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The value of $\underset{x\to 0}{lim}\frac{e^{2x}-1}{3x}$ is

If A and B are any two events, then the probability that exactly one of them occur, is

If the constant term in the binomal expansion of ${(\sqrt{x}-\frac{k}{x^2})}^{10}$ is $405$, then $\left|k\right|$ equals:

Find the value of $(4+\sqrt{5})(4-\sqrt{5})$.Is it rational ? [3 MARKS]

$\begin{array}{cccccc}& Column1& & Column2& & Column3\\ \left(I\right)& InatriangleABCsinA,sinB,sinC& \left(i\right)& Latusrectumofellipse& \left(P\right)& 2\\ & areinA.P.,then\frac{sin\left(\frac{A}{2}\right)}{sin\left(\frac{A}{2}\right)sin\left(\frac{C}{2}\right)is}& & \frac{x^2}{16}+\frac{y^2}{4}=1isequalto& & \\ \left(II\right)& In\Delta ABCif2\Delta +b^2+c^2=2bc+a^2,& \left(ii\right)& Eccentricityofhyperbola& \left(Q\right)& 3\\ & thenvalueofxwhichsatisfy{x}^2-5(sin& & \frac{(x-\sqrt{2}{)}^2}{2}-\frac{y^2}{16}=1& & \\ & A+cosA)x+25sinAcosA=0isequalto& & isequalto& & \\ \left(III\right)& In\Delta ABCifa=7,b=3,c=\sqrt{\frac{131}{2}}& \left(iii\right)& Radiusofdirectorcircleof& R& 7\\ & andmedianADmeetscircumcircleat& & \frac{x^2}{36}+\frac{y^2}{13}=1isequalto& & \\ & E,thenAEisgreaterthan& & & & \\ \left(IV\right)& Thepointofcontactofaninscribedcircleofarightangled& \left(iv\right)& Minimumvalueof|z_1-z_2|where& \left(S\right)& 4\\ & triangledividesthehypotenuseintwo& & |z_1|=2and|z_2-9|=3isequalto& & \\ & partsoflengths4and7,thenareaoftriangleisdivisibleby& & \left(wherecomplexnumbersinargandplane\right)& & \end{array}$
Which of the following is the only correct combination?

If α and β are the roots of the equation $2x^2-3x+4=0$, then the equation whose roots are ${\alpha }^2$ and ${\beta }^2$ is

The triangle PQR of area A is inscribed in the parabola $y^2=4ax$ such that P lies at the vertex of the parabola and base QR is a focal chord. The numerical difference of the ordinates of the points Q & R is

Simplify ${\left(256\right)}^{\frac{-1}{2}}$.

Choose the correct answer in the given question.
$\int e^{x}secx(1+tanx)dx$ equals
$\left(a\right)e^{x}cosx+C\left(b\right)e^{x}secx+C\left(c\right)e^{x}sinx+C\left(d\right)e^{x}tanx+C$

If $4^{th}$ term in the expansion of ${(2+\frac{3}{8}x)}^{10}$ is numerically greatest term . Then the range of $x$ is

$\int e^x\left(\frac{2+sin2x}{1+cos2x}\right)dx=$

If $\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\cdots \text{upto}\infty =\frac{{\pi }^4}{90},$ then the value of $\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\cdots \text{upto}\infty$ is

If the straight line through the point P(3, 4) makes an angle $\frac{\pi }{6}$ with the x–axis and meets the line 12x + 5y + 10 = 0 at Q then the length PQ is

1. 4.5

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):

The graph of $y=2^{1+\sin x}$ is :

If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then A =

Let $AB$ be the focal chord of a parabola $y=x^2-2x+2$ with point $A\equiv (3,5)$ . Then the slope of normal at point $B$ would be :

The value of determinant using sarrus method $\left|\begin{array}{ccc}1& 2& 3\\ 3& 2& 1\\ 2& 1& 0\end{array}\right|$ is

A rectangular hyperbola whose center is $C$, is cut by any circle of radius $r$ in four points $P,Q,R,S$. Then $C{P}^2+C{Q}^2+C{R}^2+C{S}^2=$

What will be the perpendicular distance of a point P (7, 5 , -2) from the plane -2x +7y - 4z + 5 = 0 ?

${lim}_{x\to 1}{\left\{\frac{x^3+2x^2+x+1}{x^2+2x+3}\right\}}^{\frac{1-\cos (x-1)}{(x-1{)}^2}}$

The integrating factor of the differential equation $(2x+3y^2)dy=ydx(y>0)$ is

If $f\left(x\right)=\prod _{n=1}^{20}(x-n{)}^{n^3(50-n)},$ then $\frac{f^{\prime }\left(50\right)}{f\left(50\right)}=$

Two radioactive material $A_1$ and $A_2$ have decay constants of ${\lambda }_0$ and $10{\lambda }_0$. If initially they have same number of nuclei, the ratio of number of their undecayed nuclei will be $\frac{1}{e}$ after a time

If $2sinx-cos2x=1$, then $co{s}^{2}x+co{s}^{4}x$ is equal to

A and B throw a pair of dice. If A throws 9, find B's chance of throwing a higher number.

Identify the type of differential equation