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$si{n}^{-1}[x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^2}]=$

Let f and g be real functions defined by $f\left(x\right)=\sqrt{x+2}\mathrm{and}g\left(x\right)=\sqrt{4-x^2}.\mathrm{Then},\mathrm{the}\mathrm{domain}\mathrm{of}\mathrm{the}\mathrm{function}\left(\frac{f}{g}\right)\left(x\right)\mathrm{is}$

If $a_1,a_2,a_3,....,a_n$ be an AP of non-zero terms. Prove that

$\frac{1}{a_1{a}_2}$+$\frac{1}{a_2{a}_3}$+....+$\frac{1}{a_{n-1}{a}_n}$=

$\frac{n-1}{a_1{a}_n}$

Find the value of $C_0^2+C_1^2+C_2^2.....C_n^2$, where $C_r={}^n{C}_r$

Match the following complex number with their arguements

A square with each side equal to $a$ lies above the $x$-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle $\alpha (0<\alpha <\frac{\pi }{4})$ with the positive direction of the $x$-axis. Equation of a diagonal of the square is

Four identical particles each of mass 1 kg are arranged at the corners of a square of side length 2Ö2 m. If one of the particles is removed, the shift in the centre of mass will be

The range of $f\left(x\right)=\frac{e^x}{1+\left[x\right]},x\ge 0$ is

Find the ratio in which the line segment joining the points (4,8,10) and (6,10,-8) is divided by the YZ-plane.

Find the value of $\lambda$ if (3, 4) and (-4, 3) lie on the opposite side of $x-y+\lambda$ = 0

Show that the middle term in the expansion of $(1+x{)}^{2n}$ is $\frac{1.3.5\cdots (2n-1)}{n!}.2^n.X^n,\text{where}nϵN.$

If $|1-\frac{|x|}{1+|x|}|\ge \frac{1}{2}$, then $x\in$

$\underset{x\to 0}{lim}xlo{g}_{e}x$ will be equal to ___

If the vertices of a triangle be $(a{m}_1^2,2a{m}_1),(a{m}_2^2,2a{m}_2)and(a{m}_3^2,2a{m}_3)$, then the area of the triangle is

If the tangent to the ellipse $x^2+4y^2=16$ at the point $P\left(\varphi \right)$ is normal to the circle $x^2+y^2-8x-4y=0,$ then possible values(s) of $\varphi$ is/are

For $x\in R,\text{let}\left[x\right]$ denote the greatest integer $\le x,$ then the sum of the series $[-\frac{1}{3}]+[-\frac{1}{3}-\frac{1}{100}]+[-\frac{1}{3}-\frac{2}{100}]+....+[-\frac{1}{3}-\frac{99}{100}]$ is :

If number of terms in the expansion of

$(x-2y+3z{)}^n$ are 45, then n =

A spring with spring constant K is divided into x number of equal pieces. What would be the spring constant of one small piece of spring?

Solve: ${\log }_{0.2}$ |x-3| $\ge$ 0

Out of 18 points in a plane, no three are in the same straight line except 5 points which are collinear. How many straight lines can be formed by joining them?

The intergral $\int \frac{dx}{x^2(x^4+1{)}^{3/4}}$ eauals:

Find the value of n so that $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ may be the geometric mean between a and b.

A function f is such that f(x+y) = 2 f(x).f(y). If f(x) is differentiable at x=0, then f(0) can be

Let $f\left(x\right)=\{\begin{array}{cc}{x}^2,& x\ge 0\\ ax,& x<0\end{array}$

The set of real values of $a$ such that $f\left(x\right)$ will have local minima at $x=0,$ is:

The coefficient of $x^5$ in the expansion of

$(1+x^2{)}^5$ $(1+x{)}^4$ is

f(x) = $a^x$ (where a >1) defined on f: R $\to (0,\infty )$ is -

If the lines $y=3x+1$ and $2y=x+3$ are equally inclined to the line $y=mx+4$, find the value of m.

Solve for x.

$ta{n}^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2}ta{n}^{-1}x,x>0$

If $y=si{n}^{-1}(x\sqrt{1-x}+\sqrt{x}\sqrt{1-x^2})$ then $\frac{dy}{dx}=$

Equation of the plane which passes through the point of intersection of lines $\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}$ and $\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and at the greatest distance from the point (0, 0, 0) is

If pair of straight lines $x^2-y^2+6x+4y+5=0$ are transverse and conjugate axes of hyperbola and perpendicular distance form origin to these lines represent the length of transverse and conjugate axis, then the eccentricity of hyperbola is:

Three positive numbers from an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is:

The length of common chord of the circles $(x-a{)}^2+y^2$ =$a^2$ and $x^2+(y-b{)}^2$ = $b^2$ is

If A and B are subsets of the universal set U, then show that,

$A\subset B\iff A\cup B=B$

1. 7.21

Which of the following is logically equivalent to $\sim (\sim p\Rightarrow q)?$

Which of the following Venn- diagram correctly illustrates the relationship among the following: Games, Cricket, Volleyball, Lawn Tennis, Badminton, Hockey, Games with 11 players in the final playing team and games which require a ball for being played.

Air is being pumped into a spherical balloon at a constant rate such that its radius increases constantly with respect to time according to the equation $r\left(t\right)=0.5t^2+r_{\circ },$ (where $r$ is in $\text{cm},t$ is in $\text{minute}$ and $r_{\circ }$ is the initial radius of the balloon). Then the rate of change of its volume after $2\text{minute}$ is



(Assume that the initial radius of the balloon is $2\text{cm}$)

If ${}^n{C}_2+{}^n{C}_3={}^{n+1}{C}_r$, then the minimum possible value of $r$ is

Argument of the complex number $\frac{1}{(\sqrt{3}-i{)}^{25}}$ is

Number of solutions of the equation $|x-1|+|x-2|+|x-3|=7$ is

If $f\left(\theta \right)=si{n}^2\theta +si{n}^2(\theta +\frac{2\pi }{3})+si{n}^2(\theta +\frac{4\pi }{3}),thenf\left(\frac{\pi }{15}\right)$