1 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
2 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) $\triangle =h [\frac{1}{\cos(\frac{\pi}{n})}-1]$
3 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) $S=(1-b)P+b^{2}Q$
4 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
5 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) $0<\alpha\leq\frac{1}{2}$
6 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) $\sqrt{3}\tan(\frac{\alpha}{2})+\tan(\frac{\beta}{2})=0$
7 Let $f(x)= \frac{1-x(1+\mid1-x\mid)}{\mid1-x\mid}\cos(\frac{1}{1-x})$ for x≠ 1, then A) $\lim_{x \rightarrow 1^{+}}f(x)=0$ B) $\lim_{x \rightarrow 1^{+}}f(x)$ doesnot exist C) $\lim_{x \rightarrow 1^{-}}f(x)=0$ D) $\lim_{x \rightarrow 1^{-}}f(x)$ doesnot exist
8 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) $g'(\frac{\pi}{2})=-2\pi$
9 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) $-1+\sqrt{1-y^{2}}$
10 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
11 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
12 The order of the oxidation state of the phosphorous atom in H3PO2 , H3PO4 , H3PO3 and H4P2O6 is A) $H_{3}PO_{4}>H_{3}PO_{2}>H_{3}PO_{3}>H_{4}P_{2}O_{6}$ B) $H_{3}PO_{4}>H_{4}P_{2}O_{6}>H_{3}PO_{3}>H_{3}P_{}O_{2}$ C) $H_{3}PO_{2}>H_{3}P_{}O_{3}>H_{4}P_{2}O_{6}>H_{3}P_{}O_{4}$ D) $H_{3}PO_{3}>H_{3}P_{}O_{2}>H_{3}P_{}O_{4}>H_{4}P_{2}O_{6}$
13 In a bimolecular reaction, the steric factor P was experimentally determined to be 4.5. the correct option(s) among the following is (are) A) The activation energy of the reaction is unaffected by the value of the steric factor B) The experimentally determined value of the frequency factor is higher than that predicted by Arrhenius equation C) The value of the frequency factor predicted by Arrhenius equation is higher than that determined experimentally D) Since P=4.5 the reaction will not proceed unless an effective catalyst is used
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 correct statement(s) about surface properties is (are) A) The critical temperature of ethane and nitrogen are 563 K and 126 K. respectively. The absorption of ethane will be more than that of nitrogen of the same amount of activated charcoal at a given temperature B) Cloud is an emulsion type of colloid in which liquid is dispersed phase and gas is dispersion medium C) Adsorption is accompanied by decreases in enthalpy and decreases in entropy of the system D) Brownian motion of colloidal particles does not depend on the size of the particles but depends on the viscosity of the solution
