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In a triangle $ABC,\frac{b^2\sin 2C+c^2\sin 2B}{\Delta }$ is always equal to

In a G.P. series consisting of positive terms, each term is equal to the sum of next two terms. Then the common ratio of this G.P. series is

The value of $\int \frac{dx}{(a^2+x^2{)}^{\frac{3}{2}}}$ is

Area under the curve $y=\sqrt{3x+4}$between x=0 and x=4, is

It is given that $\underset{x\to 0}{lim}\frac{e^{ax}-bx-1}{x^2}=2$. Then the value of $|a|+|b|$ is

Evalaute $\int \frac{dx}{(x^2+4)\sqrt{4x^2+1}}$ (where $C$ is integration constant)

Let $A$ and $B$ be two finite sets. The set $A$ has $480$ more subsets than $B.$ If $A\cap B$ has $2$ elements, then the number of elements in $A\cup B$ is

If x,y,z are non negative integers so that x + y + z = 12, then the

maximum value of xyz + xy + yz + zx is

In an experiment with $250$ trials, the first $150$ trials has mean $16$ and standard deviation $4.$ The overall experiment has mean $15.6$ and variance $\mathrm{13.44.}$The standard deviation of the next $100$ trials of given experiment is

The solution of differential equation $(1-xy+x^2{y}^2)dx=x^{2}dy$ is

[$C$ is constant of integration]

Let A = {9, 10, 11, 12, 13} and let $f:A\to N$ be defined by f(n) = the highest prime factor of n. Find the range of f.

Which one of the following is an eigen vector of the matrix $\left[\begin{array}{cccc}5& 0& 0& 0\\ 0& 5& 5& 0\\ 0& 0& 2& 1\\ 0& 0& 3& 1\end{array}\right]$

If $\overrightarrow{OA}=\overrightarrow{a},\overrightarrow{OB}=\overrightarrow{b},\overrightarrow{OC}=\overrightarrow{c}$,
then in $△ABC$, the equation of the median through $A$ is

The solution of the differential equation $2x^{2}y\frac{dy}{dx}=\tan \left(x^2{y}^2\right)-2x{y}^2$ given $y\left(1\right)=\sqrt{\frac{\pi }{2}},$ is

In a circle of diameter 40cm,the length of a chord is 20cm.find the length of minor archof the chord

Solution of the differential equation: $\frac{x+y\frac{dy}{dx}}{y-x\frac{dy}{dx}}=\frac{xsi{n}^2(x^2+y^2)}{y^3}$

Using integration, find the area of the region in the first quadrant enclosed by the x- axis, the line y = x and the circle $x^2+y^2=32$

Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±3), foci (0, ±5)

The value of $p^{-1}\cdot q\cdot q^{-1}\cdot r\cdot r^{-1}\cdot p$ is
(a) –1
(b) 0
(c) 1
(d) 2

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Identify carboxylic acids in the given options below.

Find the equation of the circle the end points of whose diameter are the centre of the circles
$x^2+y^2-6x-14y-1=0$
and $x^2+y^2-4x+lOy-2=0$

Find the equation of the line parallel to x-axis and having intercept -2 on y-axis.

Define a
binary operation *on the set {0, 1, 2, 3, 4, 5} as

Show that
zero is the identity for this operation and each element a ≠
0 of the set is invertible with 6 − a being the inverse
of a.

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{(e^x-e^{-x})}{2}$.

The inverse of the given function is: