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The value of ${\sum }_{r=1}^n(-1{)}^{r-1}(1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{r}){}^n{C}_r$ is equal to

Two charges $q_1$ and $q_2$ are placed at -a and a on x-axis. List-1 represent some quantities and List-2 represent their corresponding graphs. Use sign convention. In list-1 E and V represent electric field and potential and x and y represent x, y coordinates. In list-2 y-axis of graph represent E or V and x-axis of graph represent x or y coordinates.
If $q_1=+Q$ and $q_2=+Q$ match correct column.

If $\overrightarrow{a}$,$\overrightarrow{b}$ and $\overrightarrow{c}$ are vectors such that $\overrightarrow{a}+$$\overrightarrow{b}+$$\overrightarrow{c}$$=\overrightarrow{0}$ and $|\overrightarrow{a}|=7,$$|\overrightarrow{b}|=5$ and $|\overrightarrow{c}|=3,$ then the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is

Two perpendicular unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are such that $\left[\overrightarrow{r}\overrightarrow{a}\overrightarrow{b}\right]=\frac{5}{4},$ $\overrightarrow{r}\cdot (3\overrightarrow{a}+2\overrightarrow{b})=0$ and $\underset{-2\overrightarrow{r}\cdot \overrightarrow{a}}{\overset{\frac{-4}{3}\overrightarrow{r}\cdot \overrightarrow{b}}{\int }}\frac{x+1}{x^2+1}dx=\frac{\pi }{2}.$ Then which of the following is(are) CORRECT ?

If $r,k,p\in W$, then $\sum _{r+k+p=10}{}^{30}{C}_r{\cdot }^{20}{C}_k{\cdot }^{10}{C}_p$ is

The value of $\sin {765}^{\circ }+\text{cosec}(-{1110}^{\circ })-\sin {405}^{\circ }+\cot {585}^{\circ }$ is equal to

Find the distance of the point (3, 5) from the line 2 x + 3 y = 14 measured parallel to a line having slope $1/2.$

Find the principal and general solutions of the equation

Let $z,w$ be complex numbers such that $\overline{z}+i\overline{w}=0$ and $\arg zw=\pi$. Then $\arg z$ equals

The number of ways of distributing $20$ identical fruits among $5$ people, so that no one receives less than $3$ fruits is

Let a complex number $z,|z|\ne 1$ satisfy ${\log }_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1{)}^2}\right)\le 2.$ then, the largest value of $|z|$ is equal to

1. -0.954

The minimum value of the function defined by f(x) = minimum {x, x+1,2 - x } is

If $Z^2+kZ+1-2i=0$ has roots as $z_1$ and $z_2$, where $z_1=-i$ and $k\in R$, then the value of $k$ is

A tangent is drawn to parabola $y^2-4x+4=0$ at a point $P$ which cuts the directrix at the point $Q$. If a point $R$ is such that it divides $QP$ externally in ratio $1:2$, then the locus of point $R$ is

If $I(m,n)={\int }_0^1{x}^{m-1}(1-x{)}^{n-1}dx$, then $(m,n\in I,m,n\ge 0)$

Let $z=\left(\frac{\cos \theta +i\sin \theta }{\cos \theta -i\sin \theta }\right),\frac{\pi }{4}<\theta <\frac{\pi }{2},$ then $\arg \left(z\right)$ will be

Find the effective resistance between $\text{A}$ & $\text{B}$.

A river 400 m wide is flowing at a rate of 2.0 m/s. A boat is sailing at a velocity of 10 m/s with respect to the water, in a direction perpendicular to the river.

(a) Find the time taken by the boat to reach the opposite bank.

(b)How far from the point directly opposite to the starting point does the boat reach the opposite bank ?

Let $f\left(n\right)$ denote the $n^{th}$ term of the sequence $3,6,11,18,27,...$ and $g\left(n\right)$ denote the $n^{th}$ term of the sequence $3,7,13,21,...$ . Let $F\left(n\right)$ and $G\left(n\right)$ denote the sum of $n$ terms of the above sequences, respectiveley. $\underset{n\to \infty }{lim}\frac{F\left(n\right)}{G\left(n\right)}=$

If $\cos (\alpha +\beta )=\frac{3}{5},\sin (\alpha -\beta )=\frac{5}{13}$ and $0<\alpha ,\beta <\frac{\pi }{4},$ then $\tan \left(2\alpha \right)$ is equal to:

1 + 3 + 6 + 10 + 15 + ...

Determine order and degree(if defined)
of differential equation

The asymptote of the curve $y^3=x^2(x-a)$ is

The angle (in degrees) subtended at the centre of a circle of diameter $50$ cm by an arc of length $11$ cm is