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The mean of the data set comprising of 16 observations is 16. If one of the observations valued 16 is deleted and three new observations valued 3, 4 and 5 added to the data, then the mean of the resultant data is

$\frac{1+7i}{(2-i{)}^2}$=

If $\alpha =3{\sin }^{-1}\left(\frac{6}{11}\right)$ and $\beta =3{\cos }^{-1}\left(\frac{4}{9}\right)$, where the inverse trigonometric functions take only the principal values, then the correct option(s) is/are

Let $S_1$ and $S_2$ be circles of radii 1 and r (r > 1) respectively touching the coordinate axes.
Column-1: Conditions between circles $S_1$ and $S_2$
Column-2: Values of r for conditions in Column-1.
Column-3: Number of common tangents between S1 and S2 for conditions in column-1.
$\begin{array}{cccccc}& \text{Column 1}& & \text{Column 2}& & \text{Column 3}\\ \text{(I)}& S_2\text{passes through the centre}& \text{(i)}& 3& \text{(P)}& 1\\ & \text{of}{S}_1.& & & & \\ \text{(II)}& S_1\text{and}{S}_2\text{touch each other}& \text{(ii)}& 2+\sqrt{2}& \text{(Q)}& 2\\ \text{(III)}& S_1\text{and}{S}_2\text{are orthogonal}& \text{(iii)}& 2+\sqrt{3}& \text{(R)}& 3\\ \text{(IV)}& S_1\text{and}{S}_2\text{have longest}& \text{(iv)}& 3+2\sqrt{2}& \text{(S)}& 4\\ & \text{common chord}& & & & \end{array}$

Which of the following options is the only CORRECT combination?

An equation of the curve satisfying $xdy-ydx=\sqrt{x^2-y^2}$ dx and $y\left(1\right)=0$ is

Find the equation of the lines joining the origin to the points of intersection of the straight line y=3x+2 with the curve $x^2+2xy+3y^2+4x+8y-11=0$

${lim}_{x\to 0}\frac{\sqrt{1+x}-1}{x}$

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$ and $\alpha +\beta ,{\alpha }^2+{\beta }^2,{\alpha }^3+{\beta }^3$ are in G.P., where $△=b^2-4ac$, then

NSE Indices
$\begin{array}{cccc}\text{Index}& \text{Current}& \text{Previous}& \text{\% Change}\\ \text{S \& P CNX Nifty}& 3641.1& 3770.55& -3.43\%\\ \text{CNX Nifty Junior}& 6458.55& 6634.85& -2.66\%\\ \text{CNX IT}& 5100.5& 5314.05& -4.02\%\\ \text{Bank Nifty}& 5039.05& 5251.55& -4.05\%\\ \text{CNX 100}& 3519.35& 3640.35& -3.32\%\end{array}$

World Markets
$\begin{array}{cccc}\text{Index}& \text{Current}& \text{Previous}& \text{\% Change}\\ \text{NYSE Cornposite}& 8926.88& 9120.93& -2.13\%\\ \text{NASDAQ Composite}& 2350.57& 2402.29& -2.15\%\\ \text{DOW Jones IA}& 12076& 12318.6& -1.97\%\\ \text{S \& P 500}& 1377.95& 1406.6& -2.04\%\\ \text{Nikkei 225}& 16676.9& 17178.8& -2.92\%\end{array}$

The above figures are taken from the website of National Stock Exchange of India. They illustrate the movement of NSE stock indices as well as world stock indices on the date indicated.

What conclusions can you draw from the ' various movements of NSE stock indices?

Find the sum to n terms

$5^2+6^2+7^2+\dots +{20}^2$

Two persons A and B take turns in throwing a pair of dice. The first person to through 9 from both dice will be avoided the prize. If A throws first then the probability that B wins the game is

[Orissa JEE 2003]

Find $\int \frac{x^2}{(x^2+1)(x^2+4)}dx$

There are 6 boxes labeled $B_1,B_2,\dots ..B_6$. In each trial, two fair dice $D_1,D_2$ are known. If $D_1$ shows j and $D_2$ shows k, then j balls are put into the box $B_k$. After n trials, what is the probability that $B_1$ contains at most one ball?

If $y\left(\alpha \right)=\sqrt{2\left(\frac{\tan \alpha +\cot \alpha }{1+{\tan }^2\alpha }\right)+\frac{1}{{\sin }^2\alpha }}$ where $\alpha \in (\frac{3\pi }{4},\pi )$, then $\frac{dy}{d\alpha }$ at $\alpha =\frac{5\pi }{6}$ is

Let P, Q be two points on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ whose eccentric angles differ by a right angle. Tangents are drawn at P and Q to meet at R. If the chord PQ divides the joint of C and R in the ratio m: n (C being the centre of the ellipse), then find m+n(m:n is in simplified form).

If A and B are two sets such that n(A) = 70, n(B) = 60, $n(A\cup B)$= 110, then $n(A\cap B)$ is equal to

The point of trisection of the line joining the points (0, 3) and (6, -3) are

The direction cosines of the line drawn from P(–5, 3, 1) to Q(1, 5, –2) is

2+4+7+11+16+...

The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.

Let $A_1{A}_2{A}_3....A_9$ be a nine-sided regular polygon with side length $2$ units. The difference between the lengths of the diagonals $A_1{A}_5$ and $A_2{A}_4$ equals

If the points $A(2,-1,1),B(1,-3,-5)$and$C(3,-4,-4)$ form a triangle, find the circum radius for this triangle.

What are the points on y - axis whose distance from the line $\frac{x}{3}+\frac{y}{4}=1$ is 4 units?

Find the equaion of the hyperbola whose

(i) focus is (0,3), directrix is x+y-1=0 and eccentricity=2

(ii) foucs is (1,1), directrix is 3x+4y+8=0 and eccentricity=2

(iii) focus is (1,1), directrix is 2x+y=1 and 3ccentricity=$\sqrt{3}$

(iv) focus is 92,-1), firectrix is 2x+3y=1 and eccentricity =2.

(v) focus is (a,0).directrix is 2x-y+a=0 and eccentricity =$\frac{4}{3}$

(vi) focus is (2,2), directiex is x+y=9 and eccentricity =2.

Sum of values of $x$, satisfying the equation $\sqrt{3x^2+6x+7}+\sqrt{5x^2+10x+14}=4-2x-x^2,$ is

The number of distinct real values of $\lambda$ for which the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{{\lambda }^2}$ and $\frac{x-3}{1}=\frac{y-2}{{\lambda }^2}=\frac{z-1}{2}$ are coplanar is:

Let A = {x : |x-1|< 3} and B = {x : $x^2$ – 2ax + $a^2$ – 4 $\ge$ 0} be two sets. If $A\cap B=\{x:-2<x\le 1\}$, then the smallest positive integral value of ‘a’ is ___

If $A+B+C={180}^{\circ }$. Find ${\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C$ =

The income of a person is Rs. 300,000 in the first year and he receives an increase of Rs. 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.

Write the number of integral solutions of $\frac{x+2}{x^2+1}>\frac{1}{2}$

$\text{The number of values of x in the interval}[0,5\pi ]satisfyingtheequation3si{n}^{2}x-7sinx+2=0is$

Solve the following system of equations in R.

$11-5x>-4,4x+13\le -11$

Area bounded by $y^2$ = 4x and y = 2x – 4 is (in sq. units)

Let ABCD be a square. E and F be points on AC such that AE = EF = FC =AC/3. Then tan (ÐEBF) equals :

$Inany\Delta ABC,\text{prove that :}a(cosB+cosC-1)+b(cosC+cosA-1)+c(cosA+cosB-1)=0$

${lim}_{x\to 3}(\frac{1}{x-3}-\frac{3}{x^2-3x})$

Write the sum of the coefficients in the expansion of $(1-3x+x^2{)}^{111}.$

The plane passing through the point $(4,-1,2)$ and parallel to the lines $\frac{x+2}{3}=\frac{y-2}{-1}=\frac{z+1}{2}$ and $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{3}$ also passes through the point :

$\text{If cos x=k has exactly one solution in}[0,2\pi ],thenwritethevalue\left(s\right)ofk.$

The length of the major axis and the minor axis of the ellipse $2x^2+3y^2-4x-12y+13=0$ are and respectively.

How many of the following statements are correct?

1. $\int x^{2}dx=x^3$

2. $\int \text{sin}xdx=\text{cos}x$

3. $\int e^{x}dx=\ln x$

4. $\int \frac{1}{x}dx=x^0$

5. $\int \text{tanx}dx={\text{sec}}^{2}x$