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The initial rate of hydrolysis of methyl acetate (1M) by a weak acid (HA, 1M) is $1/100_{th}$ of that of a strong acid (HX, 1M), at $25_{∘}C.$ The $K_{a}$ of HA is

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Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6: 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

In the following reactions, the structure of the major product 'X' is

Let $f:RR$be a differentiable function such that $f(0),f(2π )=3andf_{prime}(0)=1.$If $g(x)=∫_{x}[f_{prime}(t)cosect−cottcosectf(t)]dtforx(0,2π ],$then $(lim)_{x0}g(x)=$

Â·If the normals of the parabola $y_{2}=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x−3)_{2}(y+2)_{2}=r_{2}$ , then the value of $r_{2}$ is

Let $P_{1}:2x+y−z=3$and $P_{2}:x+2y+z=2$be two planes. Then, which of the following statement(s) is (are) TRUE?The line of intersection of $P_{1}$and $P_{2}$has direction ratios $1,2,−1$(b) The line $93x−4 =91−3y =3z $is perpendicular to the line of intersection of $P_{1}$and $P_{2}$(c) The acute angle between $P_{1}$and $P_{2}$is $60o$(d) If $P_{3}$is the plane passing through the point $(4,2,−2)$and perpendicular to the line of intersection of $P_{1}$and $P_{2}$, then the distance of the point $(2,1,1)$from the plane $P_{3}$is $3 2 $

PARAGRAPH AThere are five students $S_{1},S_{2},S_{3},S_{4}$and $S_{5}$in a music class and for them there are five seats $R_{1},R_{2},R_{3},R_{4}$and $R_{5}$arranged in a row, where initially the seat $R_{i}$is allotted to the student $S_{i},i=1,2,3,4,5$. But, on the examination day, the five students are randomly allotted five seats. For $i=1,2,3,4,$let $T_{i}$denote the event that the students $S_{i}$and $S_{i+1}$do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T_{1}∩T_{2}∩T_{3}∩T_{4}$is$151 $(b) $101 $(c) $607 $(d) $51 $

Let $f:R→R,g:R→Randh:R→R$ be the differential functions such that $f(x)=x_{3}+3x+2,g(f(x))=xandh(g(g(x)))=x,forallx∈R.Then$ (a)g'(2)=$151 $ (b)h'(1)=666 (c)h(0)=16 (d)h(g(3))=36

Let $f:(0,∞)R$ be given by $f(x)=∫_{x1}te_{−(t+t1)}dt ,$ then (a)$f(x)$ is monotonically increasing on $[1,∞)$(b)$f(x)$ is monotonically decreasing on $(0,1)$(c)$f(2_{x})$ is an odd function of $x$ on $R$