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Let $P=\{x:x\in N\text{and}\frac{7}{x+2}>1\}$ and $Q=\{x:x\in N\text{and}|x-1|<2\}.$ Then $n\left(P\Delta Q\right)$ is

The least value of 'a' for which $\frac{4}{sinx}+\frac{1}{1-sinx}=a$ has at least one solution in the interval $(0,\pi /2)$ 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 volume of parallelopipped formed by following 3 vectors will be___cubic units.

$\overrightarrow{a}=3\hat{i}-2\hat{j}+5\hat{k}\overrightarrow{b}=2\hat{i}+2\hat{j}-\hat{k}\overrightarrow{c}=-4\hat{i}+3\hat{j}+2\hat{k}$

Let $\omega$ be a complex number such that $2\omega +1=z$ where $z=\sqrt{-3}$. If $\left|\begin{array}{ccc}1& 1& 1\\ 1& -{\omega }^2-1& {\omega }^2\\ 1& {\omega }^2& {\omega }^7\end{array}\right|=3k$, then $k$ is equal to:

If $f\left(x\right)={\int }_0^x(1+t^3{)}^{\frac{-1}{2}}$ and g (x) is the inverse of f, then the value of $\frac{g"\left(x\right)}{g^2\left(x\right)}$ is

Let $\overrightarrow{\alpha }=(\lambda -2)\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{\beta }=(4\lambda -2)\overrightarrow{a}+3\overrightarrow{b}$ be two given vectors where vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-collinear. The value of $\lambda$ for which vectors $\overrightarrow{\alpha }$ and $\overrightarrow{\beta }$ are collinear, is:

$\int si{n}^3\left(x\right).co{s}^5\left(x\right)dx$

Find the point where the graph of the function Sgn (lnx) breaks (or becomes discontinuous)

(Sgn is the Signum function)

The locus of the point which is equidistant from the points $A(2,3)$ and $B(-3,4)$ is

The number of ordered pairs $(x,y)$ satisfying $|x|+|y|=3$ and $\sin \left(\frac{\pi x^2}{3}\right)=1$ is less than equal to

If the diagonals of the parallelogram whose sides are $lx+my+n=0,lx+my+n^{\prime }=0$ and $mx+ly+n=0,mx+ly+n^{\prime }=0$ includes an angle $\theta$, then the value of $\theta$ is

The shortest distance between the lines

$\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and

$\frac{x-3}{-3}=\frac{y-7}{2}=\frac{z-6}{4}$

If $xϵR$, the solution set of the equation

$4^{-x+0.5}-{7.2}^{-x}-4<0$ is equal to

The tangent to the curve $y=x{e}^{x^2}$ is passing through the point $(1,e)$ and also passes through the point :

The new coordinates of a point (4, 5), when the origin is shifted to the point (1,-2) are

The domain of the function $\sqrt{\left(lo{g}_{0.5}x\right)}$ is

The orthocenter of the triangle formed by the lines $xy=0$ and $x+y=1$ is

If $y=\cot \theta ({\sin }^2\theta +\sin \theta \cos \theta )$, then

(where $\theta \ne n\pi ,n\in Z$)

If $\alpha ,\beta$ are complex cube roots of unity, then the value of $\frac{a+b\alpha +c\beta }{a\alpha +b\beta +c}$ can be

If $N$ is the set of all Natural Numbers, $W$ is the set of all Whole Numbers, $Z$ is the set of all Integers, $Q$ is the set of all Rational Numbers, $P$ is the set of all Irrational Numbers, $R$ is the set of all Real Numbers, then choose the correct option(s).

Equation of the line drawn through the point $(1,0,2)$ and perpendicular to the line $\frac{x+1}{3}=\frac{y-2}{-2}=\frac{z+1}{-1},$ is

${\int }_1^{a}x.a^{-\left[lo{g}_{a}x\right]}dx=\frac{e-1}{2},\text{where}a>1\text{and}[.]$ denotes the greatest integer function, then the value of $a^2$is

The set of all real numbers x for which $x^2-|x+2|+x>0$ is

Find the equation of a straight line which passes through the point (3, 4) and sum of its intercepts on the x and y axis is 14.

If f(x) is a function whose domain is symmetric about the origin, then f(x) + f(–x) is

If $z=e^{\frac{i\pi }{13}}$ then $\frac{1}{1-z}$ is equal to $(i=\sqrt{-1})$

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors such that $\overrightarrow{a}+2\overrightarrow{b}$ and $5\overrightarrow{a}-4\overrightarrow{b},$ are perpendicular to each other, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

$li{m}_{x\to 0}\frac{(e^{1/x}-1)}{(e^{1/x}+1)}$

If exactly two integers lie between the roots of the equation $x^2+ax-1=0$, then possible integral value(s) of $a$ is (are)

If the lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and$\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar, then k can have

$li{m}_{x\to 0}$ $\frac{1-cos2x}{x}$

[MMR 1983]

The equation of the hyperbola whose centre is (5, 2), vertex is (9, 2) and the length of conjugate axis is 6 is

While doing an experiment in chemistry lab, Komal found that a substance loses its moisture at a rate proportional to the moisture content. If the same substance loses half of its moisture during the first hour, when will it lose 99% ?

If $||x-2|-5|\le$ 10, then

Find the number of $5$ digit numbers using digits $0,1,2,3,4,5$ (without repetition) such that they are divisible by $5$ :

If sin$\alpha$+sin$\beta$+sin$\gamma$=0= cos$\alpha$+cos$\beta$+cos$\gamma$,

value of $si{n}^2$$\alpha$+$si{n}^2$$\beta$+$si{n}^2$$\gamma$

The number $lo{g}_{20}$ 3 lies in

If the parabola $x^2=ay$ makes an intercept of length $\sqrt{40}$ units on the line $y-2x=1$, then $a$ is equal to

If α and β are the roots of the equation $2x^2-3x+4=0$, then the equation whose roots are ${\alpha }^2$ and ${\beta }^2$ is

If $(1+ax{)}^n$ = 1 + 8x + 24 $x^2$ + ......, then the value of a and n is

If $\sqrt{\frac{x^2-4x+3}{x^2-3x+2}}\le 1,$ then the interval(s) in which $x$ can not lie, is/are