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Let ∪ = {x: x is a natural number and x ≤ 10}

A = {a: a is an even natural number and a < 10}

B = {b: b is a prime number and b ≤ 9}

Then A^I ∩ B^I is

$Generalsolutionoftan5\theta =cot2\theta is$

The interval in which the function $f\left(x\right)=3x^3+x^2-7x+13$ is decreasing is

What is the value of sin 105^o + sin 75^o?

The letters of the word ${}^{\prime }LOGARITH{M}^{\prime }$ are arranged in all possible ways. The number of arrangements in which the relative positions of the vowels and consonants are not changed is

A has no breadth.

${lim}_{x\to \frac{\pi }{4}}\frac{\sqrt{2}-cosx-sinx}{(\frac{\pi }{4}-x{)}^2}$

The lottery box contains tickets numbered 1-10. 2 tickets are drawn at random without replacement. The probability that the difference between the numbers on the ticket $>4$ is ?

Write the following relations as the sets of ordered pairs :

(i) A relation R from the set {2, 3, 4, 5, 6} to the set {1, 2, 3} defined by x = 2y.

(ii) A relation R from the set {1, 2, 3, 4, 5, 6, 7} defined by $(x,y)ϵR\iff x$ is relatively prime to y.

(iii) A relation R on the set {0, 1, 2, ...., 10} defined by 2x + 3y = 12.

(iv) A relation R from a set A = {5, 6, 7, 8} to the set B = {10, 12, 15, 16, 18} defined by $(x,y)ϵR\iff x$ divides y.

In a three-storey building, there are four rooms on the ground floor, two on the first and two on the second floor. If the rooms are to be alloted to six persons, one person occupying one room only, then number of ways in which this can be done so that no floor remains empty is

Write the value of ${lim}_{x\to 0^{-}}\left[x\right].$

Write the number of diagonals of an n-sided polygon.

If $y=a{e}^{mx}+b{e}^{-mx},then\frac{d^{2}y}{d{x}^2}=$

The value of $r$ for which ${}^{20}{C}_r{}^{20}{C}_0+{}^{20}{C}_{r-1}{}^{20}{C}_1+{}^{20}{C}_{r-2}{}^{20}{C}_2+\dots +{}^{20}{C}_0{}^{20}{C}_r$ is maximum, is :

If $2ta{n}^{-1}x=si{n}^{-1}\frac{2x}{1+x^2}$, then:

In the figure shown, find out the value of $\theta$ at this instant [assume string to be tight]:

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three unit vectors inclined with each other by an angle of $\frac{\pi }{3}$. If $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k};\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k};\overrightarrow{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$
and $A$ be the matrix of order $3\times 3$ such that
$adj\left(adjA\right)=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]$ then $|A{|}^4=$

Prove that : $\frac{(2n+1)!}{n!}=2n\mathrm{1.3.5....}(2n-1)(2n+1)$

The set of values of a for which the function $f\left(x\right)=(4a-3)(x+In5)+2(a-7)cot\left(\frac{x}{2}\right)si{n}^2\frac{x}{2}$ does not possess
critical point is

If the sum of the slopes of the normal from a point $P$ to the hyperbola $xy=c^2$ is equal to $\lambda (\lambda \in R^{+})$, then the locus of the point $P$ is

Let $C_1:x^2+y^2=1;C_2:(x-10{)}^2+y^2=1$ and $C_3:x^2+y^2-10x–42y+457=0$ be three circles. A circle C has been drawn to touch circles $C_1$ and $C_2$ externally and $C_3$ internally. Now circles $C_1,C_2$ and $C_3$ start rolling on the circumference of circle C in anticlockwise direction with constant speed. The centroid of the triangle formed by joining the centres of rolling circles $C_1,C_2$ and $C_3$ lies on

The derivative of $si{n}^{-1}\left(\frac{2x}{1+x^2}\right)$ with respect to
​$ta{n}^{-1}\left(\frac{2x}{1+x^2}\right)$ is

If $y=\frac{sin(x+9)}{cosx}$, then $\frac{dy}{dx}\text{at}x=0$ is

${lim}_{x\to \infty }{\left\{\frac{x^2+2x+3}{2x^2+x+5}\right\}}^{\frac{3x-2}{3x+2}}$

If $pϵ[-1,1]$, then the value of x for which $4x^3-3x-p=0$ has a root lies in

If |$z_1$ + $z_2$| =|$z_1$ - $z_2$| , then arg $z_1$ - arg $z_2$ =

If $\frac{\left(2n\right)!}{3!(2n-3)!}and\frac{n!}{2!(n-2)!}$ are in the ratio 44:3, find n.

In a $△ABC$, $a,b,c$ are the sides opposite to the angles $A,B,C$ respectively. The corresponding values of sides are $a=x^2,b=4x^2+x-1$ and $c=\frac{x}{3}$, where $x\in R$. If $\sin A+\sin C=2\sin B$, then $x$ can be expressed as $\frac{\alpha }{\beta }(0<\alpha ,\beta <10).$
A function is defined as $f\left(t\right)=\{\begin{array}{c}{k}_1\alpha +tt>0\\ k_2\beta +t+2t\le 0\end{array},$
where $k_1,k_2$ are integers.
If the function is continous for all $x\in R$, then $k_1+k_2$ can be equal to

The domain of definition of $f\left(x\right)=\sqrt{\frac{x+3}{(2-x)(x-5)}}$ is

The range of the polynomial $p\left(x\right)=4x^3-3x$ as $x$ varies over the interval $(-\frac{1}{2},\frac{1}{2})$ is

$\text{If}y=2^x{3}^{2x}{5}^{-x}{7}^{-x}then\frac{dy}{dx}=$

Find the interval in which the following functions are strictly incerasing or decreasing

$10-6x-2x^2$

Choose the correct answer in the following questions :
The line y=mx + 1 is a tangent to the curve $y^2=4x,$ if the value of m is
(a) $1$ (b) $2$ (c) $3$ (d) $\frac{1}{2}$

Integrate the following functions.
$\int \frac{1}{\sqrt{8+3x-x^2}}dx$

Find the coordinates of the focus and the vertex, the equations of the directrix, the axis, and length of latus rectum of the parabola $x^2=-16y.$

Let A,B,C be the feet of perpendiculars drawn from P(3,4,5) on XY,YZ,ZX planes respectively, then Coordinates of A,B and C will be .