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For any two sets $A$ and $B,$ if $(A\cup B{)}^{\prime }=A^{\prime }\cup B^{\prime },$ then

A line $y=mx+1$ intersects the circle $(x-3{)}^2+(y+2{)}^2=25$ at the points $P$ and $Q$. If the midpoint of the line segment $PQ$ has $x$-coordinate $-\frac{3}{5}$, then which one of the following options is correct?

Evaluate the Given limit:

1. 73.58

Twelve students compete in a race. In how many ways first three prizes be given?

In the expansion of (1 + a)<sup>m
+ n</sup>, prove that coefficients of
a^m
and aⁿ
are equal.

If the circle $x^2+y^2+2gx+2fy+c=0$ touches x-axis, then

Let $\{\ast \}$ and $[\ast ]$ be the fractional part function and greatest integer function repectively. If the range of the function $f\left(x\right)={\sin }^{-1}{x}^2+\left[\left\{\ln \sqrt{x-\left[x\right]}\right\}\right]+{\cot }^{-1}\left(\frac{1}{1+\sqrt{2}{x}^2}\right)$ is $(\frac{\pi }{c},\frac{a\pi }{b}),$ then

If f is a function satisfying such that , find the value of n.

Let $f\left(x\right)=x{e}^{-x}$. The maximum value of the function in the interval $(0,\infty )$ is

Find the equations to the sides of an isoceles right angled triangle the equation of whose hypotenuse is $3x+4y=4$ and the opposite vertex is the point (2, 2).

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b)ⁿ (cx + d)^m

The general solution of the differential equation $\sqrt{1+x^2+y^2+x^2{y}^2}+xy\frac{dy}{dx}=0$ is (where $C$ is a constant of integration)

Which among the following is the correct graphical representation of the quadratic polynomial $y=-2x^2+2x-0.5?$

The equation of the circle having centre at the origin and passing through the point $(3,4),$ is

The value of $\underset{n\to \infty }{lim}\frac{1}{n}\sum _{j=1}^n\frac{(2j-1)+8n}{(2j-1)+4n}$ is equal to

If $I_n=\int {\sin }^{n}xdx,n\in N,n\ge 2$ then $n{I}_n-(n-1)I_{n-2}=$

(where $C$ is integration constant)

Find x
and y, if

If $\alpha ,\beta$ are the roots of quadratic equation $x^2+3x+8=0$. Find the equation with roots $\alpha -5$ and $\beta -5$

Let$I\left(n\right)=n^n,J\left(n\right)=\mathrm{1.3.5....}(2n-1)\text{for all}(n>1),n\in N,$ then

Find the distance of the point P (-4, 3, 5) from the coordinate axes.

For the interval $[-2\pi ,2\pi ],\sin x>0$ in the interval:

Let the natural numbers be divided into groups as $\left(1\right),(2,3,4),(5,6,7,8,9),...$ and so on. Then the sum of the numbers in the $n^{th}$ group is

A die is thrown. Find the probability of getting :
(i) a prime number
(ii) 2 or 4
(iii) a multiple of 2 or 3.

Prove that
cos 4x = 1 – 8sin² x cos² x

The slope of tangent at $(2,-1)$ to the curve $x=t^2+3t-8$ and $y=2t^2-2t-5$ is :

Complete set of values of a such that $\frac{x^2-x}{1-ax}$ attains all real values is

If ${\Delta }_1=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{ccc}{a}_1+p{b}_1& b_1+q{c}_1& c_1+r{a}_1\\ a_2+p{b}_2& b_2+q{c}_2& c_2+r{a}_2\\ a_3+p{b}_3& b_3+q{c}_3& c_3+r{a}_3\end{array}\right|$ are such that ${\Delta }_2=k\cdot {\Delta }_1,$ then $k$ is