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Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that

1) all vowels occur together

2) all vowels do not occur together.

If $\underset{n\to \infty }{lim}(\frac{\sqrt{n}}{\sqrt{n^3}}+\frac{\sqrt{n}}{\sqrt{(n+2{)}^3}}+\frac{\sqrt{n}}{\sqrt{(n+4{)}^3}}+\frac{\sqrt{n}}{\sqrt{(n+6{)}^3}}+\cdots +\frac{1}{\sqrt{27n^2}})=a-\frac{1}{\sqrt{b}}$ where $a,b\in N,$ then which of the following is/are true?

Let five letter words be formed using the letters of the word REPETITIVE. Then total number of words with exactly two pairs of alike letters is

if the equation of tangent to the circle $x^2+y^2-6x+4y-12=0$ which are parallel to the line 4x+3y+5=0 is ax+by+c=0,find the value of $-(\frac{a}{b}+\frac{c}{b})$, where $\left|\frac{c}{a}\right|>6$

If one root is square of the other root of the equation $x^2+px+q=0$, then the relation between p and q is

Two isolated north poles of pole strength $16\text{Am}$ and $4\text{Am}$ respectively are at $30\text{cm}$ apart in air. The distance of the neutral point from the stronger pole is (in $\text{cm}$)

If two jobs A and B can be done independentely by m and n ways respectiveley, then the number of ways by which job A and job B can be done?

Find the value of X and Y : 2+( x+iy) = 3+i

If $c_0,c_1,c_2,\cdots c_{15}$ are the Binomial co-efficients in the expansion of $(1+x{)}^{15}$ , then the value of $\frac{c_1}{c_0}+2\frac{c_2}{c_1}+3\frac{c_3}{c_2}+\cdots +15\frac{c_{15}}{c_{14}}$ is

In a $\Delta ABC$, if $A=\frac{\pi }{4}$ and $\tan B\tan C=k$

then $k$ must satisfy

If the equation $a_1$ + $a_{2}cos2x$ + $a_{3}si{n}^{2}x$ = 1 is satisfied by every real values of x, then the number of possible values of the triplet $(a_1,a_2,a_3)$

If $f\left(x\right)=x^{11}+x^9-x^7+x^3+1$ and $f\left({\sin }^{-1}\right(\sin 8\left)\right)=\alpha ,$ where $\alpha$ is constant, then $f\left({\tan }^{-1}\right(\tan 8\left)\right)$ is equal to

If $a{e}^x+b{e}^y=c,p{e}^x+q{e}^y=d$ and ${\Delta }_1=\left|\begin{array}{cc}a& b\\ p& q\end{array}\right|,{\Delta }_2=\left|\begin{array}{cc}c& b\\ d& q\end{array}\right|,{\Delta }_3=\left|\begin{array}{cc}a& c\\ p& d\end{array}\right|$ ,where ${\Delta }_1,{\Delta }_2,{\Delta }_3$ are all positive, then the value of $(x,y)$ is:

If : a+b+c=pie

then prove that : sin2A+sin2B+sin2C= 4sinAsinBsinC

Let \(n(U) = 700 , n(A) = 200 , n(B) = 300,

n(A∩B)=100, then n(A' ∩ B')=

Poor Dolly’s T.V. has only $4$ channels, all of them quite boring, hence it is not surprising that she desires to switch (change) channel after every one minute. Find the number of ways in which she can change the channels so that she is back to her original channel for the first time after $4$ minutes.

If R is a relation defined on the set Z of integers by the rule $(x,y)ϵR\iff x^2+y^2-9,$ then write domain of R.

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea ?

How many terms of the A.P.
are
needed to give the sum –25?

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



Consider a differentiable function $f$ satisfying the relation $f(x-y+1)=\frac{f\left(x\right)}{f(y-1)}$ for all $x,y\in R$ and $f^{\prime }\left(0\right)=2,$ $f\left(0\right)=1.$



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\int f\left(x\right)dx=\frac{2f\left(x\right)}{p}+C\text{where}C\text{is constant of integration,}& \text{(P)}& 1\\ & \text{then the value of}p\text{is}& & \\ \text{(B)}& \text{The value of}\frac{d^{10}}{d{x}^{10}}\left(f\left(\frac{x}{2}\right)\right)\text{at}x=0\text{is}& \text{(Q)}& 2\\ \text{(C)}& \text{The number of solutions of the equation}f\left(x\right)=x^2\text{is}& \text{(R)}& 3\\ \text{(D)}& \text{The value of}\underset{x\to 0}{lim}\frac{f\left(x\right)-f\left(x^2\right)}{\sin x}\text{is}& \text{(S)}& 4\end{array}$



Which of the following is a CORRECT combination?

The angle between the curves $x^2+y^2=12$ and $y^2-x^2=4$ at the point $(2,-2\sqrt{2})$

Find the numbers of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, if no digit is repreated? How many to these will be even?

The equation of the circle which orthogonally intersects the circles $x^2+y^2-2x+3y-7=0$, $x^2+y^2+5x-5y+9=0$ and $x^2+y^2+7x-9y+29=0$, is

Choose the correct options for $\sin {630}^{\circ }\&\cos {810}^{\circ }.$

Which of the following step can be read as ' vast last can aim zen 16 yet 33 87 82 54 49 ' for the given input?

The number of solution of the equation $sinx+sin2x+sin3x=cosx+cos2x+cos3x,0\le x\le 2\pi$ is