1 For an isosceles prism of angle A and refractive index μ, it is found that the angle of minimum deviation of $\delta_{m}=A$. Which of the following options is/are correct? A) For the angle of incidence $i_{1}=A$, the ray inside the prism is parallel to the base of the prism B) At minimum deviation , the incident angle $i_{1}$ and refracting angle $r_{1}$ at the first refracting surface are realated by $r_{1}=(\frac{i_{1}}{2})$ C) For the prism, the emergent ray at the second surface will be tangent to the surface when the angle of incidence at the first surface is $i_{1}= \sin^{-1}[\sin A\sqrt{4\cos^{2}\frac{A}{2}-1}-cos A]$ D) For this prism, the refractive index $\mu$ and the angle prism A are related as $A=\frac{1}{2}\cos^{-1}(\frac{\mu}{2})$
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 A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is $\delta T= 0.01s$ and he measures the depth of the well to be L=20 m. Take the acceleration due to gravity g=10 ms-2 and the velocity of sound is 300 ms-1. Then the fractional error in the measurement $\frac{\delta L}{L}$is closet to A) 1% B) 5% C) 3% D) 0.2%
4 A rocket is launched normal to the surface of the earth, away from the sun, along the line joining the sun and the earth. The sun 3× 105 times heavier than the earth and is at a distance 2.5×104 times larger than the radius of the earth. The escape velocity from the earth's gravitational field is Ve=11.2 km s-1. The minimum initial velocity (Vs) required for the rocket to be able to leave the sun-earth system is closest to (Ignore the rotation and revolution of the earth and the presence of any other planet) A) $V_{s}=72 kms^{-1}$ B) $V_{s}=22 kms^{-1}$ C) $V_{s}=42 kms^{-1}$ D) $V_{s}=62 kms^{-1}$
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 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) $0<\alpha\leq\frac{1}{2}$
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 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) $a_{11}+a_{10}$
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) If a24=28 , then p+2q= A) 14 B) 7 C) 21 D) 12
10 If a chord, which is not a tangent of the parabola y2=16x has the equation 2x+y=p, and mid-point (h,k) then which of the following is(are) possible value (s) of p,h, and k? A) p=-1,h=1,k=-3 B) p=2,h=3, k=-4 C) p=-2,h=2,k=-4 D) p=5,h=4,k=-3
11 Let f:R→ R be a differentable function such that f(0)=0, $f(\frac{\pi}{2})=3$ and f ' (0)=1 if $g(x)=\int_{0}^{\frac{\pi}{2}} f'(t) cosec$ $ t f(t)-\cot $ $ t cosec$ $ t f(t)] dt$ for $x\epsilon(0,\frac{\pi}{2}]$, then $\lim_{x \rightarrow0 }g(x)=$ A) 4 B) 2 C) 1 D) 3
12 The order of basicity among the following compound is A) II>I>IV>III B) I>IV>III>I C) IV>I>II>I D) IV>I>II>III
13 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) $H_{3}PO_{3}>H_{3}P_{}O_{2}>H_{3}P_{}O_{4}>H_{4}P_{2}O_{6}$
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 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
