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The population $p\left(t\right)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{dp\left(t\right)}{dt}=0.5p\left(t\right)-450$. If $p\left(0\right)=850$, then the time at which the population becomes zero is:

Find the sum to n
terms of the series in Exercises 8 to 10 whose n^th
terms is given by

n² +
2ⁿ

The
sum and sum of squares corresponding to length x
(in
cm) and weight y

(in
gm) of 50 plant products are given below:

Which
is more varying, the length or weight?

Given that $E$ and $F$ are events such that $P\left(E\right)=0.6,P\left(F\right)=0.3$ and $P(E\cap F)=0.2$, then the value of $P\left(E/F\right)$ and $P\left(F/E\right)$ is:

The integral $\underset{1}{\overset{e}{\int }}\{{\left(\frac{x}{e}\right)}^{2x}-{\left(\frac{e}{x}\right)}^x\}{\log }_{e}xdx$ is equal to:

Let $a,b$ and $c$ be in G.P. with common ratio $r,$ where $a\ne 0$ and $0<r\le \frac{1}{2}$. If $3a,7b$ and $15c$ are the first three terms of an A.P., then the $4^{th}$ term of this A.P. is :

For $x\in R-\{0,1\},$ let $f_1\left(x\right)=\frac{1}{x}$, $f_2\left(x\right)=1-x$ and $f_3\left(x\right)=\frac{1}{1-x}$ be three given functions. If a function, $J\left(x\right)$ satisfies $(f_2\circ J\circ f_1)\left(x\right)=f_3\left(x\right)$ then $J\left(x\right)$ is equal to :

$se{c}^4\theta -se{c}^2\theta =ta{n}^4\theta +ta{n}^2\theta$

The line $x=\sqrt{3}y$ intersects with the curve $y^3-x^2-3y^2+8=0$ at the points $A,B$ and $C$. If $O$ be the origin, then $|OA.OB.OC|$ equals

If the point $(2,k)$ lies outside the circle's $x^2+y^2+x-2y-14=0$ and $x^2+y^2=13$ then range of $k$ is

If $n\left(A\right)=m,m>0$, then number of symmetric relations from $A$ to $A$ is

The value of determinant $\Delta =\left|\begin{array}{ccc}x& -x^2& -x^3\\ -x^2& x^2& -x\\ -x^3& -x& x^3\end{array}\right|,(\Delta \ne 0,x\ne 0)$ is equal to

If Rolle's theorem holds for the function $f\left(x\right)=\ln (2-x^2),x\in [-1,1]$ at the point $x=c$. Then the value of $c$ is

If in a triangle $ABC,$ $AB=5$ units, $\angle B={\cos }^{-1}\left(\frac{3}{5}\right)$ and radius of circumcircle of $△ABC$ is $5$ units, then the area (in sq. units) of $△ABC$ is

Let the equation of the plane, that passes through the point $(1,4,-3)$ and contains the line of intersection of the planes $3x-2y+4z-7=0$ and $x+5y-2z+9=0,$ be $\alpha x+\beta y+\gamma z+3=0,$ then $\alpha +\beta +\gamma$ is equal to

If f(x) = |x| + |x - 1|, write the value of $\frac{d}{dx}\left(f\right(x\left)\right)$.

The greatest integer less than or equal to $(\sqrt{2}+1{)}^6$ is

Let $y=2sinx+cos2x(0\le x\le 2\pi )$. All the points at which y is extremum are arranged in a row such that the points of maximum and minimum come alternately the number of such arrangements is :

If $n\ge 2$ is a positive integer, then the sum of the series ${}^{n+1}{C}_2+2({}^2{C}_2+{}^3{C}_2+{}^4{C}_2+\cdots +{}^n{C}_2)$ is

For $0<x_1<x_2<1,$ which of the following is correct

Find the values of x,y and z if the matrix $A=\left[\begin{array}{ccc}0& 2y& z\\ x& y& -z\\ x& -y& z\end{array}\right]$ satisfies the equation A'A=I.

For the given sequence $\log a,\log \left(ab\right),\log \left(a{b}^2\right)$, where $a,b,c>0$, the ${10}^{th}$ term is

In a $\Delta ABC,$ if $\angle A={60}^{\circ }$, then the value $(1+\frac{a}{c}+\frac{b}{c})(1+\frac{c}{b}-\frac{a}{b})=$

If ${}^{n-1}{C}_r=(k^2-3{)}^n{C}_{r+1}$, then $kϵ$

Choose the correct answer.
${\int }_0^{\frac{2}{3}}\frac{1}{4+9x^2}dx$
$\left(a\right)\frac{\pi }{6}\left(b\right)\frac{\pi }{12}\left(c\right)\frac{\pi }{24}\left(d\right)\frac{\pi }{4}$

The number of times the function $f\left(x\right)=\sum _{r=1}^{2009}\frac{r}{x-r}$ vanishes is