«mo»§#8594;«/mo»«/mover»«mo»-«/mo»«mo»(«/mo»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#8289;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mover accent=¨true¨»«mi»j«/mi»«mo»§#8594;«/mo»«/mover»«mo»,«/mo»«mover accent=¨true¨»«mi»b«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#8289;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mover accent=¨true¨»«mi»i«/mi»«mo»§#8594;«/mo»«/mover»«mo»+«/mo»«mo»(«/mo»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#8289;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mover accent=¨true¨»«mi»j«/mi»«mo»§#8594;«/mo»«/mover»«mo»,«/mo»«mover accent=¨true¨»«mi»c«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mover accent=¨true¨»«mi»k«/mi»«mo»§#8594;«/mo»«/mover»«mo»,«/mo»«mover accent=¨true¨»«mi»r«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mn»7«/mn»«mover accent=¨true¨»«mi»i«/mi»«mo»§#8594;«/mo»«/mover»«mo»+«/mo»«mover accent=¨true¨»«mi»j«/mi»«mo»§#8594;«/mo»«/mover»«mo»+«/mo»«mn»10«/mn»«mover accent=¨true¨»«mi»k«/mi»«mo»§#8594;«/mo»«/mover»«/math»'/> if <img src="data:image/png;base64,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" class="Wirisformula" alt="stack r with rightwards arrow on top equals x stack a with rightwards arrow on top plus y stack b with rightwards arrow on top plus z stack c with rightwards arrow on top" width="129" height="22" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mover accent=¨true¨»«mrow»«mi»r«/mi»«/mrow»«mo»→«/mo»«/mover»«mo»=«/mo»«mi»x«/mi»«mover accent=¨true¨»«mrow»«mi»a«/mi»«/mrow»«mo»→«/mo»«/mover»«mo»+«/mo»«mi»y«/mi»«mover accent=¨true¨»«mrow»«mi»b«/mi»«/mrow»«mo»→«/mo»«/mover»«mo»+«/mo»«mi»z«/mi»«mover accent=¨true¨»«mrow»«mi»c«/mi»«/mrow»«mo»→«/mo»«/mover»«/math»'/>, then

Solution for the question - vec(a)=(cos theta) vec(i)-(sin theta) vec(j), vec(b)=(sin theta)vec(i)+(cos theta) vec(j), vec(c)= vec(k), vec(r)=7 vec(i)+ vec(j)+1

Vec(a)=(cos theta) vec(i)-(sin theta) vec(j), vec(b)=(sin theta-Turito

Solution for the question - ut (x rarr0)(x(1+a cos x)-b sin x)//x^(3)=1 thena =,b= 3//2,1//2 '/> 2//3,5//2 '/> 5//23//2 '/> (-5)//2,(-3)//2 '/>

Ut (x rarr0)(x(1+a cos x)-b sin x)//x^(3)=1 thena =,b= -Turito

Solution for the question - (l_(()x rarr0)(x10^(x)-x)//(1-cos x)= 4log 10 '/> 2log 10 '/> 3log 10 '/> log 10 '/>

(l_(()x rarr0)(x10^(x)-x)//(1-cos x)= -Turito

<img src="data:image/png;base64,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" class="Wirisformula" alt="Cu to the power of plus 2 end exponent" width="40" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»Cu«/mi»«mrow»«mo»+«/mo»«mn»2«/mn»«/mrow»«/msup»«/math»'/> ions are more stable than <img src="data:image/png;base64,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" class="Wirisformula" alt="Cu to the power of plus" width="32" height="16" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»Cu«/mi»«mo»+«/mo»«/msup»«/math»'/> ions because  ions _________ as compared to <img src="data:image/png;base64,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" class="Wirisformula" alt="Cu to the power of plus 2 end exponent" width="40" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»Cu«/mi»«mrow»«mo»+«/mo»«mn»2«/mn»«/mrow»«/msup»«/math»'/> ions

Solution for the question - cu^(+2) ions are more stable than cul^(-) ions because ions _________as compared to cu^(+2) ions all of theseundergo extensive hydration

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$(a b c)$
$(a b c)^{(} 1 / 3)$
$(a b c)^{(} 1 / x)$
$\infty$

The correct answer is: $left parenthesis a b c right parenthesis to the power of left parenthesis 1 divided by 3 right parenthesis$

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$Cu to the power of plus 2 end exponent$ ions are more stable than $Cu to the power of plus$ ions because ions _________ as compared to $Cu to the power of plus 2 end exponent$ ions

$Cu to the power of plus 2 end exponent$ ions are more stable than $Cu to the power of plus$ ions because ions _________ as compared to $Cu to the power of plus 2 end exponent$ ions

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where a,b,c are real and non-zero=

The correct answer is:

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ions are more stable than ions because ions _________ as compared to ions

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The correct answer is: $left parenthesis a b c right parenthesis to the power of left parenthesis 1 divided by 3 right parenthesis$

$Cu to the power of plus 2 end exponent$ ions are more stable than $Cu to the power of plus$ ions because ions _________ as compared to $Cu to the power of plus 2 end exponent$ ions

$Cu to the power of plus 2 end exponent$ ions are more stable than $Cu to the power of plus$ ions because ions _________ as compared to $Cu to the power of plus 2 end exponent$ ions