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Q1 Let the roots of f(x)=x be α and β where f(x) is a quadratic polynomial. It is true that α,β also satisfies f(f(x))=x. Let the other roots of the eqn. f(f(x))=x be γ and δ. Now Correct statements are: a) If α β are ...
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Q2 *Image* ...
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A plane divides space into two halves. One half is filled with a homogeneous conducting medium and physicists work in the other. They mark the outline of a square of side a on the plane and let a current I_0 in and out at two ...
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Let f be a continuous function on [0,1] such that for all x\in[0,\,1] , \int_x^1 f(t)\ \mathrm{d}t \geq \dfrac{1-x^2}{2} Prove that \int_0^1 f^2(t)\ \mathrm{d}t \geq \dfrac{1}{3} Here, f2(x) means (f(x))2. Also, determine the ...
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A comet is moving in a parabolic path around the Sun in the same plane as the orbit of Earth (without any collision with the Earth, obviously!). Determine the maximum time that this comet can spend within the Earth's orbit, w ...
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Prove that there exists an irrational r for every natural number k(>=2) such that [rm]≡-1mod(k) for every natural number m. Here [x] is the greatest integer less than or equal to x. ...
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plz solve 1) \int_{0}^{\infty}{\frac{1}{(x^2+a^2)(x^2+b^2)}}dx 2) \int_{0}^{\pi}{log(1-6cosx+9)}dx 3) \int_{0}^{1}{\frac{x^{a-1}-x^{-a}}{(1+x)logx}}dx ...
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A circle of radius 1 rolls ( without sliding ) along the x -axis so that its centre is of the form (t , 1 ) with t increasing. A certain point P touches the x-axis at the origin as the circle rolls. As the circle rolls furthe ...
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Consider a drop of conducting liquid in gravity free space. Its radius is R while the surface tension of the liquid is T. Now the drop is supplied some charge Q. For what values of Q is the drop stable. ...
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evaluate- \sqrt{2011+2007\sqrt{2012+2008\sqrt{2013+2009\sqrt{2014+.........\infty}}}} ...
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evaluate 1) \sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3} 2) \sum_{n=1}^{\infty }\frac{1}{n^2(n+1)^2} ...
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\sum_{n= 1}^{n=\infty }{}\frac{1}{\left\{(2n- 1 )^2- ( 2m ) ^2 \right\}^2} 2. \sum_{n=1}^{n= \infty}{\frac{1}{n( 36n^2-1)}} 3. \sum_{n=1}^{n= \infty}{\frac{1}{n( 9n^2-1)}} ...
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Prove:::::::: If gcd(a,n)=1,then the integers c,c+a,c+2a,c+3a,...........................,c+(n-1)a form a complete set of reidues modulo n, for any c. ...
