Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z=x+iy \left(x, y\in \mathrm{ℝ}, i=\sqrt{-1}\right)$ of the equation $sz+t\stackrel{-}{z}+r=0$, where $\stackrel{-}{z}=x-iy$. Then, which of the following statement(s) is (are) TRUE?

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____

Considering only the principal values of the inverse trigonometric functions, the value of \(\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)\) is

The value of $\sum _{k=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{\left(k-1\right)\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{k\pi }{6}\right)}$ is equal to

Paragraph:
Consider the lines: \(L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}\)Question:
The distance of the point \((1,1,1)\) from the plane passing through the point \((-1,-2,-1)\) and whose normal is perpendicular to both the lines \(L_1\) and \(L_2\), is

For any real numbers $\alpha$ and $\beta$, let $y_{\alpha ,\beta }\left(x\right),x\in R$, be the solution of the differential equation $\frac{dy}{dx}+\alpha y=x{e}^{\beta x},y\left(1\right)=1$ Let $S=\left\{y_{\alpha ,\beta }\left(x\right) :\alpha ,\beta \in R\right\}$. Then which of the following functions belong(s) to the set $S$?

Let \(P\left(x_1, y_1\right)\) and \(Q\left(x_2, y_2\right), y_1 < 0, y_2 < 0\), be the end points of the latusrectum of the ellipse \(x^2+4 y^2=4\). The equations of parabolas with latusrectum \(P Q\) are

If \(f(x)\) is twice differentiable function such that \(f(a)=0, f(b)=2, f(c)=1, f(d)=2\), \(f(e)=0\), where \(a < b < c < d < e\), then the minimum number of zeros of \(g(x)=\left\{f^{\prime}(x)\right\}^2+f^{\prime \prime}(x) \cdot f(x)\) in the interval \([a, e]\) is

Paragraph:

The figure shows a circular loop of radius \(a\) with two long parallel wires (numbered \(1\) and \(2\) ) all in the plane of the paper. The distance of each wire from the centre of the loop is \(d\). The loop and the wires are carrying the same current \(I\). The current in the loop is in the counterclockwise direction if seen from above.



Question:

Consider \(d \gg a\), and the loop is rotated about its diameter parallel to the wires by \(30^{\circ}\) from the position shown in the figure. If the currents in the wires are in the opposite directions, the torque on the loop at its new position will be (assume that the net field due to the wires is constant over the loop)

Paragraph:

Let \(\psi_{1}:[0, \infty) \rightarrow \mathbb{R}, \psi_{2}:[0, \infty) \rightarrow \mathbb{R}, f:[0, \infty) \rightarrow \mathbb{R}\) and \(g:[0, \infty) \rightarrow \mathbb{R}\) be functions such that \(f(0)=g(0)=0\),

\(\psi_{1}(x)=e^{-x}+x, \quad x \geq 0\),

\(\psi_{2}(x)=x^{2}-2 x-2 e^{-x}+2, \quad x \geq 0\),

\(f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t, \quad x>0\)

and \(g(x)=\int_{0}^{x^{2}} \sqrt{t} e^{-t} d t, \quad x>0\)


Question:

Which of the following statements is TRUE ?

Among the following, the correct statement(s) for electrons in an atom is(are)

In the following carbocation, \(\mathrm{H} / \mathrm{CH}_3\) that is most likely to migrate to the positively charged carbon is

In a Young's double slit experiment, a combination of two glass wedges \(A\) and \(B\), having refractive indices 1.7 and 1.5, respectively, are placed in front of the slits, as shown in the figure. The separation between the slits is \(d=2 \mathrm{~mm}\) and the shortest distance between the slits and the screen is \(D=2 \mathrm{~m}\). Thickness of the combination of the wedges is \(t=12 \mu \mathrm{~m}\). The value of \(l\) as shown in the figure is 1 mm . Neglect any refraction effect at the slanted interface of the wedges. Due to the combination of the wedges, the central maximum shifts (in mm ) with respect to O by ______