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A unit vector normal to the plane through the points $\hat{i},2\hat{j}$ and $3\hat{k}$ is

A cuboidal piece of wood has dimensions $a,b$ and $c$. Its relative density is $d$. It is floating in a larger body of water such that side $a$ is vertical. It is pushed down a bit and released. The time period of SHM executed by it is

${\int }_0^{\frac{\pi }{2}}sin2xta{n}^{-1}\left(sinx\right)dx$

The eccentricity of an ellipse with it's centre at origin is $\frac{1}{2}$. If one of the directrices is $x=8$, then the equation of the ellipse is given by

If $\sqrt{{\log }_2\left(\frac{2x-3}{x-1}\right)}<1,$ then $x\in$

Solve the equation

The condition that the equation $\frac{1}{x}+\frac{1}{x+b}=\frac{1}{m}+\frac{1}{m+b}$ has real roots, that are equal in magnitude but opposite in sign, is

If $|z|=1$ , then $\arg \left(\sqrt{\frac{(1+z)}{(1-z)}}\right)$ will be

The number of points in $(-\infty ,\infty )$ for which $x^2-x\sin x-\cos x=0$ is

If $S=\sin \frac{\pi }{n}+\sin \frac{3\pi }{n}+\sin \frac{5\pi }{n}+\cdot \cdot \cdot n$ terms.

Then the value $nS$ is

The quadratic polynomial with rational coefficients for which one of the roots is 2+3i is . where 'i' is $\sqrt{-1}$

If the chords of tangents form two points $(-4,2)$ and $(2,1)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are at right angle, then the eccentricity of the hyperbola is

If $A=\{2,3,4,8,10\},$ $B=\{3,4,5,10,12\},$ $C=\{4,5,6,12,14\},$ then $(A\cup B)\cap (A\cup C)$ is

Is the sequence root 3 , root 6, root9, root12 ,........... Form an arithmetic progression . give reason.

If $a+b+c=18$ and $a^2+b^2+c^2=122$ , then find the value of ab + bc + ca

Prove that $\frac{n^{11}}{11}+\frac{n^5}{5}+\frac{n^3}{3}+\frac{n^{62}}{165}$ n is a positive integer for all $nϵN$.

The domain of the function $\sqrt{(x^2+2x+3)}+\sqrt{(1-x)}$ is

The number of 5 letter words that can be formed from letters of the word PERSON, if the repetition of letters is allowed, is

If the given expression $x^2-(5m-2)x+(4m^2+10m+25)$ can be expressed as a perfect square, then the value(s) of $m$ is/are

~[(- p)^q] is logically equivalent to

Prove that

$P(A\cup B)=P(A\cap B)+P(A\cap \overline{B})+P(\overline{A}\cap B)$

Twelve balls are distributed among three boxes. The probability that the first box will contains three balls.

If both roots of the quadratic equation $x^2+4px+6p^2+3p-2=0$ are less than $4$, then $p$ lies in the interval

If $f\left(x\right)=\underset{n\to \infty }{lim}n(x^{\frac{1}{n}}-1),$ then for $x>0,y>0,f\left(xy\right)$ is equal to

If $\int \frac{dx}{(1+\sqrt{x}{)}^{2010}}=2[\frac{1}{\alpha (1+\sqrt{x}{)}^{\alpha }}-\frac{1}{\beta (1+\sqrt{x}{)}^{\beta }}]+c$, where c is constant of integration and $\alpha ,\beta >0$, then $\alpha -\beta$ is

Let complex numbers $\alpha$ and $\frac{1}{\overline{\alpha }}$ lie on circles $(x-x_0{)}^2+(y-y_0{)}^2=r^2\text{and}(x-x_0{)}^2+(y-y_0{)}^2=4r^2,$



respectively. If $z_0=x_0+i{y}_0$ satisfies the equation $2|z_0{|}^2=r^2+2,\text{then}|\alpha |=$

The last two digits of the number $(23{)}^{14}$ are

Sum of the series $1^2+3^2+5^2+$....... upto n terms is