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The number of ways in which the letters of the word ${}^{\prime }MADHUR{I}^{\prime }$ can be arranged so that vowels always occupy the beginning, middle and end places is

$\int \frac{dx}{sinx.sin(x+\alpha )}$ is equal to

A square matrix is having p number of rows. What is ‘j’ if $a_{ij}$ is situated in the last row and second last column

The value of $2^{\frac{1}{4}}\cdot 4^{\frac{1}{8}}\cdot 8^{\frac{1}{16}}\cdot {16}^{\frac{1}{32}}\dots \dots$ is

The value of $\text{cosec}\theta +\sec (270°-\theta )-\text{cosec}(270°+\theta )+\sec (180°-\theta )$ is

If $V=\frac{4}{3}\pi r^3$,at what rate in cubic units is V increasing when r = 10 and $\frac{dr}{dt}=0.01$ ?

Let $f$ be a differentiable function from $R$ to $R$ such that $|f\left(x\right)-f\left(y\right)|\le 2|x-y{|}^{3/2}$, for all $x,y\in R$. If $f\left(0\right)=1$, then $\underset{0}{\overset{1}{\int }}{f}^2\left(x\right)dx$ is equal to :

If $\underset{0}{\overset{x}{\int }}f\left(t\right)dt=x^2+\underset{x}{\overset{1}{\int }}{t}^{2}f\left(t\right)dt$, then $f^{\prime }\left(\frac{1}{2}\right)$ is:

If three complex numbers are in A.P., then they lie on

The value of $3^{\sqrt{{\log }_{3}4}}$ is equal to

If A.M. and H.M. of the roots of a quadratic equation are $8$ and $5$ respectively, then the equation is

The value of x (x > 0) for which $\tan \left({\sec }^{-1}\frac{1}{x}\right)=\sin \left({\tan }^{-1}2\right)$ is



x (x > 0) के किस मान के लिए $\tan \left({\sec }^{-1}\frac{1}{x}\right)=\sin \left({\tan }^{-1}2\right)$ है?

The value of $\cos (\frac{\pi }{4}+A)\cos (\frac{\pi }{4}-B)-\sin (\frac{\pi }{4}+A)\sin (\frac{\pi }{4}-B)$ is

The vector equation of the plane passing through the intersection of the planes $\overrightarrow{r}\cdot (\hat{i}+\hat{j}+\hat{k})=1$ and $\overrightarrow{r}\cdot (\hat{i}-2\hat{j})=-2,$ and the point $(1,0,2)$ is

In the grid given below, we wish to go from corner $A$ to corner $B$ moving up and right only one unit at a time. The number of paths that include an edge of the shaded square is





(correct answer + 2, wrong answer - 0.50)

$\frac{d}{dx}\left\{x^{1/x}\right\}=$

If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (\alpha +2)x+\sin x}{x},& x<0\\ b,& x=0\\ \frac{(x+3x^2{)}^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}},& x>0\end{array}$ is continuous at $x=0$ then $a+2b$ is equal to :

Which is the following is true about addition of matrices.

For the curve represented implicitly as $3^x-2^y=1$, the value of $\underset{x\to \infty }{lim}\left(\frac{dy}{dx}\right)$ is

A pie chart is to be drawn for representing the following data

$\begin{array}{cc}\text{Items of expenditure}& \text{Number of families}\\ \text{Education}& 150\\ \text{Food and clothing}& 400\\ \text{House rent}& 40\\ \text{Electricity}& 250\\ \text{Miscellaneous}& 160\end{array}$

The value of the central angle for food and clothing would be

The value(s) of $x$ satisfying $|x+3|=x^2-4x-3$ is/are

If $Q$ be a point on the parabola $y^2=8x$ and $P(-2,0)$ be a point in the $xy$ plane. If the locus of the mid point of $PQ$ is a parabola, then its focus is

The total number of $6$ digit numbers that can be made using digits $1,2,3,4$, if all the digits should appear in the number at least once is

If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2+y^2+4x-6y=12$ externally at the point $(1,-1)$ then the radius of $C$ is :

If the roots of the equation $a{x}^2+bx+c=0$ be$\alpha and\beta$,, then the roots of the equation $c{x}^2+bx+a=0$ are

Let $f\left(x\right)=max(x+\left|x\right|,x-\left[x\right])$ where [x] = the greatest integer in $x\le x$. Then ${\int }_{-2}^{2}f\left(x\right)dx$ is equal to

If $g$ is the inverse of a function $f$ and $f^{\prime }\left(x\right)=\frac{1}{1+x^5}$, then $g^{\prime }\left(x\right)$ is equal to:

Let $n(A–B)=25+X,n(B–A)=2X$ and $n(A\cap B)=2X.$ If $n\left(A\right)=2\left(n\right(B\left)\right),$ then $X$ is

The value of ${\cos }^3\frac{\pi }{8}\cos \frac{3\pi }{8}+{\sin }^3\frac{\pi }{8}\sin \frac{3\pi }{8}$ is :

If z=r$e^{i\theta }$,then |$e^{iz}$|=

Normal equations to parabola $y^2=4ax$ passing through point $(5a,2a)$ are

Graph of f(x) is given. Find the graph of $f\left(x\right)-2$

The number of solutons of the equation tan x +sec x = 2cos x lying in the interval $[0,2\pi ]$ is

The sum of all values of $\theta \in (0,\frac{\pi }{2})$ satisfying ${\sin }^{2}2\theta +{\cos }^{4}2\theta =\frac{3}{4}$ is :

Let the tangent drawn at $(-1,2)$ to the circle $x^2+y^2-3x-3y-2=0$ is normal to the circle $x^2+y^2-2ay+b=0.$ If the radius $\left(r\right)$ of the second circle is such that $\left[r\right]=1,$ then

([.] denotes the greatest integer function)

Let $y=mx+{\lambda }_i,i=1,2,3,...,n$ be a family of $n$ parallel lines subjected to following conditions.

$\left(1\right)m$ being a constant

$\left(2\right)\sum _{i=1}^n{\lambda }_i=1$

A variable line through origin intersects the lines at $P_i(i=1,2,3,...,n)$ and $Q$ be a point on variable line such that $\sum _{i=1}^{n}O{P}_i=OQ$. If the locus of $Q$ is a straight line which passes through a fixed point $(a,b)\forall m\in R,$ then the value of $(3a+2b)$ is

A box, constructed from a rectangular metal sheet, is $21$ cm by $16$ cm by cutting equal squares of side $x$ cm from the corners of the sheet and then turning up the projected portions. The value of $x$ (in cm) so that volume of the box is maximum is

Acute angle bwtween the lines $ax+by+c=0$ and $x\cos \theta +y\sin \theta =c(c\ne 0)$ is ${45}^{\circ }.$ If both the lines meet with the line $y\cos \theta =x\sin \theta$ at same point, then the value of $a^2+b^2$ is

The range of $a$ for which the equation $x^2+ax-4=0$ has its smaller root in the interval $(-1,2)$ is

Let $S$ be the set of all points in $(-\pi ,\pi )$ at which the function, $f\left(x\right)$ = min $\{\sin x,\cos x\}$ is not differentiable. Then $S$ is a subset of which of the following?

If the equations $k(6x^2+3)+rx+(2x^2-1)=0$ and $6k(2x^2+1)+px+(4x^2-2)=0,k\ne 0$ have both roots common, then the value of $\frac{p}{r}$ is

If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is:

Which of the following statements are correct regarding the function f(x) = $\sqrt{x}$

The equation of tangent to the parabola $y^2=6x$ at point $(6,6)$ is

A party of $9$ persons are to travel in two vehicles, one of which will not hold more than $7$ and other not more than $4.$ The number of ways the party can travel is

The set of all points, where the function $f\left(x\right)=\frac{x}{1+\left|x\right|}$ is differentiable is