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The value of ${\cot }^2\frac{\pi }{12}-{\tan }^2\frac{\pi }{12}$ is

Find a,b and n in the expansion of $(x+b{)}^n$ if the first three terms in the expansion are 729, 7290 and 30375 respectively.

${lim}_{x\to 0}\frac{ta{n}^{-1}x-si{n}^{-1}x}{x^3}\text{is equal to}$

The locus of a point (to the right of $x=2$) whose sum of the distances from the origin and the line $x=2$ is $4$ units, is

Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of $A^2$ is $1,$ then the possible number of such matrices is :

If the inverse of the matrix

$A=\left[\begin{array}{ccc}1& 2& 2\\ 2& -1& 2\\ 2& 2& 1\end{array}\right]$ is $\frac{1}{5}\left[\begin{array}{ccc}-3& 2& 2\\ 2& -3& \alpha \\ 2& 2& -3\end{array}\right]$,

then the value of $\alpha$ is

In a triangle $ABC$, coordinates of $A$ are $(1,2)$ and the equations of the medians through $B$ and $C$ are respectively, $x+y=5$ and $x=4$. Then area of $△ABC$ (in sq. units) is :

If $x=111\dots \dots 1\left(20\text{digits}\right),$ $y=333\dots \dots 3\left(10\text{digits}\right)$ and $z=222\dots \dots 2\left(10\text{digits}\right)$, then $x-z$ is equal to

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

$Ina\Delta ABC,ifcosA=\frac{sinB}{2sinC},\text{then show that c = a.}$

Show that the line joining (2, -5) and (-2, 5) is perpendicular to the line joining (6,3) and (1, 1).

4tan^-1(1/5)-tan^-1(1/239) is equal to

If ${\sum }_{r-1}^{n}r=55$,find ${\sum }_{r-1}^n{r}^3$

The amplitude of $\frac{1}{i}$ is equal to

A locomotive of mass $m$ starts moving so that its velocity varies according to the law $v=k\sqrt{s}$ where $k$ is constant and $s$ is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first $t$ seconds after the beginning of motion.

Two forces $P$ and $2P$ are inclined at $120°$ with each other. If their resultant makes an angle $\alpha$ with their bisector then the value of $\alpha$ (in degree) is

What is the minimum value of $2^x$+ $2^{-x}$, if x is real number?

Study the following information carefully and answer the questions given below.
(i) ${}^{\prime }P÷Q^{\prime }$ means 'P is the sister of Q'.
(ii) ${}^{\prime }P\times Q^{\prime }$ means 'P is the brother of Q'.
(iii) ${}^{\prime }P-Q^{\prime }$ means 'P is the mother of Q'.
(iv) ${}^{\prime }P+Q^{\prime }$ means 'P is the father of Q'.
Which of the following means 'H is the maternal uncle of T'?

If A, B, C are three sets such that $A\subset B$, then prove that $C-B\subset C-A$

If the 2nd, 3rd and 4th terms in the expansion of $(x+a{)}^n$ and 240, 720 and 1080, find x,a,n.

Let $A$ and $B$ be two smallest sets such that $A\cup \left\{1\right\}=\{1,2,3,4\}$ and $B\cup \left\{5\right\}=\{4,5,6,7,8\}.$ If $P=A-B$ and $Q=B-A,$ then the number of relations from $P$ to $Q$ is

Find the equation of the line parallel to x-axis and passing through (3, -5).

Let $a_1,a_2,a_3,\cdots$ be an arithmetic progression with npn-zero common difference. It is given that $\sum _{i=4}^{12}{a}_i=63$ and $a_k=7$ for some $k$. Then the value of $k$ is

If $cosA=mcosB$, then $cot\frac{A+B}{2}cot\frac{B-A}{2}=$

If $e^{f\left(x\right)}=\frac{10+x}{10-x},xϵ(-10,10)$ and $f\left(x\right)=kf\left(\frac{200x}{100+x^2}\right)$, then k =

The number of solution(s) of $y=-|\log x|$ and $y=1,$ is

The values of $p$ for which the equation $x^2-6|x|+5-|p|=0$ has exactly four real roots, is

If $f\left(x\right)$ is an odd differentiable function defined on $(-\infty ,\infty )$ such that $f^{\prime }\left(3\right)=2$ , then $f^{\prime }(-3)$ equal to

The circle $x^2+y^2-4x-4y+4=0$ is inscribed in a triangle which has two of its sides along the coordinate axes. If the locus of the circumcenter of the triangle is $x+y-xy+k\sqrt{x^2+y^2}=0$, then the value of $k$ is

${lim}_{x\to \sqrt{3}}\frac{x^4-9}{x^2+4\sqrt{3x}-15}$

If A is any set , prove that :

$A⊈\varphi \iff A=\varphi$.

If $f\left(\theta \right)=2({\sec }^2\theta +{\cos }^2\theta ),$ then its
value always

The value(s) of k for which equations $3x^2+4kx+2=0and2x^2+3x-2=0$ will have a common root can be:

Write down the power set of $A=\{1,\left\{3\right\},4\}.$

Compute the following:
$\left(ii\right)\left[\begin{array}{cc}{a}^2+b^2& b^2+c^2\\ a^2+c^2& a^2+b^2\end{array}\right]+\left[\begin{array}{cc}2ab& 2bc\\ -2ac& -2ab\end{array}\right]$

The minimum number of terms required of the sequence $1,1.2,1.44,1.728,\cdots$ so that the sum of the numbers is greater than $25$, is
$(\text{Use}{\log }_{10}2=0.30,{\log }_{10}3=0.48)$

Find the shortest distance between the following lines :

$\overrightarrow{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda (2\hat{i}+3\hat{j}+4\hat{k})$ and $\overrightarrow{r}=2\hat{i}+4\hat{j}+5\hat{k}+\mu (4\hat{i}+6\hat{j}+8\hat{k})$

OR

Find the equation of the plane passing through the line of intersection of the planes 2x + y - z = 3 and 5x - 3y + 4z + 9 = 0 and is parallel to the line $\frac{x-1}{2}=\frac{y-3}{4}=\frac{5-z}{-5}.$

A shopkeeper sells three types of flower seeds $A_1,A_2$ and $A_3$ . They are sold as a mixture, where the proportions are 4 : 4 : 2, respectively. The germination rates of the three types of seeds are 45%, 60% and 35% . Calculate the probability

(i) of a randomly chosen seed to germinate.

(ii) that it will not germinate given that the seed is of type $A_3$.

(iii) that it is of the type $A_2$ given that a randomly chosen seed does not germinate.

Let R be the relation on Z defined by R = {(a, b): a, $bϵZ^2,a-b$ is an integer}. Find the domain and range of R.

Find the particular solution of the differential equation $(x-y)\frac{dy}{dx}=(x+2y)$,given that y=0 when x=1.

Using elementary transformations, find the inverse of the followng matrix.

$\left[\begin{array}{cc}4& 5\\ 3& 4\end{array}\right]$

Form point $P(8,27)$, tangent $PQ$ and $PR$ are drawn to the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$.If the angle subtended by $QR$ at origin is $\varphi$, then $\tan \varphi =$

Two concentric circles $C_1$(radius = 5cm) and $C_2$(radius = 3cm). Then the lenght of the chord of the circles $C_1$ which touches the circles $C_2$

If the distance between the foci and the distance between two directrices of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are in the ratio $3:2$, then $b:a$ is

The complete set of values of $x$ for which the inequality ${\log }_x\left(\frac{4x+5}{6-5x}\right)<-1$ holds good, is