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Methylene blue, from its aqueous solution, is adsorbed on activated charcoal at $25_{∘}C.$ For this process, the correct statement is

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Let $X=(_{10}C_{1})_{2}+2(_{10}C_{2})_{2}+3(_{10}C_{3})_{2}+¨+10(_{10}C_{10})_{2}$, where $_{10}C_{r}$, $r∈{1,2,,¨ 10}$denote binomial coefficients. Then, the value of $14301 X$is _________.

For $a>b>c>0$, if the distance between $(1,1)$ and the point of intersection of the line $ax+by−c=0$ is less than $22 $ then,

Four person independently solve a certain problem correctly with probabilities $21 ,43 ,41 ,81 ˙$Then the probability that he problem is solve correctly by at least one of them is$256235 $b. $25621 $c. $2563 $d. $256253 $

If $f(x)∣cos(2x)cos(2x)sin(2x)−cosxcosx−sinxsinxsinxcosx∣,then:$$f_{prime}(x)=0$at exactly three point in $(−π,π)$$f_{prime}(x)=0$at more than three point in $(−π,π)$$f(x)$attains its maximum at $x=0$$f(x)$attains its minimum at $x=0$

A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equation $3 x+y−6=0$ and the point D is (3 sqrt3/2, 3/2). Further, it is given that the origin and the centre of C are on the same side of the line PQ. (1)The equation of circle C is (2)Points E and F are given by (3)Equation of the sides QR, RP are

Let the curve C be the mirror image of the parabola $y_{2}=4x$ with respect to the line $x+y+4=0$. If A and B are the points of intersection of C with the line $y=−5$, then the distance between A and B is

For a point $P$in the plane, let $d_{1}(P)andd_{2}(P)$be the distances of the point $P$from the lines $x−y=0andx+y=0$respectively. The area of the region $R$consisting of all points $P$lying in the first quadrant of the plane and satisfying $2≤d_{1}(P)+d_{2}(P)≤4,$is

Perpendiculars are drawn from points on the line $2x+2 =−1y+1 =3z $ to the plane $x+y+z=3$ The feet of perpendiculars lie on the line (a) $5x =8y−1 =−13z−2 $ (b) $2x =3y−1 =−5z−2 $ (c) $4x =3y−1 =−7z−2 $ (d) $2x =−7y−1 =5z−2 $