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The value of the integral $\int \frac{5x}{(x+1)(x^2-4)}dx$ is

(where $m$ is an arbitrary constant)

Let $f\left(x\right)=\{\begin{array}{cc}-2\sin x,& x\le -m\\ A\sin x+B,& -m<x<m\\ \cos x,& x\ge m\end{array}$ be a continuous function, where $m$ is the principal solution of the equation ${\sin }^{3}x+\sin x\cos x+{\cos }^{3}x=1$. If $m>0$, then

If $(2+\sqrt{3}{)}^n=I+f,n\in N$, where $I$ is integral part and $f$ is fractional part, then $(I+f)(1-f)$ is

Let Which of the following statements is FALSE?

Which of the following is a function in their respective given domain and co-domain?

Let D, E, F be the mid points of the sides BC, CA, AB respectively of a $\Delta ABC$. Then,$\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}$ equals

Let $PQ$ be a chord of the parabola $y^2=8x$. A circle drawn with $PQ$ as diameter passes through the vertex $V$ of the parabola. If area of $\Delta PVQ=80$ square unit, then the coordinates of $P$ are

The value of integral $\underset{-2}{\overset{2}{\int }}\frac{e^x+e^{-x}}{x}dx,$ is

Two cubes have their face painted as either red or blue color. $1st$ cube has $5$ faces red and $1$ face blue. The probabilty that both the faces show same color on rolling is $\frac{1}{2}$. How many red faces are present on the $2nd$ cube.

Find the integral of the function $\frac{1}{x^2+a^2}$

${lim}_{x\to \sqrt{6}}\frac{\sqrt{5+2x}-(\sqrt{3}+\sqrt{2})}{x^2-6}$

If $nϵN,$ then find the value of $i^n+i^{n+1}+i^{n+2}+i^{n+3}$

Differentiate $e^x(x^3+\sqrt{x})$.

If $sinA=\frac{1}{2},cosB=\frac{\sqrt{3}}{2},$ where $\frac{\pi }{2}$ (i) tan(A+B)
(ii) tan(A-B)

Let $f\left(x\right)=e^x\sin x$ be a continuous function in $R.$ If $f\left(x\right)=0$ has atleast one real solution, then interval of $x$ can be

If $A,B,C$ are three events are such that $P\left(B\right)=\frac{3}{4},P(A\cap B\cap C^{\prime })=\frac{1}{3}$ and $P(A^{\prime }\cap B\cap C^{\prime })=\frac{1}{3},$ then $P(B\cap C)$ is equal to

The number of ordered triplets $(a,b,c)$ where $a,b\in W$ and $c\in N$ such that origin and the point $(1,1,4)$ lie on the same side of the plane $ax+by+cz=39$ is

A whistle '$\text{S}$' of frequency $f_0$ revolves in a circle of radius $R$ at a constant speed $v_s$. If the speed of sound in stationary medium is $v$, what is the ratio of frequency detected by a detector ${}^{\prime }{\text{D}}^{\prime }$ at rest in the two cases when source was on topmost point and when the source was at bottom most point?

Consider a $△ABC,$ whose vertices are $A(-2,3,-4),B(4,2,1)$ and $C(1,1,0),$ then the distance (in units) of its centroid from origin is

The equation of the straight line which passes through the point $P(-4,3)$ such that the portion of it between the $x$-axis and $y$-axis is divided by the point $P$ in the ratio $1:2$ respectively, is

Let $A(a,b)$ and $B(0,0)$ be two fixed points. If $M_1$ is the mid point of line segment $AB$, $M_2$ is the mid point of line segment $A{M}_1$ and $M_3$ is the mid point of line segment $A{M}_2$ and so on, then which of the following is/are true?

From the graph function f is...... in the interval I ,

A vector of magnitude $\sqrt{2}$ coplanar with the vectors $\overrightarrow{a}=\hat{i}+\hat{j}+2\hat{k}\text{and}\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}$ and perpendicular to the vector $\overrightarrow{c}=\hat{i}+\hat{j}+\hat{k},$ is