1 A block of mass M has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surfaced of a fixed table. Initially, the right edge of the block is at x=0, in a coordinate system fixed in the table. A point mass m is released from rest at the topmost point of the path as shown and slide down. When the mass loses contact with the block, its position is x and the velocity is v, At that instant, which of the following option is/are correct? A) The velocity of the point mass m is $v= \sqrt{\frac{2gR}{1+\frac{m}{M}}}$ B) The x component of displacement of the centre of mass of the block M is $-\frac{mR}{M+m}$ C) The position of the point mass is $x= -\sqrt{2}\frac{mR}{M+m}$ D) The velocity of the block M is $v= -\frac{m}{M}\sqrt{2gR}$
2 A human body has a surface area of approximately 1m2. The normal body temperature is 10K above the surrounding room temperature T0. Take the room temperature to be T0= 300K. For T0 =300K, the value of $\sigma T_0^4=460 Wm^{-2}$ (where σ is the Stefan Boltzmann constant). Which of the following options 1s/are correct? A) If the body temperature rises significantly, then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths B) If the surrounding temeprature reduces by a small amount $\triangle T_{0}<<T_{0}$, then to maintain the same body temperature the same (living) human being needs to radiate $\triangle W=4\sigma T_0^3\triangle T_{0}$ more energy per uint time C) The amount of energy radiated by the body in 1s is close to 60 J D) Reducing the exposed surface area of the body (e.g. by curling up) allows human to maintain the same body temperature while reducing the energy lost by radiation
3 An electron in a hydrogen atom undergoes a transition from an orbit with quantum number ni to another with quantum number nf . vi and vf are respectively, the initial and final potential energies of the electron. If $\frac{v_{i}}{v_{f}}=6.25$ then the smallest possible nf is A) 5 B) 8 C) 4 D) 3
4 A stationary source emits a sound of frequency f0=492 Hz. The sound is reflected by a large car approaching the source with a speed of 2 ms-1. The reflected signal is received by the source and superposed with the original. What will be the beat frequency of the resulting signal in Hz? (Given that the speed of sound in air is 330ms -1 and the car reflects the sound at the frequency it has received]. A) 4 B) 6 C) 10 D) 5
5 The instantaneous voltage at three terminals marked X, Y and Z are given by VX= V0 sinωt. $V_{Y}=V_{0}\sin(\omega t+\frac{2\pi}{3}) and$ $V_{Z}=V_{0}\sin(\omega t+\frac{4\pi}{3})$ An ideal voltmeter is configured to read rms value of the potential difference between its terminals. It is connected between points X and Y and then between Y and Z . The reading (s) of the voltmeter will be A) $V_{YZ}^{rms}=V_{0}\sqrt{\frac{1}{2}}$ B) $V_{XY}^{rms}=V_{0}\sqrt{\frac{3}{2}}$ C) independent of the choice of the two terminals D) $V_{XY}^{rms}=V_{0}$
6 Let S={1,2,3.......,9} For k=1,2,...5 , Let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N1+N2+N3+N4+N5 = A) 210 B) 252 C) 126 D) 125
7 Let α and β be non zero real numbers such $2(\cos\beta-\cos\alpha)+\cos\alpha\cos\beta=1$ . Then which of the following is/are true? A) $\sqrt{3}\tan(\frac{\alpha}{2})-\tan(\frac{\beta}{2})=0$ B) $\tan(\frac{\alpha}{2})-\sqrt{3}\tan(\frac{\beta}{2})=0$ C) $\tan(\frac{\alpha}{2})+\sqrt{3}\tan(\frac{\beta}{2})=0$ D) $\sqrt{3}\tan(\frac{\alpha}{2})+\tan(\frac{\beta}{2})=0$
8 Let X and Y the two events such $P(X)=\frac{1}{3}$ , $P(X\mid Y)= \frac{1}{2}$, $P(Y\mid X)= \frac{2}{5}$ Then A) $P(Y)= \frac{4}{15}$ B) $P(X'\mid Y)= \frac{1}{2}$ C) $P(X\cup Y)= \frac{2}{5}$ D) $P(X\cap Y)= \frac{1}{5}$
9 Let [X] be the greatest integer less than or equals to x. Then , at which of the following points(s) the function $f(x)=x\cos(\pi(x+[x]))$ discontinous? A) x=-1 B) x=1 C) x=0 D) x=2
10 The IUPAC name(s) of the following compound A) 4-methylchlorobenzene B) 4-chlorotoluene C) 1-chloro-4-methylbenzene D) 1-methyl-4-chlorobenzene
11 The correct statment(s) about the oxoacids, HClO4 and HClO, is(are) A) The central atom in both $HClO_{4}$ AND HClO is $sp^{3}$ hybridised B) $HClO_{4}$ is formed in the reaction between $Cl_{2}and H_{2}O$ C) The conjugate base of $HClO_{4}$ is weaker base than $ H_{2}O$ D) $HClO_{4}$ is more acidic than HClO because of the resonance stabilisation of its anion
12 The conductance of a 0.0015 M aqueous solution of a weak monobasic acid was determined by using a conductivity cell consisting of platinised pt electrodes. The distance between the electrodes is 120 cm with an area of cross section of 1 cm2. The conductance of this solution was found to be 5× 10-7 S. The pH of the solution is 4. The value of limiting molar conductivity ( $\wedge^{0}_{m}$) of this weak monobasic acid in aqueous solution is Z× 102 S cm-1 mol-1. The value of Z is A) 600 B) 800 C) 500 D) 400
13 The order of basicity among the following compound is A) II>I>IV>III B) I>IV>III>I C) IV>I>II>I D) IV>I>II>III
14 Which of the following combination will produce H2 gas? A) Fe metal and conc. $HNO_{3}$ B) Cu metal and conc,$HNO_{3}$ C) Au metal and NaCN(aq) in the presence of air D) Zn metal and NaOH(aq)
15 Among the following , the correct statement(s) is (are) A) $Al(CH_{3})_{3}$ has the three centre two -electron bonds in its dimeric structure B) The lewis acidity of $BCl_{3}$ is greater than that of $AlCl_{3}$ C) $AlCl_{3}$ has the three centre two electron bonds in its dimeric structure D) $BH_{3}$ has the three centre two electron bonds in its dimeric structure