1 Consider a concave mirror and a convex lens ( refractive index=1.5) of focal length 10 cm each, separated by a distance of 50cm in air (refractive index=1) as shown in the figure. An object is placed at a distance of 15 cm from the mirror. Its erect image formed by this combination has magnification M1. When the set up is kept in a medium of refractive index. $\frac{7}{6}$, the magnification becomes M2. The magnitude $\mid\frac{M_{2}}{M_{1}}\mid$ is A) 6 B) 8 C) 7 D) 2
2 A nuclear power plant supplying electrical power to a village uses a radioactive material of half-life T years as fuel.The amount of fuel at the beginning is such that the total power requirement of the village is 12.5% of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of nT years, then the value of n is A) 4 B) 2 C) 3 D) 1
3 The correct statement (s) about Cr2+ and Mn3+ is/are [atomic number of Cr=24 and Mn=25] A) $ Cr^{2+}$ is a reducing agent B) $Mn^{3+}$ is an oxidising agent C) both $Cr^{2+}$ and $Mn^{3+}$ exhibit $d^{4}$ electronic configuration D) when $Cr^{2+}$ is used as a reducing agent, the chromium ion attains $d^{5}$ electronic configuration
4 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) $P-Q=x+y-y'-(y')^{2}$
5 Let $\triangle PQR$ be a triangle . Let a=QR, b= RP and c= PQ . If |a|=12, |b|= $4\sqrt{3}$ and b.c=24, then which of the following is/are true? A) $\frac{|c|^{2}}{2}-|a|=12$ B) $\frac{|c|^{2}}{2}+|a|=30$ C) |a x b+c x a|= $ 48\sqrt{3}$ D) a.b= - 72
6 In the following circuit, the current through the resistor R(=2Ω) is I amperes. The value of I is A) 3 B) 1 C) 2 D) 0
7 An electron in an excited state of Li2+ ion has angular momentum $\frac{3h}{2\pi}$. The de Broglie wavelength of the electron in this state is $p\pi a_{0}$ (where $a_{0}$ is the Bohr radius). The value of p is A) 4 B) 3 C) 2 D) 1
8 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 $\frac{17}{6}p_{1}V_{1}$
9 Light guidance in an optical fibre can be understood by considering a structure comprising of thin solid glass cylinder of refractive index n1 surrounded by a medium of lower refractive index n2. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media n1 and n2 as shown in the figure. All rays with the angle of incidence i less than a particular value im are confined in the medium of refractive index n1. The numerical aperture (NA) of the structure is defined as sin I'm. For two structures namely S1 with $n_{1}=\frac{\sqrt{45}}{4}$ and $n_{2}=\frac{3}{2}$ and S2 with $n_{1}=\frac{8}{5}$ and $n_{2}=\frac{7}{5}$ and taking the refractive index of water to be $\frac{4}{3}$ and that to air to be 1. the correct options is/are A) NA of $S_{1}$ immersed in water is the same as that of $S_{2}$ immersed in a liquid of refractive index $\frac{16}{3\sqrt{15}}$ B) NA of $S_{1}$ immersed in liquid of refractive index $\frac{6}{\sqrt{15}}$ is the same as that of $S_{2}$ immersed in water. C) NA of $S_{1}$ placed in air is the same as that $S_{2}$ immersed in liquid of refractive index $\frac{4}{\sqrt{15}}$ D) NA of $S_{1}$ placed in air is the same as that of $S_{2}$ placed in water
10 Among the complex ions,$[Co(NH_{2}-CH_{2}-CH_{2}-NH_{2})_{2}Cl_{2}]^{+},$ $[CrCl_{2}(C_{2}O_{4})_{2}]^{3-},$ $[Fe(H_{2}O)_{4}(OH)_{2}]^{+}$,$[Fe(NH_{3})_{2}(CN)_{4}]^{-}$ $[Co(NH_{2}-CH_{2}-CH_{2}-NH_{2})_{2}(NH_{3})Cl]^{2+}$ and $[Co(NH_{3})_{4}(H_{2}O)Cl]^{2+}$ the number of complex ion(s) that show(s) cis-trans isomerism is A) 4 B) 5 C) 6 D) 8
11 Under hydrolysis conditions, the compounds used for preparation of linear polymer and for chain termination m respectively are A) $CH_{3}SiCl_{3}$ and $Si(CH_{3})_{4}$ B) $(CH_{3})_{2}SiCl_{2}$ and $(CH_{3})_{3}SiCl_{}$ C) $(CH_{3})_{}SiCl_{2}$ and $(CH_{3})_{}SiCl_{3}$ D) $SiCl_{4}$ and $(CH_{3})_{3}SiCl_{}$
12 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
13 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) $e_{1}.e_{2}=\frac{\sqrt{3}}{4}$
14 Consider the hyperbola H: x2-y2=1 and a circle S with centre N(x2,0). Suppose that H and S touch each other at a point P(x1,y1) with x1>1 and y1>0. The common tangent to H and S at P intersects the X-axis at point M. If (l,m) is the centroid of $\triangle PMN$, then the correct expression(s) is/are A) $\frac{dl}{dx_{1}}=1-\frac{1}{3x_1^2}$ for $x_{1}>0$ B) $\frac{dm}{dx_{1}}=\frac{x_{1}}{3(\sqrt{x_1^2-1}}$ for $x_{1}>0$ C) $\frac{dl}{dx_{1}}=1+\frac{1}{3x_1^2}$ for $x_{1}>0$ D) $\frac{dm}{dy_{1}}=\frac{1}{3}$ for $y_{1}>0$
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) $a=4,L=\frac{e^{4\pi}+1}{e^{\pi}+1}$
