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Statement-I : If <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator 3 x plus 4 over denominator left parenthesis x plus 1 right parenthesis squared left parenthesis x minus 1 right parenthesis end fraction equals fraction numerator A over denominator x minus 1 end fraction plus fraction numerator B over denominator x plus 1 end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis squared end fraction" width="319" height="39" style="vertical-align:-16px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/mrow»«mrow»«mo»(«/mo»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«mo»(«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mo»)«/mo»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mi»A«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mi»B«/mi»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mi»C«/mi»«mrow»«mo»(«/mo»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»'/> then A=<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJ9JREFUeNpjYMAO/hPAFINQIJ5NqSHiQHwYiDkoNWgTEBtQakgGENdSaog8EJ8EYhZKDToIxI7UiKVtlBrCBMQ3qBHAAVBvUQyWA3EiNQx6CcSCDKNgcIP/ZOJRMFiBDzViiAeIr1HDoJlAnEypQdZAvBcpwZIF2ID4ErQWocigDiDOQctCJAM9ID6KJS+SDEAVogo1DKJpjqdakUEfgwCGRTp0g/2KWQAAAGJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWZyYWM+PG1uPjc8L21uPjxtbj40PC9tbj48L21mcmFjPjwvbWF0aD7kU9SNAAAAAElFTkSuQmCC" class="Wirisformula" alt="7 over 4" width="18" height="35" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»7«/mn»«mn»4«/mn»«/mfrac»«/math»'/> Statement-II : If <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator p x plus q over denominator left parenthesis 2 x minus 3 right parenthesis squared end fraction equals fraction numerator 1 over denominator 2 x minus 3 end fraction plus fraction numerator 3 over denominator left parenthesis 2 x minus 3 right parenthesis squared end fraction" width="238" height="39" style="vertical-align:-16px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»p«/mi»«mi»x«/mi»«mo»+«/mo»«mi»q«/mi»«/mrow»«mrow»«mo»(«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mn»3«/mn»«mrow»«mo»(«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»'/> then <img src="data:image/png;base64,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" class="Wirisformula" alt="p equals 2 comma q equals 3" width="81" height="15" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»p«/mi»«mo»=«/mo»«mn»2«/mn»«mo»,«/mo»«mi»q«/mi»«mo»=«/mo»«mn»3«/mn»«/math»'/> Which of the above statements is true

Solution for the question - statement-i : ifthen a= statement-ii : ifthen which of the abovestatements is true neither i nor ii trueboth i & ii are true

Statement-i : ifthen a= statement-ii : ifthen which-Turito

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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: (x - a) (x - b) - 1 = 0. Simplifying it, we get: x2−(a+b)x−ab−1=0 Now lets find the discriminant. The formula is: D=b2-4ac  Applying it, we get: D=(a+b)2−4(ab−1) D=(a−b)2+4 D&gt;0 So now we can say that it has two real roots. Also f(a)=−1 and f(b)=−1 but b &gt; a which eventually means that a and b are distinct as coefficient of x2 is position (it is 1), minima of f(x) is between a and b. <div>Therefore, one of the roots will be in the interval of (−α,a) and the other root will be in the interval (b,α).</div>

If b &gt; a , then the equation, (x - a) (x - b) - 1 = 0, has:

Solution for the question - if b > a , then the equation, (x - a) (x - b) - 1 = 0, has: one root in '/>  both roots in'/>   both roots in'/>

If b > a , then the equation, (x - a) (x - b) - 1 = 0, has:-Turito

If <img src="data:image/png;base64,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" class="Wirisformula" alt="alpha comma beta" width="28" height="15" 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»«/math»'/> be that roots <img src="data:image/png;base64,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" class="Wirisformula" alt="4 x squared minus 16 x plus lambda equals 0" width="131" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»16«/mn»«mi»x«/mi»«mo»+«/mo»«mi»§#955;«/mi»«mo»=«/mo»«mn»0«/mn»«/math»'/> where <img src="data:image/png;base64,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" class="Wirisformula" alt="lambda element of R" width="37" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#955;«/mi»«mo»§#8712;«/mo»«mi»R«/mi»«/math»'/>,  such that <img src="data:image/png;base64,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" class="Wirisformula" alt="1 less than alpha less than 2" width="64" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mo»§#60;«/mo»«mi»§#945;«/mi»«mo»§#60;«/mo»«mn»2«/mn»«/math»'/> and <img src="data:image/png;base64,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" class="Wirisformula" alt="2 less than beta less than 3" width="63" height="15" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#60;«/mo»«mi»§#946;«/mi»«mo»§#60;«/mo»«mn»3«/mn»«/math»'/> then the number of integral solutions of λ is

Solution for the question - ifbe that rootswhere, such thatand then the number ofintegral solutions of is 3265

Ifbe that rootswhere, such thatand then the n-Turito

If α,β then the equation<img src="data:image/png;base64,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" class="Wirisformula" alt="x squared minus 3 x plus 1 equals 0" width="111" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«mo»=«/mo»«mn»0«/mn»«/math»'/> with roots <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator 1 over denominator alpha minus 2 end fraction comma fraction numerator 1 over denominator beta minus 2 end fraction" width="98" height="38" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mn»1«/mn»«/mrow»«mrow»«mi»α«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»,«/mo»«mfrac»«mrow»«mn»1«/mn»«/mrow»«mrow»«mi»β«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfrac»«/math»'/> will be

Solution for the question - if,then the equation with roots will be '/>   '/>  none of these'/>

If,then the equation with roots will be -Turito

The equation of the directrix of the conic <img src="data:image/png;base64,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" class="Wirisformula" alt="2 equals r left parenthesis 3 plus C o s invisible function application theta right parenthesis" width="113" height="16" style="vertical-align:-2px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mn»2«/mn»«mo»=«/mo»«mi»r«/mi»«mo»(«/mo»«mn»3«/mn»«mo»+«/mo»«mi»C«/mi»«mi»o«/mi»«mi»s«/mi»«mo»⁡«/mo»«mi»θ«/mi»«mo»)«/mo»«/math»'/> is

Solution for the question - the equation of the directrix of the conic 2=r(3+cos theta) is 4=rcos theta   2=rcos theta   1=rcos theta   2=3rcos theta

The equation of the directrix of the conic 2=r(3+cos theta) is -Turito

The conic with length of latus rectum 6 and eccentricity is

Solution for the question - the conic with length of latus rectum 6 and eccentricity is (6)/(r)=1+sqrt2cos theta   (6)/(r)=sqrt2+cos theta  (3)/(r)=1+sqrt2cos theta

The conic with length of latus rectum 6 and eccentricity is-Turito

For the circle <img src="data:image/png;base64,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" class="Wirisformula" alt="r equals 6 C o s space theta" width="80" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»r«/mi»«mo»=«/mo»«mn»6«/mn»«mi»C«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«/math»'/> centre and radius are

Solution for the question - for the circle r=6cos theta centre and radius are (6,0),3(6,0),6(3,0),3

For the circle r=6cos theta centre and radius are -Turito

The polar equation of the circle of radius 5 and touching the initial line at the pole is

Solution for the question - the polar equation of the circle of radius 5 and touching the initial lineat the pole is r=10 cos theta   r=5cos theta

The polar equation of the circle of radius 5 and touching the i-Turito

The circle with centre at and radius 2 is

Solution for the question - the circle with centre at and radius 2 is r^(2)-6rsin theta+5=0   r^(2)-ercos theta=5   r^(2)-grsin theta=5  r^(2)-grcos theta+5=0

The circle with centre at and radius 2 is-Turito

Statement-I : If <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator x squared plus 3 x plus 1 over denominator x squared plus 2 x plus 1 end fraction equals A plus fraction numerator B over denominator x plus 1 end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis squared end fraction text  then  end text bold italic A plus bold italic B plus bold italic C equals bold 0" width="406" height="43" style="vertical-align:-16px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»=«/mo»«mi»A«/mi»«mo»+«/mo»«mfrac»«mi»B«/mi»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mi»C«/mi»«mrow»«mo»(«/mo»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mtext»§#160;then§#160;«/mtext»«mi mathvariant=¨bold-italic¨»A«/mi»«mo»+«/mo»«mi mathvariant=¨bold-italic¨»B«/mi»«mo»+«/mo»«mi mathvariant=¨bold-italic¨»C«/mi»«mo»=«/mo»«mn mathvariant=¨bold¨»0«/mn»«/math»'/> Statement-II :If <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator x squared plus 2 x plus 3 over denominator x cubed end fraction equals A over x plus B over x squared plus C over x cubed text  then  end text A plus B minus C equals 0" width="355" height="42" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mfrac»«mo»=«/mo»«mfrac»«mi»A«/mi»«mi»x«/mi»«/mfrac»«mo»+«/mo»«mfrac»«mi»B«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«mo»+«/mo»«mfrac»«mi»C«/mi»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mfrac»«mtext»§#160;then§#160;«/mtext»«mi»A«/mi»«mo»+«/mo»«mi»B«/mi»«mo»§#8722;«/mo»«mi»C«/mi»«mo»=«/mo»«mn»0«/mn»«/math»'/> Which of the above statements is true

Solution for the question - statement-i : if statement-ii :if which of the above statements is true neither i nor ii trueboth i ii are true

Statement-i : if statement-ii :if which of the above stat-Turito

If x, y are rational number such that <img src="data:image/png;base64,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" class="Wirisformula" alt="x plus y plus left parenthesis x minus 2 y right parenthesis square root of 2 equals 2 x minus y plus left parenthesis x minus y minus 1 right parenthesis square root of 6" width="318" height="19" 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»«mo»(«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi»y«/mi»«mo»)«/mo»«msqrt»«mn»2«/mn»«/msqrt»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mi»y«/mi»«mo»+«/mo»«mo»(«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mi»y«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mo»)«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/math»'/> Then

Solution for the question - if x, y are rational number such thatthen none of thesex = 5, y = 1x = 2, y =1x and y cannot be determined

If x, y are rational number such thatthen -Turito

Number of groups having 4 boys and 1 girl = (4C4) (gC1) = g and number of groups having 3 boys and 2 girls = (4C3) (gC2) = 2g(g – 1) Thus, the number of dolls distributed = g(1) + (2)[2g (g – 1)] = 4g2 – 3g We are given 4g2 – 3g = 85 <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»'/> g = 5.

A class contains 4 boys and g girls. Every Sunday five students, including at least three boys go for a picnic to Appu Ghar, a different group being sent every week. During, the picnic, the class teacher gives each girl in the group a doll. If the total number of dolls distributed was 85, then value of g is -

Solution for the question - a class contains 4 boys and g girls. every sunday five students, includingat least three boys go for a picnic to appu ghar, a differ

A class contains 4 boys and g girls. every sunday five students-Turito

Let the number of yellow balls be x, that of black be 2x and that of green be y. Then x + 2x + y = 20 or 3x + y = 20 <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 = 20 – 3x. As 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»'/> y 20, we get 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»'/> 20 – 3x <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»'/> 20 <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»'/> 3x <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»'/> 20 or 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»'/> x <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 <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 number of ways of selecting the balls is 7.

There are three piles of identical yellow, black and green balls and each pile contains at least 20 balls. The number of ways of selecting 20 balls if the number of black balls to be selected is twice the number of yellow balls, is -

Solution for the question - there are three piles of identical yellow, black and green balls and eachpile contains at least 20 balls. the number of ways of sele

There are three piles of identical yellow, black and green ball-Turito

We have (a3 – a) + (b3 – b) + (c3 – c) = (a – 1) (a) (a + 1) + (b – 1) (b) (b + 1) + (c – 1) (c) (c + 1) Since (x – 1) x (x + 1) is divisible both 2 &amp; 3, it is divisible by 6. Thus, 6 | {(a3 – a) + (b3 – b) + (c3 – c)} <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»'/>6 | (a3 + b3 + c3) as 6|(a + b + c).

If a, b, c are the three natural numbers such that a + b + c is divisible by 6, then a3 + b3 + c3 must be divisible by -

Solution for the question - if a, b, c are the three natural numbers such that a + b + c is divisibleby 6, then a3 + b3 + c3 must be divisible by - none of these

If a, b, c are the three natural numbers such that a + b + c is-Turito

For x <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»'/> R, let [x] denote the greatest integer <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»'/> x, then value of<img src="data:image/png;base64,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" class="Wirisformula" alt="open square brackets negative fraction numerator 1 over denominator 3 end fraction close square brackets" width="45" height="42" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfenced open=¨[¨ close=¨]¨ separators=¨|¨»«mrow»«mo»-«/mo»«mfrac»«mrow»«mn»1«/mn»«/mrow»«mrow»«mn»3«/mn»«/mrow»«/mfrac»«/mrow»«/mfenced»«/math»'/>+<img 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Solution for the question - for x e r, let [x] denote the greatest integer 153135123

For x e r, let [x] denote the greatest integer -Turito

Let t = z + 1 Equation reduces to x + y + t = 25 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»'/> x <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, 12 <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»'/> y <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»'/> 18, t <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 x25 in [(x + x2 + x3 + x4 + x5) (x12 + x13 +…..+ x18) (1 + x + x2 + …..)] = Coefficient of x12 in (1 + x + x2 + x3 + x4) (1 + x + x2 + x3 + x4 + x5 + x6) (1 + x + x2 + …..) = Coefficient of x12 in <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator left parenthesis 1 minus x to the power of 6 end exponent right parenthesis over denominator left parenthesis 1 minus x right parenthesis end fraction" width="64" height="41" style="vertical-align:-14px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mo»(«/mo»«mn»1«/mn»«mo»-«/mo»«msup»«mrow»«mi»x«/mi»«/mrow»«mrow»«mn»6«/mn»«/mrow»«/msup»«mo»)«/mo»«/mrow»«mrow»«mo»(«/mo»«mn»1«/mn»«mo»-«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«/mfrac»«/math»'/> (1 – x)–1 = Coefficient of x12 in (1 – x5) (1 – x6) (1 – x)–3 = Coefficient of x12 in (1 – x5 – x6 + x11) (1 – x)–3 = 12+3–1C3–1 – 7+3–1C3–1 – 6+3–1C3–1 + 1+3–1C3–1 = 14C2 – 9C2 – 8C2 + 3C2 = 91 – 36 – 28 + 3 = 30.

The number of integral solutions of the equation x + y + z = 24 subjected to conditions that 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»'/> x <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, 12 <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»'/> y <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»'/> 18, z <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»'/> –1

Solution for the question - the number of integral solutions of the equation x + y + z = 24 subjectedto conditions that 1 none of these