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The A. M. of n observations is M. If the sum of n - 4 observations is a, then the mean of remaining 4 observations is

Which of the following limit is not in the indeterminant form ?

The value of integral $\underset{\pi /4}{\overset{3\pi /4}{\int }}\frac{x}{1+\sin x}dx$ is :

how is it written in logarithmic form

If $f\left(x\right)=\frac{t+3x-x^2}{x-4},$ where $t$ is a parameter and $f\left(x\right)$ has exactly one minimum and one maximum, then the range of values of $t$ is

For any real number $x$, let $\left[x\right]$ denotes the largest integer less than or equal to $x$. Let $f$ be a real valued function defined on the interval $[-10,10]$ by $f\left(x\right)=\{\begin{array}{c}x-\left[x\right],\text{if}\left[x\right]\text{is odd}\\ 1+\left[x\right]-x,\text{if}\left[x\right]\text{is even}\end{array}$

Then, the value of $\frac{{\pi }^2}{10}\underset{-10}{\overset{10}{\int }}f\left(x\right)\cos \pi xdx$ is

Find the values of other five trigonometric functions if , x lies in third quadrant.

Let R be a relation on the set N be defined by $\left\{\right(x,y)|x,y|N,2x+y=41\}$. Then R is

Let $(a,b)$ be a point on a circle which passes through $(-3,1)$ and touches the line $x+y=2$ at the point $(1,1).$ If maximum possible value of $a$ is $\alpha ,$ then a quadratic equation with rational coefficients whose one root is $\alpha ,$ is

The value of k, for which $(cosx+sinx{)}^2+ksinxcosx-1=0$ is an identity, is

Twelve persons are to be arranged around two round tables such that one table can accommodate seven persons and another table can accommodate five persons only.

The total number of ways in which these $12$ persons can be arranged is

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0,l^2+m^2-n^2=0$ is given by
[MP PET 1993; RPET 2001]

The number of integral values of $k$ for which $h\left(x\right)=\text{sgn}(x^2-2kx+\text{sgn}(k+2\left)\right)$ is continuous for all $x\in R$ is
[Note : $\text{sgn}\left(y\right)$ denotes the signum function of $y$.]

The number of integral values of $k$ for which $h\left(x\right)=\text{sgn}(x^2-2kx+\text{sgn}(k+2\left)\right)$ is continuous for all $x\in R$ is

[Note : $\text{sgn}\left(y\right)$ denotes the signum function of $y$.]

If $S_1,S_2,S_3--------S_n$ denotes the sum of 1, 2, 3 ----n terms of an A.P, having first term a. If $\frac{Skx}{Sx}$ k $\ne$ 1, is independent of x, then $S_1+S_2+S_3--------S_n$ =

Find the equation of the circle which touches the x-axis at a distance of 3 from the origin and cuts an intercepts of 6 on the y-axis.

A matrix of order 3 $\times$ 3 is formed by using the elements from {1, 2, ------, 9} without repetition. Then

Differentiate the
function with respect to x.

${\int }_0^{\pi }sin3xcos2xdx=$

Suppose that the foci of the ellipse are (f₁, 0) and (f₂, 0) where f₁ >0 and f₂ <0 .Let P₁ and P₂ be two parabolas with a common vertex at (0, 0) and with foci at (f₁, 0) and (2f₂, 0), respectively. Let T₁ be a tangent to P₁ which passes through (2f₂, 0) and T₂ be a tangent to P₂ which passes through (f₁, 0). If m₁ is the slope of T₁ and m₂ is the slope of T₂, then the value of is

The variance of first n natural numbers is

If $A=2si{n}^2\theta -cos2\theta$,then A lies in the interval

Parul and Vijay throw 3 dice in a single throw. It is known that Parul throws a total of 16. Find Vijay’s probability of getting a higher value.

If $A(a,0,0)$ and $B(0,b,0)$ are two vertices of a triangle whose third vertex lies on the curve $y^2=2x-3z$, then the locus of the centroid of the triangle is

Let $f\left(x\right)=\log \left({\log }_{1/3}\right({\log }_7(\sin x+a)\left)\right)$ be defined for every real values of $x$, then the range of $a$