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question_answer1) Let f be a differentiable function from R to R such that \[|f\,(x)-f(y)|\le 2|x-y{{|}^{3/2}},\] for all \[x,y,\in R\]. If \[f(0)=1\] then is equal to
question_answer2) The value of is equal to \[\frac{k}{3},\] then the value of k is
question_answer3) If \[\int\limits_{0}^{x}{f(t)\,dt={{x}^{2}}+}\int\limits_{x}^{1}{{{t}^{2}}}f(t)\,dt,\] if \[f'(1/2)\] is \[\frac{24}{{{5}^{k}}},\]then the value of k is
question_answer4) then is equal to
question_answer5)
question_answer6) Let f be a positive function. Let \[{{I}_{1}}=\int\limits_{1-k}^{k}{x\,\,f\left[ x\left( 1-x \right) \right]}\,dx,\] \[{{I}_{2}}=\int\limits_{1-k}^{k}{\,f\left[ x\left( 1-x \right) \right]}\,dx,\] where \[2k-1>0\]. Then \[\frac{{{I}_{1}}}{{{I}_{2}}}\] is
question_answer7) Let \[f:R\to R\] and \[g:R\to R\] be continuous functions. Then the value of the integral \[\int\limits_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\left[ f\left( x \right)+f\left( -x \right) \right]\,\,\left[ g\left( x \right)-g\left( -x \right) \right]}\,\,dx\] is
question_answer8) If where C is a constant of integration. If the function \[f(x)\] is equal to . Then, the value of C is
question_answer9) The value of the integral is equal to \[\frac{\pi }{4}-\frac{k}{2}\] In 2, (natural log In) then k is
question_answer10) If \[\int{{{x}^{5}}\,{{e}^{-{{x}^{2}}}}}dx=g(x)\,{{e}^{-{{x}^{2}}}}+c,\] where c is a constant of integration, if \[g\left( -1 \right)\] is equal to \[-m,\]then find m.
question_answer11) Let \[f:R\to R\] be a continuously differentiable function such that \[f(2)=6\] and \[f'(2)=\frac{1}{48.}\].
question_answer12) The total number of distinct values of \[\alpha \] such that is
question_answer13) Let \[F(x)=f(x)+f\left( \frac{1}{x} \right),\] where then \[F(e)\] is equals to
question_answer14) The value of is
question_answer15) The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\int\limits_{0}^{{{x}^{2}}}{{{\sec }^{2}}tdt}}{x\,\sin \,x}\] is
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