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${lim}_{x\to 0}\frac{xtanx}{1-cosx}$

If $2a=2\tan {10}^{\circ }+\tan {50}^{\circ }$
$2b=\tan {20}^{\circ }+\tan {50}^{\circ }$
$2c=2\tan {10}^{\circ }+\tan {70}^{\circ }$
$2d=\tan {20}^{\circ }+\tan {70}^{\circ }$,
then which of the following is/are correct ?

If $A=\left[\begin{array}{cc}1& 0\\ 1& 2\end{array}\right]$, then $A^2=$

If the eccentricity of the hyperbola$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is$\frac{5}{4}$ and $2x+3y–6=0$ is a focal chord of the hyperbola, then the length of transverse axis is equal to ____________

If $\omega$ is a complex cube root of unity, then the value of $\frac{a+b\omega +c{\omega }^2}{c+a\omega +b{\omega }^2}$ + $\frac{a+b\omega +c{\omega }^2}{b+c\omega +b{\omega }^2}$ is :

In the following hexagons, made up of two different material P and Q, current enters and leaves from points X and Y respectively. In which of the following case the magnetic field at its centre is not zero?

It is not true that ‘Roses are yellow implies violets are green’. The simplified form of this statement is

The number of values of x where the function f(x) = 2 (cos 3x + cos $\sqrt{3x}$ attains its maximum, is

The equation of normal at $(at,\frac{a}{t})$ to the hyperbola $xy=a^2$ is

Let (3, 4, -1) and (-1, 2, 3) be the end points of a diameter of a sphere.

Then, the radius of the sphere is equal to

$3x+9\ge -x+19$

If $y=ta{n}^{-1}\left(\frac{1}{1+x+x^2}\right)+ta{n}^{-1}\left(\frac{1}{x^2+3x+3}\right)+ta{n}^{-1}\left(\frac{1}{x^2+5x+7}\right)+\cdots \cdots \cdots$ n terms, then y'(0) is

The slope of the line touching both the parabolas $y^2=4x$ and $x^2=-32y$ is :

If a right circular cone, having maximum volume, is inscribed in a sphere of radius $3$ cm, then the curved surface area (in cm${}^2$) of this cone is :

If $\alpha +\beta =\frac{\pi }{2}$ and $\beta +\gamma =\alpha ,$ then tan $\alpha$ equal to

The number of ordered pairs $(x,y)$ satisfying the equation $x^2+2x\sin \left(xy\right)+1=0$ is
$(\text{where}y\in [0,2\pi \left]\right)$

The fractional part of a real number $x$ is $x-\left[x\right],$ where $\left[x\right]$ is the greatest integer less than or equal to $x.$ Let $F_1$ and $F_2$ be the fractional parts of ${(44-\sqrt{2017})}^{2017}\text{and}{(44+\sqrt{2017})}^{2017}$ respectively. Then $F_1+F_2$ lies between the numbers

Angle between two planes $a_{1}x+b_{1}x+c_{1}x+d_1=0$ & $a_{2}x+b_{2}x+c_{2}x+d_2=0$ is given by-

There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.

If $\underset{x\to \infty }{lim}\frac{1}{\ln x-x}=\frac{a}{b}$( where $b\ne 0)$ and the equation $a{x}^3+x^2+bx+1=0$ has equal roots, then the value of $|a|+|b|$ is

f(x) = $\{\begin{array}{c}{x}^{10}-1,ifx\le 1\\ x^2,ifx>1\end{array}$

Let $f:R\to R$ be such that $f\left(1\right)=3$ and $f^{\prime }\left(1\right)=6,$ Then ${lim}_{x\to 0}{\left(\frac{f(1+x)}{f\left(1\right)}\right)}^{\frac{1}{x}}$ is equal to

The coefficient of $\frac{1}{x}$ in the expansion of $(1+x{)}^n{(1+\frac{1}{x})}^n$ is

The equation of the parabola whose focus lies at the intersection point of the lines $x+y=3$ and $x-y=1$ and directrix is $x-y+5=0$

A game is played with a special fair cubic die which has one red side, two blue sides, and three green sides. The result is the colour of the top side after the die has been rolled. If the die is rolled repeatedly, the probability that the second blue result occurs on or before the tenth roll, can be expressed in the form $\frac{3^p-2^q}{3^r}$ where p, q, r are positive integers, If $p^2+q^2+r^2.=280+x$. Find x

If P $\left(A\right)=0.8,P\left(B\right)=0.5andP\left(\frac{B}{A}\right)=0.4,$ find
$P(A\cap B)$

$P\left(\frac{A}{B}\right)$

$P(A\cup B)$

The perpendicular distance from the points $\hat{i}+\hat{j}+\hat{k}and5i+5j$ to the plane given by $\overline{r}.(\hat{i}+\hat{j}+\hat{k})=10$ will be

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. $\underset{n\to \infty }{lim}\frac{F\left(n\right)}{G\left(n\right)}=$

The equation of the pair of tangents drawn from the point (4,3) to the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ is .

If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A,$ then

The distance between the directrices of the hyperbola $x=8sec\theta ,y=8tan\theta$,is

The value of $c$ in the Lagrange's mean value theorem for the function $f\left(x\right)=x^3-4x^2+8x+11$, where $x\in [0,1]$ is :

The vertices of the triangle are A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D.

Find the coordinates of D and the length AD.

If A, B, C are in A.P., then $\frac{sinA-sinC}{cosC-cosA}=$

A variable line $y=l\left(x\right)$ intersects the parabola $y=x^2$ at points $P$ and $Q$ whose $x$-coordinates are $\alpha$ and $\beta$ respectively with $\alpha <\beta$. The area of the figure enclosed by the segment $PQ$ and the parabola is always equal to $\frac{4}{3}$. The variable segment $PQ$ has its middle point as $M$. Then the value of $(\beta -\alpha )$ is greater than (or) equal to-

A circle of radius $7$ units touches the coordinate axes in the second quadrant. If the circle makes five complete rolls along the positive direction of $x-$axis, then the equation of circle in new position is
$(\text{Assume}\pi =\frac{22}{7})$

The value of $\sum _{k=1}^6(\sin \frac{2k\pi }{7}-\cos \frac{2k\pi }{7})$ is

A bag contains $30$ tokens numbered serially from $0$ to $29.$ The number of ways of selecting $3$ tokens from the bag, such that sum of numbers on them is $30$, is

By using properties of definite integrals, evaluate the integrals
${\int }_2^8|x-5|dx.$

A square piece of tin of side 18 cm is to be made into a box without top, by cutting-off square from each corner and folding up the flaps of the box. What should be the side of the square to be cut off so that the volume of the box is maximum possible?

If $2|x+2|-|x+5|\le 4$, then $x\in$

Three numbers are selected at random (without replacement) from first six positive integers. Let X denotes the largest of the three numbers obtained. Find the probability distribution of X. Also, find the mean and variance of the distribution.