A line $y=mx+1$ intersects the circle ${\left(x-3\right)}^2+{\left(y+2\right)}^2=25$ at the points $P$ and $Q.$ If the midpoint of the line segment $PQ$ has $x-$coordinate $-\frac{3}{5},$ then which one of the following options is correct?

Match the statements of Column I with these in Column II.
[Note : Here \(z\) takes values in the complex plane and \(\operatorname{Im}(z)\) and \(\operatorname{Re}(z)\) denote respectively, the imaginary part and the real part of \(z\) ]

Column I	Column II
(A)The set of points \(z\) satisfying \(|z-i| z||=|z+i| z||\) is contained in or equal to	(p)an ellipse with eccentricity \(4 / 5\)
(B) The set of points \(z\) satisfying \(|z+4|+|z-4|=0\) is contained in or equal to	(q) the set of points \(z\) satisfying \(\operatorname{Im}(z)=0\)
(C) If \(|w|=2\), then the set of points \(z=w-\frac{1}{w}\) is contained in or equal to	(r) the set of points \(z\) satisfying \(|\operatorname{Im} z| \leq 1\)
(D) If \(|w|=1\), then the set of points \(z=w+\frac{1}{w}\) is contained in or equal to	(s) the set of points satisfying \(|\operatorname{Re} z| \leq 2\)
	(t) the set of points \(z\) satisfying \(|z| \leq 3\)

Consider the equation ${\int }_1^e\frac{{\left({\log }_{e}x\right)}^{{\displaystyle \frac{1}{2}}}}{x{\left(a-{\left({\log }_{e}x\right)}^{{\displaystyle \frac{3}{2}}}\right)}^2}dx=1,a\in \left(-\infty ,0\right)\cup \left(1,\infty \right)$. Which of the following statements is/are TRUE?

A bag contains \(N\) balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For \(i=1,2,3\), let \(W_i, G_i\), and \(B_i\) denote the events that the ball drawn in the \(i^{\text {th }}\) draw is a white ball, green ball, and blue ball, respectively. If the probability \(P\left(W_1 \cap G_2 \cap B_3\right)=\frac{2}{5 N}\) and the conditional probability \(P\left(B_3 \mid W_1 \cap G_2\right)=\frac{2}{9}\), then \(N\) equals ________.

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Let \(A, B, C\) be three sets of complex numbers as defined below.
\(A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ;\) \(C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}\).
Question:
Let \(z\) be any point in \(A \cap B \cap C\). The \(|z+1-i|^2+|z-5-i|^2\) lies between

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Let \(O\) be the origin, and \(\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}\) be three unit vectors in the directions of the sides \(\overrightarrow{Q R}, \overrightarrow{R P}\), \(\overrightarrow{P Q}\), respectively, of a triangle \(P Q R\).


Question:

\(|\overrightarrow{O X} \times \overrightarrow{O Y}|=\)

Paragraph:

Let \(O\) be the origin, and \(\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}\) be three unit vectors in the directions of the sides \(\overrightarrow{Q R}, \overrightarrow{R P}\), \(\overrightarrow{P Q}\), respectively, of a triangle \(P Q R\).


Question:

If the triangle \(P Q R\) varies, then the minimum value of

\(\cos (P+Q)+\cos (Q+R)+\cos (R+P)\) is

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A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity \(\omega\) is an example of a non-inertial frame of reference.

The relationship between the force \(\vec{F}_{\text {rot }}\) experienced by a particle of mass \(m\) moving on the rotating disc and the force \(\vec{F}_{\text {in }}\) experienced by the particle in an inertial frame of reference is

\(\vec{F}_{\mathrm{rot}}=\vec{F}_{\mathrm{in}}+2 m\left(\vec{v}_{\mathrm{rot}} \times \vec{\omega}\right)+m(\vec{\omega} \times \vec{r}) \times \vec{\omega}\),

where \(\vec{v}_{\text {rot }}\) is the velocity of the particle in the rotating frame of reference and \(\vec{r}\) is the position vector of the particle with respect to the centre of the disc.

Now consider a smooth slot along a diameter of a disc of radius \(R\) rotating counter-clockwise with a constant angular speed \(\omega\) about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the \(x\)-axis along the slot, the \(y\)-axis perpendicular to the slot and the \(z\)-axis along the rotation axis \((\vec{\omega}=\omega \hat{k}) .\) A small block of mass \(m\) is gently placed in the slot at \(\vec{r}=(R / 2) \hat{i}\) at \(t=0\) and is constrained to move only along the slot.


Question:
The distance \(r\) of the block at time \(t\) is

Among \(\left[\mathrm{Co}(\mathrm{CN})_4\right]^{4-},\left[\mathrm{Co}(\mathrm{CO})_3(\mathrm{NO})\right], \mathrm{XeF}_4,\left[\mathrm{PCl}_4\right]^{+},\)\(\left[\mathrm{PdCl}_4\right]^{2-},\left[\mathrm{ICl}_4\right]^{-},\left[\mathrm{Cu}(\mathrm{CN})_4\right]^{3-}\) and \(\mathrm{P}_4\) the total number of species with tetrahedral geometry is _______

Which one of the following statement is WRONG in the context of X-rays generated from X-ray tube?

The value of the integral \(\int_{-\pi / 2}^{\pi / 2}\left(x^{2}+\ln \frac{\pi+x}{\pi-x}\right) \cos x d x\) is

A plane passes through \((1,-2,1)\) and is perpendicular to two planes \(2 x-2 y+z=0\) and \(x-y+2 z=4\), then the distance of the plane from the point \((1,2,2)\) is

An ideal monatomic gas of n moles is taken through a cycle \(W X Y Z W\) consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic \(V-T\) diagram. The volume of the gas at \(W, X\) and \(Y\) points are, \(64 \mathrm{~cm}^3, 125 \mathrm{~cm}^3\) and \(250 \mathrm{~cm}^3\), respectively. If the absolute temperature of the gas \(T_W\) at the point \(W\) is such that \(n R T_W=1 \mathrm{~J}\) ( R is the universal gas constant), then the amount of heat absorbed (in J ) by the gas along the path \(X Y\) is \(\qquad\)