1 A ring of mass M and radius R is rotating with angular speed ω about a fixed vertical axis passing through its centre O with two-point masses each of mass $\frac{M}{8}$ at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant , the angular speed of the system is $\frac{8}{9}\omega$ and one of the maases is at a distance of $\frac{3}{5}R$ from O. At this instant , the distance of the other mass from O is A) $\frac{2}{3}R$ B) $\frac{1}{3}R$ C) $\frac{3}{5}R$ D) $\frac{4}{5}R$
2 If the freezing point of a 0.01 molal aqueous solution of a cobalt (III) chloride -ammonia complex (which behaves as a strong electrolyte) is -0.0558° C, the number of chloride (s) in the coordination sphere of the complex is [Kf of water =1.86 K kg mol-1] A) 1 B) 2 C) 3 D) 4
3 Copper is purified by electrolytic refining of blister copper. The correct statement(s0 about this process is/are A) impure Cu strip is used as cathode B) acidified aqueous $CuSO_{4}$ is used as elecrolyte C) pure Cu deposits at cathode D) impurities settle as anode -mud
4 The minimum number of times a fair coin needs to be tossed, so that the probability of getting atleast two heads is atleast 0.96, is A) 8 B) 6 C) 10 D) 4
5 Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then, the value of $\frac{m}{n}$ is A) 4 B) 5 C) 6 D) 3
6 Consider the family of all circles whose centres lie on the straight line y=x, If this family of circles is represented by the differential equation $Py"+Qy'+1=0$ , where P,Q are the functions of x, y and y' (here, $y'=\frac{dy}{dx},y''=\frac{d^{2}y}{dx^{2}})$, then which of the following statement (s) is /are true? A) P=y+x B) P=y-x C) $P+Q=1-x+y+y'+(y')^{2}$ D) $P-Q=x+y-y'-(y')^{2}$
7 In plotting stress versus strain curves for two materials P and Q, a student by m take puts strain on the y-axis and stress on the x-axis as shown in the figure. Then, the correct statement is /are A) P has more tensile strength than Q B) P is more ductile than Q C) P is more brittle than Q D) The Young's modulus of P is more that that of Q
8 A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity . If P(r) is the pressure at r(r<R) , then the correct options is /are A) P(r=0)=0 B) $\frac{P(r=\frac{3R}{4})}{P(r=\frac{2R}{3})}=\frac{63}{80}$ C) $\frac{P(r=\frac{3R}{5})}{P(r=\frac{2R}{5})}=\frac{16}{21}$ D) $\frac{P(r=\frac{R}{2})}{P(r=\frac{R}{3})}=\frac{20}{27}$
9 An ideal monoatomic gas is confined ina horizontal cylinder by a spring-loaded piston (as shown in the figure). Initially, the gas is at temperature T1, pressure P1 and volume V1 and the spring is in its relaxed state. The gas is then heated very slowly to temperature T2, pressure P2 and volume V2 . during this process the piston moves out by a distance of x, Ignoring the friction between the piston and the cylinder, the correct statements is/are A) If $V_{2}=2V_{1}$ and $T_{2}=3T_{1}$ , then the energy stored in the spring is $\frac{1}{4}p_{1}V_{1}$ B) If $V_{2}=2V_{1}$ and $T_{2}=3T_{1}$ , then the change in internal energy $3p_{1}V_{1}$ C) If $V_{2}=3V_{1}$ and $T_{2}=4T_{1}$, then the work done by the gas $\frac{7}{3}p_{1}V_{1}$ D) If $V_{2}=3V_{1}$ and $T_{2}=4T_{1}$, then the heat supplied to the gas is $\frac{17}{6}p_{1}V_{1}$
10 A fission reaction is given by $_{92}^{236}U\rightarrow_{54}^{140}Xe +_{38}^{94} Sr+x+y$ , where x and y are two paricles. Considering $_{92}^{236}U$ to be at rest, the kinetic energies of the products are denoted bt Kxe , Ksr , Kx (2MeV) and Ky (2 MeV), respectively, Let the binding energies per nucleon of $_{92}^{236}U$ . $_{54}^{140}xe$ and $_{38}^{94}sr$ be 7.5 MeV , 8.5 MeV and 8.5 MeV, respectively. Considering different vconservation laws , the correct options is/are A) x=n,y=n , $K_{sr}=129 MeV,K_{xe}=86MeV$ B) x=p, y=e, $K_{sr}=129 MeV,K_{xe}=86MeV$ C) x=p, y=n, $K_{Sr}=129 MeV,K_{Xe}=86MeV$ D) x=n,y=n, $K_{Sr}=86 MeV,K_{Xe}=129MeV$
11 Three moles of B2H6 are completely reacted with methanol. The number of moles of boron containing product formed is A) 4 B) 2 C) 5 D) 6
12 Suppose that p,q, and r three non-coplanar vectors in R3. Let the components of a vector s along p,q and r be 4,3 and 5 , respectively. If the components of this vector s along (-p+q+r) (p-q+r) and (-p-q+r) are x,y and z respectively, then the value of 2x+y+z is A) 8 B) 9 C) 12 D) 4
13 Let $f'(x)=\frac{192x^{3}}{2+\sin^{4}\pi x}$ for all x ε R with $f(\frac{1}{2})=0$ If $m\leq\int_{1/2}^{1} f(x) dx\leq M,$ , then the possible values of m and M are A) m=13,N=24 B) $m=\frac{1}{4},M=\frac{1}{2}$ C) m=-11,M=0 D) m=1,M=12
14 Let E1 and E2 be two ellipse whose centres are at the origin. The major axes of E1 and E2 lie along the lie X-axis and Y-axis, respectively. Let S be the circle $x^{2}+(y-1)^{2}=2$ . The straight-line x+y=3 touches the curve S. E1 and E2 at P,Q and R, respectively. Suppose that PQ=PR= $\frac{2\sqrt{2}}{3}$. If e1 and e2 are eccentricities of E1 and E2 respectively, then the correct expression(s) is/are A) $e_1^2+e_2^2=\frac{43}{40}$ B) $e_{1}.e_{2}=\frac{\sqrt{7}}{2\sqrt{10}}$ C) $|e_1^2-e_2^2|=\frac{5}{8}$ D) $e_{1}.e_{2}=\frac{\sqrt{3}}{4}$
15 The options (s) with the values of a and L that satidfy the equation $\frac{\int_{0}^{4\pi} e^{t}(\sin^{6}at+\cos^{4}at)dt}{\int_{0}^{\pi} e^{t}(\sin^{6}at+\cos^{4}at)dt} =L$ is\are A) $a=2,L=\frac{e^{4\pi}-1}{e^{\pi}-1}$ B) $a=2,L=\frac{e^{4\pi}+1}{e^{\pi}+1}$ C) $a=4,L=\frac{e^{4\pi}-1}{e^{\pi}-1}$ D) $a=4,L=\frac{e^{4\pi}+1}{e^{\pi}+1}$
