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 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
4 A source of constant voltage V is connected to a resistance R and two ideal inductors L1 and L2 through a switch S as shown. There is no mutual inductance between the two inductors. The switch S is initially open. At t=0, the switch is closed and current begins to flow. Which of the following options is/are correct? A) After a long time the current through $L_{1}$ will be $\frac{V}{R}\frac{L_{2}}{L_{1}+L_{2}}$ B) After a long time , the current through $L_{2}$ will be $\frac{V}{R}\frac{L_{1}}{L_{1}+L_{2}}$ C) The ratio of the currents through $L_{1}$ and $L_{2}$ is fixed at all times (t>0) D) At t=0 the current through the resistance $R is\frac{V}{R}$
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 If $f:R\rightarrow R$ is twice differentablr function such that f''(x)>0, for all xε R, and $f(\frac{1}{2})=\frac{1}{2}$ , f(1)=1, then A) $f''(1)\leq0$ B) $f'(1)>1$ C) $0\lt f'(1) \le \frac{1}{2}$ D) $\frac{1}{2}\lt f'(1) \le 1$
7 Three randomly choosen non negative intergers x,y and z are found to satisfy the equation x+y+z=10 , Then the probability that z is even is A) $\frac{1}{2}$ B) $\frac{36}{55}$ C) $\frac{6}{11}$ D) $\frac{5}{11}$
8 The equation of the plane passing through the point(1,1,1) and perpendicular to the planes 2x+y-2z=5 and 3x-6y-2z=7 A) 14x+2y-15z=1 B) -14x+2y+15z=3 C) 14x-2y+15z=27 D) 14x+2y+15z=31
9 If f:R→ R is differentiable function such that f'(x) >2f(x) for all x ε R, and f(0)=1 then A) $f(x)\gt e^{2x}in (0,\infty)$ B) $f(x)\lt;e^{2x}in (0,\alpha)$ C) f(x) is increasing in $(0,\infty)$ D) f(x) decresing in $(0,\infty)$
10 Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR,RP,PQ respectively of triangle PQR. I f the triangle PQR varies, then the minimum value of $\cos(P+Q)+\cos(Q+R)+\cos(R+P)$ is A) $-\frac{3}{2}$ B) $\frac{3}{2}$ C) $\frac{5}{3}$ D) $-\frac{5}{3}$
11 If 2x-y+1=0, is a tangent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{16}=1$ , then which of the following CANNOT be sides of a right angled triangle? A) a.4,1 B) 2a,4,1 C) a,4,2 D) 2a,8,1
12 For how many values of p, the circle $x^{2}+y^{2}+2x+4y-p=0$ and the coordinate axes have exactly three common points? A) 0 B) 1 C) 2 D) 3
13 For a solution formed by mixing liquids L and M, the vapor pressure of L plotted against the mole fraction of M in solution is shown in the following figure. Here XL and XM represent mole fractions of L and M respectively, In the solution. The correct statement(s) applicable to this system is (are)→ A) The point Z represents vapour pressure of pure liquid M and Raoult's law is obeyed from $X_{L}$=0 AND $X_{L}$=1 B) Attractive intermolecular interactions between L-L in pure liquid L and M -M in pure liquid M are stronger than those between L-M when mixed in solution C) The point Z represents vapor pressure of pure liquid M and raoult's law is obeyed when $X_{L}$$\rightarrow$ 0 D) The point Z represents vapor pressure of pure liquid L and Raoult's law is obeyed when $X_{L}$$\rightarrow$ 1
14 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
15 In a bimolecular reaction, the steric factor P was experimentally determined to be 4.5. the correct option(s) among the following is (are) A) The activation energy of the reaction is unaffected by the value of the steric factor B) The experimentally determined value of the frequency factor is higher than that predicted by Arrhenius equation C) The value of the frequency factor predicted by Arrhenius equation is higher than that determined experimentally D) Since P=4.5 the reaction will not proceed unless an effective catalyst is used
