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Insert two numbers between 3 and 81 so that the resulting sequences is G.P.

The equation to the hyperbola having its eccentricity 2 and the distance between its foci is 8, is,

Find the area bounded by the curve $y=e^{-x}$, the X-axis and the Y-axis

If ${}^n{P}_r=5040({}^{n-1}{C}_{r-1}+{}^{n-1}{C}_r)$,then r =

Figure shows a relation between set P and Q. Write the relation R

Solve for x, $2si{n}^{3}x=cosx$.

Using $V_n$ method, if $T_n$ = (3n - 1)(3n + 2), Find the value of K if $V_n$ - $V_{n-1}$ = K $T_n$ __

The perpendicular distance from (1,2) to the straight line 12x+5y=7 is

Two wires AC and BC are tied at C of small sphere of mass 5 kg, which revolves at a constant speed v in the horizontal circle of radius 1.6 m. Find the minimum value of v.

For positive integers n₁,n₂ the value of the expression $(1+i{)}^{n_1}$+$(1+i^3{)}^{n_1}$+$(1+i^5{)}^{n_2}$ + $(1+i^7{)}^{n_2}$ where i=$\sqrt[2]{2-1}$ is a real number if and only if

If a root of the equations $x^2+px+q=0$ and $x^2+\alpha x+\beta =0$ is common, then its value will be ( where p ≠ α and q ≠ β )

Distance between the lines represented by the equation $x^2+2\sqrt{3}xy+3y^2-3x-3\sqrt{3}y-4=0$ is

If the sum of first m terms of an AP is the same as the sum of its first n terms, show that the sum of its (m +n) terms is zero.

Or

The sum of n terms of two arithmetic progressions are in the ratio $(3n+8)$ : $(7n+15)$. Find the ration of their 12th terms.

Find the principal and general solutions of the following equations.
cot x =$-\sqrt{3}$

Which of the following can be an alternate representation of variance, where x_i being the midpoint of class intervals, fi being frequency of class interval and $\overline{x}$ the mean,N = ${\sum }_{i=1}^n{f}_i$

The value of m for which coordinates (3,5), (m,6) and $(\frac{1}{2}$, $\frac{15}{2})$ are collinear.

Find ${}^n{C}_0$ - 2${}^n{C}_1$ + 3${}^n{C}_2$ - 4${}^n{C}_3$ .....................+ $(-1{)}^n$ (n+1) ${}^n{C}_n$

$\underset{x\to \infty }{lim}(1+\frac{1}{x}{)}^x=$

If a₁, a₂, a₃.....a_n are in H.P., then $\frac{a_1}{a_2+a_3+......a_n}$ . $\frac{a_2}{a_1+a_3+.....+a_n}$,...... $\frac{a_n}{a_1+a_2+.......+a_{n-1}}$

are in

Find the mean deviation about the median for the given data.

$\begin{array}{ccccccc}{x}_i& 5& 7& 9& 10& 12& 15\\ f_i& 8& 6& 2& 2& 2& 6\end{array}$

Prove by the principle of mathematical induction that $1+2+3+...+n<\frac{(2n+1{)}^2}{8}$, for all $n\in N.$

Or

Using the principle of mathematical induction, prove that

$1.3+{2.3}^2+{3.3}^3+...+n{.3}^n=\frac{(2n-1)3^{n+1}+3}{4}.$ for all $n\in N.$

The sum of three vectors shown in figure is zero. Find the magnitudes of the vectors $\overrightarrow{OB}$ and $\overrightarrow{OC}$

The harmonic mean of 2 numbers is 4. Their arithmetic mean A and geometric mean G satisfy the relation 2A + $G^2$ = 27. The sum of the numbers is __

$\theta$ ∈ [0, 2$\pi$] and $z_1,z_2,z_3$ are three complex numbers such that they are collinear and

$(1+|sin\theta |)z_1$ + $(|cos\theta |-1)z_2$ - $\sqrt{2}$$z_3$ =0. If at least one of the complex numbers $z_1,z_2,z_3$ is

non-zero then number of possible values of $\theta$ is

Equation of the straight line passing through the point of intersection of the lines $3x+4y=7,x-y+2=0$ and having slope $3$ is

If (1+i)(1+2i)(1+3i)......(1+ni) = a+ib, then

2.5.10.....(1+$n^2$) is equal to

Calculate the mean from the following data :

$\begin{array}{cccccc}\text{C.I}& 10-20& 10-30& 10-40& 10-50& 10-60\\ \text{f}& 5& 17& 31& 41& 49\end{array}$

Three vertices of a parallelogram ABCD are A(3,-1,2), B(1,2,4) and C(-1,1,2). Find the fourth vertex .

The variance of first $50$ even natural numbers is:

If ${\log }_2(5\times 2^x+1),{\log }_4(2^{1-x}+1)$ and $1$ are in A.P., then $x$ equals

If G = {7, 8} and H = {5, 4, 2], find $G\times H$ and $H\times G.$

The statement $p\Rightarrow (q\Rightarrow p)$ is equivalent to

The general term of the sequence $7,12,17,22,27,\dots$is

The minimum value of $f\left(x\right)=|x-1|+|x-2|+|x-3|$ is

The equation of the line passing through the points $(4,-2)$ and $(-3,3)$ is

How many numbers lying between 10 and 1000 can be formed from the

digits 1, 2, 3, 4, 5, 6, 7, 8, 9 (repetition is allowed)

If ${{32}^{32}}^{32}$ is divided by 7 , then the remainder is

Find the sum of the series $4+4^2+4^3+...{.4}^{100}+{4.4}^{101}$

The sum of the digits in the unit place of all numbers formed with the help of 3,4,5,6 taken all at a time is

The sum of the series $6+13+22+33+\dots \dots$ upto $20$ terms is

Let $z_1=2-i,z_2=-2+i$. Find

$\left(i\right)Re\left(\frac{z_1{z}_2}{\overline{z_1}}\right)$

$\left(ii\right)Im\left(\frac{1}{z_1\overline{z_1}}\right)$

Using first principle, find the derivative of $tan\sqrt{x}.$

A committee of 12 members is to be formed from 9 women and 8 men. In how many ways can this be done, if at least five women have to be included in a committee?

Convert each of the complex numbers in he polar form:

$1-i$

Make correct statements by filling in the symbols $\subset$ or $\subset ̸$ in the blank spaces:

(i) {2, 3, 4} ..................... {1, 2, 3, 4, 5}

(ii) {a, b, c} ..................... {b, c, d}

(iii) {x : x is a student of Class XI of your school} ......... {x : x student of your school}

(iv) {x : x is a circle in the plane} ........... {x : x is a circle in the same plane with radius 1 unit}

(v) {x : x is a triangle in a plane} ........... {x : x is a rectangle in the same plane}

(vi) {x : x is an equilateral triangle in a plane} ............ {x : is a triangle in the same plane}

(vii) {x : x is an even natural number} .......... {x : x is an integer}

If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P., then the common ratio of the G.P. is

If the arithmetic mean of the number $x_1,x_2,x_3.........,x_n$ is $\overline{x}$ , then the arithmetic mean of numbers$a{x}_1,+b,a{x}_2+b,a{x}_3+b,.............a{x}_n+b,$
where a, b are two constants would be

A conic section is defined by the equations x = - 1 + sect y = 2 + z tant. The coordinates of the foci are,