Let $P$ and $Q$ be distinct points on the parabola $y^2=2x\mathrm{ }$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\mathrm{\Delta }OPQ$ is $3\sqrt{2},\mathrm{ }$then which of the following is (are) the coordinates of $P$?

The option(s) with the value of $a$ and $L$ that satisfy the following equation is(are) $\frac{\underset{0}{\overset{4\pi }{\int }}{e}^t\left({\sin }^{6}at+{\cos }^{4}at\right) dt}{\underset{0}{\overset{\pi }{\int }}\left({\sin }^{6}at+{\cos }^{4}at\right) dt}=L ?$

Tangent is drawn at any point \(P\) of a curve which passes through \((1,1)\) cutting \(X\)-axis and \(Y\)-axis at \(A\) and \(B\), respectively. If \(A P: B P=3: 1\), then

Let $f\left(x\right)=x+{\log }_{e}x-x{\log }_{\mathrm{e}}x,x\in \left(0, \infty \right).$
$•$   Column 1 contains information about zeros of $f\left(x\right), f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right).$
$•$   Column 2 contains information about the limiting behaviour of $f\left(x\right), f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right)$ at infinity.
$•$   Column 3 contains information about increasing-decreasing nature of $f\left(x\right)$ and $f^{\prime }\left(x\right).$
 

Column 1	Column 2	Column 3
(I) $f\left(x\right)=0$ for some $x\in \left(1, e^2\right)$	(i) $\underset{x\to \infty }{\lim }f\left(x\right)=0$	(P) $f$ is increasing in (0, 1)
(II) $f^{\prime }\left(x\right)=0$ for some $x in \left(1, e\right)$	(ii) $\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$	(Q) $f$ is decreasing in $\left(e, e^2\right)$
(III) $f^{\prime }\left(x\right)=0$ for some $x\in \left(0, 1\right)$	(iii) $\underset{x\to \infty }{\lim }{f}^{\prime }\left(x\right)=-\infty$	(R) $f^{\prime }$ is increasing in (0, 1)
(IV) $f^{\prime \prime }\left(x\right)=0$ for some $x\in \left(1, e\right)$	(iv) $\underset{x\to \infty }{\lim }{f}^{\prime \prime }\left(x\right)=0$	(S) $f^{\prime }$ is decreasing in $\left(e, e^2\right)$

Which of the following options is the only Correct combination?

Let \(A B C\) be a triangle such that \(\angle A C B=\frac{\pi}{6}\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to \(A, B\) and \(C\) respectively. The value(s) of \(x\) for which \(a=x^2+x+1, b=x^2-1\) and \(c=2 x+1\) is (are)

For \(x>0, \lim _{x \rightarrow 0}\left((\sin x)^{1 / x}+\left(\frac{1}{x}\right)^{\sin x}\right)\) is

Let $\mathrm{S}$ be the sample space of all $3\times 3$ matrices with entries from the set $\left\{0,\mathrm{ }1\right\}.$ Let the events ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ be given by
${\mathrm{E}}_1=\left\{\mathrm{A}\in \mathrm{S}:\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{A}=0\right\}$ and
${\mathrm{E}}_2=\{\mathrm{A}\in \mathrm{S}:$ sum of entries of $\mathrm{A}$ is $7\}.$
If a matrix is chosen at random from $\mathrm{S},$ then the conditional probability $\mathrm{P}\left({\mathrm{E}}_1|{\mathrm{E}}_2\right)$ equals____

For spherical symmetrical charge distribution, variation of electric potential with distance from centre is given in diagram. Given that:
\(V=\frac{q}{4 \pi \varepsilon_0 R_0} \text { for } r \leq R_0\)
and \(V=\frac{q}{4 \pi \varepsilon_0 r}\) for \(r \geq R_0\)
Then which option(s) are correct

A straight line \(L\) through the point \((3,-2)\) is inclined at an angle \(60^{\circ}\) to the line \(\sqrt{3} x+y=1\). If \(L\) also intersects the \(X\)-axis, then the equation of \(L\) is

Four point charges, each of \(+q\), are rigidly fixed at the four corners of a square planar soap film of side \(a\). The surface tension of the soap film is \(\gamma\). The system of charges and planar film are in equilibrium, and \(a=k\left[\frac{q^2}{\gamma}\right]^{1 / N}\), where \(k\) is a constant. Then, \(N\) is

A few electric field lines for a system of two charges \(Q_1\) and \(Q_2\) fixed at two different points on the \(x\)-axis are shown in the figure. These lines suggest that

Paragraph :
When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.
Question :
If the radius of the opening of the dropper is \(r\), the vertical force due to the surface tension on the drop of radius \(R\) (assuming \(r< < R\) ) is

Paragraph:
Tangents are drawn from the point \(P(3,4)\) to the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) touching the ellipse at points \(A\) and \(B\).Question:
The orthocentre of the triangle \(P A B\) is