1 In the circuit shown L= 1μ H, C= 1μ F, and R= 1 kΩ. They are connected in series with an AC source V= V0 sinωt as shown. Which of the following options is/are correct? ω A) At $\omega\sim0$ the current flowing through the circuit becomes nearly zero B) The frequency at which the current will be in phase with the voltage is indepedent of R C) The current will be in phase with the voltage if$\omega =10^{4} rads^{-1}$ D) At $\omega >>10^{6}rads^{-1}$ the circuit behaves like a capacitor
2 A block M hangs vertically at the bottom end of a uniform rope constant mass per unit length. The top end of the rope is attached to a fixed rigid support at O. A transverse wave pulse (pulse 1) of wavelength λ0 is produced at point O on the rope. The pulse takes time TOA to reach point A. If the wave pulse of wavelength λ 0 is produced at point A (pulse 2) without disturbing the position of M it takes time TAO to reach point O. Which of the following options is/are correct? A) The time $T_{AO}=T_{OA}$ B) The wavelength of the pulse 1 becomes longer when it reaches point A C) The velocity of any pulse along the rope is independent of its frequency and wavelength D) The velocity of the two pulses (pluses1 and pulse 2) are the same at the midpoint of rope
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 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 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
6 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
7 If $f:R\rightarrow R$ is twice differentablr function such that f''(x)>0, for all xε R, and $f(\frac{1}{2})=\frac{1}{2}$ , f(1)=1, then A) $f''(1)\leq0$ B) $f'(1)>1$ C) $0\lt f'(1) \le \frac{1}{2}$ D) $\frac{1}{2}\lt f'(1) \le 1$
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 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 The IUPAC name(s) of the following compound A) 4-methylchlorobenzene B) 4-chlorotoluene C) 1-chloro-4-methylbenzene D) 1-methyl-4-chlorobenzene
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 Which of the following combination will produce H2 gas? A) Fe metal and conc. $HNO_{3}$ B) Cu metal and conc,$HNO_{3}$ C) Au metal and NaCN(aq) in the presence of air D) Zn metal and NaOH(aq)
14 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
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
