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The mean of a set of $30$ observations is $75$. If each observation is multiplied by a non-zero number $\lambda$ and then each of them is decreased by $25$, their mean remains the same. Then $\lambda$ is equal to

The power of P(1,2) with respect to the circle $x^2+y^2+4x+2y+3$ = 0 is double the power of P with respect to $x^2+y^2+2x+y+k$ = 0 then the value of k is

The minimum value of $ata{n}^{2}x+bco{t}^{2}x$ equals the maximum value of a $si{n}^2\theta +bco{s}^2\theta wherea>b>0,$ then

The number of real values of x for which the equality $|3x^2+12x+6|=5x+16$ holds good is

How many triangles can be formed by joining four points on a circle?

Mode of the data $3,2,5,2,3,5,6,6,5,3,5,2,5$ is

If $x=e^{y+e^{y+e^{y+e^{y+\cdots \infty }}}}$

Parametric form of a straight line passing through $(4,5)$ and making an angle ${60}^{\circ }$ with $x$-axis in the positive direction is

(where $\lambda$ be any parameter)

If a set $A$ has $n$ elements, then the total number of subsets of $A$ is

A variable plane moves so that the sum of reciprocals of its intercepts on the three coordinate axes is constant $\lambda .$ It passes through a fixed point, which has coordinates

The value of when m is equal to

Match the following by appropiately matching the lists based on the information given in Column I and Column II.

​​​​​​$\begin{array}{cc}Column1& Column2\\ \text{(a) The probability of a bomb hitting a bridge is}& \\ \frac{1}{2}.\text{Two direct hits are needed to destroy it.}& \\ \text{The number of bombs recquired so that the}& \\ \text{probability of the bridge being destroyed is}& \\ \text{greater than}0.9\text{can be}& \text{(p)}4\\ \text{(b) A bag contains}2\text{red,}3\text{white,}5\text{black balls, a}& \\ \text{ball is drawn its color is noted and replaced}& \\ \text{The number of times, a ball can be drawn}& \\ \text{so that the probability of getting a red ball}& \\ \text{for the first time is atleast}1/2& \text{(q)}6\\ \text{(c) A drawer contains a mixture of red socks}& \\ \text{and blue socks, at most}17\text{in all. It so}& \\ \text{happens that when two socks are selected}& \\ \text{randomly without replacement, there is a}& \\ \text{probability of exactly}1/2\text{that both are red}& \\ \text{or both are blue. Then number of red}& \\ \text{socks in drawer can be}& \text{(r)}7\\ \text{(d) There are two red, two blue, two white and}& \\ \text{certain number (greater than}0\text{) of green}& \\ \text{socks in a drawer. If two socks are taken at}& \\ \text{random from the drawer without}& \\ \text{replacement, the probability that they}& \\ \text{are of same color is}\frac{1}{5}\text{, then the number of}& \\ \text{green socks are}& \text{(s)}10\end{array}$

The solution set of the equation $\sin 3x+\cos 2x=-2$ is

(where $n\in Z$)

If $A$ and $B$ are finite sets and $A\subset B,$ then

The equation of the circle circumscribing the triangle formed by the line $x+y=6,2x+y=4$ and $x+2y=5$, is

If $\cos (4y-3x-2)-\cos (4y+3x+2)=2+2\ln (k^4-255)$ and $\cos (4y-3x-2)+\cos (4y+3x+2)=2k+8$ have real solutions $(x,y)$, then the range of $k$ is

If $p=2$ and $q=-5,$ then the distance between $p$ and $q$ along the number line is

If $\alpha =\cos \left(\frac{8\pi }{11}\right)+i\sin \left(\frac{8\pi }{11}\right),$ then $Re(\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4+{\alpha }^5)$ is equal to

Find the value of ${\log }_{e}144$.

Which of the following hold good?

Let $P=\left[a_{ij}\right]$ be a 3 $\times$ 3 matrix and let $Q=\left[b_{ij}\right]$, where $b_{ij}=2^{i+j}{a}_{ij}for1\le i,j\le 3$. If the determinant of P is 2, then the determinant of the matrix Q is

If a tangent to a parabola $y^2=4ax$ makes an angle of $\frac{\pi }{3}$ with the axis of the parabola. Then point of contact(s) is/are

A team of $12$ railway station masters is to be divided into two groups of $6$ each, one for day duty and the other for night duty. The number of ways in which this can be done if two specified persons $A$ and $B$ should not be included in the same group is

The value of ${\sin }^{-1}\sin \frac{36\pi }{7}+{\cos }^{-1}\sin \frac{39\pi }{7}$ is

In a certain city only two newspapers A and B are published, it is known that 25% of the city population reads A and 20% reads B, while 8% reads both A and B. It is also known that 30% of those who read A but not B look into advertisements and 40% of those who read B but not A look into advertisements while 50% of those who read both A and B look into advertisements. If a person is chosen at random from the population, what is the percentage probability that he/she reads advertisements?

If $x=111\dots \dots 1\left(20\text{digits}\right),$ $y=333\dots \dots 3\left(10\text{digits}\right)$ and $z=222\dots \dots 2\left(10\text{digits}\right)$, then $x-z$ is equal to

If $|4x-3|=|x+5|$, then $x$ is/are

If $A=\{a,c,e\}\text{and}B=\{a,b,c,d,e,f\}$ then the value of $A\Delta B$ is

A fair die is rolled. The probability that the first time 1 occurs at an even throw, is

The sum of real roots of the equation $(x-1)(x-2)(3x-1)(3x-2)=8x^2$ is

If ${\int }_0^{\alpha }\frac{dx}{1-\cos \alpha \cos x}=\frac{A}{\sin \alpha }+B(\alpha \ne 0)$ the values of A and B are

If $z_1$ and $z_2$ are two $n^{\text{th}}$ roots of unity, then $\arg \left(\frac{z_1}{z_2}\right)$ is a multiple of

If $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k},$ $\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}$ is a unit vector perpendicular to the vector $\overrightarrow{a}$ and coplanar with $\overrightarrow{a}$ and $\overrightarrow{b},$ then direction cosines of a vector which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{c}$ are

If a curve passes through the point $(1,-2)$ and has slope of the tangent at any point $(x,y)$ on it as $\frac{x^2-2y}{x}$, then the curve also passes through the point :

If f(y) =$e^y$, g(y) = y ; y > 0 and F(t) = ${\int }_0^{t}f(t-y)$ g(y) dy, then

Given f(x) defined on f: R $\to$ R is a periodic function with the fundamental period 2 then f(3) is not equal to -

In throwing a fair die find the probability of the event 'a number less than or equal to 4 turns up'

A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is 'a', the closest approach between two atoms in metallic crystal will be

Find the equation to the director circle of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

The mean deviation about the mean for the following data :

$5,6,7,8,6,9,13,12,15$ is :

Let $A$ and $B$ be two independent events such that $P\left(A\right)=\frac{1}{3}$ and $P\left(B\right)=\frac{1}{6}$. Then, which of the following is TRUE ?

Find the general value of ${log}_e$(i).