1 The position vector r of particle of mass m is given by the following equation $r(t)=\alpha t^{3}\hat{i}+\beta t^{2}\hat{j}$ where , $\alpha= \frac{10}{3}ms^{-3}$, $\beta=5 ms^{-2}$ and m=0.1 kg .At t = 1s ,which of the following statement is (are ) true about the particle? A) The velocity v is given by $v=(10\hat{i}+10\hat{j})ms^{-1}$ B) The angular momentum L with respect to the origin is given by $L=(5/3)\hat{k}Nms$ C) The force f is given by $F=(\hat{i}+2\hat{j})N$ D) The torque $\tau$ with respect to the origin is given by $\tau=-\frac{20}{3}\hat{k}Nm$
2 Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principle quantum number n. where n >>1. Which of the following statement(s) is (are) true? A) Relative change in the radii of two consecutive orbitals does not depend on Z B) Relative change in the radii of two consecutive orbitals varies as 1/n C) Relative change in the energy of two consecutive orbitals varies as $1/n^{3}$ D) Relative change in the angular momenta of two consecutive orbitals varies as 1/n
3 Two inductors L1 (inductance lmH, internal resistance 3Ω) and k (inductance 2 mH, internal resistance 4 Ω), and a resistor R (resistance 12 Ω) are all connected in parallel across a 5V battery. The circuit is switched on at time t = 0. The ratio of the maximum to the minimum current (Imax , Imin ) drawn from the battery is A) 8 B) 7 C) 6 D) 9
4 A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated (P) by the metal. The sensor has a scale that displays \log(P/P_{0}) where P0 is a constant. When the metal surface is at a temperature of 4870C, the sensor shows a value 1. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to 27670C? A) 10 B) 8 C) 5 D) 9
5 An accident in a nuclear laboratory resulted in deposition of a certain amourt of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use? A) 64 B) 90 C) 108 D) 120
6 Consider two identical galvanometer and two identical resistors with resistance R. If the internal resistance of the galvanometers Rc < R/2, which of the following statement(s) about anyone is (are) true? A) The maximum voltage range is obtained when all the components are connected in series B) The maximum voltage range is obtained when the two resistors and one galvanometer are connected in series, and the second galvanometer is connected in parallel to the first galvanometer C) The maximum current range is obtained when all the components are connected in parallel D) The maximum current range is obtained when the two galvanometers are connected in series, and the combination is connected in parallel with both the resistors
7 A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position x0. Consider two cases : (i) when the block is at x0 and (ii) when the block is at x=x0 +A . In both the cases, a particle with mass m (<M) is softly placed on the block after which they stick to each other. Which of the following statement(s) is (are) true about the motion after the mass m is placed on the mass M? A) The amplitude of oscillation in the first case changes by a factor of $\sqrt{\frac{M}{m+M}}$ whereas in the second case it remains unchanged B) The final time period of oscillation in both the cases is same C) The total energy decreases in both the cases D) The instantaneous speed at $x_{0}$ of the combined masses decreases in both the cases.
8 Light of wavelength $\lambda_{ph}$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is Φ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de-Broglie wavelength of the electrons passing through the anode is $\lambda_{e}$ . which of the following statement (s) is (are) true ? A) $\lambda_{e}$ increases at the same rate as $\lambda_{ph}$ for $\lambda_{ph}$ &amp;lt; $\frac{hc}{\phi}$ B) $\lambda_{e}$ is approximately halved, if d is doubled C) $\lambda_{e}$ decreases with increase in $\phi$ and o $\lambda_{ph}$ D) For large potential difference $(V\gg \phi/e),\lambda e$ is approximately halved if V is made four times
9 Let P be the image of the point (3,1,7) with respect to the plane x-y+z=3 Then, the equation of the plane passing through P and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is A) x+y-3z=0 B) 3x+z=0 C) x-4y+7z=0 D) 2x-y=0
10 Let a,b ε R and $a^{2}+b^{2}\neq 0$ . Suppose $S=( z\epsilon C: z=\frac{1}{a+ibt},t\epsilon R, t\neq0)$ where$i=\sqrt{-1}$ . If z=x+iy and z ε S, then (x,y) lies on A) the circle with radius $\frac{1}{2a}$ and centre $(\frac{1}{2a},0)$ for a>0, $b\neq 0$ B) the circle with radius $-\frac{1}{2a}$ and centre $(-\frac{1}{2a},0)$ for $a<0,b\neq 0$ C) the X-axis for $a\neq0, b=0$ D) the Y-axis for $a=0, b\neq0$
11 Football teams $T_{1}$ and $T_{2}$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_{1}$ winning, drawing and losing a game against $T_{2}$ are $\frac{1}{2},\frac{1}{6}and \frac{1}{3}$ ,respectively. Each team gets 3 points for a win, 1 point for a draw and 0 points for a loss in a game. Let X and Y denote the total points scored by teams $T_{1}$ and $T_{2}$., respectively, after two games P(X>Y) is A) $\frac{1}{4}$ B) $\frac{5}{12}$ C) $\frac{1}{2}$ D) $\frac{7}{12}$
12 The increasing of atomic radii of the following group 13 elements is A) Al<Ga<In<Tl B) Ga<Al<In<Tl C) Al<In<Ga<Tl D) Al<Ga<Tl<In
13 On complete hydrogenation, natural rubber produces A) ethylene-propylene copolymer B) vulcanised rubber C) polypropylene D) polybutylene
14 A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include almost one boy, the number of ways of selecting the team is A) 380 B) 320 C) 260 D) 95
15 Let $-\frac{\pi}{6}<\theta <-\frac{\pi}{12}$ . Suppose $\alpha_{1}$ and $\beta_{1}$ are the roots of the equation $x^{2}-2x\sec\theta+1=0,$ and $\alpha_{2}$ and $\beta_{2}$ are roots of the equation $x^{2}+2x\tan\theta-1=0$. If $\alpha_{1}>\beta_{1}$ and $\alpha_{2}>\beta_{2}$ , $\alpha_{1}+\beta_{2}$ equals A) $2(\sec \theta-\tan \theta)$ B) $2\sec \theta$ C) $-2\tan \theta$ D) 0