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The range of values of $x$ which satisfies the inequation ${\log }_{1/6}(x^2-3x+2)+1<0$ is

A G.P. has even number of terms . If the sum of all the terms is $5$ times the sum of the terms occupying the odd places, then the common ratio of the G.P. is

The circle $x^2+y^2-6x-4y+9=0$ bisects the circumference of the circle $x^2+y^2-(\lambda +4)x-(\lambda +2)y+(5\lambda +3)=0if\lambda$ is equal to

The length of the latus rectum of the ellipse $5x^2+9y^2=45$ is

For $n>0,{\int }_0^{2\pi }\frac{xsi{n}^{2n}x}{si{n}^{2n}x+co{s}^{2n}x}dx=$

One or more options can be correct.

Find the coefficient of $x^{100}$ in $(1-x{)}^{-3}$.

If ${log}_e$(A+iB) = $(ilog\sqrt{3}+\frac{\pi }{4})$ .Find the value of A and B.

The sum ${\sum }_{i=0}^m(\frac{10}{i})(\frac{20}{m-i}),where(\frac{p}{q})=0ifp>q,$ is maximum when m is equal to

Let $PQ$ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,a>0$.

If chord $PQ$ subtends an angle $\theta$ at the vertex of $y^2=4ax$, then tan $\theta$ is equal to

$\text{The value of}\underset{n\to \infty }{lim}\frac{\sqrt[4]{4n^5+2}-\sqrt[3]{3n^2+1}}{\sqrt[5]{5n^4+2}-\sqrt[2]{2n^3+1}}is$

If x is real, the expression $\frac{x+2}{2x^2+3x+6}$ takes all value in the interval

Find the sum of the series $\frac{1}{2}+\frac{1}{3^2}+\frac{1}{2^3}+\frac{1}{3^4}+\frac{1}{2^5}+\frac{1}{3^6}+\dots \infty$

The solution of the differential equation $\frac{dy}{dx}+\frac{3x^2}{1+x^3}y=\frac{si{n}^{2}x}{1+x^3}$ is

If the circle $x^2+y^2+2\lambda x=0,\lambda \in R$ touches the parabola $y^2=4x$ externally, then

$\underset{x\to \infty }{lim}{\left(\frac{3x^2+2x+1}{x^2+x+2}\right)}^{\frac{6x+1}{3x+2}}\text{is equal to}$

The value of ${\log }_5{\log }_3\sqrt{\sqrt[5]{59}}$ is

Let $k$ be an integer such that the triangle with vertices $(k,–3k),(5,k)\text{and}(–k,2)$ has area $28$ sq. units. Then the orthocentre of this triangle is at the point:

If $\sin \theta =\frac{3}{5},\cos \varphi =\frac{12}{13}$ where $\theta ,\varphi \in (0,\pi /2)$, then the value of $\tan (\theta +\varphi )$ is

If $\sum _{r=0}^n{\left(\frac{{}^n{C}_{r-1}}{{}^n{C}_r+{}^n{C}_{r-1}}\right)}^3=\frac{25}{24},$ then the value of $n$ is

$\frac{1}{2}$, $\frac{1}{4}$, - - - - form a H.P The ratio of 5th term to 7th term is

If the coefficients of $x^3$ and $x^4$ in the expansion of $(1+ax+b{x}^2$) $(1–2x{)}^{18}$ in powers of $x$ are both zero, then $(a,b)$ is equal to

If, in a $G.P.$ of $3n$ terms, $S_1$ denotes the sum of the first $n$ terms, $S_2$ the sum of the second block of $n$ terms and $S_3$ the sum of the last $n$ terms, then $S_1,S_2,S_3$ are in

If arg (z) < 0, then arg (-z)-arg (z) equals

If $\left|\frac{|x|-2}{7-2|x|}\right|=1$, then $x$ can be

The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+(1+x{)}^{23}+\cdots +(1+x{)}^{30}$

If arg(z) = ${\log }_e{i}^i$, then the value of complex number z is

The phrase 'inter alia' meaning 'among other things' is one of the many Latin expression commonly used in English.

Find out what these Latin phrases mean.

1.Prima face

2. ad hoc

3. in camera

4.ad infinitum

5.mutatis multanis

6.tabula rasa

The mean of $5$ observation is $6$ and the varience is $\mathrm{6.80.}$ If three of the five observation are $8,5$ and $10.$ Then the remaining two observation are

${\int }_0^{\frac{\pi }{2}}(cosx-sinx)e^{x}dx=$

The length of the perpendicular from the origin to the plane passing through three non-collinear points $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ is

Find the coefficient of abcd in the expansion of $(a+b+c+d{)}^4$

If $\sum _{r=1}^n{T}_r=\frac{n}{8}(n+1)(n+2)(n+3)$, and $\sum _{r=1}^n\frac{1}{T_r}=\frac{n^2+3n}{4\sum _{k=1}^{p}k}$, then $p$ is equal to

The formula for Fisher’s method is _______.

In a triangle $ABC$ we define

$x=tan\frac{B-C}{2}tan\frac{A}{2},y=tan\frac{C-A}{2}tan\frac{B}{2}$ and $z=tan\frac{AB}{2}tan\frac{C}{2}$

Then the value of $x+y+z$ (in terms of $x,y,z$) is

In h(x) = f(x) + f(-x), then h (x) has got an extreme value at a point where f'(x) is

1. 4

Let P(6, 3) be a point on hyperbola

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$

If the normal at the point P intersect the x - axis at (9, 0), then the eccentricity of the hyperbola is

Figure shows the curve $y=x^2$.Find the area of the shaded part between $x=0$ and $x=6$

If relation R is defined as "Is of the same color" on set of objects. Then the total number of equivalence classes on the set with respect to R is equal to.

An infinite G.P. has first term $x$ and sum $5.$ Then which of the following is true

If an equilateral triangle is inscribed in a parabola $y^2=12x$ with one vertex at the vertex of the parabola, then its height is

If $A=\{1,2,3\},B=\{4,5,6,7,8\},$ $C=\{4,8,12,16,20\},$ then $n\left[\right(A\times B)\cup (A\times C\left)\right]=$

Let $p,q,r$ be three statements such that the truth value of $(p\wedge q)\to (\sim q\vee r)$ is $F$. Then the truth values of $p,q,r$ are respectively: