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Fourth term of ${(1-\frac{2x}{3})}^{\frac{3}{4}}$ is :

There are $15$ two bed room flats in a building and $10$ two bed room flats in other building and $8$ two bed room flats in a third building. The number of choices a customer will have for buying a flat is

If x<1 and y=x+x^2+x^3+..... show that: x=y/1+y

A line passing through $P(1,2)$ inclined at an angle $\theta$ with positive $x-$axis cuts the circle $x^2+y^2-6x-8y+24=0$ at $A$ and $B$ such that $PA+PB=4\sqrt{2}.$ Then the value of $\cos \theta +\sin \theta$ is

Find the roots of the equation $2x^2-5x+3=0$, by factorization.

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

The value of $\underset{-4}{\overset{4}{\int }}(x^3\sec x+3)\sqrt{16-x^2}dx$ is

If $\beta$ is one of the angles between the normals to the ellipse, $x^2+3y^2=9$ at the points $(3\cos \theta ,\sqrt{3}\sin \theta )$ and $(-3\sin \theta ,\sqrt{3}\cos \theta )$; $\theta \in (0,\frac{\pi }{2});$ then $\frac{2\cot \beta }{\sin 2\theta }$ is equal to

If $f\left(x\right)=2e^x-a{e}^{-x}+(2a+1)x-3$ is an increasing function for all real values of $x,$ then

If $\int \frac{1}{25{\cos }^{2}x+9{\sin }^{2}x}dx=\frac{1}{A}{\tan }^{-1}\left(B\tan x\right)+C,$ then the value of $AB$ is

(where $A,B$ are fixed constants and $C$ is integration constant)

If ${\cot }^{-1}\left(\alpha \right)={\cot }^{-1}\left(2\right)+{\cot }^{-1}\left(8\right)+{\cot }^{-1}\left(18\right)+{\cot }^{-1}\left(32\right)+\cdots$upto $100$ terms, then $\alpha$ is:

If $A=\left[\begin{array}{ccc}1& 2& x\\ 4& -1& 7\\ 2& 4& -6\end{array}\right]$ is a singlular matrix, then x =

Determine order and degree (when defined) of differential equations.
$\frac{d^{2}y}{d{x}^2}=cos3x+sin3x$

Let $x_1,x_2,...,x_n$ be n observations . Let $y_1=ax+b$ for i = 1, 2 ,...., n , where a and b are standard deviation of y1 is 15, the values of a and b are

If sin $\theta$, cos $\theta$ are the roots of the equation $x^2-\sqrt{2}x+\frac{1}{2}=0$, then $\theta$ is equal to

The equation of the circle with origin as centre passing the vertices of an equilateral triangle whose median

is of length 3a is

Solve the equation x² + 3x + 5 = 0

If $z=r{e}^{i\theta }$, then find the value of $|e^{iz}|$

If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p - q, q - r, r - s are in G.P.

If n is a natural number then ${\left(\frac{n+1}{2}\right)}^n$ ≥ n ! is true

when

If $\sqrt{9x^2+6x+1}<2-x,$ then integral values of $x$ is/are

How many five-digit number licence plates can be made if
(i) first digit cannot be zero and the repetition of digits is not allowed.
(ii) the first-digit cannot be zero, but the repetition of digits is not allowed?

The value of $\cos \left[{\tan }^{-1}\right(\tan 4\left)\right]$ is

If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are

Find a
particular solution of the differential equation,
given that y = 0 when x = 0

If the pairs of lines $x^2+2xy+a{y}^2=0$ and $a{x}^2+2xy+y^2=0$ have exactly one line in common then the joint equation of the other two lines is given by

Using
properties of determinants, prove that:

If the points (1, 1, p)
and (−3, 0, 1) be equidistant from the plane
,
then find the value of p.