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The sum of two geometric mean's inserted between $3$ and $81$ is

Let $-\frac{\pi }{6}<\theta <-\frac{\pi }{12}$. Suppose ${\alpha }_1$ and ${\beta }_1$ are the roots of the equation $x^2-2xsec\theta +1=0$, ${\alpha }_2$ and ${\beta }_2$ are the roots of the equation $x^2+2xtan\theta -1=0$. If ${\alpha }_1>{\beta }_1$ and ${\alpha }_2>{\beta }_2$, then ${\alpha }_1+{\beta }_2$ equals

The probability distribution of the random variable X is given below

X: 0 1 2 3

$P\left(X\right):k\frac{k}{2}\frac{k}{4}\frac{k}{8}$Then the value of k is

If three distinct natural numbers are chosen randomly from the first $100$ natural numbers, then the probability that all three of them are divisible by both $2$ and $3,$ is

The value of ${lim}_{n\to \infty }{\sum }_{r=0}^n\frac{{}^n{C}_r}{n^r(r+3)}=e-x,$ then x is___.

Let , $f\left(x\right)=x^2+5x+6$, then the number of real roots of $\left(f\right(x){)}^2+5f(x)+6-x=0$ is

There are n lines in a plane, no two of which are parallel and no three of them are concurrent. Let plane be divided in $U_n$ parts by these n lines. Then

${lim}_{x\to 0}(\cos x+a\sin bx{)}^{\frac{1}{x}}$

If A.M. and H.M. of the roots of a quadratic equation are $8$ and $5$ respectively, then the equation is

The letters of the word $\text{COCHIN}$ are permuted and all permutations are arranged in an alphabatical order as in english dictionary. The number of words that appear before the word $\text{COCHIN}$ is

If $x=\sec \theta -\cos \theta$ and $y={\sec }^n\theta -{\cos }^n\theta$, then ${\left(\frac{dy}{dx}\right)}^2$ is equal to

Using properties of determinants prove the following questions.

$\left|\begin{array}{ccc}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{array}\right|=1$

If $\beta$ is one of the angles between the normals to the ellipse, $x^2+3y^2=9$ at the points $(3\cos \theta ,\sqrt{3}\sin \theta )$ and $(-3\sin \theta ,\sqrt{3}\cos \theta )$; $\theta \in (0,\frac{\pi }{2});$ then $\frac{2\cot \beta }{\sin 2\theta }$ is equal to :

For the curve $f\left(x\right)=(x-5{)}^2$, applying mean value theorem on $[4,6]$, the tangent at ______ is parallel to the chord joining $A(4,1),B(6,1)$.

Mode of the data $3,2,5,2,3,5,6,6,5,3,5,2,5$ is

The value of $\frac{cos3\theta }{2cos2\theta -1}$ is equal to

Write the argument of -i.

If $(1+2x+3x^2{)}^{10}=a_0+a_{1}x+a_2{x}^2+...+a_{20}{x}^{20},$ then $a_1$ equals

The end points of the latus rectum of a parabola having vertex at origin are $(4,8)$ and $(4,-8)$ then the equation of the parabola is

The difference between fourth and first term of a $G.P.$ is $52$ and sum of the first three terms is $26$. If $T_n$ and $S_n$ represent $n^{th}$ term and sum to $n$ terms respectively of the $G.P.$, then which of the following is/are true

Area of the region bounded by the curve $y=\cos x,x=0,$ and $x=\pi$ is

${lim}_{x\to 0}\frac{x(e^x-1)}{1-\cos x}$

The locus of the point of intersection of the tangents to the parabola $y^2=4ax$ which makes angles ${\theta }_1$ and ${\theta }_2$ with its axis so that $\cot {\theta }_1+\cot {\theta }_2=k$ is

Let $\ast$ be a binary operation on the set Q of rational number as follows:
$\left(i\right)a\ast b=a-b$

$\left(ii\right)a\ast b=a^2+b^2$

$\left(iii\right)a\ast b=a+ab$

$\left(iv\right)a\ast b=(a-b{)}^2$

$\left(v\right)a\ast b=\frac{ab}{4}$

$\left(vi\right)a\ast b=a{b}^2$
Find which of the binary operation are commutative and which are associative?

Find the integrals of the functions.
$\int ta{n}^{4}xdx.$

Find the angle between the following pairs of lines :

$r=2\hat{i}-5\hat{j}+\hat{k}+\lambda (3\hat{i}+2\hat{j}+6\hat{k})andr=7\hat{i}-6\hat{k}+\mu (\hat{i}+2\hat{j}+2\hat{k})$

$r=3\hat{i}+\hat{j}-2\hat{k}+\lambda (\hat{i}-\hat{j}-2\hat{k})andr=2\hat{i}-\hat{j}-56\hat{k}+\mu (3\hat{i}-5\hat{j}-4\hat{k})$

The system of equations ax + y + z = a –1, x + ay + z = a –1, x + y + az = a –1 has no solution, if a is

The possible values of the expression $\frac{1}{x^2-2x+5}$

The value of a for which the quadratic equation 2x^2 - (a^3+8a-1)X + a^2-4a =0 possess roots of opposite signs are given by

a>0

a>5

4<a<8.5

0<a<4

What is the orthogonal trajectory of $x^2-y^2=c$?

The intercept form of the line $3x-4y+12=0$ is

The foot of the perpendicular drawn from the point $(1,8,4)$ on the line joining the points $(0,–11,4)$ and $(2,–3,1)$ is

If $px+qy+r=0$ is tangent of the parabola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ then

Simplify √40×√90=

Five sticks of length 1,3,5,7 and 9 feet are given. Three of these sticks and selected at random. What is the probability that selected sticks form a triangle.

A train 125 m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is:

In a certain code, the word ROAD is coded as WTFI. Following the same rule of coding, what should be the word for the code GTFY?

$\frac{1}{1+x^{\left(b-a\right)}+x^{(c-a)}}.\frac{1}{1+x^{\left(a-b\right)}+x^{\left(c-b\right)}}.\frac{1}{1+x^{\left(b-c\right)}+x^{\left(a-c\right)}}=?$

A bag contains 4 white, 5 red and 6 black balls. If two balls are drawn at

random, then the probability that one of them is white is

The integral form of exponential growth equation is

For a>0, b>0, c>0, which of the following hold good?

Equation of the line passing through the point $(2,3,1)$ and parallel to the line of the intersection of the plane $x-2y-z+5=0\text{and}x+y+3z=6$ is