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Appropriately matching the information given in the three columns of the following table.

$\begin{array}{cccccc}& \text{Column 1}& & \text{Column 2}& & \text{Column 3}\\ \text{(I)}& \text{If}a,b,cϵR-\left\{0\right\}\text{such that}a\ne b\ne c& & & & \\ & \text{and}\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\text{and}A=\left[\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right],\text{then}& \left(i\right)& \text{A is singular matrix}& \text{(P)}& |adjA|=|A{|}^2\\ \text{(II)}& \text{If}\alpha ,\beta ,\gamma ϵR,\text{and}& & & & \\ & A=\left[\begin{array}{ccc}1& \cos (\alpha -\beta )& \cos (\alpha -\gamma )\\ \cos (\beta -\alpha )& 1& \cos (\beta -\gamma )\\ \cos (\gamma -\alpha )& \cos (\gamma -\beta )& 1\end{array}\right],\text{then}& \left(ii\right)& \text{A is singular matrix}& \left(Q\right)& adj\left(adjA\right)=|A|A\\ \left(III\right)& \text{If}\omega \ne 1\text{be cube root of unity and}& \left(iii\right)& \text{A is non-singular matrix}& \left(R\right)& |A|\text{is equal to}\\ & A=\left[\begin{array}{ccc}1+2{\omega }^{100}+{\omega }^{200}& {\omega }^2& 1\\ 1& 1+{\omega }^{101}+2{\omega }^{202}& \omega \\ \omega & {\omega }^2& 2+{\omega }^{100}+2{\omega }^{200}\end{array}\right]& & & & \text{is equal to minimum value of}\\ & ,then& & & & {\cos }^{-1}(x-\frac{1}{x})\\ & & & & & +{\cos }^{-1}\left(\frac{y^2}{y+1}\right)\\ & & & & & +{\cos }^{-1}(z^2+z+1)\\ & & & & & \text{(where}x,y,z\text{are real numbers)}\\ \text{(IV)}& \text{If}a,b,cϵR-\left\{0\right\}\text{such that}a\ne b\ne c,\text{and}& & & & \\ & A=\left[\begin{array}{ccc}0& (a-b{)}^3& (a-c{)}^3\\ (b-a{)}^3& 0& (b-c{)}^3\\ (c-a{)}^3& (c-b{)}^3& 0\end{array}\right],\text{then}& \left(iv\right)& \text{Invertible}& \left(S\right)& |A^{-1}|=\frac{1}{|A|}\end{array}$

Which of the following is only correct combination ?

If 2^a =3^b =6^c then show that c = ab÷a+b

f(x) = sin(x) defined on f: $[\frac{-\pi }{2},\frac{\pi }{2}]$ $\to$ $[-1,1]$ is -

$\int \frac{\cos (x-a)}{\cos (x-b)}dx=Ax+B\ln |\cos (x-b)|+C$,
then $B/A=$

Integrate the function.
$\int xta{n}^{-1}xdx.$

If both the roots of the quadratic equation $x^2$-2kx+$k^2$+k-5=0 are less than 5, then k lies in the interval.

Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that $Y\subseteq X,Z\subseteq XandY\cap Z$ is empty is:

If $y+b=m_1(x+a),y+b=m_2(x+a)$ are two tangents to the parabola $y^2=4ax,$ then

When P is a natural number, then $P^{n+1}+(P+1{)}^{2n-1}$ is divisible by

100√25÷√25+x =50 then find the value of x

Good evening sir/ma'am,

I recently watches a IIT NEET video under the topic differentiation,

There was this problem "How can a string of length L be made into a rectangle so as to maximize the area of the rectangle"

I understood the solution and how it was solved but I wanted to ask that how does it matter, if the length of the string remains constant, any shape made with the same string of length L, doesn't the area of all those will be equal?

On 1st February, 2016, Ravi sold goods to Mohan for Rs. 18,000; Rs. 3,000 were paid by Mohan immediately and for the balance he accepted three months bill drawn upon him by Ravi. On the date of maturity of the bill Mohan requested Ravi to cancel the old bill and draw a new bill upon him for a period of 2 months. He further agreed to pay interest in cash to Ravi 12 % per annum. Ravi agreed to Mohan's request and cancelled the old bill and drew a new bill. The new bill was met on maturity by Mohan.
Pass necessary journal entries in the books of Ravi.

In $\Delta$ ABC having vertices $A(acos{\theta }_1,asin{\theta }_1),B(acos{\theta }_2,asin{\theta }_2)$ and $C(acos{\theta }_3,asin{\theta }_3)$ is equilateral, then which of the followings is/are true?

What universal set (s) would you propose for each of the following?

(i) The set of right triangles

(ii) The set of isosceles triangles.

Number of middle terms in the expansion of $(a+b{)}^{21}$ is:

If a leap year is selected at random, what is the chance that it will contain 53 Tuesday?

If for any $2\times 2$ square matrix A, A(adj.A)=$\left[\begin{array}{cc}8& 0\\ 0& 8\end{array}\right]$,then write the value of |A|.

If $PSQ$ is the focal chord of the parabola $y^2=8x$ such that $SP=6$ then the lenght of $SQ$ is, where $S$ is the focus of the parabola

Match the following by appropriately matching the lists based on the information given in Column I and Column II.
A bag contains some white and some black balls, all combinations being equally likely. The total number of balls in the bag is 12. Four balls are drawn at random from the bag at random without replacement.
$\begin{array}{cc}\text{Column I}& \text{Column II}\\ a.\text{Probability that all the four balls are black is equal to}& p.\frac{14}{33}\\ b.\text{If the bag contains 10 black and 2 white balls, then}& \\ \text{the probability that all four balls are black is equal to}& q.\frac{1}{5}\\ c.\text{If all the four balls are black, then the probability}& \\ \text{that the bag contains 10 black balls is equal to}& r.\frac{70}{429}\end{array}$

If $x_1,x_2,x_3,......,x_n$ are in A.P. whose common difference is a, then the value of sin $\alpha$(sec $x_1$ sec $x_2$ + sec $x_2$ sec $x_3$ +....... + sec $x_{n-1}$, sec $x_n$)=

Now, as you know the product of roots is 30, let those roots be a, b, c $ϵR$. The minimum value of 10a + 9b + 10c is __

The numerically greatest term in the expansion of $(3-5x{)}^{15}$ when $x=\frac{1}{5}$ is

The number of roots of the equation x + 2tanx = $\frac{\pi }{2}$ in the interval [0, 2$\pi$]

Each of these questions are based on the information given below

Which of the following pair is diagonally opposite to each other ?

Find the real part of the complex number $(1-i{)}^{(1+i)}$

If the lines $\overline{r}=\overline{a}+\lambda (\overline{b}\times \overline{c})$ and $\overline{r}=\overline{b}+\mu (\overline{c}\times \overline{a})$ intersect each other then the condition is

Let $X,Y$ be two events such that $P\left(X\right)=\frac{1}{2},P\left(X|Y\right)=\frac{2}{3},P\left(Y|X\right)=\frac{5}{6}$ then

If the minor axis of an ellipse forms an equilateral triangle with one vertex of the ellipse then e =

The area of the region

$\{x,y):0\le y\le x^2+1,0\le y\le x+1,0\le x\le 2\}$

(in sq. units)

In a $\Delta ABC,tanA+tanB+tanC=6$ and tanA.tanB = 6, then tanA, tanB, tanC are ?

The co-ordinates of the foot of perpendicular drawn from point P(1, 0, 3)to the join of points A(4, 7, 1)and B(3, 5, 3)is
[RPET 2001]

Area bounded by $y=f^{-1}\left(x\right)$ and tangent and normal drawn to it at the points with abscissae $\pi$ and $2\pi$, where f(x) = sinx - x is

Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation $x^2–2x+3=0$ where $\alpha \ne \beta .$ For $n=0,1,2,...,$ let $a_n=p{\alpha }^n+q{\beta }^n,$ then $a_9=$

If $x=e^{y+e^y+e^{y+...\infty }}$ ,then $\frac{dy}{dx}$ is

If in $△ABC,\angle A=\frac{\pi }{4}$ and $\tan B\tan C=p$, then the possible set of value(s) of $p$ is/are

Let$f\left(x\right)=sinx+cosx$ , $g\left(x\right)=x^2-1$ . Thus $g\left(f\right(x\left)\right)$ is invertible for $xϵ$

The value of $6^{\text{th}}$ term from the beggining of ${(2^{{\log }_2\sqrt{9^{x-1}+7}}+\frac{1}{2^{\frac{1}{5}{\log }_2\left(3^{x-1}+1\right)}})}^7=84$ then value of $x$ is

Write the value of ${lim}_{x\to \infty }\frac{sin{x}^{\circ }}{x}$

If $y=\sec \left({\tan }^{-1}x\right)$, then $\frac{dy}{dx}$ at $x=1$ is equal to :

If $\theta =\frac{\pi }{2^{100}+1},$ then $\cos \theta \cos 2\theta \cos 2^2\theta \cdots \cos 2^{99}\theta$ is

If the foci of an ellipse subtend a right angle at either extremity of its minor axis, then its eccentricity