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Let f(x) = 9^x ÷ 9^x + 3 then f(x) + f(1 - x) =

In the plot of the function below. Which is the point at which the discontinuity is of removable type?

If $y={\sin }^{-1}\left(x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^2}\right\}\mathrm{and}0<x<1,\mathrm{then}\frac{\mathrm{dy}}{\mathrm{dx}}$is

Which of the following option is correct?

$$\begin{array}{cc}\text{List I}& \text{List II}\\ \begin{array}{c}\text{(A) The minimum value of the expression}\\ \frac{{\sec }^4\alpha }{{\tan }^2\beta }+\frac{{\sec }^4\beta }{{\tan }^2\alpha }\forall \alpha ,\beta \in (0,\pi /2)\text{is}\end{array}& \text{(1)}1\\ \begin{array}{c}\text{(B) If}{z}_1,z_2\text{are two non-zero complex numbers}\\ \text{satisfying the equation}\left|\frac{z_1+z_2}{z_1-z_2}\right|=1\text{, then}\\ \text{the value of}\frac{z_1}{z_2}+\overline{\left(\frac{z_1}{z_2}\right)}\text{is}\end{array}& \text{(2)}2\\ \begin{array}{c}\text{(C) Number of integers in the range of function}\\ y=\frac{x^2+x+3}{x^2+2x+6},x\in R\text{is}\end{array}& \text{(3)}0\\ \begin{array}{c}\text{(D) If}y=(3{)}^{(-3{)}^{1/(1-x)}},\text{then}\underset{x\to 1^{+}}{lim}y\text{is}\end{array}& \text{(4)}8\end{array}$$

By method of mathematical induction, the value of $1^2+3^2+5^2\cdots +(2n-1{)}^2=\frac{n}{x}(2n-1\left)\right(2n+1)$ is true for all $n\in N$

Then $x$ is:

If z = x + 3i, then value of ${\int }_2^4\left[arg\left|\frac{z-i}{z+i}\right|\right]dx$ is ___. $|\cdot |$ represents modulus of a complex numbers

The order and degree of the differential equation: (y")² + (y")³ + (y')⁴ + y⁵ = 0 is:

A square is inscribed in the circle $x^2+y^2-2x+4y-93=0$ with its sides parallel to the coordinate axes. The coordinates of its vertices are

The value of ${\cos }^{-1}\left[\frac{1}{\sqrt{2}}(\cos \frac{9\pi }{10}-\sin \frac{9\pi }{10})\right]$ is

If a $5$ digit number is made using all the digits from $1,3,4,6,8$, such that all the digits of number from $1$st position to $5$th should not be in increasing order, then the position of number ${}^{\prime \prime }{63184}^{\prime \prime }$ after listing all the numbers formed in ascending order is

The value of the definite integral $\underset{0}{\overset{\pi /2}{\int }}\sqrt{\tan x}dx$ is

Let $\left[t\right]$ denote the greatest integer $\le t$. If for some $\lambda \in R-\{0,1\}$,$\underset{x\to 0}{lim}\left|\frac{1-x+\left|x\right|}{\lambda -x+\left[x\right]}\right|=L$, then $L$ is equal to

The degree of the differential equation ${(1+\frac{dy}{dx})}^{\frac{3}{2}}=\frac{d^{2}y}{d{x}^2}$ is

You are given $8$ balls of different colours (black, white, ...). The number of ways in which these balls can be arranged in a row so that the two balls of particular colour (say red and white) may never come together, is

If A + B + C = 0, then prove that $\left|\begin{array}{ccc}1& cosC& cosB\\ cosC& 1& cosA\\ cosB& cosA& 1\end{array}\right|=0$

The numbers P, Q and R for which the function

$f\left(x\right)=P{e}^{2x}+Q{e}^x+Rx$ satisfies the conditions

$f\left(0\right)=-1,f^{{}^{\prime }}\left(log2\right)=31$ and ${\int }_0^{log4}\left[f\right(x)-Rx]dx=\frac{39}{2}$ are

given by

Find the equation of tangent at $x=0$ on the curve $y=x{e}^x$

Consider the relation $4l^2-5m^2+6l+1=0$, where $l,m\in R.$ If the line $lx+my+1=0$ touches a fixed circle, then the centre of that circle is:

The value of $(1+\omega )(1+{\omega }^2)(1+{\omega }^3)(1+{\omega }^4)\cdots (1+{\omega }^{3n})$ is

The mean of $100$ items is $49$. It was found that three items which should have been $60,70,80$, were wrongly read as $40,20,50$ respectively. The corrected mean is

A region in $xy$-plane is bounded by $y=0$ and $y=\sqrt{25-x^2}$. If a point $(a,a+1)$ lies in the interior of the region, then :

The probability that ${13}^{\text{th}}$ day of a randomly chosen month is a Friday is

The angle of elevation of the top of a hill from a point is $\alpha$. After walking b towards the top up of a slope inclined at an angle $\beta$ to the horizon, the angle of elevation of the top becomes $\gamma$. Then the height of hill is

An integer is chosen at random from first 200 positive integers. Find the

probability that the integer is divisible by 6 or 8

Let $A=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]$, B=$\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]$
Find the value of the following:
A-B

The number of three digit numbers $\overline{abc}$ such that the arithmetic mean of $b$ and $c$ and the square of their geometric mean are equal, is

The equation of the tangent at the vertex of the parabola $x^2+4x+2y=0$ is