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The treatment of $CH_{3}MgXwithCH_{3}C≡C−H$ produces

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If P and Q are the points of intersection of the circles $x_{2}+y_{2}+3x+7y+2p5=0$and $x_{2}+y_{2}+2x+2yp_{2}=0$, then there is a circle passingthrough P, Q and $(1,1)$for

A parabola has the origin as its focus and the line $x=2$as the directrix. Then the vertex of the parabola is at

Let $f:N→Y$be a function defined as $f(x)=4x+3$, where $Y={y∈N:y=4x+3$for some $x∈N}$. Show that f is invertible and its inverse is

Amount of oxalic acid present in a solution can be determined by its titration with $KMnO_{4}$ solution in the presence of $H_{2}SO_{4}$. The titration gives unsatisfactory result when carried out in the presence of HCl, because HCl

If $∣∣ z−z4 ∣∣ =2$, then the maximum value of $∣Z∣$is equal to

In context with the industrial preparation of hydrogen from water gas $(CO+H_{2})$, which of the following is the correct statement ?

The largest interval lying in $(−2π ,2π )$for which the function \displaystyle{\left[{f{{\left({x}\right)}}}={4}^{{-{{x}}}}^{2}+{{\cos}^{{-{1}}}{\left(\frac{{x}}{{2}}-{1}\right)}}+{\log{{\left({\cos{{x}}}\right)}}}\right]} is defined, is

The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is