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An infinite number of tangents can be drawn from (1, 2) to the circle $x^2+y^2-2x-4y+\lambda =0,$ then $\lambda =$

If $p,p^{\prime }$ denote the lengths of the perpendiculars from the focus and the centre of an ellipse whose semi major axis is of length $a$ units on a tangent at a point on the ellipse and $r$ denotes the focal distance of the point, then

Find the equation of the circle circumscribing the triangle formed by the lines $L_1=0,L_2=0$and $L_3=0$

Choose a proper fraction out of the following-

The value of $\underset{0}{\overset{2\pi }{\int }}\frac{x{\sin }^{8}x}{{\sin }^{8}x+{\cos }^{8}x}dx$ is equal to :

$li{m}_{\pi \to \infty }{\sum }_{k=1}^n\frac{k}{n^2+k^2}$ is equals to [Roorkee 1999]

Which of the following quantities are vectors.

A set of integers is given as (3,6,8,14,17). What is the probability that a triangle can be constructed.?

The equation of the tangents to the ellipse $3x^2+4y^2=12$ which are parallel to the line $2x-y+5=0$ are

If ln(a+c) , ln(a-c) , ln(a-2b+c) are in A.P, then

Of the 25 questions in a unit, a student has worked out only 20. In a sessional test of that unit, two questions were asked by the teacher, The probability that the student can solve both the questions correctly, is

Let $x_1,x_2,\dots ,x_{100}$ be $100$ observations such that $\sum _{i=1}^{100}{x}_i=0,$ $\sum _{1\le i<j\le 100}|x_i{x}_j|=80000$ and mean deviation from their mean be $5.$ Then their standard deviation is

Solution of the differential equation $\frac{\sqrt{x}dx+\sqrt{y}dy}{\sqrt{x}dx-\sqrt{y}dy}=\sqrt{\frac{y^3}{x^3}}$ is given by

Number of ways of selecting none or more of $10$ identical things is

In a game, a man wins Rs. $1000$ if he gets an even number $\ge 4$ on a fair die and loses Rs. $200$ for getting any other number on the die. If he decides to throw the die until he wins or maximum of three times, then his expected gain/loss (in Rupees) is

If $\sqrt{1-c^2}=nc-1$ for all permissible values of $c$ and $n$, where $z=e^{i\theta },$ then $\frac{c}{2n}(1+nz)(1+\frac{n}{z})$ is equal to

Which of the following graphs represents $f\left(x\right)=|x-2|-|x+6|$?

The plane XOZ divides the join of (1, -1, 5) and (2, 3, 4) in the ratio $\lambda :1$ then $\lambda$ is [JET 1988]

Let a, b, c be such that (b+c) ≠0. If

Then, the value of 'n' is

If $\frac{(1+4p)}{4},\frac{(1-p)}{2},\text{and}\frac{(1-2p)}{2}$ are the probabilities of three mutually exclusive events , then the value of p is

If a couple dance competition happens between $7$ married couples, then the number of such possible pairs that can be made such that no couple dances in the same pair is

Equation of common tangent of $y=x^2,y=-x^2+4x-4$ is

The range of $x$ satisfying $3^x+2^{2x}\ge 5^x$ is

A square matrix A is said to be a symmetric matrix if

Which of the following cases will not lead to non trivial solutions in case of system of linear equations according to Cramer's rule Convention given D != 0 ?

If $f\left(x\right)$ is a quadratic polynomial such that graph of $y=f\left(x\right)$ touches at $(4,0)$ and intersects the positive $y-$axis at $4$, then which of the following is/are correct?

The equation of the tangent to the curve $y=1-e^{x/2}$ at the point of intersection with the y- axis is

Derivative of the function, $f\left(x\right)=2\ln {\left(x\right)}^5+2\ln {\left(x\right)}^3+\ln {\left(x\right)}^2-\ln \left(x\right)+2x$ is

$\left[\begin{array}{ccc}3& -1& 2\\ -3& 1& 2\\ -6& 2& 4\end{array}\right]$What is the rank of the matrix.

If $5x-3\ge 3x-5$ and $x\in R^{-}$, then $x\in$

If matrix $A$ is given by $A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right],$ then the determinant of $A^{2005}-6A^{2004}$ is

Tangents are drawn from the origin to the curve y = sin x, then their point of contact lie on the curve

Find the Derivative of Sin x.

The value of $x$, for which the $6th$ term in the expansion ${\{2^{{\log }_2\sqrt{(9^{x-1}+7)}}+\frac{1}{2^{\frac{1}{5}{\log }_2(3^{x-1}+1)}}\}}^7\text{is}84$, is equal to

Let R be an equivalence relation on a finite set A having n elements. Then the number of ordered pairs in R is

Five persons entered the lift cabin on the ground floor of an 8 floor house. Suppose that each of them independently and with equal probability, can leave the cabin at any floor beginning with the first. Find out the probability of all five persons leaving at different floors.

A car goes 5 km east, 3km south, 2km west and 1km north. The magnitude of the resultant displacement will be ___ km at an angle of $ta{n}^{-1}$(- / -) with the positive x-axis (where x axis is in the east direction) in the clock wise direction.

If $0\le x\le 3\pi ,0\le y\le 3\pi$ and $\cos x\sin y=1$, then the possible number of values of the ordered pair $(x,y)$ is

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three non-coplanar vectors, then $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\cdot \left[\right(\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}+\overrightarrow{c}\left)\right]$ equals

The value of $P$ for which the equation $(P^3-3P^2+2P)x^2+(P^3-P)x+P^3+3P^2+2P=0$ has both the roots at infinity is

If ${\log }_{2}x+{\log }_{8}x+{\log }_{64}x=3$, then the value of $x$ is

In a race between Achilles and tortoise, people assigned probability to Achilles winning and tortoise winning. These probability pairs are listed below. How many of these pairs satisfy the axiomatic approach, assuming only two results are tortoise wins and Achilles wins.