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$ta{n}^{-1}2+ta{n}^{-1}3$ =

If equation of a plane is $\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=1$ the find the intercept made by this plane on z - axis?

$\begin{array}{cccc}& List-I& & List-II\\ P& P\left(t\right)=(\frac{1}{t^2+1},\frac{t}{t^2+1}).IfP\left(\alpha \right),B\left(\beta \right).C\left(\gamma \right)& 1.& 0\\ & \text{are the vertices of an equilateral triangle}& & \\ & (\alpha ,\beta ,gamma>0)\text{and its centroid is}(a,b)& & \\ & \text{then 2a+b=}\dots \dots & & \\ Q.& \text{If a complex number z satisfying}|z-2+i|\le 1,& 2.& 1^2\\ & \text{then the maximum distance of origin from}& & \\ & 4+i(2-z)is\dots \dots & & \\ R.& \text{Consider the curve}xy=25!\text{such that}(\alpha ,\beta )\text{is a}& 3.& 2^2\\ & \text{point on the curve},\alpha ,\beta ϵN.\text{Then the number of}& & \\ & \text{distinct ordered pairs}(\alpha ,\beta )\text{such that HCF}& & \\ & (\alpha ,\beta )=1is{2}^{k}thenk=\dots \dots & & \\ S.& If{2}^{(1+cos\pi x)lo{g}_2^5}+2^{x^2-1}+2^{2(1-|x|)}=3,& 4.& 3^2\\ & \text{then sum of the roots is}\dots \dots & & \end{array}$

Find the lines through the point (0, 2) making angles $\frac{\pi }{3}and\frac{2\pi }{3}$ with the x-axis. Also, find the lines parallel to them cutting the y-axis at a distance of 2 units below the origin.

A point traversed half of the distance with a velocity $v_0$. The remaining part of the distance was covered with velocity $v_1$ for one-third of the remaining time and with velocity $v_2$ for the remaining time. The mean velocity of the point averaged over the whole time of motion

If $A(\alpha ,\beta )=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0\\ -\sin \alpha & \cos \alpha & 0\\ 0& 0& e^{\beta }\end{array}\right]$, then

The contrapositive of $a\to (\sim b\to c)$ is

Directions:- Study the following information carefully to answer the question given below.

Vikas, Uday and Tushar are seated around a regular hexagonal table facing the centre. Amar, Bimla and Chetan are also seated around the same table but one of them is facing outside. Vikas is second to the left of Chetan. Uday is second to the right of Amar. Bimla is the third to the left of Tushar. Amar is seated next to Vikas.

Who two are facing the same direction?

Sum of series ${\cot }^{-1}\left(\frac{5}{\sqrt{3}}\right)+{\cot }^{-1}\left(\frac{9}{\sqrt{3}}\right)+{\cot }^{-1}\left(\frac{15}{\sqrt{3}}\right)+{\cot }^{-1}\left(\frac{23}{\sqrt{3}}\right)+....\infty =$

If $cos(A+B)sin(C-D)=cos(A-B)sin(C+D)$, then write the value $tanAtanBtanC$

The solution of $\frac{d^{3}y}{d{x}^3}-8\frac{d^{2}y}{d{x}^2}=0$ satisfying $y\left(0\right)=\frac{1}{8},y^1\left(0\right)=0$ and $y^2\left(0\right)=1$ is [here $y^n\left(x\right)=\frac{d^{n}y}{d{x}^n}]$

Find the sum of the following series:
${\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{2}{9}+{\tan }^{-1}\frac{4}{33}+....+{\tan }^{-1}\frac{2^{n-1}}{1+2^{2n-1}}$

Let $X$ be a set containing $10$ elements and $P\left(X\right)$ be its power set. If $A$ and $B$ are picked up at random from $P\left(X\right)$, with replacement, then the probability that $A$ and $B$ have equal number of elements, is :

Consider the equation $(m^2+1)x^2-3x+(m^2+1{)}^2=0.$ Let $p$ be the least value of product of roots and $q$ be the greatest value of sum of roots of the equation. Then the sum of an infinitely decreasing G.P. whose first term is equal to $p+2$ and the common ratio is $\frac{2}{q},$ is

The area bounded by $y=\sin \left(\frac{\pi }{2}x\right)$, $x=0,y=0$ and $x=\frac{4}{3}$ is

If $y=|\cos x|+|\sin x|$ then $\frac{dy}{dx}$ at $x=\frac{2\pi }{3}$ is:

If ${\left(\frac{a}{b}\right)}^{\frac{1}{3}}+{\left(\frac{b}{a}\right)}^{\frac{1}{3}}=4$, then the acute angle $\left(\theta \right)$ of intersection of the parabolas $y^2=4ax$ and $x^2=4by$ at a point other than the origin is

Let $a_1,a_2,a_3,\dots$ be a G.P. such that $a_1<0,a_1+a_2=4$ and $a_3+a_4=16$. If $\sum _{i=1}^9{a}_i=4\lambda$, then $\lambda$ is equal to :

If ${\theta }_1$ and ${\theta }_2$ be respectively the smallest and the largest values of $\theta$ in $(0,2\pi )-\left\{\pi \right\}$ which satisfy the equation, $2{\cot }^2\theta -\frac{5}{\sin \theta }+4=0$, then $\underset{{\theta }_1}{\overset{{\theta }_2}{\int }}{\cos }^{2}3\theta d\theta$ is equal to :

Let $y=\frac{1}{1+x+\ln x}$, Then

If $f\left(x\right)=\int \frac{5x^8+7x^6}{(x^2+1+2x^7{)}^2}dx$ , if f(0) = 0, then the value of f(1) is

In $△ABC$ with usual notations, if $2a^2+4b^2+c^2-4ab-2ac=0$, then $\cos A+\cos B+\cos C$ is equal to:

If the angle between the two lines represented by $2x^2+5xy+3y^2+6x+7y+4=0$ is ${\tan }^{-1}m$, then m =

Given that P (3, 2, -4), Q(5, 4, -6) and R (9, 8, -10) are collinear. Find the

ratio in which Q divides PR.

System of equation

$x+3y+2z=6$

$x+\lambda y+2z=7$

$x+3y+2z=\mu$ has

If three positive real numbers $a,b,c$ are in A.P. such that $abc=4$, then the minimum value of $b$ is

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

By using the concept of slope, show that the points (-2, -1), (4, 0), (3, 3) and (-3, 2) are the vertices of a parallelogram.

If $3sin\theta +5cos\theta =5$, then write the value of $5sin\theta -3cos\theta .$

Find the value of other five trigonometric functions if sec x = $\frac{13}{5}$, and x lies in fourth quadrant.

$\int \sqrt{5-2x+x^2}dx$

If $1<x<\sqrt{2}$, then number of solutions of the equation $ta{n}^{-1}(x-1)+ta{n}^{-1}x+ta{n}^{-1}(x+1)=ta{n}^{-1}3x$, is

Evaluate the following definite integrals as limit of sums.
${\int }_a^{b}xdx.$

Find the coefficient of:

$\left(i\right)x^{10}$ in the expansion of ${(2x^2-\frac{1}{x})}^{20}$

$\left(ii\right)x^7$ in the expansion of ${(x-\frac{1}{x^2})}^{40}$

$\left(iii\right)x^{-15}$ in the expansion of ${(3x^2-\frac{a}{3x^3})}^{10}$

$\left(iv\right)x^9$ in the expansion of ${(x^2-\frac{1}{3x})}^9$

$\left(v\right)x^m$ in the expansion of ${(x+\frac{1}{x})}^n$

(vi)x in the expansion of $(1-2x^3+3x^5){(1+\frac{1}{x})}^8$

$\left(vii\right)a^5{b}^7$ in the expansion of $(a-2b{)}^{12}.$

(viii)x in the expansion of $(1-3x+7x^2)(1-x{)}^{16}$

Show that $|a|b+|b|a$ is perpendicular to $|a|b-|b|a$ for any two non-zero vectors a and b.

$\text{If in any}\Delta ABC,\angle C={105}^{\circ },\angle B={45}^{\circ },a=2,\text{then find b.}$

If $\sin A+{\sin }^{2}A=1$ and $a{\cos }^{12}A+b{\cos }^{10}A+c{\cos }^{8}A+d{\cos }^{6}A-1=0$ then $a+b+c+d=$

$\text{If}y=acos\left(logx\right)+bsin\left(logx\right)then{x}^2{y}_2+x{y}_1=$

Suppose A is a non-singular matrix such that $A^3-3A^2+6A-I=0.Then,A^{-1}$=___