1 A circular insulated copper wire loop is twisted to form two loops of areas A and 2A as shown in the figure. At the point of crossing the wires remain electrically insulated from each other. The entire loop lies in the plane (of the paper). A uniform magnetic field B points into the plane of the paper. At t=0 the loop starts rotating about the common diameter as an axis with a constant angular velocity ω in the magnetic field. Which of the following options is/are correct? A) the emf induced in the loop is proportional to the sum of area of the two loops, B) The rate of change of the flux is maximum when the plane of the loops perpendicular to plane of the paper C) The net emf induced due to both the loops is proportional to $\cos\omega t$ D) The amplitude of the maximum net emf induced due to the loops is equal to the amplitude of maximum emf induced in the smaller loop alone.
2 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
3 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) $R^{\frac{2}{3}}$
4 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%
5 A uniform magnetic field B exists in the region between x=0 and $x=\frac{3R}{2}$ (region 2 in the figure)pointing normally into the plane of the paper. A particle with charge +Q and momentum p directed along X-axis enters region 2 from region 1 at point P1(y=-R) which of the following option(s) is/are correct? A) when the particle re-enters region 1 through the longest possible path in region 2 the magnitude of the change in its linear momentum between point $P_{1}$ and the farthest point from Y-axis is $\frac{p}{\sqrt{2}}$ B) For $B=\frac{8}{13}\frac{p}{QR}$ , the particle will enter region 3 through the point $P_{2}$ on X-axis C) For B>$\frac{2}{3}\frac{p}{QR}$ , the particle will re enter region 1 D) For a fixed B, particles of same range Q and same velocity v, the distance between the point $P_{1}$ , and the point of re entry into region 1 is inversely proportional to the mass of the particle
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 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 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 $(0,\infty)$
9 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) $\sqrt{3}\tan(\frac{\alpha}{2})+\tan(\frac{\beta}{2})=0$
10 If $g(x)=\int_{\sin x}^{\sin(2x)} \sin^{-1}(t)dt$ , then A) $g'(-\frac{\pi}{2})=2\pi$ B) $g'(-\frac{\pi}{2})=-2\pi$ C) $g'(\frac{\pi}{2})=2\pi$ D) $g'(\frac{\pi}{2})=-2\pi$
11 Let a,b,x and y be real numbers such that a-b=1 and y≠ o. If the complex number z=x+iy satisfies $Im(\frac{az+b}{z+1})=y$ , then which of the following is(are) possible value(s) of x? A) $1-\sqrt{1+y^{2}}$ B) $-1-\sqrt{1-y^{2}}$ C) $1+\sqrt{1+y^{2}}$ D) $-1+\sqrt{1-y^{2}}$
12 If a chord, which is not a tangent of the parabola y2=16x has the equation 2x+y=p, and mid-point (h,k) then which of the following is(are) possible value (s) of p,h, and k? A) p=-1,h=1,k=-3 B) p=2,h=3, k=-4 C) p=-2,h=2,k=-4 D) p=5,h=4,k=-3
13 For the following compounds, the correct statement(s) with respect to nucleophilic substitution reaction is(are) A) Compound IV undergoes inversion of configuration B) The order of reactivity for I, III, and IV is :IV >I >III C) I and III follow $S_{N}1$ mechanism D) I and II follow $S_{N}1$ mechanism
14 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
15 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) $ZnO,Al_{2}O_{3},PbO,PbO_{2}$