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1. 140

$8sin\frac{x}{8}cos\frac{x}{2}cos\frac{x}{4}cos\frac{x}{8}$ is equal to

If E and $E_2$ are independent even, write the value of $P(E_1\cup E_2)\cap (\overline{E}\cap {\overline{E}}_2)$.

$se{c}^2\left[co{t}^{-1}\left(\frac{1}{2}\right)\right]+cose{c}^2\left[ta{n}^{-1}\left(\frac{1}{3}\right)\right]=$

If $I_m=\underset{0}{\overset{m\pi }{\int }}x\left|\sin x\right|e^{\cos 4x}dx$ and $J_n=\underset{0}{\overset{n\pi }{\int }}\left|\cos x\right|e^{\cos 4x}dx,$ where $m,n\in Z,$ then

(Here, $\left[k\right]$ denotes the greatest integer less than or equal to $k$.)

If centroid of the tetrahedron , whose vertices are given by (0,0,0), (a, 2, 3),(1, b, 2) and (2, 1, c) be (1, 2, –1), then distance of P(a,b,c) from origin is equal to

Range of the function $f\left(x\right)=|x-1|+|x-2|+|x+1|+|x+2|$ where $xϵ[-2,2]$, is

The line $x+y=a$, meets the axis of $x$ and $y$ at $A$ and $B$ respectively. A triangle $AMN$ is inscribed in the triangle $OAB$, $O$ being the origin, with right angle at $N$, $M$ and $N$ lie respectively on $OB$ and $AB$. If the area of the triangle $AMN$ is $\frac{3}{8}$ of the area of the triangle $OAB$, then $\frac{AN}{BN}$ is equal to

Let $R_1$ be a relation from $A=\{1,3,5,7\}$ to $B=\{2,4,6,8\}$ and $R_2$ be another relation from $B$ to $C=\{1,2,3,4\}$ as defined below:



(i) An element $x$ in $A$ is related to an element $y$ in $B$ (under $R_1$) if $x+y$ is divisible by 3.



(ii) An element $x$ in $B$ is related to an element $y$ in $C$ (under $R_2$) if $x+y$ is even but not divisible by 3.



Which is the composite relation $R_1{R}_2$ from$A$ to $C$ ?

1. 2.08

A natural number has prime factorization given by $n=2^x{3}^y{5}^z$, where $y$ and $z$ are such that $y+z=5$ and $y^{-1}+z^{-1}=\frac{5}{6},y>z$. Then the number of odd divisors of $n$, including $1$, is:

The value of determinant using Bagula method $\left|\begin{array}{ccc}3& -2& 3\\ 0& 2& 0\\ 2& -1& 3\end{array}\right|$ is

The perpendicular distance (in units) from origin to the obtuse angular bisector between the lines $\frac{x-1}{-1}=\frac{y+1}{2}=\frac{z-1}{1}$ and $\frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-1}{2}$ is :

The value of integral $\underset{0}{\overset{18.7}{\int }}\left\{x\right\}dx$ is equal to

(where $\{\cdot \}$ denotes the fractional part function )

Discuss applicability of Rolle's Theorem for x + |x| in interval [-1, 1]

If the given expression $x^2-(5m-2)x+(4m^2+10m+25)$ can be expressed as a perfect square, then the value(s) of $m$ is/are

Sum of the first n terms of the series
$\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\cdots$ is equal to

$\sqrt{-8-6i}$=

How is n!(n+1)= (n+1)!

If $k$ is a real constant and $A,B,C$ are variable angles such that $\left(\sqrt{k^2-4}\right)\tan A+k\tan B+\left(\sqrt{k^2+4}\right)\tan C=6k,$ then the minimum value of ${\tan }^{2}A+{\tan }^{2}B+{\tan }^{2}C$ is

${\int }_3^{10}\left[ln\right[x\left]\right]dx$= ___,where [.] is GIF.

Find the number of permutations of the letters of the word ALLAHABAD.

Four identical dice are rolled once. Probability that atleast 3 different numbers appear on them is

The domain of the definition of the function $\sqrt{{\log }_{10}\left(\frac{5x-x^2}{4}\right)}$ is

If $\alpha ,\beta \ne 0$ and $f\left(n\right)={\alpha }^n+{\beta }^n$ and $\left|\begin{array}{ccc}3& 1+f\left(1\right)& 1+f\left(2\right)\\ 1+f\left(1\right)& 1+f\left(2\right)& 1+f\left(3\right)\\ 1+f\left(2\right)& 1+f\left(3\right)& 1+f\left(4\right)\end{array}\right|=K(1-\alpha {)}^2(1-\beta {)}^2(\alpha -\beta {)}^2$, then K is equal to

${lim}_{x\to 0}\frac{\sqrt{2}-\sqrt{1+cosx}}{x^2}$

If the slope of one of the lines represented by $4a{x}^2+xy+4y^2=0$ is the square of the other then a equals :

Insert five numbers between 8 and 26 so that the resulting sequence is an A.P.

The value of $(1+\cos \frac{\pi }{8})(1+\cos \frac{3\pi }{8})(1+\cos \frac{5\pi }{8})(1+\cos \frac{7\pi }{8})$ is ____________.

The angle between the lines x = 2y = - 3z and - 4x = 6y = - z is: