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The value of $\underset{x\to 1}{lim}[\frac{x-2}{x^2-x}-\frac{1}{x^3-3x^2+2x}]$ is

For all permissible values of $A,2A$, following holds true.

$\left(i\right)\cot A+\tan A=\frac{1}{\sin A\cos A}=2\text{cosec}2A\left(ii\right)\cot A-\tan A=\frac{{\cos }^{2}A-{\sin }^{2}A}{\sin A\cos A}=2\cot 2A\left(iii\right)2\cot A=2(\text{cosec}2A+\cot 2A)\Rightarrow \text{cosec}2A+\cot 2A=\cot A$



Also to evaluate a series of form $f\left(x\right)+f\left(2x\right)+f\left(4x\right)+\cdots +f\left(2^{n}x\right)$ when $f\left(x\right)$ can be expressed as $g\left(x\right)-g\left(2x\right)$, we can use the following technique,

$f\left(x\right)+f\left(2x\right)+f\left(4x\right)+\cdots +f\left(2^{n}x\right)=\left(g\right(x)-g(2x\left)\right)+\left(g\right(2x)-g(4x\left)\right)+\cdots \left(g\right(2^{n}x)-g(2^{n+1}x\left)\right)=g\left(x\right)-g\left(2^{n+1}x\right)$



Based on the above information, solve the following questions for all permissible values of $x$.



The value of $\text{cosec}2x+\text{cosec}4x+\text{cosec}8x+\text{cosec}16x+\text{cosec}32x$

Which of the following option(s) is/are correct for $0<x<\frac{\pi }{2}$

Examine the following functions for continuity :

(a) f(x) = x - 5

(b) $f\left(x\right)=\frac{1}{x-5},x\ne 5$

(c) $f\left(x\right)=\frac{x^2-25}{x+5},x\ne -5$

(d) $f\left(x\right)=|x-5|$.

Whenever horses $a,b,c$ race together, their respective probabilities of winning the race are $0.3,0.5$ and $0.2$ respectively. If they race three times the probability that "the same horse wins all the three races" is $p$ and the probability that $a,b,c$ each wins one race is $q$, then the value of $\frac{p}{q}$ is (Assume no dead heat)

If $m$ is the slope of a tangent to the curve $e^{2y}=1+4x^2$ then

The locus of points of trisection of double ordinate of $y^2=4ax$ is

cos²
2x
– cos²
6x
= sin 4x
sin
8x

If the equation $|x+1|+|x-3|=k$ has exactly two solution then $k$ lies in

$\underset{-\pi /4}{\overset{\pi /4}{\int }}|\tan x|dx$ equals to

Let $\sum _{i=1}^n{\alpha }_i=a{n}^2+bn,$ where $a,b$ are constants and ${\alpha }_1,{\alpha }_2,{\alpha }_3\in \{1,2,3,\dots ,9\}.$ If $25{\alpha }_1,37{\alpha }_2,49{\alpha }_3$ are three-digit numbers, then the value of $\left|\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\\ 5& 7& 9\\ 25{\alpha }_1& 37{\alpha }_2& 49{\alpha }_3\end{array}\right|$ is

If x + y = 6 and x - y = 2 , then x =___.

The minimum value of $ata{n}^{2}x+bco{t}^{2}x$ equals the maximum value of a $si{n}^2\theta +bco{s}^2\theta wherea>b>0,$ then

If $y=e^{\sin \sqrt{x}},$ then $\frac{dy}{dx}=$

sin(n + 1)x sin ( n+2 )x + cos(n+1)x cos (n+2)x = cos x

If cos $\alpha =\frac{3}{5}$ and cos $\beta =\frac{5}{13}$ and $\alpha ,\beta$ are in first quadrant then which of the following is/are true?

The point(s) on the curve $3y=6x-5x^3$ except origin at which the normal drawn also passes through origin is/are

If $\underset{1}{\overset{2}{\int }}(2-\frac{1}{x})dx=a-b\ln 2,$ then which of the following is/are true ?

In the expansion of $(x+y{)}^{25}$,
$1^{st}$ term from the end = $(26-p{)}^{th}$ term from the beginning
$2^{nd}$ term from the end = $(26-q{)}^{th}$ term from the beginning
$3^{rd}$ term from the end = $(26-r{)}^{th}$ term from the beginning
${10}^{st}$ term from the end = $(26-s{)}^{th}$ term from the beginning
Find the value of p+q+r+s

If $a=1+b+b^2+b^3+...$ to $\infty$, then write b in terms of a given that $|b|<1$.

The tangent to the curve $y=e^{kx}$ at the point $(0,1)$ meets the $x-$axis at $(a,0)$ where $a\in [-2,-1]$, then $k\in$

Out of $100$ bicycles, ten bicycles have puncture. What is the probability of not having any punctured bicycle in a sample of $5$ bicycles ?

Find the integrals of the functions.
$\int cos2x$ cos 4x cos 6x dx.

Find the equation of tangent to parabola $y^2=4axat(a{t}^2,2at)$

The results of an election are shown in the table below.







The difference between the fraction of votes received by Richard and Charles is

The locus of the point of intersection of two perpendicular tangents to the circle $x^2+y^2=a^2$ is