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For the complex number $z=\sqrt{3}i-i-1-\sqrt{3},$ the correct option(s) is/are

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{3x^2+mx+n}{x^2+1}.$ If the range of $f$ is $[-4,3),$ then the value of $m^2+n^2$ is

Let $a,b$ and $c$ be the $7^{\text{th}},{11}^{\text{th}}$ and ${13}^{\text{th}}$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $\frac{a}{c}$ is equal to :

How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4 ?

Middle term in the expansion of $(1+3x+3x^2+x^3{)}^6$ is

The solution y(x) of the differential equation $\frac{d^{2}y}{d{x}^2}=sin3x+e^x+x^2$ when $y_1\left(0\right)=1$ and y(0) = 0 is

The ring $M_1$ and block $M_2$ are held in the position shown in figure. Now the system is released. If $M_1$ $>$ $M_2$ find $\frac{v_1}{v_2}$ when the ring $M_1$ has descended along the smooth fixed vertical rod by the distance $y=h$ .

Which of the following statements is/are correct?

1. The locus of the point of intersection of two perpendicular tangents is called director circle of the given circle.

2. The director circle of a circle is concentric circle.

3. The radius of director circle of a circle is equal to the radius of original circle.C

Write the contrapositive of the following statement.

'If you are born in India, then you are a citizen of India'.

If $2{\sin }^{2}x+3\sin x-2>0$ and $x^2-x-2<0,$ then $x$ lies in the interval

If a function f(x) is defined on [1,4] $\to$ [1,7] and given that f(3) = 5 and its inverse exists then find $f\left(f^{-1}\right(5\left)\right)$.

Evaluate $\int \frac{x\mathrm{dx}}{(x-1)(x^2+4)}$

If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint $(h,k)$, then which of the following is(are) possible value(s) of $p,h\text{and}k$ ?

The latitude and departure of a line AB are +78 m and -45.1 m, respectively. The whole circle bearing of the line AB is:

The nearest point on the circle $x^2+y^2-6x+4y-12=0$ from the point $P(-5,4)$ is $Q(\alpha ,\beta ),$ then the value of $\alpha +\beta$ is

If the function f(x)= x³-3ax²+b is strictly increasing derivative for x > 0, then which of the following is always true?

If $A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],$ then which one of the following statements is not correct ?

The set of values of $a$ for which $a^2-a-6<0$ and $\underset{x\to \infty }{lim}{a}^x$ does not exist is/are

Let $I_1={\int }_{se{c}^{2}z}^{2-ta{n}^{2}z}xf\left(x\right(3-x\left)\right)dxandlet{I}_2={\int }_{se{c}^{2}z}^{2-ta{n}^{2}z}f\left(x\right(3-x\left)\right)dx$
where 'f' is a continuous function and 'z' is any real number, then $\frac{I_1}{I_2}=$

Find the intervals in which the function f given by $f\left(x\right)=x^3+\frac{1}{x^3},x\ne 0$ is
decreasing

The least positive integer n for which $\sqrt[3]{3n+1}-\sqrt[3]{3n}<\frac{1}{12}$ is

For $x>0$, $\int \frac{\sqrt{x-1}}{x\sqrt{x+1}}dx$ is equal to (where $c$ is constant of integration)

The principal value of $si{n}^{-1}tan\left(\frac{-5\pi }{4}\right)$ is

Let $g_i:[\frac{\pi }{8},\frac{3\pi }{8}]\to R,i=1,2$ and $f:[\frac{\pi }{8},\frac{3\pi }{8}]\to R$ be functions such that $g_1\left(x\right)=1,g_2\left(x\right)=|4x-\pi |$ and $f\left(x\right)={\sin }^{2}x,$ for all $x\in [\frac{\pi }{8},\frac{3\pi }{8}].$

If $S_i=\underset{\pi /8}{\overset{3\pi /8}{\int }}f\left(x\right)\cdot g_i\left(x\right)dx,i=1,2,$ then the value of $\frac{48S_2}{{\pi }^2}$ is

A whistle revolves in a circle with angular velocity $\omega =20rad/s$ using a string of length $50cm$. If the actual frequency of sound from the whistle is $385Hz$, then the minimum frequency heard by the observer far away from the centre is (velocity of sound $v=340m/s$)

Three numbers are chosen at random without replacement from $\{1,2,3,\cdots ,10\}$. The probability that, the smallest of the chosen number is $3$ or the greatest one is $7,$ is equal to