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Let $f\left(x\right)=\frac{\sqrt{x-1+2\sqrt{x-2}}}{\sqrt{x-2}-1}x$. Then

Numbers are selected at random, one at a time, form the two digit numbers 00,01,02, ........, 99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, find the probability that the event E occurs at least 3 times.

If the line y = mx + c be a tangent to the circle $x^2+y^2=a^2$, then the point of contact is

Which of the following is the least?

The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to the line $\frac{x}{2}=\frac{y}{3}=\frac{z-1}{-6}$ is:

Which of the following inequation is equal to the given inequality 6x - 2 < 5x + 3?

The graph of $f\left(x\right)=2x^2-3x+2$ is

The area included by the curve $y=\ln x,x-$axis and the two ordinates at $x=\frac{1}{e}$ and $x=e$ is

Solution of the differential equation $(y\ln x-1)ydx=xdy,x>0$ is:

(where $C$ is integration constant)

Find two positive numbers x and
y such that x + y = 60 and xy³
is maximum.

If $y\left(x\right)$ is the solution of the differential equation
$(x+2)\frac{dy}{dx}=x^2+4x–9,x\ne –2$
$\text{and}y\left(0\right)=0,\text{then}y(–4)$is equal to :

In the following cases,
determine whether the given planes are parallel or perpendicular, and
in case they are neither, find the angles between them.

(a)

(b)

(c)

(d)

(e)

The approximate value of $f\left(5.001\right)$ where $f\left(x\right)=x^3-7x^2+15,$ is:

The value of $\underset{1}{\overset{2}{\int }}(3x^2\tan \frac{1}{x}-x{\sec }^2\frac{1}{x})dx$ is

The number of bacteria
in a certain culture doubles every hour. If there were 30 bacteria
present in the culture originally, how many bacteria will be present
at the end of 2^nd hour, 4^th hour and n^th
hour?

If A is a square matrix of order 4 and the value of |A| is equal to 2. Then the value of |Adj(A)| is,

The value of $co{s}^2{48}^{\circ }-si{n}^2{12}^{\circ }$ is

The quantity of water in a tank is decreasing at a constant rate through out the time due to outflow of water. Draw a line diagram depicting the course of the remaining water over time and select the same from the given option as there is no inflow of water.

If $\sin \theta =n\sin (\theta +2\alpha )$, then the value of $\tan (\theta +\alpha )$ is

A triangle has a vertex at $(1,2)$ and the mid points of the two sides through it are $(–1,1)$ and $(2,3)$. Then the centroid of this triangle is :

The complex number $\frac{2^n}{(1+i{)}^{2n}}+\frac{(1+i{)}^{2n}}{2^n},\forall n\in I$ is equal to

Arrange the following function in ascending order of their fundamental period.

Let $a,b,c$ be the side-lengths of a triangle, and $l,m,n$ be the lengths of its medians. Put $K=\frac{l+m+n}{a+b+c}$. Then, as $a,b,c$ vary, $K$ can assume every value in the interval

Let $f,g$ and $h$be real valued functions defined on the interval [0,1] by$f\left(x\right)=e^{x^2}+e^{-x^2},g\left(x\right)=x{e}^{x^2}+e^{-x^2}$ and $h\left(x\right)=x^2{e}^{x^2}+e^{-x^2}$. If a, b and c denote, respectively, the absolute maximum of f, g and h on [0,1], then

The relation
f is
defined by

The
relation g
is defined by

Show
that f is
a function and g is
not a function.

The differential equation of all the lines in the xy-plane is

The set of value(s) of $a$ for which $x^2+ax+{\sin }^{-1}(x^2-4x+5)+{\cos }^{-1}(x^2-4x+5)=0$ has at least one solution is

If the equations $x^2+2x+3=0anda{x}^2+bx+c=0$ , a , b , c $\hat{I}$ R , have a common root, then a: b: c is

The distance between the line $x=2+t,$ $y=1+t,$ $z=\frac{-1}{2}-\frac{t}{2}$ where $t\in R$ and the plane $\overrightarrow{r}\cdot (\hat{i}+2\hat{j}+6\hat{k})=10$ is

If $f\left(x\right)=1+\frac{1}{x}\underset{1}{\overset{x}{\int }}f\left(t\right)dt$ for $x>0,$ then the value of $f\left(e^{-1}\right)$ is

Evaluate the definite integrals.
${\int }_0^{\frac{\pi }{4}}\frac{sinxcosx}{co{s}^{4}x+si{n}^{4}x}dx$