Turito Academy

Test Prep

Global form fields

Required number of ways <img src="data:image/png;base64,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" class="Wirisformula" alt="not stretchy sum subscript r equals 2 end subscript superscript 5 end superscript 5 C subscript 5 minus r end subscript" width="98" height="28" style="vertical-align:-7px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mrow»«msubsup»«mo stretchy=¨false¨»∑«/mo»«mrow»«mi»r«/mi»«mo»=«/mo»«mn»2«/mn»«/mrow»«mrow»«mn»5«/mn»«/mrow»«/msubsup»«mrow»«mn»5«/mn»«msub»«mrow»«mi»C«/mi»«/mrow»«mrow»«mn»5«/mn»«mo»-«/mo»«mi»r«/mi»«/mrow»«/msub»«/mrow»«/mrow»«/math»'/> D(r) <img src="data:image/png;base64,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" class="Wirisformula" alt="equals sum from r equals 2 to 5 of   fraction numerator 5 factorial over denominator r factorial left parenthesis 5 minus r right parenthesis factorial end fraction r factorial times open curly brackets 1 minus fraction numerator 1 over denominator 1 factorial end fraction plus fraction numerator 1 over denominator 2 factorial end fraction minus fraction numerator 1 over denominator 3 factorial end fraction plus horizontal ellipsis plus fraction numerator left parenthesis negative 1 right parenthesis squared over denominator r factorial end fraction close curly brackets" width="387" height="45" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«munderover»«mo»§#8721;«/mo»«mrow»«mi»r«/mi»«mo»=«/mo»«mn»2«/mn»«/mrow»«mn»5«/mn»«/munderover»«mo»§#8202;«/mo»«mfrac»«mrow»«mn»5«/mn»«mo»!«/mo»«/mrow»«mrow»«mi»r«/mi»«mo»!«/mo»«mo»(«/mo»«mn»5«/mn»«mo»§#8722;«/mo»«mi»r«/mi»«mo»)«/mo»«mo»!«/mo»«/mrow»«/mfrac»«mi»r«/mi»«mo»!«/mo»«mo»§#8901;«/mo»«mfenced open=¨{¨ close=¨}¨ separators=¨|¨»«mrow»«mn»1«/mn»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»1«/mn»«mo»!«/mo»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mo»!«/mo»«/mrow»«/mfrac»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»3«/mn»«mo»!«/mo»«/mrow»«/mfrac»«mo»+«/mo»«mo»§#8230;«/mo»«mo»+«/mo»«mfrac»«mrow»«mo»(«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/mrow»«mrow»«mi»r«/mi»«mo»!«/mo»«/mrow»«/mfrac»«/mrow»«/mfenced»«/math»'/> <img src="data:image/png;base64,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" class="Wirisformula" alt="equals sum from r equals 2 to 5 of   fraction numerator 5 factorial over denominator left parenthesis 5 minus r right parenthesis factorial end fraction" width="104" height="45" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«munderover»«mo»§#8721;«/mo»«mrow»«mi»r«/mi»«mo»=«/mo»«mn»2«/mn»«/mrow»«mn»5«/mn»«/munderover»«mo»§#8202;«/mo»«mfrac»«mrow»«mn»5«/mn»«mo»!«/mo»«/mrow»«mrow»«mo»(«/mo»«mn»5«/mn»«mo»§#8722;«/mo»«mi»r«/mi»«mo»)«/mo»«mo»!«/mo»«/mrow»«/mfrac»«/math»'/>=<img src="data:image/png;base64,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" class="Wirisformula" alt="open curly brackets fraction numerator 1 over denominator 2 factorial end fraction – fraction numerator 1 over denominator 3 factorial end fraction plus.... plus fraction numerator left parenthesis negative 1 right parenthesis to the power of r end exponent over denominator r factorial end fraction close curly brackets" width="179" height="37" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfenced open=¨{¨ close=¨}¨ separators=¨|¨»«mrow»«mfrac»«mrow»«mn»1«/mn»«/mrow»«mrow»«mn»2«/mn»«mo»!«/mo»«/mrow»«/mfrac»«mo»–«/mo»«mfrac»«mrow»«mn»1«/mn»«/mrow»«mrow»«mn»3«/mn»«mo»!«/mo»«/mrow»«/mfrac»«mo»+«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo»+«/mo»«mfrac»«mrow»«mo»(«/mo»«mo»-«/mo»«mn»1«/mn»«msup»«mrow»«mo»)«/mo»«/mrow»«mrow»«mi»r«/mi»«/mrow»«/msup»«/mrow»«mrow»«mi»r«/mi»«mo»!«/mo»«/mrow»«/mfrac»«/mrow»«/mfenced»«/math»'/> = 10 + 20 + (60 – 20 + 5) + (60 – 20 + 5 – 1) = 10 + 20 + 45 + 44 = 119.

Ravish write letters to his five friends and address the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes -

Solution for the question - ravish write letters to his five friends and address the correspondingenvelopes. in how many ways can the letters be placed in the e

Ravish write letters to his five friends and address the corres-Turito

PSAT

One-On-One Tutoring

Your gateway to the top global universities

Coding

Discover traits & skills, get a clear pathway to academic success.

Discovery Program

x1 + x2 + x3 + x4 + x5 + x6  17 When 1 <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> xi <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> 6, i = 1, 2, 3, …..6 Let x7 be a variable such that x1 + x2 + x3 + x4 + x5 + x6 + x7 = 17 Clearly x7 <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 0 Required number of ways = Coefficient of x17 in (x1 + x2 + ….. + x6)6 (1 + x + x2 + …..) = Coefficient of x11 in <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIgAAAAsCAYAAAC+N/CqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAeOiuv/QAAA6JJREFUeNrtXD1oFFEQfoSQQo6AhENEbCwkSBA7kSAiWASxTSFiIYEgVhY2IiJiY2URJI2IBAlpxELERizsRYKFiBDEwiJcIyIix8E6c5nisrvP23nv3d7M3gx8cD+5vfe+993Mt5s365yFL04D3gP+AjKCBcUNwOMGz2+N5uiLE4CPgAWTQjHOArYB0w2eI87tE+Cc5/0N4sEiFwcA3wCnJqAsLNBcWyXvdQBXAN8BXwHLJo29eABYVzr2kLKwRnPORw/wBDALmKHH1yZdHG3AT8ARpeMPKQuHAb9o7oOBWWNq4HmLsslExx3Ac8XjDy0LTwF3c69hxjg48ByzyIcGm/phpr0fn5m/wONE7LaQSYaWhUXAl9xrJykjzZKhxcVfGpOpr4PnYaa9P4gO86CYbVYFXR+IKQu7gPnca5iNdui95TGa+rp4/p9p7//SNgMPLEUg3LIwGJuBJrROU18Hzz7T3k+n14UOfAXw0LM4K4Floew7NoSb+joE4jPt7jXgouCB4+nroVzGKzOFVcpCWaCQ3gg39XVl6jLT3vcfbcEDxwV8RI8vAN4lPn4rwINxTb0WgZSZdvcnZ/BSDzyrgGHxllw2uvm5xKRMEQecM4uOqzfq9HoF054pGPhNQJe8xiiiy/jbGFOvQSAF0y5dIJj2XlCZuSxgAWJMvQaBFEy7ZIEcA7yiaw4tOnWdG/MCxJh6DQIpmHapAmnTQAcFcWlEZw+cecSYeg0CKZj2LHDAXKPJCbzQ9dJznWGLcY1jFAsQY+ql8VzJtNt2Oh4Hk8BX1wRiAqk8RxOICcQEYgIxgZhAxiSQrCGoSyCN48MyiGUQE4gJxARiAjGBmEBMICYQcXz0EhxQWhsENzj7QXoTwM8+PlL880laGwQnph1vR1kIX5r4KfDxw+1vGagrVUsJ3E7A2UIYw1emkQ/cALOkRCBV2yA4gZt/OLvaY/jSwE+Bj2cuXfe6pDaIqrFKHFSNGL408FPg46pLt0tLYxvEFnPBY/jSwE+BD3TYuzUKRFobBNbbeeYZyW6D+SnlA7u7zygyqanaIM67kkahChHKl3R+vHzccnttdxoEkrINAkvF7YDPhfIlnR8vH5iGUjQja2qDOAr4Hfj5UL4k8zOUj3su/k45mtog8JYR9yPGE8KXZH6G8oGq2wms69raIPCmL94bplQMDl/S+anMx6Jr/n1SZ8hkpujObwJfbD7wQsl6gwWCZSFlb612vrx8/APMF7nELlcxGAAAAVp0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bXN1cD48bWZlbmNlZCBzZXBhcmF0b3JzPSJ8Ij48bWZyYWM+PG1yb3c+PG1uPjE8L21uPjxtbz4tPC9tbz48bXN1cD48bWk+eDwvbWk+PG1uPjY8L21uPjwvbXN1cD48L21yb3c+PG1yb3c+PG1uPjE8L21uPjxtbz4tPC9tbz48bWk+eDwvbWk+PC9tcm93PjwvbWZyYWM+PC9tZmVuY2VkPjxtbj42PC9tbj48L21zdXA+PG1mZW5jZWQgc2VwYXJhdG9ycz0ifCI+PG1mcmFjPjxtbj4xPC9tbj48bXJvdz48bW4+MTwvbW4+PG1vPi08L21vPjxtaT54PC9taT48L21yb3c+PC9tZnJhYz48L21mZW5jZWQ+PC9tYXRoPlSFsb0AAAAASUVORK5CYII=" class="Wirisformula" alt="open parentheses fraction numerator 1 minus x to the power of 6 end exponent over denominator 1 minus x end fraction close parentheses to the power of 6 end exponent open parentheses fraction numerator 1 over denominator 1 minus x end fraction close parentheses" width="136" height="44" style="vertical-align:-14px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«msup»«mrow»«mfenced separators=¨|¨»«mrow»«mfrac»«mrow»«mn»1«/mn»«mo»-«/mo»«msup»«mrow»«mi»x«/mi»«/mrow»«mrow»«mn»6«/mn»«/mrow»«/msup»«/mrow»«mrow»«mn»1«/mn»«mo»-«/mo»«mi»x«/mi»«/mrow»«/mfrac»«/mrow»«/mfenced»«/mrow»«mrow»«mn»6«/mn»«/mrow»«/msup»«mfenced separators=¨|¨»«mrow»«mfrac»«mrow»«mn»1«/mn»«/mrow»«mrow»«mn»1«/mn»«mo»-«/mo»«mi»x«/mi»«/mrow»«/mfrac»«/mrow»«/mfenced»«/math»'/> = Coefficient of x11 in (1– 6C1 x6 + 6C2 x12……) (1 – x)–7 = Coefficient of x11 in (1 – x)–7 – 6C1 × coefficient of x5 in (1 – x)–7 = 11+7–1C7–1 – 6C1 × 7+5–1C7–1 = 17C6 – 6 × 11C6 = 9604.

In how many ways can we get a sum of at most 17 by throwing six distinct dice -

Solution for the question - in how many ways can we get a sum of at most 17 by throwing six distinctdice - none of these9604

In how many ways can we get a sum of at most 17 by throwing six-Turito

3x + y + z = 24, x <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 0, y <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 0, z <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 0 Let x = k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAKCAYAAAC0VX7mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAADlJREFUeNpjYCANlAAxHwMVQTUQPwRia1I0/ScStwIxE7VcOhdq6FJqurCDGi4kKwwJxTIPwyhABgA3dxtvdcXJJQAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjFEMjs8L21vPjwvbWF0aD5WOBRoAAAAAElFTkSuQmCC" class="Wirisformula" alt="rightwards double arrow" width="20" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8658;«/mo»«/math»'/> y + z = 24 – 3k …(1) <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAKCAYAAAC0VX7mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAADlJREFUeNpjYCANlAAxHwMVQTUQPwRia1I0/ScStwIxE7VcOhdq6FJqurCDGi4kKwwJxTIPwyhABgA3dxtvdcXJJQAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjFEMjs8L21vPjwvbWF0aD5WOBRoAAAAAElFTkSuQmCC" class="Wirisformula" alt="rightwards double arrow" width="20" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8658;«/mo»«/math»'/> 24 – 3k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 0 <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAKCAYAAAC0VX7mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAADlJREFUeNpjYCANlAAxHwMVQTUQPwRia1I0/ScStwIxE7VcOhdq6FJqurCDGi4kKwwJxTIPwyhABgA3dxtvdcXJJQAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjFEMjs8L21vPjwvbWF0aD5WOBRoAAAAAElFTkSuQmCC" class="Wirisformula" alt="rightwards double arrow" width="20" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8658;«/mo»«/math»'/> k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> 8 <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAKCAYAAAC0VX7mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAADlJREFUeNpjYCANlAAxHwMVQTUQPwRia1I0/ScStwIxE7VcOhdq6FJqurCDGi4kKwwJxTIPwyhABgA3dxtvdcXJJQAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjFEMjs8L21vPjwvbWF0aD5WOBRoAAAAAElFTkSuQmCC" class="Wirisformula" alt="rightwards double arrow" width="20" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8658;«/mo»«/math»'/> 0 <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> 8 For fixed value of k the number of solutions of (1) is 24–3k+2–1C2–1 = 25–3kC1 = 25 – 3k Hence number of solutions <img src="data:image/png;base64,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" class="Wirisformula" alt="not stretchy sum subscript k equals 0 end subscript superscript 8 end superscript left parenthesis 25 minus 3 k right parenthesis" width="117" height="28" style="vertical-align:-7px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mrow»«msubsup»«mo stretchy=¨false¨»∑«/mo»«mrow»«mi»k«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«mrow»«mn»8«/mn»«/mrow»«/msubsup»«mrow»«mo»(«/mo»«mn»25«/mn»«mo»-«/mo»«mn»3«/mn»«mi»k«/mi»«mo»)«/mo»«/mrow»«/mrow»«/math»'/>= 25 × 9 – <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator 3 cross times 8 cross times 9 over denominator 2 end fraction" width="72" height="35" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mn»3«/mn»«mo»×«/mo»«mn»8«/mn»«mo»×«/mo»«mn»9«/mn»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/mfrac»«/math»'/>= 225 – 108 = 117.

The number of non negative integral solutions of equation 3x + y + z = 24

Solution for the question - the number of non negative integral solutions of equation 3x + y + z = 24 none of these97

The number of non negative integral solutions of equation 3x + -Turito

Any divisor of 25 · 37 · 53 · 72 is of the type of 2l 3m 5n 7p, where 0<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/>  l <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> 5, 0 <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> m <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> 7, 0  n  3 and 0 <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> p <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> 2 Hence the sum of the divisors = (1 + 2 + …… + 25) (1 + 3 + ……. + 37) (1 + 5 + 52 + 53) (1 + 7 + 72) = <img src="data:image/png;base64,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" class="Wirisformula" alt="open parentheses fraction numerator 2 to the power of 6 end exponent minus 1 over denominator 2 minus 1 end fraction close parentheses open parentheses fraction numerator 3 to the power of 8 end exponent minus 1 over denominator 3 minus 1 end fraction close parentheses open parentheses fraction numerator 5 to the power of 4 end exponent minus 1 over denominator 5 minus 1 end fraction close parentheses open parentheses fraction numerator 7 to the power of 3 end exponent minus 1 over denominator 7 minus 1 end fraction close parentheses" width="268" height="43" style="vertical-align:-14px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfenced separators=¨|¨»«mrow»«mfrac»«mrow»«msup»«mrow»«mn»2«/mn»«/mrow»«mrow»«mn»6«/mn»«/mrow»«/msup»«mo»-«/mo»«mn»1«/mn»«/mrow»«mrow»«mn»2«/mn»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mrow»«/mfenced»«mfenced separators=¨|¨»«mrow»«mfrac»«mrow»«msup»«mrow»«mn»3«/mn»«/mrow»«mrow»«mn»8«/mn»«/mrow»«/msup»«mo»-«/mo»«mn»1«/mn»«/mrow»«mrow»«mn»3«/mn»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mrow»«/mfenced»«mfenced separators=¨|¨»«mrow»«mfrac»«mrow»«msup»«mrow»«mn»5«/mn»«/mrow»«mrow»«mn»4«/mn»«/mrow»«/msup»«mo»-«/mo»«mn»1«/mn»«/mrow»«mrow»«mn»5«/mn»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mrow»«/mfenced»«mfenced separators=¨|¨»«mrow»«mfrac»«mrow»«msup»«mrow»«mn»7«/mn»«/mrow»«mrow»«mn»3«/mn»«/mrow»«/msup»«mo»-«/mo»«mn»1«/mn»«/mrow»«mrow»«mn»7«/mn»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mrow»«/mfenced»«/math»'/> =<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAN8AAAAnCAYAAACPOJ1YAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAbSkFbcgAABO1JREFUeNrtnV1kHFEUx6+KVRUhKiKiQlRFVYSqiFgRKqLyEGFVRFSFqIiqvvQpjyX6VNWXylP0IURFRUWIqoiIJaqqKkJUH/pQJQ9RNSJsz5Ez3Iz5uDtzZ+be2fPjkF2zu/PPmXPuOXfu3hUiHfrBtsEcsBqZbXSDrYH9AzsF2wEri+IxZal/gnCvN/TZF7AR0UDcBPsMdstyHd/AJsEukeHffwrmK/TRbsGCT6YH7KetJz8Htljna5ZzGCEW6Vx16sBAuyE9vkYBKQqgDWkF2wPrzDH40tLmgknz2FBtofSSc0SMi3aKMs4hWCUjsbt0zrp0jIH9ABuk0XydStE80K0NeQc2IJVqokDakKtgr8EeGqhN6YV90uNBctiJVE9P+bzuDGwJrAWsRH/r+AfgKLRAn+vHberLonQMg21QL+dQgnhJzpJpI+fjSL4K9gqsOSUn6dJWCzGZZ2DTntcJi7XVAvq+6ZSDK662UPpIvMw29T3NUm+3S8/JHNJw79Ksqe5+CzYbcaHskOAwHZ/AJigxyMdteo5b8zTrPRSEaaBLm2oQqVzApmkLokIJ3gsm/+f02cIgbZG8AJtXOK4L7KvnuSXqJ1zwIt/XKDhM6GNPj6CqQ9CI7h3BvTgpZ9Kk2mopfK7Jfmuni/tyyDGOYdoi2axj0sTxqdWXKfM0Ud09mpFQ7GE+xtCBvdyB5zksJfqlx6NxSoiMtdkafHH9tu4pTb2UNSd+HdqURoGSwnEDAeUB9oJHYL9TmHAJE1ryjGBROtqpH8VS+Z7PqL5BycWhkrMjRyeqaLM1+Or1G/KI+q2gktqhIO4yTFskZwrH4FBfpYmYLIm6UE4VdHh7nSfCDJJqc28un1DieJriJFEefpMTY5UqK9u0JQ4+7Onei3xWD+h0IuoYJ0cOJTynembj0tbWRE3+ApXTPQXShuDk37CmaylrbYnKzm4KvOspCc+6fBE0OlQNz6BxtY3QxVoUbRUa1U1Ca9m5FVBOYgbF2cwrhgr1NrdbdZTFjuFOZG3nt7AOIiZZTNcWid9UbztNOjQZLHSezj1Mhx+9NOkiCqgN128eFUTbuCGjeBJtkfjd5PyQsHfIQqjKzdptKl2aKJNi74DLyB4UQNsaZVp3MfgoBd5EAbQhKyLf5WI6tCmxJy6uS0valKbRHMsELeXx6ihTInFvIeCKlzEDnKdDW4VGcJywOKZK5U5BtCF466rVsKCLo02pFNsT9pDWAl3Wxtp0a1MCb2QuWiASa+rZAuhgbY2jjWEYhmEYhmGypsbGxpaJMQzDMAzDMAzDMAzDMIydqG65GBdcl6pjBg3XgK7SeeKaV9wfZYjdx9iM6paLccD3/K4h+GYp2NwdulsoQeyw+5ii4bflYhzegM0kDD5MBvvsEqaRSPrtcixn3W9I1xIG8CS7g2kUgrZcVKVEI2eXhuDDbRk62CVMI6Cy5eIKWRD4NRt5m4Wo4At7P4d6Pfm3CPeF3kkhhskd1S0Xw9YA9or6f6Mh7P3wefw9Rdw02N1aAxMDfiN+jl3GFAFdWy5Wfd4jSdmJtxY6fZ7HfVR+sdsY29G55aLuFfMb4uKvSsmcsusYm8liy8UkIx+WlvcDEkaV3cfYTBZbLiYJPpw5xYUAM1KCwF9nwtnUu+w+xmay+FJl0vdqo7L4rzjfdhCDscyuS5f/Cui291zFq9kAAAHPdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtcm93Pjxtbz4oPC9tbz48bXN1cD48bW4+MjwvbW4+PG1uPjY8L21uPjwvbXN1cD48bW8+JiN4MjAxMzs8L21vPjxtbj4xPC9tbj48bW8+KTwvbW8+PG1vPig8L21vPjxtc3VwPjxtbj4zPC9tbj48bW4+ODwvbW4+PC9tc3VwPjxtbz4tPC9tbz48bW4+MTwvbW4+PG1vPik8L21vPjxtbz4oPC9tbz48bXN1cD48bW4+NTwvbW4+PG1uPjQ8L21uPjwvbXN1cD48bW8+LTwvbW8+PG1uPjE8L21uPjxtbz4pPC9tbz48bW8+KDwvbW8+PG1zdXA+PG1uPjc8L21uPjxtbj4zPC9tbj48L21zdXA+PG1vPi08L21vPjxtbj4xPC9tbj48bW8+KTwvbW8+PC9tcm93Pjxtcm93Pjxtbj4yPC9tbj48bW8+JiN4Qjc7PC9tbz48bW4+NDwvbW4+PG1vPiYjeEI3OzwvbW8+PG1uPjY8L21uPjwvbXJvdz48L21mcmFjPjwvbWF0aD6gj03lAAAAAElFTkSuQmCC" class="Wirisformula" alt="fraction numerator left parenthesis 2 to the power of 6 end exponent – 1 right parenthesis left parenthesis 3 to the power of 8 end exponent minus 1 right parenthesis left parenthesis 5 to the power of 4 end exponent minus 1 right parenthesis left parenthesis 7 to the power of 3 end exponent minus 1 right parenthesis over denominator 2 times 4 times 6 end fraction" width="223" height="39" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mo»(«/mo»«msup»«mrow»«mn»2«/mn»«/mrow»«mrow»«mn»6«/mn»«/mrow»«/msup»«mo»–«/mo»«mn»1«/mn»«mo»)«/mo»«mo»(«/mo»«msup»«mrow»«mn»3«/mn»«/mrow»«mrow»«mn»8«/mn»«/mrow»«/msup»«mo»-«/mo»«mn»1«/mn»«mo»)«/mo»«mo»(«/mo»«msup»«mrow»«mn»5«/mn»«/mrow»«mrow»«mn»4«/mn»«/mrow»«/msup»«mo»-«/mo»«mn»1«/mn»«mo»)«/mo»«mo»(«/mo»«msup»«mrow»«mn»7«/mn»«/mrow»«mrow»«mn»3«/mn»«/mrow»«/msup»«mo»-«/mo»«mn»1«/mn»«mo»)«/mo»«/mrow»«mrow»«mn»2«/mn»«mo»·«/mo»«mn»4«/mn»«mo»·«/mo»«mn»6«/mn»«/mrow»«/mfrac»«/math»'/>.

Sum of divisors of 25 ·37 ·53 · 72 is –

Solution for the question - sum of divisors of 25 37 53  72 is none of these26  38  54  73  1

Sum of divisors of 25 37 53  72 is -Turito

The length of the perpendicular from the pole to the straight line <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator 6 square root of 2 over denominator r end fraction equals C o s space theta plus S i n space theta" width="163" height="39" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»6«/mn»«msqrt»«mn»2«/mn»«/msqrt»«/mrow»«mi»r«/mi»«/mfrac»«mo»=«/mo»«mi»C«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»+«/mo»«mi»S«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«/math»'/> is

Solution for the question - the length of the perpendicular from the pole to the straight line(6sqrt2)/(r)=cos theta+sin theta is 3sqrt2 units  3 units

The length of the perpendicular from the pole to the straight l-Turito

The condition for the lines <img src="data:image/png;base64,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" class="Wirisformula" alt="c subscript 1 over r equals a subscript 1 c o s space theta plus b subscript 1 s i n space theta" width="177" height="36" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msub»«mi»c«/mi»«mn»1«/mn»«/msub»«mi»r«/mi»«/mfrac»«mo»=«/mo»«msub»«mi»a«/mi»«mn»1«/mn»«/msub»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»+«/mo»«msub»«mi»b«/mi»«mn»1«/mn»«/msub»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«/math»'/> and <img src="data:image/png;base64,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" class="Wirisformula" alt="c subscript 2 over r equals a subscript 2 c o s space theta plus b subscript 2 s i n space theta" width="177" height="36" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msub»«mi»c«/mi»«mn»2«/mn»«/msub»«mi»r«/mi»«/mfrac»«mo»=«/mo»«msub»«mi»a«/mi»«mn»2«/mn»«/msub»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»+«/mo»«msub»«mi»b«/mi»«mn»2«/mn»«/msub»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«/math»'/> to be perpendicular is

Solution for the question - the condition for the lines (c_(1))/(r)=a_(1)cos theta+b_(1)sin theta and(c_(2))/(r)=a_(2)cos theta+b_(2)sin theta to be perpendicul

The condition for the lines (c_(1))/(r)=a_(1)cos theta+b_(1)sin-Turito

If f : R →R; f(x) = sin x + x, then the value of <img src="data:image/png;base64,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" class="Wirisformula" alt="not stretchy integral subscript 0 end subscript superscript pi end superscript blank" width="28" height="41" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mrow»«msubsup»«mo stretchy=¨false¨»∫«/mo»«mrow»«mn»0«/mn»«/mrow»«mrow»«mi»π«/mi»«/mrow»«/msubsup»«mrow/»«/mrow»«/math»'/> (f-1 (x)) dx, is equal to

Solution for the question - if f : r r; f(x) = sin x + x, then the value of (f-1 (x)) dx, is equalto none of these'/>  p2

If f : r r; f(x) = sin x + x, then the value of (f-1 (x)) dx-Turito

The polar equation of the straight line with intercepts 'a' and 'b' on the rays <img src="data:image/png;base64,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" class="Wirisformula" alt="theta equals 0" width="38" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mi»θ«/mi»«mo»=«/mo»«mn»0«/mn»«/math»'/> and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADEAAAAfCAYAAABOOiVaAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAATRJrTQAAAAW5JREFUeNpjYBhcIAuIvwHxfzy4nGGQg4VALAvEa4A4AEkcxPdhGGLgLtQzMHAfiMWHkgdYoEkKmf9lqMWCJRDvR+PvHWqeiATi+Uj8cGheGVJgGhAnIvGbgXgKqYZYA/E6IH4KxAeB2HwAMrUBEr8ViHcDsQoQcxFjgDnUA0xQvijUUHoBULF6Dk3MFIg/AfFiYg25AMTSaGI7gFhvqKRFF7QMBQPLgdhrqHhiNhAHYREHJS83HHr+E4HpCq7hcciQqC2ZoJmHWHFqgP9UwChAHEetaAptfDEMheQEK1rRQT1axTOoAQ+0eEUGgtAym20oVfdXgDgPytYA4uNDsQ2vB42NX0B8CYj9GEYBWcAaWpB8ggYmKFCjh5onDkKb4jxQvhYQH4WKDWkgD03eQx78GOoesIQmqSELOID4JDTDD0kAqmw34GlBD3qgBPWAylD1gAa0b8M1VD0Aak2vYoAMmA1ZsAUaE0MakNUvAQDOHXEYscbGvAAAAIJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWk+JiN4M0I4OzwvbWk+PG1vPj08L21vPjxtZnJhYz48bWk+JiN4M0MwOzwvbWk+PG1uPjI8L21uPjwvbWZyYWM+PC9tYXRoPqlyXD0AAAAASUVORK5CYII=" class="Wirisformula" alt="theta equals fraction numerator pi over denominator 2 end fraction" width="49" height="31" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mi»θ«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»π«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/mfrac»«/math»'/> respectively is

Solution for the question - the polar equation of the straight line with intercepts 'a' and 'b' on therays theta=0 and theta=(pi)/(2) respectively is r(a cos theta-b sin theta)=ab

The polar equation of the straight line with intercepts 'a' and-Turito

The polar equation of the straight line parallel to the initial line and at a distance of 4 units above the initial line is

Solution for the question - the polar equation of the straight line parallel to the initial line and ata distance of 4 units above the initial line is r cos theta=-4

The polar equation of the straight line parallel to the initial-Turito

The polar equation of <img src="data:image/png;base64,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" class="Wirisformula" alt="x to the power of 3 end exponent plus y to the power of 3 end exponent equals 3" width="80" height="19" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«msup»«mrow»«mi»x«/mi»«/mrow»«mrow»«mn»3«/mn»«/mrow»«/msup»«mo»+«/mo»«msup»«mrow»«mi»y«/mi»«/mrow»«mrow»«mn»3«/mn»«/mrow»«/msup»«mo»=«/mo»«mn»3«/mn»«/math»'/> axy is

Solution for the question - the polar equation of x^(3)+y^(3)=3 axy is r(cos^(3)theta-sin^(3)theta)=a cos 2theta  r(cos^(3)theta-sin^(3)theta)=3a cos theta sin theta

The polar equation of x^(3)+y^(3)=3 axy is -Turito

Let y = p + 1 and z = q + 2. Then x <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 0, p <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 0, q <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 0 and x + y + z = 15 <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAKCAYAAAC0VX7mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAADlJREFUeNpjYCANlAAxHwMVQTUQPwRia1I0/ScStwIxE7VcOhdq6FJqurCDGi4kKwwJxTIPwyhABgA3dxtvdcXJJQAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjFEMjs8L21vPjwvbWF0aD5WOBRoAAAAAElFTkSuQmCC" class="Wirisformula" alt="rightwards double arrow" width="20" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8658;«/mo»«/math»'/> x + p + q = 12 <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAKCAYAAABSfLWiAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAACBJREFUeNpjYCAM/kMxRYAqhowC/AEZAJVbThdDBgYAAO75C+VyngYIAAAAUHRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbz4mI3gyMjM0OzwvbW8+PC9tYXRoPhpvPpoAAAAASUVORK5CYII=" class="Wirisformula" alt="therefore" width="17" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8756;«/mo»«/math»'/> The reqd. number of values of (x, y, z) and hence of (x, p, q) = No. of non-negative integral solutions of x + p + q= 12 = Coeff. of x12 in (x0 + x1 + x2 + ……)3 = Coeff. of x12 in (1 – x)–3 = Coeff. of x12 in [2C0 + 3C1 x + 4C2 x2 + ….] = 14C12 = <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator 14 factorial over denominator 2 factorial 12 factorial end fraction" width="48" height="35" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mn»14«/mn»«mo»!«/mo»«/mrow»«mrow»«mn»2«/mn»«mo»!«/mo»«mn»12«/mn»«mo»!«/mo»«/mrow»«/mfrac»«/math»'/> = <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator 14 cross times 13 over denominator 2 end fraction" width="65" height="35" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mn»14«/mn»«mo»×«/mo»«mn»13«/mn»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/mfrac»«/math»'/> = 91.

If x, y, z are integers and x <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 0, y <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 1, z <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 2, x + y + z = 15, then the number of values of the ordered triplet (x, y, z) is -

Solution for the question - if x, y, z are integers and x >= 0, y >= 1, z >= 2, x + y + z =15, then the number of values of the ordered triplet (x, y, z) is - none of these

If x, y, z are integers and x >= 0, y >= 1, z >= 2, x-Turito

If <img src="data:image/png;base64,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" class="Wirisformula" alt="alpha not equal to beta comma alpha 2 equals 5 alpha minus 3 comma beta 2 times equals 5 beta minus 3 comma" width="241" height="15" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#945;«/mi»«mo»§#8800;«/mo»«mi»§#946;«/mi»«mo»,«/mo»«mi»§#945;«/mi»«mn»2«/mn»«mo»=«/mo»«mn»5«/mn»«mi»§#945;«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«mo»,«/mo»«mi»§#946;«/mi»«mn»2«/mn»«mo»§#8901;«/mo»«mo»=«/mo»«mn»5«/mn»«mi»§#946;«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«mo»,«/mo»«/math»'/>then the equation whose roots are <img src="data:image/png;base64,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" class="Wirisformula" alt="alpha divided by beta straight &amp; beta divided by alpha" width="79" height="17" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#945;«/mi»«mo»/«/mo»«mi»§#946;«/mi»«mi mathvariant=¨normal¨»§#38;«/mi»«mi»§#946;«/mi»«mo»/«/mo»«mi»§#945;«/mi»«/math»'/>

Solution for the question - ifthen the equation whose roots are '/>   '/>   '/>  none of these

Ifthen the equation whose roots are -Turito

A quadratic equation, or sometimes just quadratics, is a polynomial equation with a maximum degree of two. It takes the following form: ax² + bx + c = 0 where a, b, and c are constant terms and x is the unknown variable. Now we have given  p, q <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAMCAYAAABr5z2BAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAALV/ZLFgAAAGNJREFUeNpjYMANNIB4AhCfA+JvQPwLiP8zEAmCgPgKEGcAsTEQszGQAGShtnIxkAkmAXEAAwXgKRAzUWIALLBwYdq7YAKlYQCKhTNAzEOJIeFI6cAAiFnIMQQ9Jf4hJSUSDQC2pRnou82QIwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjIwODs8L21vPjwvbWF0aD4h58XOAAAAAElFTkSuQmCC" class="Wirisformula" alt="element of" width="16" height="12" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8712;«/mo»«/math»'/> {1, 2, 3, 4}, the equation given is : px2 + qx + 1 = 0 <div>Now we know that for real roots, the discriminant is always greater than or equal to 0, so we have:</div>
<div>D=b2-4ac, applying this, we get: q2−4p ≥ 0 ⇒ q2 ≥ 4p Now the set includes 4 terms, putting each, we get:</div>
<div>For p=1,q2 ≥ 4</div>
<div>q = 2,3,4</div>
<div> </div>
<div>For p=2,q2≥8</div>
<div>q = 3,4</div>
<div> </div>
<div>For p=3,q2≥12</div>
<div>q = 4</div>
<div> </div>
<div>For p=4,q2≥16</div>
<div>q = 4</div>
<div>So here we can see that total 7 seven solutions are possible so 7 equations can be formed. </div>

Let p, q <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAMCAYAAABr5z2BAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAALV/ZLFgAAAGNJREFUeNpjYMANNIB4AhCfA+JvQPwLiP8zEAmCgPgKEGcAsTEQszGQAGShtnIxkAkmAXEAAwXgKRAzUWIALLBwYdq7YAKlYQCKhTNAzEOJIeFI6cAAiFnIMQQ9Jf4hJSUSDQC2pRnou82QIwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjIwODs8L21vPjwvbWF0aD4h58XOAAAAAElFTkSuQmCC" class="Wirisformula" alt="element of" width="16" height="12" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8712;«/mo»«/math»'/> {1, 2, 3, 4}. Then number of equation of the form px2 + qx + 1 = 0, having real roots, is

Solution for the question - let p, q in {1, 2, 3, 4}. then number of equation of the form px2 + qx +1 = 0, having real roots, is 87915

Let p, q in {1, 2, 3, 4}. then number of equation of the form-Turito

A quadratic equation, or sometimes just quadratics, is a polynomial equation with a maximum degree of two. It takes the following form: ax² + bx + c = 0 where a, b, and c are constant terms and x is the unknown variable. Now we have given a &gt; 0, b &lt; 0 and c &lt; 0, the equation is ax2 + bx + c = 0. Let the roots be α and β, where β&gt;α, then: α + β = -b/a &gt;0 as a &gt; 0, b &lt; 0. αβ = c/a as a &gt; 0, c &lt; 0. Now that the roots are of opposite signs, so β &gt; 0 and α &lt; 0. So: ∣α∣ &lt; β as α + β &gt; 0. So therefore: 0 &lt; ∣α∣ &lt; β

ax2 + bx + c = 0 has real and distinct roots null. Further a &gt; 0, b &lt; 0 and c &lt; 0, then –

Solution for the question - ax2 + bx + c = 0 has real and distinct roots null. further a > 0, b < 0 andc < 0, then |alpha|+|beta|=|(b)/(a)| '/> alpha+beta

Ax2 + bx + c = 0 has real and distinct roots null. further a > -Turito

A quadratic equation, or sometimes just quadratics, is a polynomial equation with a maximum degree of two. It takes the following form: ax² + bx + c = 0 where a, b, and c are constant terms and x is the unknown variable. Now we have given the equation as <img src="data:image/png;base64,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" class="Wirisformula" alt="r squared c o s space 2 theta equals a squared" width="103" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mi»§#952;«/mi»«mo»=«/mo»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/math»'/>. Now we know that: <img src="data:image/png;base64,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" class="Wirisformula" alt="cos space 2 theta equals cos squared theta minus sin squared theta" width="172" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mi»§#952;«/mi»«mo»=«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»§#952;«/mi»«mo»§#8722;«/mo»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»§#952;«/mi»«/math»'/>. So applying this, we get: <img src="data:image/png;base64,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" class="Wirisformula" alt="r squared left parenthesis cos squared theta minus sin squared theta right parenthesis equals a squared" width="165" height="18" style="vertical-align:-2px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«mo»(«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»§#952;«/mi»«mo»§#8722;«/mo»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»§#952;«/mi»«mo»)«/mo»«mo»=«/mo»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/math»'/> Now lets substitute x=rcosθ and y=rsinθ, we get: <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKsAAAAuCAYAAABXlelyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAVrfl2dQAABH9JREFUeNrtnV9k1WEYx3/OxUwSk0kmMUkyRzeZ2UUiycwxI9nFZHaTLrK7SSaZSLKLdJPJJBkzk8xEZiaZmMzMJLrooosZXUzmmHF6HnsO7369Z7/3vO/7+9f5fnjU7/dr5zzv9zzP8/6Z8xQEZlTE9sg+k50L0iErfuTB37xp5Z0C2V2yr/AjN/7mTSvvSVbOyMDKOUv2MrRKNsmukC1nYDBZ8SMP/uZNKy9JdlpKc3vKAzDxo5tsjuyXfFCdGff3qGmxkbTykmTnyeZl8Gli4keniF+Q61ayHxn2N65gzZtWXhKcf3CB7EQGBmDixxpZW+jeB7JiRv2FVnUm2UOyUbI7ZKuyVhiVZzzoCxFv0C0lm3/uJ9lg6Hkv2abs8vjPkuY1OIuWyHaVXaGKiR/XyKY096fJemIQ1kW3qPHqKmv1+qZUwLLofiYHWnH1fkS2JXHAiXLWJslmZM3yTFN6KxpTuUS2TdYnDnFmvA1NNSxsh1x3yHVX6HVWawSxqR/MJFm/5j5Pdddj+ABcdIsab61g5UB9qnzQQ5q1XVxaVQysFlwtn5O1SJzMSMzUm2TB9xqOmzBLdi/iea+m0r4L3duVgbiweYSIp2IIVhfdTMarC9armoq1l3GtBiQ4VV5rKnhk8BdEOFt2Iso2P28K3WuS+yp9sqgecJhmduq471otXHUzGW/FcMNVSUgrW5Y0M+myZhkQSae8mC0Vy+e6anBcpqdNWU7UA1eDRc39y1LdfeOqm8l44wpWW61sE3tXOXGoJsUf2xI95SD4tqfKqnI/qP+3F9VjGN0maCiGYHXVzWS8cQVr0lptazbkVr+d4kXvsIMjkwZr1pJmCpyLmKb2LKrUWuhei4jSFMMH4KqbyXjjCtakteJNaLNyPWY72805HlW0y3FVv4jO53avQlML7/4vynVRrruPeM1hyyl2Q0kc3lWuaDZ3vnDVzWS8cQVr0lqNk72U+Dgrp0ULNi/0OxT1NvDx1ReyfcnYkmb3/02qx4bmyEJdD+3LQGx+61OU9+f3WTc4GnLBVTeT8cYZrElqxTyQE4Ex0W2rRhwAkDlaIQEAAAAAAAAAAAAAAAAAAABoLNBKCOQOtBJC4uQOtBLKXgIDDWgllO0EBkKeWgk1cuI0PHlqJdTIiYOKGuSrlVCjJs5/DX8H6Inm/nhw+EtxRh0zHOHvb93S3B8he5ygH9VNk5eWN8AvvItVO3HwV3JfhP5NPe1ibLkdHLTHUTkWHHQ/OWnhRyZa3gC/3CCbkL9z067FlPzoCf5tNcPfZx9L2A9vLW9APHyUHe2aUsVssa1o/L4bynW1Z2hzwlp4a3kD4mFE1mbFlP1Qe0hNiF9JJ423ljfAP9x0YlaCYyBlXz4FB0dA7VJVCyn44K3lDfALB8V72chwa5lVD8sAF3htyE0XuHPHYEo+eGt5A/zRKjtaNTi5g8qblJcjfIS1nqIP3lreAD9wEy7u2dSmeTYtJwRp0CdryVLK+qDlDYiEg3Qlo7MQAIfg6bYLMoCsw/9BxjxkACp/AWnJvEm7IWeaAAABt3RFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtc3VwPjxtaT5yPC9taT48bW4+MjwvbW4+PC9tc3VwPjxtc3VwPjxtaT5jb3M8L21pPjxtbj4yPC9tbj48L21zdXA+PG1pPiYjeDNCODs8L21pPjxtbz4mI3gyMjEyOzwvbW8+PG1zdXA+PG1pPnI8L21pPjxtbj4yPC9tbj48L21zdXA+PG1zdXA+PG1pPnNpbjwvbWk+PG1uPjI8L21uPjwvbXN1cD48bWk+JiN4M0I4OzwvbWk+PG1vPj08L21vPjxtc3VwPjxtaT5hPC9taT48bW4+MjwvbW4+PC9tc3VwPjxtc3BhY2UgbGluZWJyZWFrPSJuZXdsaW5lIi8+PG1zdXA+PG1pPng8L21pPjxtbj4yPC9tbj48L21zdXA+PG1vPi08L21vPjxtc3VwPjxtaT55PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz49PC9tbz48bXN1cD48bWk+YTwvbWk+PG1uPjI8L21uPjwvbXN1cD48L21hdGg+FP2AiwAAAABJRU5ErkJggg==" class="Wirisformula" alt="r squared cos squared theta minus r squared sin squared theta equals a squared
x squared minus y squared equals a squared" width="171" height="46" style="vertical-align:-25px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»§#952;«/mi»«mo»§#8722;«/mo»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»§#952;«/mi»«mo»=«/mo»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«mspace linebreak=¨newline¨/»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/math»'/>

The cartesian equation of <img src="data:image/png;base64,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" class="Wirisformula" alt="r squared c o s space 2 theta equals a squared" width="103" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mi»§#952;«/mi»«mo»=«/mo»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/math»'/> is

Solution for the question - the cartesian equation of r^(2)cos 2theta=a^(2) is x^(2)-y^(2)=a^(2) '/>   x^(2)+y^(2)=a^(2) '/>   x^(2)+y^(2)+a^(2)=0'/>

The cartesian equation of r^(2)cos 2theta=a^(2) is -Turito

A quadratic equation, or sometimes just quadratics, is a polynomial equation with a maximum degree of two. It takes the following form: ax² + bx + c = 0 where a, b, and c are constant terms and x is the unknown variable. Now we have given the equation as <img src="data:image/png;base64,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" class="Wirisformula" alt="r c o s invisible function application left parenthesis theta minus alpha right parenthesis equals p" width="112" height="17" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mi»r«/mi»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»⁡«/mo»«mo»(«/mo»«mi»θ«/mi»«mo»-«/mo»«mi»α«/mi»«mo»)«/mo»«mo»=«/mo»«mi»p«/mi»«/math»'/>. Now we know that: <img src="data:image/png;base64,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" class="Wirisformula" alt="cos space left parenthesis space theta space minus space alpha space right parenthesis equals cos space theta space cos space alpha space plus space sin space theta space sin space alpha" width="306" height="16" style="vertical-align:-2px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mo»§#160;«/mo»«mi»§#945;«/mi»«mo»§#160;«/mo»«mo»)«/mo»«mo»=«/mo»«mi»cos«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»§#160;«/mo»«mi»cos«/mi»«mo»§#160;«/mo»«mi»§#945;«/mi»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mi»§#945;«/mi»«/math»'/>. So applying this, we get: <img src="data:image/png;base64,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" class="Wirisformula" alt="r left parenthesis cos space theta space cos space alpha space plus space sin space theta space sin space alpha right parenthesis equals p" width="239" height="17" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»r«/mi»«mo»(«/mo»«mi»cos«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»§#160;«/mo»«mi»cos«/mi»«mo»§#160;«/mo»«mi»§#945;«/mi»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mi»§#945;«/mi»«mo»)«/mo»«mo»=«/mo»«mi»p«/mi»«/math»'/> Now lets substitute x=rcosθ and y=rsinθ, we get: <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPIAAAApCAYAAAAPkXRIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAATRJrTQAAABUVJREFUeNrtnX+EHVcUx8f746laZa2KWBWiqipWqVprVZWoWGutsCp/REWoqqroPxVVVVWqav9YUSqioqpUraqKUlWxqkpE1FoVVkVV/wj5I39ErDA9x35X5k3uz9m5d+al3w/HZmbezLv3nHvOPfe+906KghCyXz4WeYNqIGT8+VVkhmrwMy+yLvKPyGWRWaokK+WY2LTsSD/PiWxwmLiZhcEHOH5cZJtqGWtHTmXTskMdbcChiYVrItO1cz/+D1OZkjbtNW9hvdwp74u8I/K6yBWRuziupkGXcf6GyMna/YsiWyI7+LtkeI/DIr+I3MGgDBmYR0W+MJz/WmQhIHXros3KlMhFvLf+nahc0+OVzI7ssm9IH0vL8Qpm0rvQ9RMJbZq7nbF2nBP5OZFvvivygci/aIvq4aDpxd9grfIpFFblWZGbIstIhZ4S+aqWJqmSjuD4CI7nas+5YnEWF+dFjhvOa1r2suO+Lts8xObHAv6tSj+Ha4dErnYwI7vsG9JHk4PoIP4EfVJOwUlS2TR3O2PtqK+57WiXT1y220bgncR4XkXge4DrFuUq3yJtKBzXFw2z3Xe1c3fQkBi2HB0/4GlTV21+T+RsLajsrf80mLzQgSO77BvSR5ODvFQ7N0B2k8qmudvZxI47Rfuo7Z6pnXvMFDQGUJKN27jRdX0YEJ2WEeFOBHZgYIlwA0fk67rNyt+1FKzADKCzzWeRjts0isfYN6SPZWBQKRPaNGc7m9qxbUe22e4Rk75mkTY0nQnKiE5NILXaQrrr4oBlzfE8ZtSih222fQxxWuQ3w8DIMSP77BvSx7YcZD82zdnOJnZMkVrbbGdcj5+wbD7scbOl2a3K2YC14t5HFKbF/ynPvV21ecWydrmI2aTowJF99g3pY1sOsh+b5mxnEzum2Oyy2e5tpP4jrCHSuDYnfOvNJUMatO5JGXxpiEa9a7VzkzDgMGBDpYs2Hzc84zwGqUb4JztwZJ99Q/rYloPsx6Y529nEjm9iU61N1gwBTvX0R2HYtV4v/Nv+N9A5VaB+/nehlhZtVxbkMziedzzzdEC6p2xWHPJppDWLAfd11WbNAv5C2jeJaPo5rl0yRdEMjuyzb0gf21x7NrVpznY2sWOKL4SsY7PrKI6ncc6YvdzC4tmF7tj9LnIPEbU+m6kh/kSU3MTsZlsn3IMyDgZ0ZAbvt4MoFPNRUFdtPoRBdrUY/ZzxVfQhNz77hvSxTUduatPc7YyxY6qvaN7CuLxeGcfLBSEkCSl+NDGEIxNCMqDr4tcSPHfBs2dDCBkDNIVfoxoIIYQQQgghhBBCCCGEEEIIIYQQQgjpKSVVQAgdmRA6DiHjhP5Q21Rc+8PCX12C9aN30cIFrxjOnxH5yHJP32s15yCqbjPxoz+grpYj1SoE5zz3sH70ffTXKvVSL48Wuz8Kn7Lc0/daza42tVHhU4mq20z8HIMCFS0tElJIjPWj77OAQVmfbVzlfcahVnNqgus2k3B+Enmx2C0pMhXwetaPHk1ZNyvHe/8xmavMzjjUak5JVN1mEs4ZRPaZCCdocv1hrB9d1AblKvTpo8+1mlMHv6i6zSSMeaTCqxEzI+tHj7KBDazDmI0HEff2sVZzaqLqNhM/OvC+x+bMBDZhQlJr1o9+0CmWsN482SDN7Fut5tRE1W0mbnQtd6nmuLoh9WVgAGD96NGlyYWiWdnbPtZqTk1U3WZiZwjFTRuuaRp2LOAZrB89mm2UxcNRqzkHrNtMeskSNo5I2CTCus2kl+isOkc1BMG6zaSX6LfWfqAagmHd5pb4D86bnrIfqVawAAACYnRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtaT5yPC9taT48bWk+Y29zPC9taT48bW8+JiN4QTA7PC9tbz48bWk+JiN4M0I4OzwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPmNvczwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPiYjeDNCMTs8L21pPjxtbz4mI3hBMDs8L21vPjxtbz4rPC9tbz48bW8+JiN4QTA7PC9tbz48bWk+cjwvbWk+PG1pPnNpbjwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPiYjeDNCODs8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5zaW48L21pPjxtbz4mI3hBMDs8L21vPjxtaT4mI3gzQjE7PC9taT48bW8+KTwvbW8+PG1vPj08L21vPjxtaT5wPC9taT48bXNwYWNlIGxpbmVicmVhaz0ibmV3bGluZSIvPjxtaT54PC9taT48bW8+JiN4QTA7PC9tbz48bWk+Y29zPC9taT48bW8+JiN4QTA7PC9tbz48bWk+JiN4M0IxOzwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPis8L21vPjxtbz4mI3hBMDs8L21vPjxtaT55PC9taT48bW8+JiN4QTA7PC9tbz48bWk+c2luPC9taT48bW8+JiN4QTA7PC9tbz48bWk+JiN4M0IxOzwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPj08L21vPjxtbz4mI3hBMDs8L21vPjxtaT5wPC9taT48L21hdGg+flESHwAAAABJRU5ErkJggg==" class="Wirisformula" alt="r cos space theta space cos space alpha space plus space r sin space theta space sin space alpha right parenthesis equals p
x space cos space alpha space plus space y space sin space alpha space equals space p" width="242" height="41" style="vertical-align:-22px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»r«/mi»«mi»cos«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»§#160;«/mo»«mi»cos«/mi»«mo»§#160;«/mo»«mi»§#945;«/mi»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mi»r«/mi»«mi»sin«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mi»§#945;«/mi»«mo»)«/mo»«mo»=«/mo»«mi»p«/mi»«mspace linebreak=¨newline¨/»«mi»x«/mi»«mo»§#160;«/mo»«mi»cos«/mi»«mo»§#160;«/mo»«mi»§#945;«/mi»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mi»y«/mi»«mo»§#160;«/mo»«mi»sin«/mi»«mo»§#160;«/mo»«mi»§#945;«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»p«/mi»«/math»'/>

The castesian equation of <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHAAAAARCAYAAAAFSCi0AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAOJ5y/mQAAAxBJREFUeNrtWU1kXFEUfkaMLMYQUTGiQlRFFhEiRkV2oypqRJhFREWUioqKCFER1UWJqC5GlYqKLKJEzTJCVVTEbCqiqqpURVUXJSIqajzSc+R79Xp7zp2pzI03+g4f8+559+/8nzeeF026TVj0ok+LOGtMIeohlBvovDs4c0whgfQqvAFCifCV8JqQjcB5+wjbsdpOqRcKlCgL5SXwfIHwKSLn3oYi/3taIkwpvD1CuzG2GZHwdScKOfseYY4wSXhD+InngFIQMIevCryhJcRnj3hG+IG5q4RmYZ95wj7BJ5wYe7BCBoU5OcKKMP6cMORIHq24Q3CXVIjHz4XQ8xXCK0c6YXndJ3zDWbYIGenldSjnIaHT4PHhdwkPCGlCkjBLuA4+K/ItYQIhLg3hLgnKW8Z8iY4UHs8ZEcbZiK46EFwSoXwIv1loj8HrgCzM94+UtU5qgEbrSBNzkDHL9hFk+xd9VIQUlMvFKqFvxhjrxJpmgZKzrOMr4+8tl29zoMAFwl0jNwf5dk2JEhUH52D5dRtjaclYWLPHyiLM+26ES4lvhkt+PjTGCrDe/D8oMKFYd8Ji9We1/C9GyPRQ9bK3P1Hm1FuBmk6apXtnESYk6q9SJmtltLZmBu9vCUKSQmibkl/4XC/OsS24if40pYTceodQTX5ivh1VigQPeaBkuXAesdqkaeRMT7Eizpk3jPGX6PWk9kFK8BMOFFhQcsyqpcVxUcRoOplBiP+DirAwzSI/WDa6RtgQ3J/zVpdlHlesYzW0ESm0EGFqQShOOlDgiGAwyzAW9sxLwpwpoWA7KxUFA03C8DNSNWcrx/fgTU1IotMhT2lC9RpUpO1Yb96yXj8U3FpjI/8OvZYHoyiH9qs38f0+Ey7DUNgLnoK3IVm/o0a+hCImZ8hVjDoHSs8W0EUkcR8WYHrrIBRSgbdKmxwi3vuI7Zp3loXmvAdGVMH+ecc9cQfOuGv0e+PY/zw+pbFOhqFEH/cfboSvMfHH7NNQedDIn9Qmvcb4O4nz3i0H61YrHGOKOI1X+XDym34BAu3Otf0ZVHoAAADKdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1pPnI8L21pPjxtaT5jPC9taT48bWk+bzwvbWk+PG1pPnM8L21pPjxtbz4mI3gyMDYxOzwvbW8+PG1vPig8L21vPjxtaT4mI3gzQjg7PC9taT48bW8+LTwvbW8+PG1pPiYjeDNCMTs8L21pPjxtbz4pPC9tbz48bW8+PTwvbW8+PG1pPnA8L21pPjwvbWF0aD7zXc8BAAAAAElFTkSuQmCC" class="Wirisformula" alt="r c o s invisible function application left parenthesis theta minus alpha right parenthesis equals p" width="112" height="17" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mi»r«/mi»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»⁡«/mo»«mo»(«/mo»«mi»θ«/mi»«mo»-«/mo»«mi»α«/mi»«mo»)«/mo»«mo»=«/mo»«mi»p«/mi»«/math»'/> is

Solution for the question - the castesian equation of r cos(theta-alpha)=rho is x cos alpha+y cos alpha=rho '/>   x cos alpha-y sin alpha=rho '/>  x cos alpha+y sin alpha=rho '/>