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In how many of the distinct permutations of the letters in MISSISSIPPI do the four I's not come together ?

Given,A={1,2,{1,2}}, Is the element{1,2} of A , is a subset of itself??

Solve:

log (5x + 19) = log( 7x + 3)

The number of real roots of $(x-1{)}^4+(x+1{)}^4=16$ is :

If A = {2, 3, 5, 6}, B = {4, 8, 15, 17} a R b $\Rightarrow$ a divides b. Find the relation R.

Which direction is K facing?

I. N is initially facing North and if he turns to his left, he will be facing the opposite direction as that of K.

II. S who is facing East is to the right of K.

If ${\cos }^{-1}x-{\cos }^{-1}\frac{y}{2}=\alpha$ where $-1\le x\le 1,$ $-2\le y\le 2,x\le \frac{y}{2},$ then for all $x,y,4x^2-4xy\cos \alpha +y^2$ is equal to :

The value of expression $\frac{1-4sin{10}^{o}sin{70}^o}{2sin{10}^o}$ is

Find the sum of the series
${\sum }_n^{r=0}(-1{)}^r{}^n{C}_r[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+\frac{{15}^r}{2^{4r}}\cdots uptomterms]$

sin (n+1)x cos(n+2)x-cos(n+1)x sin(n+2)x=

A curve is represented parametrically by the equations $x=f\left(t\right)=a^{\ln \left(b^t\right)}$ and $y=g\left(t\right)=b^{-\ln \left(a^t\right)};a,b>0$ and $a\ne 1,b\ne 1$ where $t\in R$.

The value of $\frac{d^{2}y}{d{x}^2}$ at the point where $f\left(t\right)=g\left(t\right)$ is

If the line $x-1=0$ is the directrix of the parabola $y^2-kx+8=0$, then one of the values of $k$ is

The variance of 15 observations is 4. If each observation is increased by 9, find the variance of the resulting observations.

The maximum value of Z=x+y,subject to:$2x+y⩽50,x+2y⩽40,x⩾0,y⩾0$ is

If $\overrightarrow{a}=3\hat{j}-4\hat{k}$
$\overrightarrow{b}=2\hat{i}-\hat{j}+2\hat{k}$
and $\overrightarrow{c}$ is directed along the direction parallel to angle bisector of $\overrightarrow{a}$ and $\overrightarrow{b}$ and the projection of $\overrightarrow{c}$ on $\overrightarrow{d}$ is of magnitude $\frac{10}{9}$ where $\overrightarrow{d}=4\hat{i}-4\hat{j}+7\hat{k}$ then $|\overrightarrow{c}{|}^2$ is

If the eccentricity of the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\text{is}d\frac{5}{4}\text{and}2x+3y-6=0$
is the focal chord of the hyperbola, then the length of transverse axis is equal to

If $\sqrt{1-x^{2n}}+\sqrt{1-y^{2n}}=a(x^n-y^n)$ then $\sqrt{\frac{1-x^{2n}}{1-y^{2n}}}.\frac{dy}{dx}=$

The range of $x$ for which $x^2-|x+2|+x>0$ is

The value of ${\cos }^3({60}^{\circ }-A)-{\cos }^3({60}^{\circ }+A)$ is

A square ABCD of diagonal 2a is folded along the diagonal AC so that the planes DAC and BAC are at right angle. The shortest distance between DC and AB is

Area bounded by the curves $y=e^x,y=lo{g}_{e}x$ and the lines x = 0, y = 0, y = 1 is

Of the 25 questions in a unit, a student has worked out only 20. In a sessional test of that unit, two questions were asked by the teacher, The probability that the student can solve both the questions correctly, is

The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is

The value of $\int \frac{e^x(1+x)dx}{{\cos }^2\left(e^{x}x\right)}$

8 coins are tossed simultaneously. The probability of getting at least 6 heads is

[AISSE 1985; MNR 1985; MP PET 1994]

The value of the expression $(2+\sqrt{2}{)}^4$ lies between

If n ∈ N, then $x^{2n-1}+y^{2n-1}$ is divisible by

Which of the following hold good? If n is a +ve integer, then

A rod of length I sides with its ends on two perpendicular lines. Then, the locus of its midpoint is

The general solution of $y^{2}dx+(x^2-xy+y^2)dy=0$ is

[EAMCET 2003]

If x is positive, the sum of infinity of the series $\frac{1}{1+x}-\frac{1-x}{(1+x{)}^2}+\frac{(1-x{)}^2}{(1+x{)}^2}-\frac{(1-x{)}^3}{(1+x{)}^4}+....$ is

The number of ways to select two different natural numbers which are less than or equal to 100 and differ by at most 10 is

Let $F_1(x_1,0)$ and $F_2(x_2,0),$ for $x_1<0$ and $x_2>0,$ be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1.$ Suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.

The orthocentre of the triangle $F_{1}MN$ is

Prove that $cos\alpha +cos(\alpha +\beta )+cos(\alpha +2\beta )+.....+cos(\alpha +(n-1\left)\beta \right)$

$=\frac{cos\{\alpha +\frac{n-1}{2}\beta \}sin\left(\frac{n\beta }{2}\right)}{sin\frac{\beta }{2}}$ for all n $ϵ$ N.

The function $f\left(x\right)=x^2(x-2{)}^2$

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:

Number of integral values of x satisfying ${\left(\frac{3}{4}\right)}^{6x+10-x^2}<\left(\frac{27}{64}\right)$ is ___.

If tan A +cot A =4, then $ta{n}^{4}A+co{t}^{4}A$ is equal to

In what direction a line be drawn through the point (1, 2) so that its point of intersection with the line x + y = 4 will be at a Distance of $\frac{\sqrt{6}}{3}$

$\text{The smallest positive angle which satisfies the quation}2si{n}^2\theta +\sqrt{3}cos\theta +1=0is$

If $y=1+t^4$ and $x=3t^3+t$ then what is $\frac{dy}{dx}$

Prove that:
(i) $cos3A+cosSA+cos7A+cos15A4cos4AcosSAcos6A$
(ii) $cosA+cos3A+cos5A+cos7A=4cosAcos2Acos4A$
(iii) $sinA+sin2A+sin4A+sin5A4cos\frac{A}{2}cos\frac{3}{A}sin3A$
(iv) $sin3A+sin2A+sinA-4sinAcos\frac{A}{2}cos\frac{3A}{2}$
(v) $cos{20}^{\circ }cos{100}^{\circ }+cos{100}^{\circ }cos{140}^{c}os200°=-\frac{3}{4}$

(vi) $sin\frac{\theta }{2}sin\frac{7\theta }{2}+sin\frac{3\theta }{2}sin\frac{11\theta }{2}=sin2\theta sin5\theta$
(vii) $sin\frac{\theta }{2}-cos3\theta cos=\frac{9\theta }{2}=sin7\theta sin8\theta$

Simplify:
$\left|\begin{array}{ccc}3x& -x+y& -x+z\\ x-y& 3y& z-y\\ x-z& y-z& 3z\end{array}\right|$

Whats the length of chord intercepted by the parabola $y^2=4xonthelinex+y=0$