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 ?

Let \(A_n=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\ldots+(-1)^{n-1}\)\(\left(\frac{3}{4}\right)^n\) and \(B_n=1-A_n\). Find a least odd natural number \(n_0\), so that \(B_n>A_n, \forall n \geq n_0\).

The set of values of \(\theta\) satisfying the inequation \(2 \sin ^2 \theta-5 \sin \theta+2>0\), where \(0 < \theta < 2 \pi\), is

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

Three boys and two girls stand in a queue. The probability that the number of boys ahead of every girl is atleast one more than the number of girls ahead of her is

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 unit vector perpendicular to both \(L_1\) and \(L_2\) is

Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be unit vectors such that \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\). Which one of the following is correct?

Look at the drawing given in the figure, which has been drawn with ink of uniform line-thickness.

The mass of ink used to draw each of the two inner circles, and each of the two line segments is \(m\). The mass of the ink used to draw the outer circle is \(6 \mathrm{~m}\). The coordinates of the centres of the different parts are, outer circle 10 , \(0)\), left inner circle \((0,0)\), right inner circle \((a, a)\), vertical line \((0,0)\) and horizontal line \((0,-a)\). The \(y\)-coordinate of the centre of mass of the ink in this drawing is

Paragraph:

Consider an evacuated cylindrical chamber of height \(h\) having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius \(r \ll h\). Now a high voltage source \((HV)\) is connected across the conducting plates such that the bottom plate is at \(+V_{0}\) and the top plate at \(-V_{0} .\) Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)



Question:

The average current in the steady state registered by the ammeter in the circuit will be

Paragraph:
The coordination number of \(\mathrm{Ni}^{2+}\) is 4 .
\(\mathrm{NiCl}_2+\mathrm{KCN}\) (excess) \(\rightarrow A\) (cyano complex)
\(\mathrm{NiCl}_2+\) conc. \(\mathrm{HCl}\) (excess) \(\rightarrow B\) (chloro complex)
Question:
The hybridization of \(A\) and \(B\) are

A uniform circular disc of mass $\text{50 kg}$ and radius $\text{0.4 m}$ is rotating with an angular velocity of ${\text{10 rad s}}^{-1}$ about its own axis, which is vertical. Two uniform circular rings, each of mass $\text{6.25 kg}$ and radius $\text{0.2 m}$, are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity $\left({in\; rad\; s}^{-1}\right)$ of the system is

Match the complexes in Column I with their properties listed in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Column I	Column II
A. \(\text O _2^{-} \longrightarrow \text{O} _2\) \(+~\text O _2^{2-}\)	P. Redox reaction
B. \(\text{CrO} _4^{2-}+\text H ^{+} \longrightarrow\)	Q. One of the products has trigonal planar structure
C. \(\text{MnO} _4^{-}+ \text{NO} _2^{-}\) \(+~ \text H ^{+} \longrightarrow\)	R. Dimeric bridged tetrahedral metal ion
D. \(\text{NO} _3^{-}+ \text H _2 \text{SO} _4\) \(+~ \text{Fe} ^{2+} \longrightarrow\)	S. Disproportionation

The graph between object distance \(u\) and image distance \(v\) for a lens is given below. The focal length of the lens is