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Find the maximum and minimum values, if any, of the following function given by,

$f\left(x\right)=9x^2+12x+2$

Write the value of the expression
$\frac{1-4sin{10}^{\circ }sin{70}^{\circ }}{2sin{10}^{\circ }}$

The number of elements in the power set of $A$, where $A=\{x:x\in W\text{and}{x}^3-1\le 7\}$

If two adjacent sides of a square are represented by the vectors $x\hat{i}+\hat{j}+4\hat{k}$ and $y\hat{i}+3\hat{j},$ then $\left(xy\right)=$

Let $L$ be a line passing through the point of intersection of the lines $x+2y+1=0$ and $2x+3y-1=0$. The locus of the circumcentre of the triangle formed by $L$ and coordinate axes is

Evaluating ${\int }_0^{\frac{\pi }{2}}\frac{co{s}^{2}xdx}{1+3si{n}^{2}x}$

Let $g\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt,$ where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3}\le f\left(t\right)\le 1$ for all $t\in [0,1]$ and $0\le f\left(t\right)\le \frac{1}{2}$ for all $t\in (1,3]$. The largest possible interval in which $g\left(3\right)$ lies is :

Analyze the given table:





The correct equation for the given table is .

The number of values of x lying in the interval (-$\pi$, $\pi$) which satisfy the equation

$8^{1+|cosx|+|co{s}^{2}x|+...............+\infty }$=64 is

The middle term of expansion of ${(\frac{10}{x}+\frac{x}{10})}^{10}$

The plane containing the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z-1}{3}$ and also containing its projection on the plane $2x+3y-z=5$, contains which one of the following points ?

In $\Delta ABC,\overrightarrow{AB}=\hat{i}+3\hat{j}-2\hat{k},\overrightarrow{AC}=3\hat{i}-\hat{j}-2\hat{k}.$

If the bisector of $\angle BAC$ meets $BC$ at $D$ and $G$ is the centroid of $\Delta ABC$, then $|\overrightarrow{GD}|=$

Let $f\left(x\right)$ and $g\left(x\right)$ be two bijective functions, where $f\left(x\right)=3x+5$ and $g\left(x\right)=\frac{e^x}{3}-2$ , then $\left(fog\right(x){)}^{-1}$ is equal to

Find the distance between planes 2x + 5y + 4z - 3 = 0 and 2x + 5y + 4z- 5 = 0.

If the product of all solutions of the equation $\frac{\left(2019\right)x}{2020}=(2019{)}^{{\log }_x\left(2020\right)}$ can be expressed in the lowest form as$\frac{m}{n}$, then the value of $m-n$ is

Football teams $T_1$ and $T_2$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_1$ winning, drawing and losing a game against $T_2$ are $\frac{1}{2},\frac{1}{6}$ and $\frac{1}{3},$ respectively. Each team gets $3$ points for a win, $1$ point for a draw and $0$ point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T_1$ and $T_2,$ respectively, after two games.



$P(X>Y)$ is

If $A(\ne O)$ is a skew-symmetric matrix of order $2\times 2$ which is formed with $0$ and $i,$ then the determinant of matrix is

(where $i=\sqrt{-1}$)

In a $△ABC,$ if 2s = a + b + c and (s-b)(s-c)=x $si{n}^2\frac{A}{2},$ then x = [MP PET 1992]

Consider the quadratic equation $(c-5)x^2-2cx+(c-4)=0$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0,2)$ and another root lies in the interval $(2,3)$. The number of elements in $S$ is

Let $y=y\left(x\right)$ be solution of the differential equation ${\log }_e\left(\frac{dy}{dx}\right)=3x+4y,$ with $y\left(0\right)=0$. If $y(-\frac{2}{3}{\log }_{e}2)=\alpha {\log }_{e}2,$ then the value of $\alpha$ is equal to

If a line is tangent at one point and normal at another point on the curve $x=4t^2+3,y=8t^3-1$, then slope(s) of such a line is/are

A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are drawn at random from the box without replacement. The probability of drawing 2 green balls and one blue ball is

(a) $\frac{3}{28}$
(b) $\frac{2}{21}$
(c) $\frac{1}{28}$
(d) $\frac{167}{168}$

N characters of information are held on magnetic tape, in batches of x character each; the batch processing time is $\alpha +\beta x^2$ seconds, $\alpha ,\beta$ are constants. The optimum value of x for fast processing is

If $\sin x+\cos x=\frac{1}{5},$ then sum of the possible value(s) of $|12\tan x|$ is equal to

What is the probability of selecting a square on a chess board, which is not colored uniformly?