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 block M hangs vertically at the bottom end of a uniform rope constant mass per unit length. The top end of the rope is attached to a fixed rigid support at O. A transverse wave pulse (pulse 1) of wavelength λ0 is produced at point O on the rope. The pulse takes time TOA to reach point A. If the wave pulse of wavelength λ 0 is produced at point A (pulse 2) without disturbing the position of M it takes time TAO to reach point O. Which of the following options is/are correct? A) The time $T_{AO}=T_{OA}$ B) The wavelength of the pulse 1 becomes longer when it reaches point A C) The velocity of any pulse along the rope is independent of its frequency and wavelength D) The velocity of the two pulses (pluses1 and pulse 2) are the same at the midpoint of rope
3 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
4 Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density ρ remains uniform throughout the volume. The rate of fractional change in density $(\frac{1}{\rho}\frac{d\rho}{dt})$ is constant. The velocity v of any point of the surface of the expanding sphere is proportional to A) R B) $\frac{1}{R}$ C) $R^{2}$ D) $R^{\frac{2}{3}}$
5 A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is $\delta T= 0.01s$ and he measures the depth of the well to be L=20 m. Take the acceleration due to gravity g=10 ms-2 and the velocity of sound is 300 ms-1. Then the fractional error in the measurement $\frac{\delta L}{L}$is closet to A) 1% B) 5% C) 3% D) 0.2%
6 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 $R is\frac{V}{R}$
7 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
8 Let $f(x)= \frac{1-x(1+\mid1-x\mid)}{\mid1-x\mid}\cos(\frac{1}{1-x})$ for x≠ 1, then A) $\lim_{x \rightarrow 1^{+}}f(x)=0$ B) $\lim_{x \rightarrow 1^{+}}f(x)$ doesnot exist C) $\lim_{x \rightarrow 1^{-}}f(x)=0$ D) $\lim_{x \rightarrow 1^{-}}f(x)$ doesnot exist
9 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
10 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
11 The color of the X2 molecules of group 17 elements changes gradually from yellow to violet down the group. This is due to A) decrease in $\pi^{*}-\sigma^{*}$ gap down the group B) decrease in ionisation energy down the group C) the physical state of $X_{2}$ at room temperature changes from gas to solid down the group D) decreases in HOMO-LUMO gap down the group
12 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)
13 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
14 The option (s) with only amphoteric oxides is (are) A) $NO,B_{2}O_{3},PbO,SnO_{2}$ B) $Cr_{2}O_{3},CrO,SnO,PbO$ C) $Cr_{2}O_{3},BeO,SnO,SnO_{2}$ D) $ZnO,Al_{2}O_{3},PbO,PbO_{2}$
15 The correct statement(s) about surface properties is (are) A) The critical temperature of ethane and nitrogen are 563 K and 126 K. respectively. The absorption of ethane will be more than that of nitrogen of the same amount of activated charcoal at a given temperature B) Cloud is an emulsion type of colloid in which liquid is dispersed phase and gas is dispersion medium C) Adsorption is accompanied by decreases in enthalpy and decreases in entropy of the system D) Brownian motion of colloidal particles does not depend on the size of the particles but depends on the viscosity of the solution