Consider the lines $L_1\text{:}\frac{x-1}{2}=\frac{y}{-1}=\frac{z+3}{1}\text{,}{L}_2\text{:}\frac{x-4}{1}=\frac{y+3}{1}=\frac{z+3}{2}$ and the planes $P_1\text{:}7x+y+2z=3\text{,}{P}_2\text{:}3x+5y-6z=4\text{.}$ Let $ax+by+cz=d$ be the equation of the plane passing through the point of intersection of L₁ and L₂, and perpendicular to planes P₁ and P₂.
Match List- I with List - II and select the correct answer using the code given below the lists :
 

	List I		List II
A.	a =	P.	13
B.	b =	Q.	-3
C.	c =	R.	1
D.	d =	S.	-2

If $f:R\to R$ is a twice differentiable function such that $f^{\prime \prime }\left(x\right)>0$ for all $x\in R$ and $f\left(\frac{1}{2}\right)=\frac{1}{2}, f\left(1\right)=1$ then

Let \(a=3 \sqrt{2}\) and \(b=\frac{1}{5^{1 / 6} \sqrt{6}}\). If \(x, y \in \mathbb{R}\) are such that
\(\begin{gathered}3 x+2 y=\log _a(18)^{\frac{5}{4}} \\2 x-y=\log _b(\sqrt{1080})\end{gathered}\)
and then \(4 x+5 y\) is equal to . ________.

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Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The real number \(s\) lies in the interval

Let $Q$ be the cube with the set of vertices $\left\{\left(x_1, x_2, x_3\right)\in {\mathrm{ℝ}}^3 : x_1, x_2, x_3\in \left\{0, 1\right\}\right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $\left(0, 0, 0\right)$ and $\left(1, 1, 1\right)$ is in $S$. For lines ${\ell }_1$ and ${\ell }_2$, let $d\left({\ell }_1, {\ell }_2\right)$ denote the shortest distance between them. Then the maximum value of $d\left({\ell }_1, {\ell }_2\right)$, as ${\ell }_1$ varies over $F$ and ${\ell }_2$ varies over $S$, is

Let $\overline{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of complex numbers, the number of distinct roots of the equation $\overline{z}-z^2=i\left(\overline{z}+z^2\right)$ is _______.

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and more over the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

Let $E_1=${$x\in \mathrm{ℝ}: x\ne 1$ and $\frac{x}{x-1}>0$} and $E_2=${$x\in E_1: {\sin }^{-1}\left({\log }_e\left(\frac{x}{x-1}\right)\right)$ is a real number} (Here the inverse trigonometric function ${\sin }^{-1}\left(x\right)$ assumes values in $\left[-\frac{\mathrm{\pi }}{2}. \frac{\mathrm{\pi }}{2}\right]$.)
Let $f:E_1\to \mathrm{ℝ}$ be the function defined by $f\left(x\right)={\log }_e\left(\frac{x}{x-1}\right)$
And $g:E_2\to \mathbb{R}$ be the function defined by $g\left(x\right)={\sin }^{-1}\left({\log }_e\left(\frac{x}{x-1}\right)\right)$ .

LIST-I	LIST-II
A. The range of $f$ is	P. $\left(-\infty ,\frac{1}{1-e}\right]\cup \left[\frac{e}{e-1}, \infty \right)$
B. The range of $g$ contains	Q. (0, 1)
C. The domain of $f$ contains	R. $\left[-\frac{1}{2},\frac{1}{2}\right]$
D. The domain of $g$ is	S. $\left(-\infty ,0\right)\cup \left(0, \infty \right)$
	T. $\left(-\infty ,\frac{e}{e-1}\right]$
	U. $\left(-\infty ,0\right)\cup \left(\frac{1}{2},\frac{e}{e-1}\right]$

The correct option is:

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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)

The value of \(b\) for which the equations \(x^2+b x-1=0, x^2+x+b=0\) have one root in common is

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)

A thermodynamic system is taken from an initial state $\mathrm{i}$ with internal energy ${\mathrm{U}}_{\mathrm{i}}\mathrm{}=100\mathrm{}\mathrm{J}$ to the final state f along two different paths iaf and ibf, as schematically shown in the figure. The work done by the system along the paths af, ib and bf are ${\mathrm{w}}_{\mathrm{a}\mathrm{f}}=\mathrm{}200\mathrm{}\mathrm{J},\mathrm{}{\mathrm{W}}_{\mathrm{i}\mathrm{b}}\mathrm{}=\mathrm{}50\mathrm{}\mathrm{J}$ and ${\mathrm{W}}_{\mathrm{b}\mathrm{f}}\mathrm{}=\mathrm{}100\mathrm{}\mathrm{J}$ respectively. The heat supplied to the system along the path iaf, ib and bf are iaf ib ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}},\mathrm{}{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}$ and ${\mathrm{Q}}_{\mathrm{b}\mathrm{f}}$ respectively. If the internal energy of the system in the state b is ${\mathrm{U}}_{\mathrm{b}}=200\mathrm{}\mathrm{J}$ and ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}}=500\mathrm{}\mathrm{J}$ , the ratio $\frac{{\mathrm{Q}}_{\mathrm{b}\mathrm{f}}}{{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}}\mathrm{}$ is

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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 equation of the locus of the point whose distance from the point \(P\) and the line \(A B\) are equal, is