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Find the equation of the lines having slope -1 that are tangent to the curve $y=\frac{1}{x-1},x\ne 1$

$$\begin{array}{cc}\text{Trigonometric Equation}& \text{Principal Solutions}\\ 1.\sin x=\frac{\sqrt{3}}{2}& A.\frac{5\pi }{6}\\ 2.\tan x=\frac{-1}{\sqrt{3}}& B.\frac{5\pi }{12}\\ 3.\sec 2x=\frac{-2}{\sqrt{3}}& C.\frac{7\pi }{12}\\ 4.\cos 3x=\left(\frac{-1}{2}\right)& D.\frac{\pi }{3}\\ & E.\frac{2\pi }{9}\\ & F.\frac{4\pi }{9}\\ & G.\frac{11\pi }{6}\\ & H.\frac{2\pi }{3}\end{array}$$

The equation of the parabola with focus (0,0) and directrix x+y=4 is

In the given figure, if $A_1$ is the area of the $\Delta AOB$ and $A_2$ is the area of the parabolic region $AOB$ with $x-$axis, then the ratio $\frac{A_1}{A_2}$ is equal to

${lim}_{n\to \infty }{(1+\frac{x}{n})}^n$

From any point on the line $y=x+4$, tangents are drawn to the auxiliary circle of the ellipse $x^2+4y^2=4.$ If $P,Q$ are the points of contact and $A,B$ are the corresponding points of $P$ and $Q$ on the ellipse respectively, then the locus of the midpoint of $AB$ is

The value of integral $\underset{0}{\overset{\theta }{\int }}\ln (1+\tan \theta \tan x)dx,\theta \in (0,\frac{\pi }{2})$ is equal to

A tangent PT is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3},1)$. A straight line L, perpendicular to PT is a tangent to the circle $(x-3{)}^2+y^2=1$

A common tangent of the two circles is

Express the complex numbers in the form of a + ib:

$i^9+i^{19}$

The value of $\underset{n\to \infty }{lim}\prod _{r=1}^n{\left(\frac{n^2+r^2}{n^2}\right)}^{\frac{1}{n}}$ is:

Find the shortest
distance between the lines whose vector equations are

If

and A_ij is Cofactors of a_ij,
then value of Δ is given by

If the extremities of the latus rectum of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is $(\alpha ,\beta )$, then the distance between the point $P(1,1)$ and $(\alpha ,\beta )$, when $\alpha >0$ is/are

Let $x_n,y_n,z_n,w_n$ denote $n^{\text{th}}$ term of four different arithmetic progressions with positive terms. If $x_4+y_4+z_4+w_4=8$ and $x_{10}+y_{10}+z_{10}+w_{10}=20,$ then maximum possible value of $x_{20}\cdot y_{20}\cdot z_{20}\cdot w_{20}$ is

सूरज आसमान में दौड़ क्यों लगाता होगा?

The determinant of the matrix$\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right|$ is___

Let $\alpha \ne \beta$, ${\alpha }^2+3=5\alpha$ and ${\beta }^2=5\beta -3$. The quadratic equation whose roots are $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$ will be:

Let $z_1$ satisfy the condition $|z-3|=2$ and $z_2$ satisfy the equation $|z-1|+|z+1|=3$. If $|z_1-z_2{|}_{min}=m$ and $|z_1-z_2{|}_{max}=M$, then which of the following is/are correct?

Mark the correct alternative in each of the following :

In the sides of a triangle are in the ratio $1:\sqrt{3}:2,$ then the measure of its greatest angle is

If $p\to (\sim p\vee \sim q)$ is false, then the truth values of $p\text{and}q$ are respectively:

A question paper consisting of $10$ question is divided into $3$ parts with $5,3,2$ questions respectively. A candidate is to answer $6$ questions without neglecting a question from any part. The number of ways in which he can answer the paper is

The sum of two positive numbers is $6$ times their geometric mean. Then the ratio of these two numbers is

The condition on $a$ and $b$, such that the portion of the line $ax+by-1=0$, intercepted between the lines $ax+y=0$ and $x+by=0$, subtends a right angle at the origin is

$\text{Write the solution set of the equation}(2cos\theta +1)(4cos\theta +5)=0intheinterval[0,2\pi ].$

$IfA=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right],thenfornϵN,A^n=$