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$li{m}_{\pi \to \infty }\frac{1^{\rho }+2^{\rho }+3^{\rho }+\cdots +n^{\rho }}{n^{\rho +1}}=$ [AIEEE 2002]

The expression $(p\wedge \sim q)\vee q\vee (\sim p\wedge q)$ is equivalent to

Given an example of a map

(i) which is one-one but not onto.

(ii) which is not one-one but onto.

(iii) which is neither one-one nor onto.

An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a balls is drawn at random. What is the probability that the second ball is red?

Solve the following for $x:si{n}^{-1}(1-x)-2si{n}^{-1}x=\frac{\pi }{2}$

OR

Show that: $2si{n}^{-1}\left(\frac{3}{5}\right)-ta{n}^{-1}\left(\frac{17}{31}\right)=\frac{\pi }{4}$

Let $I_1={\int }_0^{\frac{\pi }{4}}{x}^{2008}(tanx{)}^{2008}dx,I_2={\int }_0^{\frac{\pi }{4}}{x}^{2009}(tanx{)}^{2009}dxand{I}_3={\int }_0^{\frac{\pi }{4}}{x}^{2010}(tanx{)}^{2010}dx$

The point on the curve $y^2=8x$ on which the abscissa and ordinate changes at the same rate is

Locus of the point which divides double ordinates of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$ in the ratio $1:2$ internally is

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x-5y=20 to the circle x²+y²=9 is

A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions ?

If $f_{n-1}\left(x\right)=\ln \left(f_n\right(x\left)\right)\forall n\in N$ and $f_0\left(x\right)=x-1$, then $\frac{d}{dx}(f_n\left(x\right))$ is

The graph of $y=(x-1{)}^2+2$ is

Pratima, Ekta and Shikha are partners in a firm. Their profit sharing ratio is 5 : 3 : 2. However, Shikha is guaranteed a minimum amount of Rs 20,000 as share of profit every year. Any deficiency arising on the account shall be met by Ekta. The profits for the two years ending 31st March, 2016 and 2017 are Rs 80,000 and Rs 1,20,000 respectively. Prepare Profit and Loss Appropriation Account for two years.

If ${\cos }^{3}x\sin 2x=a_1\sin x+a_2\sin 2x+\cdots +a_n\sin nx$ $\forall x\in R$, where $a_n\ne 0$, then which of the following is/are correct ?

The integral $\int \frac{e^{3{\log }_{e}2x}+5e^{2{\log }_{e}2x}}{e^{4{\log }_{e}x}+5e^{3{\log }_{e}x}-7e^{2{\log }_{e}x}}dx,x>0,$ is equal to :
(where c is a constant of integration)

If three numbers $(x,y,z)=(\frac{2}{3},\frac{7}{6},\frac{8}{9}),$ then the H.C.F. of $x,y,z$ is

Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$Ina\Delta ABC,ifsinAandsinB\text{are the roots of the equation}{c}^2{x}^2-c(a+b)x+ab=0,thenfind\angle C.$

Represent to solution set of each of the following in equations graphically in two dimensional plane :

y>2x-8

if a coin is tossed three times (or three coins are tossed together), then describe the sample space for this experiment.

If the vectors $\overrightarrow{\alpha }=\hat{i}+a\hat{j}+a^2\hat{k},\overrightarrow{\beta }=\hat{i}+b\hat{j}+b^2\hat{k}\text{and}\overrightarrow{\gamma }=\hat{i}+c\hat{j}+c^2\hat{k}$ are three non-coplanar vectors and $\left|\begin{array}{ccc}a& a^2& 1+a^3\\ b& b^2& 1+b^3\\ c& c^2& 1+c^3\end{array}\right|=0$ , then the value of $abc$ is

If |z - 5i| = |z + 5i|, then find the locus of z.

A function $f:R-\left\{\right(2k-1\left)\pi \right\}\to R$ is defined as, $f\left(x\right)=\frac{1}{\sqrt{m^2-n^2}}\ln \left(\frac{\sqrt{m+n}+\sqrt{m-n}\tan \left(x/2\right)}{\sqrt{m+n}-\sqrt{m-n}\tan \left(x/2\right)}\right)$ where $k,m,n\in Z^{+}$ and $n<m$
If $f^{\prime }\left(\frac{\pi }{3}\right)=\frac{1}{9}$, then which of the following ordered pairs of $(m,n)$ is/are correct?

According to LMVT, if a function f(x) is continuous on [a, b] and differentiable on the interval (a, b) then which of the following option should be correct for some value c from the interval (a,b)?( c can take any value from the interval (a,b) )

The equation of circle passing through the origin and cutting off equal intercepts of $4$ units on the lines $xy=0$ can be

If a circle of constant radius $3k$ passes through the origin $O$ and meets co-ordinate axes at $A$ and $B$, then the locus of the centroid of the triangle $OAB$ is

Number of points of intersection of diagonals of polygon with $2009$ sides which are situated inside the polygon.

Determine order and degree (when defined) of differential equations.
y''+2y'+sin y=0.

If $A=\left|\begin{array}{ccc}1& 2& 0\\ -2& -1& -2\\ 0& -1& 1\end{array}\right|$, then find the value of $A^{-1}$
Using $A^{-1}$, solve the system of linear equations x - 2y = 10, 2x- y - z = 8 and -2y + z = 7

Find the integrals of the functions.
$\int$ sin x sin 2x sin 3x dx

Let f be a function defined on [a, b] such that f'(x) > 0, for all $xϵ$ [a, b]. Then prove that f is an increasing function on [a, b].

For the matrix $A=\left[\begin{array}{cc}3& 2\\ 1& 1\end{array}\right]$, find the numbers a and b such that $A^1+aA+bI=0$

In a triangle ABC, 3sinA+4cosB=6 and 4sinB+3cosA=1.Find measure of angle C.

The domain of the function $f\left(x\right)={\left[{\log }_{10}\left(\frac{5x-x^2}{4}\right)\right]}^{1/2}$ is

If $y=\log |x|,x\ne 0$ then find $\frac{dy}{dx}=$

If x - 3 = 9, then x is equal to:

Prooved it , 2 sin^-1x = cos ^-1 x

If $x,y,z$ are real numbers then the range of $x^2+4y^2+9z^2-6yz-3xz-2xy$ is

It is a differentiation problem the question is like this

If Y is equal to X square + 2X divided by 3X -4 Then DY by DX is equal to?

By using quotient rule please solve the problem !

Answer for this is 3x square -8X-8 divided by (3X-4) whole square please explain every step in detail

Thank you byju's