A clock has a continuously moving second's hand of $$0.1\,\text{m}$$ length. The average acceleration of the tip of the hand (in units of $$\text{ms}^{-2}$$) is of the order of:
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A clock has a continuously moving second's hand of $$0.1\,\text{m}$$ length. The average acceleration of the tip of the hand (in units of $$\text{ms}^{-2}$$) is of the order of:
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An insect is at the bottom of a hemispherical ditch of radius $$1\,\text{m}$$. It crawls up the ditch but starts slipping after it is at height $$h$$ from the bottom. If the coefficient of friction between the ground and the insect is $$0.75$$, then $$h$$ is: $$(g = 10\,\text{m s}^{-2})$$
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If the potential energy between two molecules is given by $$U = \frac{A}{r^6} + \frac{B}{r^{12}}$$, then at equilibrium, separation between molecules, and the potential energy are:
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Shown in the figure is a hollow ice-cream cone (it is open at top). If its mass is M, radius of its top is R and height, H, then its moment of inertia about its axis is:
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Four point masses, each of mass $$m$$, are fixed at the corners of a square of side $$l$$. The square is rotating with angular frequency $$\omega$$, about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about the axis is:
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A satellite is in an elliptical orbit around a planet $$P$$. It is observed that the velocity of the satellite when it is farthest from the planet is 6 times less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is:
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Molecules of an ideal gas are known to have three translational degrees of freedom. The gas is maintained at a temperature of T. The total internal energy, U of a mole of this gas, and the value of $$\gamma = \left(\frac{C_p}{C_v}\right)$$ are given, respectively, by:
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An object of mass $$m$$ is suspended at the end of a massless wire of length $$L$$ and area of cross-section, A. Young modulus of the material of the wire is $$Y$$. If the mass is pulled down slightly its frequency of oscillation along the vertical direction is:
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A sound source S is moving along a straight track with speed $$v$$, and is emitting sound of frequency $$v_0$$. An observer is standing at a finite distance, at the point O, from the track. The time variation of frequency heard by the observer is best represented by: ($$t_0$$ represents the instant when the distance between the source and observer is minimum)
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Charges $$Q_1$$ and $$Q_2$$ are at points A and B of a right-angled triangle OAB. The resultant electric field at point O is perpendicular to the hypotenuse, then $$Q_1/Q_2$$ is proportional to:
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For the given input voltage waveform $$V_{in}(t)$$, the output voltage waveform $$V_0(t)$$, across the capacitor is correctly depicted by:
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A particle of charge $$q$$ and mass $$m$$ is moving with a velocity $$-v\hat{i}$$ $$(v \neq 0)$$ towards a large screen placed in the $$Y-Z$$ plane at distance $$d$$. If there is a magnetic field $$\vec{B} = B_0\hat{k}$$, the minimum value of $$v$$ for which the particle will not hit the screen is:
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An electron is moving along $$+x$$ direction with a velocity of $$6 \times 10^6\,\text{ms}^{-1}$$. It enters a region of uniform electric field of $$300\,V/cm$$ pointing along $$+y$$ direction. The magnitude and direction of the magnetic field set up in this region such that the electron keeps moving along the $$x$$ direction will be:
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An AC circuit has $$R = 100\,\Omega$$, $$C = 2\,\mu F$$ and $$L = 80\,\text{mH}$$, connected in series. The quality factor of the circuit is:
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A point like object is placed at distance of $$1\,\text{m}$$ in front of a convex lens of focal length $$0.5\,\text{m}$$. A plane mirror is placed at a distance of $$2\,\text{m}$$ behind the lens. The position and nature of the image formed by the system is:
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In the figure below, $$P$$ and $$Q$$ are two equally intense coherent sources emitting radiation of wavelength $$20\,\text{m}$$. The separation between P and Q is $$5\,\text{m}$$ and the phase of P is ahead of that of Q by $$90^\circ$$. A, B and C are three distinct point of observation, each equidistant from the midpoint of PQ. The intensities of radiation at A, B, C will be in the ratio:
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An electron, a doubly ionized helium ion $$(He^{++})$$ and proton are having the same kinetic energy. The relation between their respective de-Broglie wavelength $$\lambda_e$$, $$\lambda_{He^{++}}$$ and $$\lambda_p$$ is:
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You are given that $$^7_3\text{Li} = 7.0160\,\text{u}$$, Mass of $$^4_2\text{He} = 4.0026\,\text{u}$$ and Mass of $$^1_1\text{He} = 1.0079\,\text{H}$$. When $$20\,\text{g}$$ of $$^7_3\text{Li}$$ is converted into $$^4_2\text{He}$$ by proton capture, the energy liberated, (in kWh), is: [Mass of nucleon $$= 1\,\text{GeV}/c^2$$]
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Identify the correct output signal $$Y$$ in the given combination of gates (as shown $$n$$) for the given inputs $$A$$ and $$B$$:
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A screw gauge has 50 divisions on its circular scale. The circular scale is 4 units ahead of the pitch scale marking, prior to use. Upon one complete rotation of the circular scale, a displacement of $$0.5\,\text{mm}$$ is noticed on the pitch scale. The nature of zero error involved and the least count of the screw gauge, are respectively:
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The density of a solid metal sphere is diameter. The maximum error in the density of the sphere is $$\left(\frac{x}{100}\right)\%$$. If the relative errors in measuring the mass and the diameter are $$6.0\%$$ and $$1.5\%$$ respectively, the value of $$x$$ is ___
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Two bodies of the same mass are moving with the same speed, but in different directions in a plane. They have a completely inelastic collision and move together thereafter with a final speed which is half of their initial velocities of the two bodies (in degree) is ___
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Initially a gas of diatomic molecules is contained in a cylinder of volume $$V_1$$ at a pressure $$P_1$$ and temperature $$250\,\text{K}$$. Assuming that $$25\%$$ of the molecules get dissociated causing a change in number of moles. The pressure of the resulting gas at temperature $$2000\,\text{K}$$, when contained in a volume $$2V_1$$ is given by $$P_2$$. The ratio $$P_2/P_1$$ is ___
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Suppose that intensity of a laser is $$\left(\frac{315}{\pi}\right)\,\text{W m}^{-2}$$. The rms electric field, in units of $$\text{V m}^{-1}$$ associated with this source is close to the nearest integer is ___ ($$\varepsilon_0 = 8.86 \times 10^{-12}\,\text{C}^2\,\text{N m}^{-2}$$; $$c = 3 \times 10^8\,\text{m s}^{-1}$$)
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A part of a complete circuit is shown in the figure. At some instant, the value of current I is $$1\,\text{A}$$ and it is decreasing at a rate of $$10^2\,\text{As}^{-1}$$. The value of the potential difference $$V_P - V_Q$$, (in volts) at that instant is___
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A solution of two components containing $$n_1$$ moles of the 1st component and $$n_2$$ moles of the 2nd component is prepared. $$M_1$$ and $$M_2$$ are the molecular weights of component 1 and 2 respectively. If $$d$$ is the density of the solution in $$\text{g mL}^{-1}$$, $$C_2$$ is the molarity and $$x_2$$ is the mole fraction of the 2nd component, then $$C_2$$ can be expressed as:
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The variation of equilibrium constant with temperature is given below:
Temperature: , Equilibrium Constant:
$$T_1 = 25^\circ\text{C}$$ $$K_1 = 10$$
$$T_2 = 100^\circ\text{C}$$, $$K_2 = 100$$
The values of $$\Delta H^\circ$$, $$\Delta G^\circ$$ at $$T_1$$ and $$\Delta G^\circ$$ at $$T_2$$ (in $$\text{kJ mol}^{-1}$$) respectively, are close to [use $$R = 8.314\,\text{JK}^{-1}\text{mol}^{-1}$$]
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For the reaction
$$\text{Fe}_2\text{N}(s) + \frac{3}{2}\text{H}_2(g) \rightleftharpoons 2\text{Fe}(s) + \text{NH}_3(g)$$
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Arrange the following solutions in the decreasing order of pOH:
(A) 0.01 M HCl
(B) 0.01 M NaOH
(C) 0.01 M $$\text{CH}_3\text{COONa}$$
(D) 0.01 M NaCl
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Among the sulphates of alkaline earth metals, the solubilities of $$\text{BeSO}_4$$ and $$\text{MgSO}_4$$ in water, respectively, are:
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Which of the following compounds shows geometrical isomerism?
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Consider the following reactions:
'A' is:
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The major product obtained from the following reaction is:
The major products of the following reaction are:
The major product of the following reaction is:
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Consider the following reactions:
$$A \to P1;\; B \to P2;\; C \to P3;\; D \to P4$$
The order of the above reactions are a, b, c and d, respectively. The following graph is obtained when log[rate] vs log[conc.] are plotted:
Among the following, the correct sequence for the order of the reactions is:
Kraft temperature is the temperature:
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The correct statement with respect to dinitrogen is:
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The presence of soluble fluoride ion upto 1 ppm concentration in drinking water, is:
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The INCORRECT statement is:
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The set that contains atomic numbers of only transition elements, is:
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The lanthanoid that does NOT show +4 oxidation state is:
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The species that has a spin-only magnetic moment of 5.9 BM, is: ($$T_d$$ = tetrahedral)
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The increasing order of $$pK_b$$ values of the following compounds is:
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Consider the Assertion and Reason given below.
Assertion (A): Ethene polymerized in the presence of Ziegler Natta Catalyst at high temperature and pressure is used to make buckets and dustbins.
Reason (R): High density polymers are closely packed and are chemically inert. Choose the correct answer from the following:
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A spherical balloon of radius $$3\,\text{cm}$$ containing helium gas has a pressure of $$48 \times 10^{-3}\,\text{bar}$$. At the same temperature, the pressure, of a spherical balloon of radius $$12\,\text{cm}$$ containing the same amount of gas will be............ $$\times 10^{-6}\,\text{bar}$$.
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In an estimation of bromine by Carius method, $$1.6\,\text{g}$$ of an organic compound gave $$1.88\,\text{g}$$ of AgBr. The mass percentage of bromine in the compound is........ (Atomic mass, Ag = 108, Br = 80 $$\text{g mol}^{-1}$$)
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The elevation of boiling point of $$0.10\,\text{m}$$ aqueous $$\text{CrCl}_3 \cdot x\text{NH}_3$$ solution is two times that of $$0.05\,\text{m}$$ aqueous $$\text{CaCl}_2$$ solution. The value of $$x$$ is............ [Assume 100% ionisation of the complex and $$\text{CaCl}_2$$, coordination number of Cr as 6, and that all $$\text{NH}_3$$ molecules are present inside the coordination sphere]
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Potassium chlorate is prepared by the electrolysis of KCl in basic solution
$$6\text{OH}^- + \text{Cl}^- \to \text{ClO}_3^- + 3\text{H}_2\text{O} + 6e^-$$. If only 60% of the current is utilized in the reaction, the time (rounded to the nearest hour) required to produce $$10\,\text{g}$$ of $$\text{KClO}_3$$ using a current of $$2\,\text{A}$$ is........ (Given: F = 96,500 C/mol; molar mass of $$\text{KClO}_3$$ = 122 g mol$$^{-1}$$)
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The number of $$\text{Cl} = \text{O}$$ bonds in perchloric acid is, $$n$$.........
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If $$\alpha$$ and $$\beta$$ be two roots of the equation $$x^2 - 64x + 256 = 0$$. Then the value of $$\left(\frac{\alpha^3}{\beta^5}\right)^{1/8} + \left(\frac{\beta^3}{\alpha^5}\right)^{1/8}$$ is:
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The region represented by $$\{z = x + iy \in \mathbb{C} : |z| - \text{Re}(z) \leq 1\}$$ is also given by the inequality:
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Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?
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Let $$a, b, c, d$$ and $$p$$ be non-zero distinct real numbers such that $$(a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) = 0$$. Then:
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If $$\{p\}$$ denotes the fractional part of the number $$p$$, then $$\left\{\frac{3^{200}}{8}\right\}$$ is equal to:
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A ray of light coming from the point $$(2, 2\sqrt{3})$$ is incident at an angle $$30^\circ$$ on the line $$x = 1$$ at the point A. The ray gets reflected on the line $$x = 1$$ and meets $$x$$-axis at the point B. Then, the line AB passes through the point:
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Let $$L_1$$ be a tangent to the parabola $$y^2 = 4(x+1)$$ and $$L_2$$ be a tangent to the parabola $$y^2 = 8(x+2)$$ such that $$L_1$$ and $$L_2$$ intersect at right angles. Then $$L_1$$ and $$L_2$$ meet on the straight line:
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Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $$\frac{x^2}{4} + \frac{y^2}{2} = 1$$ from any of its foci?
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The negation of the Boolean expression $$p \vee (\sim p \wedge q)$$ is equivalent to:
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If $$\sum_{i=1}^{n}(x_i - a) = n$$ and $$\sum_{i=1}^{n}(x_i - a)^2 = na$$, $$(n, a \gt 1)$$, then the standard deviation of $$n$$ observations $$x_1, x_2, \ldots, x_n$$ is:
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Let m and M be respectively the minimum and maximum values of $$\begin{vmatrix} \cos^2 x & 1 + \sin^2 x & \sin 2x \\ 1 + \cos^2 x & \sin^2 x & \sin 2x \\ \cos^2 x & \sin^2 x & 1 + \sin 2x \end{vmatrix}$$. Then the ordered pair (m, M) is equal to:
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The values of $$\lambda$$ and $$\mu$$ for which the system of linear equations $$x + y + z = 2$$, $$x + 2y + 3z = 5$$, $$x + 3y + \lambda z = \mu$$ has infinitely many solutions, are respectively:
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If $$f(x+y) = f(x)f(y)$$ and $$\sum_{x=1}^{\infty} f(x) = 2$$, $$x, y \in N$$, where $$N$$ is the set of all natural numbers, then the value of $$\frac{f(4)}{f(2)}$$ is:
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The position of a moving car at time $$t$$ is given by $$f(t) = at^2 + bt + c$$, $$t > 0$$, where $$a$$, $$b$$ and $$c$$ are real numbers greater than 1. Then the average speed of the car over the time interval $$[t_1, t_2]$$ is attained at the point:
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If $$I_1 = \int_0^1 (1-x^{50})^{100}\,dx$$ and $$I_2 = \int_0^1 (1-x^{50})^{101}\,dx$$ such that $$I_2 = \alpha I_1$$ then $$\alpha$$ equals to:
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$$\lim_{x \to 1}\left(\frac{\int_0^{(x-1)^2} t\cos t^2\,dt}{(x-1)\sin(x-1)}\right)$$
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The area (in sq. units) of the region $$A = \{(x, y) : |x| + |y| \leq 1,\; 2y^2 \geq |x|\}$$
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The general solution of the differential equation $$\sqrt{1 + x^2 + y^2 + x^2y^2} + xy\frac{dy}{dx} = 0$$ (where C is a constant of integration)
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The shortest distance between the lines $$\frac{x-1}{0} = \frac{y+1}{-1} = \frac{z}{1}$$ and $$x + y + z + 1 = 0$$, $$2x - y + z + 3 = 0$$ is:
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Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference is:
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The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be $$45^\circ$$. After walking a distance of $$80$$ meters towards the top, up a slope inclined at angle of $$30^\circ$$ to the horizontal plane the angle of elevation of the top of the hill becomes $$75^\circ$$. Then the height of the hill (in meters) is_____.
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Set $$A$$ has $$m$$ elements and set $$B$$ has $$n$$ elements. If the total number of subsets of $$A$$ is 112 more than the total number of subsets of $$B$$, then the value of $$m \cdot n$$ is___.
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} x^5\sin\left(\frac{1}{x}\right) + 5x^2, & x < 0 \\ 0, & x = 0 \\ x^5\cos\left(\frac{1}{x}\right) + \lambda x^2, & x > 0 \end{cases}$$. The value of $$\lambda$$ for which $$f''(0)$$ exists, is___.
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Let $$AD$$ and $$BC$$ be two vertical poles at $$A$$ and $$B$$ respectively on a horizontal ground. If $$AD = 8\,\text{m}$$, $$BC = 11\,\text{m}$$, $$AB = 10\,\text{m}$$; then the distance (in meters) of a point M lying in between AB from the point A such that $$MD^2 + MC^2$$ is minimum, is___.
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If $$\vec{a}$$ and $$\vec{b}$$ are unit vectors, then the greatest value of $$\sqrt{3}|\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|$$ is___.
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