A vector $$\vec{A}$$ is rotated by a small angle $$\Delta\theta$$ radians $$(\Delta\theta \ll 1)$$ to get a new vector $$\vec{B}$$. In that case $$\left|\vec{B} - \vec{A}\right|$$ is:
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A vector $$\vec{A}$$ is rotated by a small angle $$\Delta\theta$$ radians $$(\Delta\theta \ll 1)$$ to get a new vector $$\vec{B}$$. In that case $$\left|\vec{B} - \vec{A}\right|$$ is:
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A beaker contains a fluid of density $$\rho$$ $$\frac{kg}{m^3}$$, specific heat $$S$$ $$\frac{J}{kg \cdot ^\circ C}$$ and viscosity $$\eta$$. The beaker is filled up to height h. To estimate the rate of heat transfer per unit area $$\left(\frac{Q}{A}\right)$$ by convection when beaker is put on a hot plate, a student proposes that it should depend on $$\eta$$, $$\left(\frac{S\Delta\theta}{h}\right)$$ and $$\left(\frac{1}{\rho g}\right)$$ when $$\Delta\theta$$ (in $$^\circ C$$) is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for $$\left(\frac{Q}{A}\right)$$ is:
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If electronic charge $$e$$, electron mass $$m$$, speed of light in vacuum $$c$$ and Planck's constant $$h$$ are taken as fundamental quantities, the permeability of vacuum $$\mu_0$$ can be expressed in units of:
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From the top of a 64 metres high tower, a stone is thrown upwards vertically with the velocity of 48 m/s. The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration $$g = 32$$ m/s$$^2$$, is:
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A large number $$(n)$$ of identical beads, each of mass $$m$$ and radius $$r$$ are strung on a thin smooth rigid horizontal rod of length $$L(L \gg r)$$ and are at rest at random positions. The rod is mounted between two rigid supports (see the figure below). If one of the beads is now given a speed $$v$$, the average force experienced by each support after a long time is (assume all collisions are elastic):
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A particle is moving in a circle of radius $$r$$ under the action of a force $$F = \alpha r^2$$ which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy = 0 for $$r = 0$$):
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A uniform thin rod AB of length $$L$$ has linear mass density $$\mu(x) = a + \frac{bx}{L}$$, where $$x$$ is measured from A. If the CM of the rod lies at a distance of $$\left(\frac{7}{12}L\right)$$ from A, then $$a$$ and $$b$$ are related as:
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A particle of mass 2 kg is on a smooth horizontal table and moves in a circular path of radius 0.6 m. The height of the table from the ground is 0.8 m. If the angular speed of the particle is 12 rad s$$^{-1}$$, the magnitude of its angular momentum about a point on the ground right under the center of the circle is:
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Which of the following most closely depicts the correct variation of the gravitation potential, $$V(r)$$ with distance $$r$$ due to a large planet of radius $$R$$ and uniform mass density? (figures are not drawn to scale)
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An experiment takes 10 min to raise the temperature of water in a container from 0$$^\circ$$C to 100$$^\circ$$C and another 55 min to convert it totally into steam by a heater supplying heat at a uniform rate. Neglecting the specific heat of the container and taking specific heat of the water to be 1 cal (g$$^\circ$$C)$$^{-1}$$, the heat of vaporization according to this experiment will come out to be:
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Using equipartition of energy, the specific heat (in J kg$$^{-1}$$ K$$^{-1}$$) of Aluminium at high temperature can be estimated to be (atomic weight of Aluminium = 27)
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A pendulum with the time period of 1 s is losing energy due to damping. At a certain time, its energy is 45 J. If after completing 15 oscillations its energy has become 15 J, then its damping constant (in s$$^{-1}$$) will be
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A cylindrical block of wood (density = 650 kg m$$^{-3}$$), of base area 30 cm$$^2$$ and height 54 cm, floats in a liquid of density 900 kg m$$^{-3}$$. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly):
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A source of sound emits sound waves at frequency $$f_0$$. It is moving towards an observer with fixed speed $$v_s$$ $$(v_s < v)$$, where $$v$$ is the speed of sound in air. If the observer were to move towards the source with speed $$v_0$$, one of the following two graphs (A and B) will give the correct variation of the frequency $$f$$ heard by the observer as $$v_0$$ is changed.
The variation of $$f$$ with $$v_0$$ is given correctly by:
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A wire of length $$L = 20$$ cm is bent into a semi-circular arc and the two equal halves of the arc are uniformly charged with charges $$+Q$$ and $$-Q$$ as shown in the figure. The magnitude of the charge on each half is $$|Q| = 10^3\varepsilon_0$$, where $$\varepsilon_0$$ is the permittivity of free space. The net electric field at the centre $$O$$ is
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An electric field $$\vec{E} = \left(25\hat{i} + 30\hat{j}\right)$$ N C$$^{-1}$$ exists in a region of space. If the potential at the origin is taken to be zero then the potential at $$x = 2$$ m, $$y = 2$$ m is:
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In the figure is shown a system of four capacitors connected across a 10 V battery. The charge that will flow from switch S when it is closed is:
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In the electric network shown, when no current flows through the 4 $$\Omega$$ resistor in the arm EB, the potential difference between the points A and D will be:
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A short bar magnet is placed in the magnetic meridian of the earth with North Pole pointing north. Neutral points are found at a distance of 30 cm from the magnet on the East-West line, drawn through the middle point of the magnet. The magnetic moment of the magnet in Am$$^2$$ is close to: (Given $$\frac{\mu_0}{4\pi} = 10^{-7}$$ in SI units and $$B_H$$ = Horizontal component of earth's magnetic field = $$3.6 \times 10^{-5}$$ Tesla.)
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A wire carrying current $$I$$ is tied between points $$P$$ and $$Q$$ and is in the shape of a circular arc of radius $$R$$ due to a uniform magnetic field $$B$$ (perpendicular to the plane of the paper, as shown in the figure) in the vicinity of the wire. If the wire subtends an angle $$2\theta_0$$ at the center of the circle (of which it forms an arch) then the tension in the wire is:
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Two long straight parallel wires, carrying (adjustable) currents $$I_1$$ and $$I_2$$, are kept at a distance $$d$$ apart. If the force $$F$$ between the two wires is taken as 'positive' when the wires repel each other and 'negative' when the wires attract each other, the graph showing the dependence of $$F$$, on the product $$I_1 I_2$$, would be:
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The AC voltage across a resistance can be measured using a:
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For the LCR circuit, shown here, the current is observed to lead the applied voltage. An additional capacitor $$C'$$, when joined with the capacitor C present in the circuit, makes the power factor of the circuit unity. The capacitor $$C'$$, must have been connected in:
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For plane electromagnetic waves propagating in the $$+z$$-direction, which one of the following combinations gives the correct possible direction for $$\vec{E}$$ and $$\vec{B}$$ field respectively?
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A thin convex lens of focal length $$f$$ is put on a plane mirror as shown in the figure. When an object is kept at a distance $$a$$ from the lens-mirror combination, its image is formed at a distance $$\dfrac{a}{3}$$ in front of the combination. The value of $$a$$ is:
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In a Young's double slit experiment with light of wavelength $$\lambda$$, the separation of slits is $$d$$ and distance of screen is $$D$$ such that $$D \gg d \gg \lambda$$. If the Fringe width is $$\beta$$, the distance from point of maximum intensity to the point where intensity falls to half of the maximum intensity on either side is:
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Unpolarized light of intensity $$I_0$$ is incident on surface of a block of glass at Brewster's angle. In that case, which one of the following statements is true?
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The de-Broglie wavelength associated with the electron in the $$n = 4$$ level is:
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Let $$N_\beta$$ be the number of $$\beta$$ particle emitted by 1 gram of Na$$^{24}$$ radioactive nuclei having a half life of 15 h. In 7.5 h, the number $$N_\beta$$ is close to $$[N_A = 6.023 \times 10^{23}$$ mole$$^{-1}]$$
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The value of the resistor, $$R_S$$, needed in the DC voltage regulator circuit shown here, equals:
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A 2 V battery is connected across AB as shown in the figure. The value of the current supplied by the battery when in first case battery's positive terminal is connected to A and in second case when positive terminal of battery is connected to B will respectively be:
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In the following reaction:
$$A + 2B + 3C \rightleftharpoons AB_2C_3$$
6.0 g of A, $$6.0 \times 10^{23}$$ atoms of B and 0.036 mol of C reacted and formed 4.8 g of compound $$AB_2C_3$$. If the atomic mass of A and C are 60 and 80 amu, respectively. What is the atomic mass of B in amu? (Avogadro number = $$6 \times 10^{23}$$)
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An element X shows +3, oxidation state in its compounds. Out of the four compounds given below, choose the incorrect formula for the element X.
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At temperature T, the average kinetic energy of any particle is $$\frac{3}{2}kT$$. The de Broglie wavelength follows the order:
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Molecule AB has a bond length of 1.617 $$\mathring{A}$$ and a dipole moment of 0.38 D. The fractional charge on each atom (absolute magnitude) is: $$(e_0 = 4.802 \times 10^{-10}$$ esu$$)$$
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Which compound exhibits maximum dipole moment among the following?
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When does a gas deviate the most from its ideal behavior?
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The increase of pressure on ice $$\rightleftharpoons$$ water system at constant temperature will lead to:
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Which physical property of di-hydrogen is wrong?
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Which of the alkaline earth metal halides given below is essentially covalent in nature?
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The number of structural isomers for $$C_6H_{14}$$ is:
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Match the organic compounds in column - I with the Lassaigne's test result in column - II appropriately:
Column - I Column - II
A. Aniline i. Red colour with FeCl$$_3$$
B. Benzene sulfonic acid ii. Violet color with sodium nitroprusside
C. Thiourea iii. Blue color with acidic solution of FeSO$$_4$$
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Which of the following pairs of compounds are positional isomers?
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Addition of phosphate fertilizers to water bodies causes:
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Determination of the molar mass of acetic acid in benzene using freezing point depression is affected by:
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At 298 K, the standard reduction potentials are 1.51 V for $$MnO_4^- | Mn^{2+}$$, 1.36 V for $$Cl_2|Cl^-$$, 1.07 V for $$Br_2|Br^-$$, 0.54 V for $$I_2|I^-$$. At pH = 3, permanganate is expected to oxidize: $$\left(\frac{RT}{F} = 0.059\right)$$
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$$A + 2B \rightarrow C$$, the rate equation for the reaction is given as Rate = k[A][B]. If the concentration of A is kept the same but that of B is doubled what will happen to the rate itself?
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For the equilibrium, $$A(g) \rightleftharpoons B(g)$$, $$\Delta H$$ is $$-40$$ kJ/mol. If the ratio of the activation energies of the forward $$(E_f)$$ and reverse $$(E_b)$$ reactions is $$\frac{2}{3}$$ then:
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Under ambient conditions, which among the following surfactants will form micelles in aqueous solution at lowest molar concentration?
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Calamine is an ore of:
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Chlorine water on standing loses its color and forms:
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Which of the following compounds has a P - P Bond?
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Which of the following statements is/are false?
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Which of the following complex ions has electrons that are symmetrically filled in both $$t_{2g}$$ and $$e_g$$ orbitals?
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When concentrated HCl is added to aqueous solution of $$CoCl_2$$, its colour changes from reddish pink to deep blue. Which complex ion gives blue colour in reaction?
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What is the major product expected from the following reaction?
Where D is an isotope of hydrogen.
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In the reaction sequence $$2CH_3CHO \xrightarrow{OH^-} A \xrightarrow{\Delta} B$$; the product B is:
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Which one of the following structures represents the neoprene polymer?
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Which artificial sweetener contains chlorine?
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Accumulation of which of the following molecules in the muscles occurs as a result of vigorous exercise?
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If the two roots of the equation, $$(a-1)(x^4 + x^2 + 1) + (a+1)(x^2 + x + 1)^2 = 0$$ are real and distinct, then the set of all values of $$a$$ is equal to
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If $$z$$ is a non-real complex number, then the minimum value of $$\frac{Im\ z^5}{(Im\ z)^5}$$ is (Where $$Im\ z$$ = Imaginary part of $$z$$)
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Let $$A = \{x_1, x_2, \ldots, x_7\}$$ and $$B = \{y_1, y_2, y_3\}$$ be two sets containing seven and three distinct elements respectively. Then the total number of functions $$f : A \rightarrow B$$ that are onto, if there exist exactly three elements $$x$$ in $$A$$ such that $$f(x) = y_2$$, is equal to:
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If in a regular polygon the number of diagonals is 54, then the number of sides of this polygon is:
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The sum of the 3$$^{rd}$$ and the 4$$^{th}$$ terms of a G.P. is 60 and the product of its first three terms is 1000. If the first term of this G.P. is positive, then its 7$$^{th}$$ term is:
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If $$\sum_{n=1}^{5}\frac{1}{n(n+1)(n+2)(n+3)} = \frac{k}{3}$$, then $$k$$ is equal to:
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The term independent of $$x$$ in the binomial expansion of $$\left(1 - \frac{1}{x} + 3x^5\right)\left(2x^2 - \frac{1}{x}\right)^8$$ is
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If $$\cos \alpha + \cos \beta = \frac{3}{2}$$ and $$\sin \alpha + \sin \beta = \frac{1}{2}$$ and $$\theta$$ is the arithmetic mean of $$\alpha$$ and $$\beta$$, then $$\sin 2\theta + \cos 2\theta$$ is equal to:
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A straight line $$L$$ through the point $$(3, -2)$$ is inclined at an angle of $$60^\circ$$ to the line $$\sqrt{3}x + y = 1$$. If $$L$$ also intersects the X-axis, then the equation of $$L$$ is:
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If a circle passing through the point $$(-1, 0)$$ touches y-axis at $$(0, 2)$$, then the x-intercept of the circle is
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If the incentre of an equilateral triangle is $$(1, 1)$$ and the equation of its one side is $$3x + 4y + 3 = 0$$, then the equation of the circumcircle of this triangle is:
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If $$PQ$$ be a double ordinate of the parabola, $$y^2 = -4x$$, where $$P$$ lies in the second quadrant. If $$R$$ divides $$PQ$$ in the ratio 2 : 1, then the locus of $$R$$ is
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If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is:
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Consider the following statements:
P: Suman is brilliant
Q: Suman is rich
R: Suman is honest
The negation of the statement, "Suman is brilliant and dishonest if and only if Suman is rich" can be equivalently expressed as
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Let 10 vertical poles standing at equal distances on a straight line, subtend the same angle of elevation $$\alpha$$ at a point $$O$$ on this line and all the poles are on the same side of $$O$$. If the height of the longest pole is $$h$$ and the distance of the foot of the smallest pole from $$O$$ is $$a$$; then the distance between two consecutive poles, is
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If $$A$$ is a $$3 \times 3$$ matrix such that $$|5 \cdot adj A| = 5$$, then $$|A|$$ is equal to
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If $$\begin{vmatrix} x^2+x & x+1 & x-2 \\ 2x^2+3x-1 & 3x & 3x-3 \\ x^2+2x+3 & 2x-1 & 2x-1 \end{vmatrix} = ax - 12$$, then $$a$$ is equal to:
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Let $$k$$ be a non-zero real number. If $$f(x) = \begin{cases} \frac{(e^x - 1)^2}{\sin\left(\frac{x}{k}\right)\log\left(1 + \frac{x}{4}\right)}, & x \neq 0 \\ 12, & x = 0 \end{cases}$$ is a continuous function at $$x = 0$$, then the value of $$k$$ is
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The equation of a normal to the curve, $$\sin y = x\sin\left(\frac{\pi}{3} + y\right)$$ at $$x = 0$$, is:
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Let $$k$$ and $$K$$ be the minimum and the maximum values of the function $$f(x) = \frac{(1+x)^{0.6}}{1+x^{0.6}}$$ in $$[0, 1]$$, respectively, then the ordered pair $$(k, K)$$ is equal to:
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If $$\int \frac{\log\left(t + \sqrt{1+t^2}\right)}{\sqrt{1+t^2}} dt = \frac{1}{2}(g(t))^2 + c$$, where c is a constant, then $$g(2)$$ is equal to
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Let $$f : R \rightarrow R$$ be a function such that $$f(2-x) = f(2+x)$$ and $$f(4-x) = f(4+x)$$, for all $$x \in R$$ and $$\int_0^2 f(x)dx = 5$$. Then the value of $$\int_{10}^{50} f(x)dx$$ is
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Let $$f : (-1, 1) \rightarrow R$$ be a continuous function. If $$\int_0^{\sin x} f(t) dt = \frac{\sqrt{3}}{2}x$$, then $$f\left(\frac{\sqrt{3}}{2}\right)$$ is equal to:
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The solution of the differential equation $$ydx - (x + 2y^2)dy = 0$$ is $$x = f(y)$$. If $$f(-1) = 1$$, then $$f(1)$$ is equal to
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In a parallelogram $$ABCD$$, $$\left|\overrightarrow{AB}\right| = a$$, $$\left|\overrightarrow{AD}\right| = b$$ and $$\left|\overrightarrow{AC}\right| = c$$. $$\overrightarrow{DB} \cdot \overrightarrow{AB}$$ has the value:
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A plane containing the point $$(3, 2, 0)$$ and the line $$\frac{x-1}{1} = \frac{y-2}{5} = \frac{z-3}{4}$$ also contains the point
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The shortest distance between the z-axis and the line $$x + y + 2z - 3 = 0 = 2x + 3y + 4z - 4$$, is
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If the mean and the variance of a binomial variate $$X$$ are 2 and 1 respectively, then the probability that $$X$$ takes a value greater than or equal to one is:
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If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the probability that the triangle is of maximum area given that it is an isosceles triangle, is:
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