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If $\int \frac{\sqrt{1-x^2}}{x^4}dx=A\left(x\right){\left(\sqrt{1-x^2}\right)}^m+C$, for a suitable chosen integer $m$ and a function $A\left(x\right)$, where $C$ is a constant of integration, then ${\left(A\right(x\left)\right)}^m$ equals :

${lim}_{x\to 0}\frac{xcosx+sinx}{x^2+tanx}$

Argument of $z$ if $z=\sin \left(\frac{\pi }{5}\right)+i(1-\cos \left(\frac{\pi }{5}\right))$ is

The locus of z satisfying the inequality $lo{g}_{\frac{1}{3}}$|z+1| > $lo{g}_{\frac{1}{3}}$|z-1| is

If $n$ integers taken at random are multiplied together, then the probability that the last digit of the product is $1,3,7$ or $9$ is:

Differentiate the following functions with respect to x :

$\frac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}}$

1. 33.36

Find
the coefficient of x⁵
in (x +
3)⁸

If $\alpha ={\tan }^{-1}\frac{1}{2}+{\tan }^{-1}\frac{1}{3},$ $\beta ={\cos }^{-1}\frac{2}{3}+{\cos }^{-1}\frac{\sqrt{5}}{3}$ and $\gamma ={\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)+\frac{1}{2}{\cos }^{-1}\left(\cos \frac{2\pi }{3}\right),$ then the value of $\tan \alpha -\tan \left(\frac{\beta }{2}\right)+\sqrt{3}\tan \left(\frac{\gamma }{4}\right)$ is equal to

If $x=r\sin A\sin B,y=r\cos A\sin B,z=r\cos B,$ then $x^2+y^2+z^2$ is equal to

${lim}_{x\to 0}$ $\frac{1-{cos}^{3}x}{sin3x.sin5x}$​ =

Q57. Probability that Ria can solve a problem is 1/2 and the probability that Tina can solve a problem is 2/3. If both attempt a question, then what is the probability that the problem is solved?

रिया द्वारा एक प्रश्न को हल करने की प्रायिकता 1/2 है और टीना द्वारा एक प्रश्न को हल करने की प्रायिकता 2/3 है। यदि दोनों एक प्रश्न को हल करना आरंभ करती हैं, तो प्रश्न के हल हो जाने की प्रायिकता क्या है?

If α and β(α$<$β) are two different real roots of the equation a$x^2$ + bx +c = 0, then

Let the acute angle bisector of the two planes $x-2y-2z+1=0$ and $2x-3y-6z+1=0$ be the plane $P.$ Then which of the following points lies on $P?$

Let the observations $x_i(1\le i\le 10)$ satisfy the equations, $\sum _{i=1}^{10}(x_i-5)=10$ and $\sum _{i=1}^{10}(x_i-5{)}^2=40$. If $\mu$ and $\lambda$ are the mean and the variance of observations, $(x_1-3),(x_2-3),...,(x_{10}-3)$, then the correct option(s) is/are

The probability of a number ‘n’ showing in a throw of a die marked 1 to 6 is proportional to ‘n’. If p represents the probability that the die shows 3, then the value of $\frac{1}{p}$ is___

If [x] denotes the integral part of x and f(x)=[n+psin x], 0<x<π , n belongs to integers and p is a prime number, then the number of points , where f(x) is not differentiable is

a) p-1

b) p

c) 2p-1

d)2p+1

The domain of $f\left(x\right)=\sqrt{\cos x}+\sqrt{4-x^2}$ is

Find a
particular solution of the differential equation,
given that y = – 1, when x = 0 (Hint: put x
– y = t)

If $f\left(x\right)$ satisfies the relation $f\left(x\right)-\lambda \underset{0}{\overset{\frac{\pi }{2}}{\int }}\sin x(\cos t\cdot f(t\left)\right)dt=\sin x,\lambda >2,$ then $f\left(x\right)$ decreases in which of the following interval?

The pair of straight lines joining the origin to the points of intersection of the line $y=2\sqrt{2}x+c$ and the circle $x^2+y^2=2$ are at right angles, if

If circles $x^2+y^2+24x-10y+a=0$ and $x^2+y^2-36=0$ have no point in common, then range of values of $a$ is

The solution set of $(x-1{)}^{99}(x+1{)}^{100}\le 0$ is

$Ina\Delta ABC,\text{prove that}acosA+bcosB+ccosC=2asinBsinC$

If the length of the tangents from $(a,b)$ to the circles $x^2+y^2-4x-5=0$ and $x^2+y^2+6x-2y+6=0$ are equal then $10a-2b$ is equal to

Let $A$ and $B$ be two sets containing $2$ elements and $4$ elements respectively. The number of subsets of $A\times B$ having $3$ or more elements is :

If a quadratic expression cuts the x-axis at $x=\alpha ,\beta$ and $(p,q)$ is it's vertex, then $p=$

A class consists of 10 boys and 8 girls. Three students are selected at random. What is the probability that the selected group has

(i) all boys?

(ii) all girls?

(iii) 1 boy and 2 girls?

(iv) at least one girls?

(v) at most one girls?