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 flat plane is moving normal to its plane through a gas under the action of a constant force F. The gas is kept at very low pressure. The speed of the plane v is much less than the average speed u of the gas molecules. Which of the following options is/are true? A) At a later time the external force F balances the resistive force B) The plate will continue to move with constant non-zero acceleration at all times. C) The resistive force experienced by the plate is proportional to v D) The pressure difference between the leading and trailing faces of the plate is proportinal to uv
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 Three vectors P, Q, and R are shown in the figure. Let S be any point on the vector R. The distance between the point P and S is b[R]. The general relation among vectors P, Q and S is A) $S=(1-b^{2})P+bQ$ B) $S=(b-1)P+bQ$ C) $S=(1-b^{})P+bQ$ D) $S=(1-b)P+b^{2}Q$
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 wheel of radius R and mass M is placed at the bottom of a fixed step of height R as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $\tau$ about an axis normal to the plane of the paper passing through the point Q. Which of the following options is/are correct ? A) If the force is applied normal to the circumference at point P then $\tau$ is zero B) if the force is applied tangentially at point S then $\tau\neq 0$ but the wheel never climbs the step C) If the force is applied at point P tangentially , then $\tau$ decreases continously as the wheel climbs D) If the force is applied normal to the circumfernce at point X , then $\tau $ is constant
7 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
8 if y=y(x) satisfies the differential equation $8\sqrt{x}(\sqrt{9+\sqrt{x}})dy=(\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1}$ dx,x >0 and $y(0)=\sqrt{7}$ , then y(256)= A) 16 B) 3 C) 9 D) 80
9 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) $\frac{5}{11}$
10 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
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 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
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 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
15 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