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The solution of the differential equation $\frac{dy}{dx}=\frac{1}{xy[x^2\sin y^2+1]}$ is

(where $C$ is integration constant)

Find the sum of the series $4-5x+7x^2-9x^3+11x^4-13x^5+.....\infty$

What is the conjugate hyperbola of the hyperbola

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

If tan(A - B)=1, sec (A + B)= $\frac{2}{\sqrt{3}}$, then the smallest positive value of B is

In the following example find the distance of each of the given points from the corresponding given plane:

$\begin{array}{cc}point& Plane\\ (3,-2,1)& 2x-y+2z+3=0\end{array}$

Let the equation of two concentric circles is $x^2+y^2-2x+4y+\lambda =0$. If a chord of first circle is tangent to second circle and normal to a circle passing through centre of these circles and having a centre at $(2,-1)$ is also touching first circle. Then length of chord with respect to first circle is:

If $\left\{x\right\}$ represents the fractional part of $x,$ then $\left\{\frac{5^{200}}{8}\right\}$ is

The probability that a certain beginner at golf gets good shot if he uses correct club is $\frac{1}{3},$ and the probability of a good shot with an incorrect club is $\frac{1}{4}.$ In his bag there are 5 different clubs only one of which is correct for the good shot. If he chooses a club at random and take a stroke, the probability that he gets a good shot is

A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of 5 steps, he is one step away from the starting point.

OR

Suppose a girl throws a die. If she gets a 1 or 2, she tosses a coin three times and notes the number of "tails". If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a 'head' or 'tail' is obtained. If she obtained exactly one "tail", what is the probability that she threw 3, 4, 5 or 6 with the die?

Find out the wrong number in the series given below :

$2,3,6,15,42,122$

The real function f : $[0,\infty )\to$ R is defined by $f\left(x\right)=x^2$, then f is

If $I=\underset{0}{\overset{x}{\int }}\frac{2^t}{2^{\left[t\right]}}dt,x\in R^{+}$, where $[\cdot ]$ denotes the greatest integer function, then the value of $I$ is

If $\sec x-\tan x=0.5,$ then select the correct statement(s).

If the product of the roots of the quadratic equation $m{x}^2-2x+(2m-1)=0$ is $3$, then the value of $m$ is

$(ab{)}^n=a^n{b}^n$ for all n $nϵN$.

Find the values of x in each of the following

$\left(i\right)lo{g}_{2}x=3$

$\left(ii\right)lo{g}_{81}x=\frac{3}{2}$

1. 73

Let $b_i>1\text{for}i=1,2,...,101.$ Suppose ${\log }_e{b}_1,{\log }_e{b}_2,...,{\log }_e{b}_{101}$ are in Arithmetic Progression (A.P.) with the common difference ${\log }_{e}2.$ Suppose $a_1,a_2,...,a_{101}$ are in A.P. such that $a_1=b_1$ and $a_{51}=b_{51}.$ If $t=b_1+b_2+\cdots +b_{51}\text{and}s=a_1+a_2+\cdots +a_{51},$ then

For the differential equation in given question find the general solution.​​​​

$e^{x}tanydx+(1-e^x)se{c}^{2}ydy=0$

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

If $m$ is a positive integer, then $\left[\right(\sqrt{3}+1{)}^{2m}]+1$ is divisible by

(where [.] denotes the greatest integer function)

The distance of line $3y-2z-1=0=3x-z+4$ from the point $(2,-1,6)$ is

Which of the following equations are consistent?

Roots of the agebraic equation $x^3+x^2+x+1=0$ are

The point P(1,1) is translated parallel to 2x=y in the first quadrant through a unit distance.The coordinates of nee position of P is

The equation of the circle having one of its diameter as end points of foci of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$, is

${lim}_{x\to 0}9$