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$2ta{n}^{-1}\left[\sqrt{\frac{a-b}{a+b}}tan\frac{\theta }{2}\right]=$

[Dhanbad Engg. 1976]

$r^{th}$ term in the expansion of $(a+2x{)}^n$ is

Let a,b,c be the sides of a triangle where a$\ne b\ne c$ and $\lambda ϵ$R.If the roots of the equation $x^2+2(a+b+c)x+3\lambda (ab+bc+ca)=0$ are real. then

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{In an A.P., the series containing 99 terms, the sum of all}& \text{(P)}& 5010\\ & \text{the odd numbered terms is}2550.\text{The sum of all the}99\text{terms}& & \\ & \text{of the A.P. is}& & \\ \text{(B)}& f\text{is a function for which}f\left(1\right)=1\text{and}f\left(n\right)=n+f(n-1)& \text{(Q)}& 5049\\ & \text{for each natural number}n\ge 2\text{. The value of}f\left(100\right)\text{is}& & \\ \text{(C)}& \text{Suppose}f\left(n\right)={\log }_2\left(3\right)\cdot {\log }_3\left(4\right)\cdot {\log }_4\left(5\right)\dots {\log }_{n-1}\left(n\right).& \text{(R)}& 5050\\ & \text{Then the sum}\sum _{k=2}^{100}f\left(2^k\right)\text{equals}& & \\ \text{(D)}& \text{Concentric circles of radii}1,2,3,\dots ,100\text{cms}\text{are drawn. The}& \text{(S)}& 5100\\ & \text{interior of the smallest circle is coloured red and the annular}& & \\ & \text{regions are coloured alternately green and red, so that no}& & \\ & \text{two adjacent regions are of the same colour. The total area}& & \\ & \text{of the green regions in sq. cm is}k\pi .\text{Then}k\text{equals}& & \\ & & \text{(T)}& 5030\end{array}$



Which of the following is the only CORRECT combination?

Football teams $T_1$ and $T_2$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_1$ winning, drawing and losing a game against $T_2$ are $\frac{1}{2},\frac{1}{6}$ and $\frac{1}{3},$ respectively. Each team gets $3$ points for a win, $1$ point for a draw and $0$ point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T_1$ and $T_2,$ respectively, after two games.



$P(X=Y)$ is

If $\left[\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right]$ ; then for all



$\theta \in (\frac{3\pi }{4},\frac{5\pi }{4}),det\left(A\right)$ lies in the interval :

If $\cos B$ is the geometric mean of $\sin A$ and $\cos A$, where $0<A,B<\frac{\pi }{2}$, then the value(s) of $\cos 2B$ is/are

Given orthocentre $\overline{H}$ and circumcentre $\overline{C}$ for a triangle as $2\hat{i}+3\hat{j}and4\hat{i}+5\hat{k}$.Then the centroid of triangle can be given by,

If $\tan {40}^{\circ }+2\tan {10}^{\circ }=\cot x$, where $x\in (0,\pi /2)$, then the possible value of $x$ is

If a hyperbola passes through the point P $(\sqrt{2},\sqrt{3})$ and has foci $(\pm 2,0),$ then the tangent to this hyperbola at P also passes through the point

If $\alpha$ and $\beta$ are the roots of the equation $375x^2-25x-2=0,$ then $\underset{n\to \infty }{lim}\sum _{r=1}^n{\alpha }^r+\underset{n\to \infty }{lim}\sum _{r=1}^n{\beta }^r$ is equal to :

${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{e^x.se{c}^{2}xdx}{e^{2x}-1}$is equal to

Find the medians of following data sets.

Set I -- {3, 7, 2, 11, 8}

Set II -- {-1, 0, 1, 7, 11, 4}

In the expansion of ${(1-x-x^2+x^3)}^6,$ the sum of the coefficients of $x$ is

If tan ${35}^{\circ }$= k, then the value of $\frac{tan{145}^{\circ }-tan{125}^{\circ }}{1+tan{145}^{\circ }tan{125}^{\circ }}$=

The real values of $x$ satisfying ${\log }_{0.5}\left(\frac{x+1}{x+2}\right)\le 1$

If the Cartesian coordinates of a point are $(-3,-\sqrt{3})$, then the polar coordinates are

If

$f\left(x\right)=\{\begin{array}{cc}xsin\frac{1}{x}& x\ne 0\\ 0& x=0\end{array},$

then $\underset{x\to 0}{lim}f\left(x\right)=$

One vertex of the equilateral triangle with centriod at origin and one side as $x+y-2=0$ is

The equations of the tangnets to the ellipse $4x^2+3y^2=5$ which are perpendicular to the line $3x-y+7=0$ are :

If $z=-2+2\sqrt{3}i,$ then $z^{2n}+2^{2n}{z}^n+2^{4n}$ may be equal to

Which of the following is/are the definition of a simple event?

If a, b, c $ϵ$ R and a $\ne$ 0, c > 0, the graph of $f\left(x\right)$ = $a{x}^2+bx+c$ for which $f\left(x\right)=0$ has only imaginary roots, will look like

The equation of the pair of straight lines through origin, each of which makes as angle $\alpha$ with the line y = x, is

If $I=\sum _{k=1}^{98}\underset{k}{\overset{k+1}{\int }}\frac{k+1}{x(x+1)}dx,$ then

If two circles $x^2+y^2-2ax+c^2=0$ and $x^2+y^2-2by+c^2=0$ touch each other externally, then

Which among the following expressions are equivalent $(\forall n\in I)$.

xy plane divides the line joining the points $(2,4,5)$and$(-4,3,-2)$ in the ratio

The total number of ways in which $20$ different pearls of two colours can be set alternately on a necklace, there being $10$ pearls of each colour, is

For the equation $x^2+bx+c=0,$ if $1+b+c=0$ for all $b,c\in R,$ then roots are

If $|\sin x+\cos x|=|\sin x|+|\cos x|$, where $\sin x\ne 0,\cos x\ne 0$, then in which quadrant does $x$ lie?

Let $p,q,r$ be the roots of $x^3+2x^2+3x+3=0$, then which of following is/are correct?

Find the total number of 9 digit numbers which have all different digits.

If the tangents on the ellipse $4x^2+y^2=8$ at the point $(1,2)$ and $(a,b)$ are perpendicular to each other, then $a^2$ is equal to:

In a multiple choice question there are four alternative answers of which one or more than one is or are correct. A candidate will get marks on the question only if he ticks all correct answers. The candidate decides to tick answers at random. If he is allowed up to three chances to answer the question, the probability that he will get marks on it is given by

If the line $y-\sqrt{3}x+3=0$ cuts the curve $y^2=x+2$ at $A$ and $B,P$ is a point on the line whose ordinate is $0.$ Then $|PA.PB|=$

The sum of roots of the polynomial equation $(x-1)(x-2)(x-3)=2(x-2)(x-3)$ is

For an equilateral triangle, one side lies on $x+y=6$ and the $3^{rd}$ vertex is mirror image of the origin in the mirror $x+y=6$. Then the coordinates of other two vertices are

Identify the graph of $-|-x+3|$

The equation of straight line passing through the point $(3,6)$ and cutting $y=\sqrt{x}$ orthogonally is

Two people leave the same city to different destinations at the same time. Person $A$ leaves due east at a constant speed of $3\sqrt{2}$ kmph and person $B$ leaves due south east at a constant speed of $6$ kmph. How fast is the distance between them incresing $4$ hours later?

Graph of y = f(x) is given. Find the graph of |y| = f(x).

Which of the following describes a conic?

Here S is a fixed point and P is the moving point. 'e' is the eccentricity of the conic

The equation of the line which makes an angle of ${15}^{\circ }$ with positive $x$-axis and cuts-off an intercept of $3$ unit on the negative $y$-axis is

Find the value of ${\tan }^{-1}\left(\tan \left(-6\right)\right).$