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Which of the following function are monotonic in the interval (0,1) ?

Consider set of observations $x_1,x_2,x_3,\cdots ,x_{101}$. It is given that $x_1<x_2<x_3\cdots <x_{100}<x_{101}$, then the mean deviation of the set of observations about $k$ is minimum when $k$ equals

Consider two families $A$ and $B.$ Suppose there are $4$ men, $4$ women and $4$ children in family $A$ and $2$ men, $2$ women and $2$ children in family $B.$ The recommended daily amount of calories is $2400$ for a man, $1900$ for a woman, $1800$ for a child and $45$ grams of proteins for a man, $55$ grams for a woman and $33$ grams for a child.

The requirement of calories and proteins for each person is given by matrix $R$ and the number of family members in each family is given by matrix $F.$



If $F^T$ represents the transpose of matrix $F,$ then $R+100F^T$ is equal to

The median $\overrightarrow{AD}$ of $△ABC$ is bisected at $E$ and $\overrightarrow{BE}$ is producted to meet the side $\overrightarrow{AC}$ in $F.$ Then the ratio $\overrightarrow{AF}:\overrightarrow{FC}$ is

The value of $\frac{\cos {105}^{\circ }+\sin {105}^{\circ }}{\cos {105}^{\circ }-\sin {105}^{\circ }}$ is

A bag initially contains one red and two blue balls. A trial consists of selecting a ball at random, noting its colour and replacing it together with an additional ball of the same colour. Three such trials are made. Let probability of event listed in column I is $\frac{\alpha }{\beta }$, where $\alpha$ and $\beta$ are coprime numbers. Match them with Column II
$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{(I) Atleast one blue is drawn}& \text{(P)}\beta -\alpha =1\\ \text{(I) Exactly one blue is drawn}& \text{(Q)}\beta -\alpha =4\\ \text{(III) Probability that drawn balls}& \text{(R)}\alpha +\beta =6\\ \text{are all red given that all drawn}& \\ \text{balls are same colour}& \\ \text{(IV) Atleast one red balls is drawn}& \text{(S)}\beta \text{is even}\\ & \text{(T)}\beta \text{is odd}\end{array}$

Which of the following is only CORRECT combination?

The value of $cos\left(\frac{1}{2}co{s}^{-1}\left(\frac{1}{8}\right)\right)$ is

In figure $A,B,C$ and $D$ are centre of 4 circles that each have a radius of length one unit. If a point is selected at random from the interior of a square $ABCD$. Then the probability that it will be chosen from shaded region is

$Ifxcos\theta =ycos(\theta +\frac{2\pi }{3})=zcos(\theta +\frac{4\pi }{3}),thenthevalueof\frac{1}{x}+\frac{1}{y}+\frac{1}{z}isequalto$

The standard equation of the circle whose parametric equation are $x=5-5\sin t$ and $y=4+5\cos t$ is

If $\lambda$ be the ratio of the roots of the quadratic equation in $x,3m^2{x}^2+m(m-4)x+2=0,$ then the least value of $m$ for which $\lambda +\frac{1}{\lambda }=1$, is :

Which of the following function(s) will have atleast one real root in $[0,\frac{\pi }{2}]$ ?

If $X=\{x:x\text{is a solution of}{x}^2+6x+9=0\},$ then which of the following is correct?

Which of the values cannot be the value of sin θ?

From the adjoining venn diagram, find $(A\cup B)\cap C$

If $\omega$ is a complex cube root of unity , then ${(1+{\omega }^2-\omega )}^6$ is equal to

The value of $\int \frac{x+5}{(x-2{)}^2}dx$ is :

If circles with radii $a$ units and $b$ units touch each other externally and the angle between their direct common tangents is $\theta$, where $a>b\ge 2$, then the value of $\sin \theta$ is

If $\theta$ is the acute angle between $y^2=x^3$ and $y=2x^2-1$ at $(1,1),$ then $\tan \theta$ is equal to

Differentiate given problems w.r.t.x.
$(sinx-cosx{)}^{(}sinx-cosx)$

The inverse of $f\left(x\right)={(5-(x-8{)}^5)}^{\frac{1}{3}}$ is

The number of solution(s) of the equation $3^{|x|}(|2-|x||)=1$ is

If two distinct chords of a parabola $y^2=4ax$, passing through (a, 2a) are bisected on the line $x+y=1$, then length of the latus-rectum can be

${lim}_{x\to 1}\frac{\sqrt{5x-4}-\sqrt{x}}{x-1}$

Match the equivalent pairs of multiplication and division equations.

Let a,b,c,be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x= cy +bz, y=az + cx and z = bx +ay. Then , a² + b²+ c²+ 2abc is equal to

How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times ?

Let $f:R^{+}\to \{-1,0,1\}$ defined by $f\left(x\right)=sgn(x-x^4+x^7-x^8-1)$ where sgn denotes signum function. Then $f\left(x\right)$ is