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The $n^{\text{th}}$ term of the series
$1+4+13+40+121+364+...$ is :

Let $C$ be the set of all complex numbers. Let

$S_1=\{z\in C:|z-2|\le 1\}$ and

$S_2=\{z\in C:z(1+i)+\overline{z}(1-i)\ge 4\}$. Then the maximum value of ${|z-\frac{5}{2}|}^2$ for $z\in S_1\cap S_2$ is equal to

If $y={\log }_7(2x-3),$ then $\frac{dy}{dx}=$

If A(vector) = 3i(cap) + 5j(cap)

We sometimes use 3 + 5 = 8 and sometimes

√(3² + 5²) = 34

explain elaborately where to simply add and where to find resultant

If $tan\theta$ = $\frac{sin\alpha -cos\alpha }{sin\alpha +cos\alpha }$, then $sin\alpha +cos\alpha$ and

$sin\alpha -cos\alpha$ must be equal to

The value of $\left(\begin{array}{c}30\\ 0\end{array}\right)\left(\begin{array}{c}30\\ 10\end{array}\right)-\left(\begin{array}{c}30\\ 1\end{array}\right)\left(\begin{array}{c}30\\ 11\end{array}\right)+\left(\begin{array}{c}30\\ 2\end{array}\right)\left(\begin{array}{c}30\\ 12\end{array}\right)+\cdots +\left(\begin{array}{c}30\\ 20\end{array}\right)\left(\begin{array}{c}30\\ 20\end{array}\right)$

1. 10

The value of ${\sec }^{-1}\left(\sec 3\right)+{\cos }^{-1}\left(\cos 12\right)+{\text{cosec}}^{-1}\left(\text{cosec}6\right)+{\cot }^{-1}\left(\cot 10\right)$ is equal to

Two identical particles of charge $q$ each are connected by a massless spring of force constant $K$. They are placed over a smooth horizontal surface and the spring is unstretched. If the initial length of the spring is $r$ and the maximum extension of the spring is $r$ when the constrained is removed, then the value of $K$ is



$($Neglect gravitational effect$)$

The Taylor expansion of sin $x$ about $x=\frac{\pi }{6}$ by

A value of $\theta$ for which $\frac{2+3isin\theta }{1-2isin\theta }$ is purely imaginary, is:

Find the ratio in which the line segment joining the points, (2, -1, 3) and (-1, 2, 1) is divided by the plane $x+y+z=5$.

Let $x=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac{1}{2}$. If $P(1,\beta ),\beta >0$ is a point on this ellipse, then the equation of the normal to it at $P$ is

If $S_n={\cot }^{-1}\left(\frac{5}{\sqrt{3}}\right)+{\cot }^{-1}\left(\frac{9}{\sqrt{3}}\right)+{\cot }^{-1}\left(\frac{15}{\sqrt{3}}\right)+{\cot }^{-1}\left(\frac{23}{\sqrt{3}}\right)+\cdots$ upto $n$ terms, then

If y = f$(x^2+2)$ and $f^{\prime }$(3) =5, then $\frac{dy}{dx}$ at $x=1$ is _______.

The points $C$ and $D$ on a semicircle with $AB$ as diameter are such that $AC=1,CD=2$, and $DB=3$. Then the length of $AB$ lies in the interval

Show that ${lim}_{x\to 0}\frac{1}{x}$ does not exist.

The quadratic equation whose roots are $-4$ and $6$ is given by

The sum of non-integeral roots of the equation $x^4-3x^3-2x^2+3x+1=0$ is

The graph of $f\left(x\right)=e^{x^2}$, where $e>1$

Negation of the statement $\sim p\to (q\vee r)$ is