1 For an isosceles prism of angle A and refractive index μ, it is found that the angle of minimum deviation of $\delta_{m}=A$. Which of the following options is/are correct? A) For the angle of incidence $i_{1}=A$, the ray inside the prism is parallel to the base of the prism B) At minimum deviation , the incident angle $i_{1}$ and refracting angle $r_{1}$ at the first refracting surface are realated by $r_{1}=(\frac{i_{1}}{2})$ C) For the prism, the emergent ray at the second surface will be tangent to the surface when the angle of incidence at the first surface is $i_{1}= \sin^{-1}[\sin A\sqrt{4\cos^{2}\frac{A}{2}-1}-cos A]$ D) For this prism, the refractive index $\mu$ and the angle prism A are related as $A=\frac{1}{2}\cos^{-1}(\frac{\mu}{2})$
2 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.
3 A drop of liquid of radius R=10-2m having surface tension $S= \frac{0.1}{4\pi}$ Nm-1 divides itself into K identical drops. In this process the total change in the surface energy $\triangle U= 10^{-3}$ J . If $K=10^{\alpha}$, then the value of $\alpha$ is A) 5 B) 7 C) 6 D) 3
4 Consider regular polygons with the number of sides n=3,4,5 ..... as shown in the figure. The center of mass of all the polygons is at height h from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is Δ. Then, Δ depends on n and h as A) $\triangle =h\sin^{2}(\frac{\pi}{n})$ B) $\triangle =h\sin^{}(\frac{2\pi}{n})$ C) $\triangle =h\tan^{2}(\frac{\pi}{2n})$ D) $\triangle =h [\frac{1}{\cos(\frac{\pi}{n})}-1]$
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 rigid uniform bar AB of length L is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is θ. Which of the following statements about its motion is/are correct? A) Instantaneous torque about the point in contact with the floor is proportional to $\sin\theta$ B) The trajectory of the point A is parabola C) The midpoint of the bar will fall vertically downward D) When the bar makes an angle $\theta$ with the vertical, the displacement of its mid point from intial position is proportional to $(1-\cos\theta)$
7 How many 3x3 matrices M with entries from {0,1,2} are there, for which the sum of the diagonal entries of MTM is 5? A) 198 B) 162 C) 126 D) 135
8 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) $\frac{5}{11}$
9 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
10 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
11 If the line $x=\alpha$ divides the area of region R={(x,y) $\in$ R2 : x3≤ y≤ x, o≤x≤1 } into two equal parts, then A) $2\alpha^{4}-4\alpha^{2}+1=0$ B) $\alpha^{4}+4\alpha^{2}-1=0$ C) $\frac{1}{2}<\alpha<1$ D) $0<\alpha\leq\frac{1}{2}$
12 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
13 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
14 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
15 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
