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Find the value of $\underset{x\to \frac{\pi }{2}}{lim}\frac{\sin x}{x}$

Team of four students is to be selected from a total of 12 students for a quiz competition. In how many ways it can be selected if three particular students should be included and one particular student doesn't want to be in the team.

Find the value of ${}^n{C}_0$ +${}^n{C}_3$ +${}^n{C}_6$_{............................}

If the function 'a' is$x^3$. which of the following could be the function b?

If $\frac{cos(x-y)}{cos(x+y)}+\frac{cos(7+t)}{cos(7-t)}=0$ then tan x tan y tan 7 tan t=

In a triangle PQR, R = $\frac{\pi }{2}$ If tan $\frac{P}{2}$and tan$\frac{Q}{2}$are the roots of the equation \( ax^2\) +bx+c=0 (a$\ne$0), then:

If $f\left(x\right)=\sqrt{2x+3}$, Then $f^{\prime }\left(x\right)=$___

Then eccentricity of the Ellipse $4x^2+y^2-8x-2y+1$ = 0

The $K_{SP}$ of Agl is $1.5\times {10}^{-16}$. On mixing equal volumes of the following

solutions, precipitation will occur only with

The sum of the series $1^2$.2 + $2^2$.3 + $3^2$.4 + ........ to n terms is

If $(0,\pm 4)$ and $(0,\pm 2)$ be the foci and vertices of a hyperbola, then its equation is

In triangle ABC, right angled at B, if one angle is ${45}^{\circ }$, find the value of sin A, cosC, cot A and tan C respectively.

The abscissae of A and B are the roots of the equation $x^2+2ax-b^2=0$ and their coordinates are the roots of the

equation y^2 + 2py - q^2 = 0. The equation of the circle with AB as diameter

if a=$\sqrt[2]{22i}$ then which of the following is correct

Let N be the set of all natural numbers. Let $R=\left\{\right(a,b):a,bϵN\text{and}2a+b=10\}$. Show that R is a binary relation on N. Find its domain, range and co-domain.

Find the equation of normal having slope 1 to the parabola
$y^2=4x$

Find the coefficient of $x^{11}$ in the expansion $(1-6x^6)(1-x{)}^{-7}$

If $P_k=\frac{1-x^{k+1}}{1-x}$, the number of terms in the product $P_1.P_2....P_n$ is

Find the first integral term in the expansion of $(\sqrt{3}+\sqrt[3]{32}{)}^9$

(Commultative laws) For any two sets a and B, prove that:
$I.A\cup B=B\cup A$ [Commutative law for union of sets]

$II.A\cap B=B\cap A$ [Commutative law for intersection of sets]

If $tan\theta =\frac{sin\alpha -cos\alpha }{sin\alpha +cos\alpha }thensin\alpha +cos\alpha andsin\alpha -cos\alpha$
and must be equal to

The scores of two batsmen A and B in five innings during a certain match are :

$\begin{array}{cccccc}\text{A}& 32& 28& 47& 63& 71\\ \text{B}& 19& 31& 48& 53& 67\end{array}$

Identify the better player. Who is more consistent?

${}^n{C}_1$ + ${}^n{C}_3$ + ${}^n{C}_5$..............=

Find the imaginary part of the complex number $(1+i{)}^{(1-i)}$

Given that the 4th term in the expansion of ${(2+\frac{3x}{8})}^{10}$ has the maximum numerical value, then x lies in the interval

Find a point on the $x$-axis, which is equidistant from the points (5, 4) and (2, 3).

If sinx + $si{n}^{2}x$ = 1 Then the value of $co{s}^{12}x+3co{s}^{10}x+3co{s}^{8}x+co{s}^{6}x$ is equal to

A chord BC is drawn in a parabola $y^2=4x$ such that x cordinate of C is twice that of B. Also OB makes ${45}^{\circ }$ with x-axis. What is the projecton of BC on y-axis?

If A and B are any two sets, then $A\cup (A\cap B)$ = ___.

Prove that

$\left(i\right)log12=log3+log4$

$\left(ii\right)log50=log2+2log5$

$\left(iii\right)log(1+2+3)=log1+log2+log3$

Consider the two statements P: He is intelligent and Q: He is strong. Then the symbolic form of the statement "It is not true that he is intelligent or strong" is

Three numbers a, b, c are in G.P. Their sum is 26 and their product is 216. Find the common ratio [integer value].

Find the coefficient of correlation between X and Y.

$\begin{array}{ccc}& \text{X Series}& \text{Y Series}\\ \text{No. of items}& 30& 30\\ \text{Standard deviation}& 3& 2\end{array}$
Summation of product of deviations of X and Y series from their respective means = 150

How many of the following statements are correct?

If AB and OC are straight lines and the angle made by OC with OB is ${30}^o$, then $\theta ={150}^o$

(2)

The angle made by OA with respect to positive x-axis is ${120}^{o}or-{240}^o$.

Box I contain three cards bearing numbers 1,2,3; box II contains five cards bearing numbers 1,2,3,4,5; and box III contains seven cards bearing numbers 1,2,3,4,5,6,7. A card is drawn from each of the boxes. Let $x_i$ be the number on the card drawn from the $i^{th}$ box $i=1,2,3.$

The probability that $x_1,x_2$ and $x_3$ are in arithmetic progression, is ?

Suppose a population A has 100 observations 101, 102, . . . . 200 and another population B has 100 observations 151, 152, . . . . 250. If $V_A$ and $V_B$ represent the variances of the two populations, respectively then $\frac{V_A}{V_B}$ is

To fill 3 vacancies there are 25 candidates of which five are from scheduled caste. If all 3 of the vacancies are reserved for scheduled caste, then the number of ways in which the selection can be made __________.

If p: The earth is round, q : 3 + 4 =7, then $\sim p\vee \sim q$ is

The sum of the coefficients of even powers of x in the expansion of $(1+x+x^2+x^3{)}^5$ is equal to:

The dimensions of $({\mu }_0{ϵ}_0{)}^{\frac{-1}{2}}$ are

As the temperature is raised from ${20}^{\circ }C$ to ${40}^{\circ }C$. The average kinetic energy of neon atoms changes by a factor of which of the following:

In a triangle ABC, if sin $AsinB=\frac{ab}{c^2},$ then the triangle is