1 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.
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 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) $\triangle =h [\frac{1}{\cos(\frac{\pi}{n})}-1]$
4 Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density ρ remains uniform throughout the volume. The rate of fractional change in density $(\frac{1}{\rho}\frac{d\rho}{dt})$ is constant. The velocity v of any point of the surface of the expanding sphere is proportional to A) R B) $\frac{1}{R}$ C) $R^{2}$ D) $R^{\frac{2}{3}}$
5 A point charge +Q is placed just outside an imaginary hemispherical surface of radius R as shown in the figure. Which of the following statement is/are correct? A) The electric flux passing through the curved surface of the hemisphere is $-\frac{Q}{2\epsilon_{0}}(1-\frac{1}{\sqrt{2}})$ B) The component of the electric field normal to the flat surface is constant over the surface C) Total flux through the curved and the flat surface is $\frac{Q}{\epsilon_{0}}$ D) The circumference of the flat surface is an equipotential
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 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
8 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
9 Let p,q be integers and let α ,β be the roots of the equation $x^{2}-x-1=0$ where α ≠β, For n=0,1,2...... Let $a_{n}=p\alpha^{n}+q\beta^{n}$ ( If a and b are rational numbers and $a+b\sqrt{5}=0$, then a=0=b) a12= A) $a_{11}+2a_{10}$ B) $2a_{11}+a_{10}$ C) $a_{11}-a_{10}$ D) $a_{11}+a_{10}$
10 Let p,q be integers and let α ,β be the roots of the equation $x^{2}-x-1=0$ where α ≠β, For n=0,1,2...... Let $a_{n}=p\alpha^{n}+q\beta^{n}$ ( If a and b are rational numbers and $a+b\sqrt{5}=0$, then a=0=b) If a24=28 , then p+2q= A) 14 B) 7 C) 21 D) 12
11 Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR,RP,PQ respectively of triangle PQR. I f the triangle PQR varies, then the minimum value of $\cos(P+Q)+\cos(Q+R)+\cos(R+P)$ is A) $-\frac{3}{2}$ B) $\frac{3}{2}$ C) $\frac{5}{3}$ D) $-\frac{5}{3}$
12 Let f: R→ (0,1) be a continuous function. Then, which of the following function (s) has (have) the value zero at some point in the interval (0,1)? A) $e^{x}-\int_{0}^{x} f(t) \sin t dt $ B) $f(x)+\int_{0}^{\frac{\pi}{2}} f(t)\sin t dt$ C) $x-\int_{0}^{\frac{\pi}{2}-x} f(t)\cos t dt$ D) $x^{9}-f(x)$
13 The correct statment(s) about the oxoacids, HClO4 and HClO, is(are) A) The central atom in both $HClO_{4}$ AND HClO is $sp^{3}$ hybridised B) $HClO_{4}$ is formed in the reaction between $Cl_{2}and H_{2}O$ C) The conjugate base of $HClO_{4}$ is weaker base than $ H_{2}O$ D) $HClO_{4}$ is more acidic than HClO because of the resonance stabilisation of its anion
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
