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x is a rational number satisfying (1-x) (1+x+x²+x³+x⁴) = .

Then 1+x+x²+x³+x⁴+X⁵ is

1 + 4 + 13 + 40 + 121 + ...

$\frac{x}{5}<\frac{3x-2}{4}-\frac{5x-3}{5}$

If $A+B=\frac{\pi }{3}$ and $cosA+cosB=1$
then find the value of $cos\frac{A-B}{2}$

Prove that:

$4cos\theta (\frac{\pi }{3}+\theta )cos(\frac{\pi }{3}-\theta )=cos3\theta$

Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.

Let an ellipse and a hyperbola have same foci. If the length of conjugate axis of the hyperbola is equal to the length of minor axis of the ellipse, then the value of $\frac{1}{e_1^2}+\frac{1}{e_2^2}$ is $(e_1$ and $e_2$ denote the eccentricities of the two conics$)$

For which of the following curves, the line $x+\sqrt{3}y=2\sqrt{3}$ is the tangent at the point $(\frac{3\sqrt{3}}{2},\frac{1}{2})$ ?

Gravitational field in a region is given by vector $(4\hat{i}+\hat{j})$. Work done by this field is zero when particle is moved along line

The polynomial $x^6+4x^5+3x^4+2x^3+x+1$ is divisible by (where $\omega$ is one of the imaginary cube roots of unity)

Complete solution set $\left[{\cot }^{-1}x\right]+2\left[{\tan }^{-1}x\right]=0$, where $[.]$ denotes the greatest integer function, is

Which of the following functions are monotonically decreasing functions ?

The distance between the pair of parallel lines represented by $x^2+4xy+4y^2+3x+6y-4=0$ is ___ units

If $|{\log }_{2}x+1|+|1-({\log }_{2}x{)}^2|=|{\log }_{2}x+({\log }_{2}x{)}^2|,$ then the true set of values of $x$ is $\left\{\lambda \right\}\cup [\mu ,\infty ).$ Then

If $\tan \left(\frac{\pi }{2}\sin \theta \right)=\cot \left(\frac{\pi }{2}\cos \theta \right)$, then $\sin (\theta +\frac{\pi }{4})$ can be

1. undefined
2. undefined
3. undefined
4. undefined

If $f\left(x\right)=log\left(\frac{1+x}{1-x}\right)$, then $f\left(\frac{2x}{1+x^2}\right)$ is equal to

Given the function $f\left(x\right)=e^x+\ln (x+1)-ax$, where $a\in R$. If there exists two distinct roots $x_1,x_2(x_1<x_2)$ of $f^{\prime }\left(x\right)=0$, then

If the sum of m terms of an AP is equal to sum of n terms of AP then sum of m+n terms js

If $(a\sec \theta ,b\tan \theta )$ and $(a\sec \varphi ,b\tan \varphi )$ are the ends of a focal chord of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the value of $\tan \frac{\theta }{2}\cdot \tan \frac{\varphi }{2}$ can be equal to :

Be the position of the point (-3,-2) with respect to the circle whose equation is x2+ y2-3x+2y-19=0

Length of the sub-tangent at any point P(x, y) on the parabola $y^2=4ax$ equals _____ the abscissa of the point P

The difference between the maximum and minimum value of the expression $y=|x-9|-|x+2|$ is

A box contains I white and 3 identical black balls.
Two balls are drawn at random in succession without replacement.
Write the sample space for this experiment.

The term without x in the expansion of $(2x-\frac{1}{2x^2})$ is

Find the intervals in which the function f given by $f\left(x\right)=\frac{4sinx-2x-xcosx}{2+cosx}$ is
increasing

Find the intervals in which the function f given by $f\left(x\right)=\frac{4sinx-2x-xcosx}{2+cosx}$ is
decreasing

Let n be a fixed positive integer. Define a relation R in Z as follows $\forall a,b\in Z$, aRb if and only if a - b is divisible by n. Show that R is an equivalence relation.

In a non-zero G.P., if $T_{p-1}+T_{p+1}=3T_p,$ where $T_n$ denotes the $n^{th}$ term of the G.P., then the common ratio of the G.P. can be

Find the equations of tangent & normal at parametric point 'P' of the parabola $y^2=4ax.$

If there is a term containing $x^{2r}in{(x+\frac{1}{x^2})}^{n-3}$, then :

If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

Which one of the following curves cuts the parabola $y^2=4ax$ at right angles

If a = cos(2pi/7)+isin(2pi/7), L = a+a²+a⁴ and M = a³+a⁵+a⁶, then L,M are roots of the equation _________

In how many ways 4 persons can occupy 10 chairs in a row ,if no two sit on adjacent chairs.

Find the equation of a plane which is at a distance $3\sqrt{3}$ units from origin and the normal to which is equally inclined to coordinate axis.

If $6P\left(A\right)=8P\left(B\right)=14P(A\cap B)=1$, then $P\left(A^{\prime }|B\right)=$_______

Let $f\left(x\right)=e^{co{s}^{-1}sin(x+\frac{\pi }{3})}$,then