If the length of the pendulum in pendulum clock increases by 0.1%, then the error in time per day is:
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If the length of the pendulum in pendulum clock increases by 0.1%, then the error in time per day is:
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The angle between vector $$\left(\vec{A}\right)$$ and $$\left(\vec{A} - \vec{B}\right)$$ is:
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Match List - I with List - II:
List - I List - II
a. Magnetic induction i. $$ML^2T^{-2}A^{-1}$$
b. Magnetic flux ii. $$M^0L^{-1}A$$
c. Magnetic permeability iii. $$MT^{-2}A^{-1}$$
d. Magnetization iv. $$MLT^{-2}A^{-2}$$
Choose the most appropriate answer from the options given below:
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A particle of mass $$m$$ is suspended from a ceiling through a string of length $$L$$. The particle moves in a horizontal circle of radius $$r$$ such that $$r = \frac{L}{\sqrt{2}}$$. The speed of particle will be:
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A bomb is dropped by a fighter plane flying horizontally. To an observer sitting in the plane, the trajectory of the bomb is a:
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The solid cylinder of length 80 cm and mass $$M$$ has a radius of 20 cm. Calculate the density of the material used if the moment of inertia of the cylinder about an axis $$CD$$ parallel to $$AB$$ as shown in figure is 2.7 kg m$$^2$$.
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Two blocks of masses 3 kg and 5 kg are connected by a metal wire going over a smooth pulley. The breaking stress of the metal is $$\frac{24}{\pi} \times 10^2$$ N m$$^{-2}$$. What is the minimum radius of the wire?
(take g = 10 m s$$^{-2}$$)
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The temperature of equal masses of three different liquids $$x$$, $$y$$ and $$z$$ are 10°C, 20°C and 30°C respectively. The temperature of mixture when $$x$$ is mixed with $$y$$ is 16°C and that when $$y$$ is mixed with $$z$$ is 26°C. The temperature of mixture when $$x$$ and $$z$$ are mixed will be:
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A refrigerator consumes an average 35 W power to operate between temperature -10°C to 25°C. If there is no loss of energy then how much average heat per second does it transfer?
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A cylindrical container of volume $$4.0 \times 10^{-3}$$ m$$^3$$ contains one mole of hydrogen and two moles of carbon dioxide. Assume the temperature of the mixture is 400 K. The pressure of the mixture of gases is:
[Take gas constant as 8.3 J mol$$^{-1}$$ K$$^{-1}$$]
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The two thin coaxial rings, each of radius $$a$$ and having charges $$+Q$$ and $$-Q$$ respectively are separated by a distance of $$s$$. The potential difference between the centres of the two rings is:
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A parallel-plate capacitor with plate area $$A$$ has separation $$d$$ between the plates. Two dielectric slabs of dielectric constant $$K_1$$ and $$K_2$$ of same area $$\frac{A}{2}$$ and thickness $$\frac{d}{2}$$ are inserted in the space between the plates. The capacitance of the capacitor will be given by:
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An electric bulb of 500 W at 100 V is used in a circuit having a 200 V supply. Calculate the resistance $$R$$ to be connected in series with the bulb so that the power delivered by the bulb is 500 W.
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If you are provided a set of resistances, 2 $$\Omega$$, 4 $$\Omega$$, 6 $$\Omega$$ and 8 $$\Omega$$. Connect these resistances to obtain an equivalent resistance of $$\frac{46}{3}$$ $$\Omega$$.
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In the given circuit the AC source has $$\omega = 100$$ rad s$$^{-1}$$. Considering the inductor and capacitor to be ideal, what will be the current $$I$$ flowing through the circuit?
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A light beam is described by $$E = 800 \sin\omega\left(t - \frac{x}{c}\right)$$. An electron is allowed to move normal to the propagation of light beam with a speed of $$3 \times 10^7$$ m s$$^{-1}$$. What is the maximum magnetic force exerted on the electron?
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The de-Broglie wavelength of a particle having kinetic energy $$E$$ is $$\lambda$$. How much extra energy must be given to this particle so that the de-Broglie wavelength reduces to 75% of the initial value?
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At time $$t = 0$$, a material is composed of two radioactive atoms $$A$$ and $$B$$, where $$N_A(0) = 2N_B(0)$$. The decay constant of both kind of radioactive atoms is $$\lambda$$. However, $$A$$ disintegrates to $$B$$ and $$B$$ disintegrates to $$C$$. Which of the following figures represents the evolution of $$\frac{N_B(t)}{N_B(0)}$$ with respect to time $$t$$?
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Four NOR gates are connected as shown in figure. The truth table for the given figure is:
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A transmitting antenna at top of a tower has a height of 50 m, and the height of receiving antenna is 80 m. What is the range of communication for the line of sight (LOS) mode?
[use radius of the earth = 6400 km]
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The acceleration due to gravity is found up to an accuracy of 4% on a planet. The energy supplied to a simple pendulum of known mass $$m$$ to undertake oscillations of time period $$T$$ is being estimated. If time period is measured to an accuracy of 3%, the accuracy to which $$E$$ is known is _________ %
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The coefficient of static friction between two blocks is 0.5 and the table is smooth. The maximum horizontal force that can be applied to move the blocks together is _________ N (take $$g = 10$$ m s$$^{-2}$$)
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Two simple harmonic motions are represented by the equations $$x_1 = 5\sin\left(2\pi t + \frac{\pi}{4}\right)$$ and $$x_2 = 5\sqrt{2}(\sin 2\pi t + \cos 2\pi t)$$. The amplitude of the second motion is _________ times the amplitude in the first motion.
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Two waves are simultaneously passing through a string and their equations are:
$$y_1 = A_1 \sin k(x - vt)$$, $$y_2 = A_2 \sin k(x - vt + x_0)$$. Given amplitudes $$A_1 = 12$$ mm and $$A_2 = 5$$ mm, $$x_0 = 3.5$$ cm and wave number $$k = 6.28$$ cm$$^{-1}$$. The amplitude of resulting wave will be _________ mm.
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If the maximum value of accelerating potential provided by a radio frequency oscillator is 12 kV. The number of revolution made by a proton in a cyclotron to achieve one sixth the speed of light is:
[$$m_p = 1.67 \times 10^{-27}$$ kg, $$e = 1.6 \times 10^{-19}$$ C, Speed of light = $$3 \times 10^8$$ m s$$^{-1}$$]
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A coil in the shape of an equilateral triangle of side 10 cm lies in a vertical plane between the pole pieces of permanent magnet producing a horizontal magnetic field 20 mT. The torque acting on the coil when a current of 0.2 A is passed through it and its plane becomes parallel to the magnetic field will be $$\sqrt{x} \times 10^{-5}$$ Nm. The value of $$x$$ is _________
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A circular coil of radius 8.0 cm and 20 turns is rotated about its vertical diameter with an angular speed of 50 rad s$$^{-1}$$ in a uniform horizontal magnetic field of $$3.0 \times 10^{-2}$$ T. The maximum emf induced in the coil will be _________ $$\times 10^{-2}$$ volt (rounded off to the nearest integer).
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An object is placed at a distance of 12 cm from a convex lens. A convex mirror of focal length 15 cm is placed on another side of the lens at 8 cm as shown in the figure. The image of the object coincides with the object.
When the convex mirror is removed, a real and inverted image is formed at a position. The distance of the image from the object will be _________ cm
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A source of light is placed in front of a screen. The intensity of light on the screen is $$I$$. Two Polaroids $$P_1$$ and $$P_2$$ are so placed in between the source of light and screen that the intensity of light on the screen is $$\frac{I}{2}$$. Then the $$P_2$$, should be rotated by an angle of _________ (degrees) so that the intensity of light on the screen becomes $$\frac{3I}{8}$$.
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For the given circuit, the power across zener diode is _________ mW.
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The class of drug to which chlordiazepoxide with above structure belongs is:
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The interaction energy of London forces between two particles is proportional to $$r^x$$, where r is the distance between the particles. The value of x is:
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The bond order and magnetic behaviour of $$O_2^-$$ ion are, respectively:
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Heavy water is used for the study of reaction mechanism.
Reason (R): The rate of reaction for the cleavage of O-H bond is slower than that of O-D bond.
Choose the most appropriate answer from the options given below:
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Barium carbonate is insoluble in water and is highly stable.
Reason (R): The thermal stability of the carbonates increases with increasing cationic size.
Choose the most appropriate answer from the options given below:
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Which one of the following compounds is not aromatic?
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Consider the given reaction, Identify X and Y:
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R)
Assertion (A) : Photochemical smog causes cracking of rubber.
Reason (R): Presence of ozone, nitric oxide, acrolein, formaldehyde and peroxyacetyl nitrate in photochemical smog makes it oxidizing.
Choose the most appropriate answer from the options given below:
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The sol given below with negatively charged colloidal particles is:
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Given below are two statements:
Statement I : Sphalerite is a sulphide ore of zinc and copper glance is a sulphide ore of copper.
Statement II : It is possible to separate two sulphide ores by adjusting proportion of oil to water or by using depressants in a froth flotation method.
Choose the most appropriate answer from the options given below:
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The number of non-ionisable hydrogen atoms present in the final product obtained from the hydrolysis of PCl$$_5$$ is:
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Chalcogen group elements are:
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Arrange the following Cobalt complexes in the order of increasing Crystal Field Stabilization Energy (CFSE) value.
Complexes:
Choose the correct option:
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Indicate the complex/complex ion which did not show any geometrical isomerism:
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Which one of the following phenols does not give colour when condensed with phthalic anhydride in presence of conc. H$$_2$$SO$$_4$$?
The number of stereoisomers possible for 1,2-dimethyl cyclopropane is:
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Match List - I with List - II:
List - I (Chemical Reaction) List - II (Reagent used)
a. CH$$_3$$COOCH$$_2$$CH$$_3$$ $$\rightarrow$$ CH$$_3$$CH$$_2$$OH i. CH$$_3$$MgBr/H$$_3$$O$$^+$$ (one equivalent)
b. CH$$_3$$COOCH$$_3$$ $$\rightarrow$$ CH$$_3$$CHO ii. H$$_2$$SO$$_4$$/H$$_2$$O
c. CH$$_3$$C$$\equiv$$N $$\rightarrow$$ CH$$_3$$CHO iii. DIBAL-H/H$$_2$$O
d. iv. SnCl$$_2$$, HCl/H$$_2$$O
Choose the most appropriate match:
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Sucrose is a disaccharide and a non-reducing sugar.
Reason (R): Sucrose involves glycosidic linkage between $$C_1$$ of $$\beta$$-glucose and $$C_2$$ of $$\alpha$$-fructose.
Choose the most appropriate answer from the options given below:
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100 mL of Na$$_3$$PO$$_4$$ solution contains 3.45 g of sodium. The molarity of the solution is _________ $$\times 10^{-2}$$ mol L$$^{-1}$$. (Nearest integer)
[Atomic Masses - Na : 23.0u, O : 16.0u, P : 31.0u]
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For water $$\Delta_{vap}H = 41$$ kJ mol$$^{-1}$$ at 373 K and 1 bar pressure. Assuming that water vapour is an ideal gas that occupies a much larger volume than liquid water, the internal energy change during evaporation of water is _________ (kJ mol$$^{-1}$$):
[Use: R = 8.3 J mol$$^{-1}$$ K$$^{-1}$$]
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The equilibrium constant $$K_c$$ at 298 K for the reaction A + B $$\rightleftharpoons$$ C + D is 100. Starting with an equimolar solution with concentrations of A, B, C and D all equal to 1M, the equilibrium concentration of D is _________ $$\times 10^{-2}$$ M. (Nearest integer)
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In the sulphur estimation, 0.471 g of an organic compound gave 1.44 g of barium sulfate. The percentage of sulphur in the compound is _________ (Nearest integer) (Atomic Mass of Ba = 137u)
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The reaction rate for the reaction
$$[PtCl_4]^{2-} + H_2O \rightleftharpoons [Pt(H_2O)Cl_3]^- + Cl^-$$
was measured as a function of concentrations of different species. It was observed that
$$\frac{-d[PtCl_4]^{2-}}{dt} = 4.8 \times 10^{-5}[PtCl_4]^{2-} - 2.4 \times 10^{-3}[Pt(H_2O)Cl_3]^-][Cl^-]$$
where square brackets are used to denote molar concentrations.
The equilibrium constant K$$_c$$ = X (Nearest integer). Value of $$\frac{1}{X}$$ is _________
$$K_c = X$$ (Nearest integer)
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A chloro compound "A".
(i) forms aldehydes on ozonolysis followed by the hydrolysis.
(ii) when vaporized completely 1.53 g of A, gives 448 mL of vapour at STP. The number of carbon atoms in a molecule of compound A is _________
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83 g of ethylene glycol dissolved in 625 g of water. The freezing point of the solution is _________ K. (Nearest integer)
[Use: Molal Freezing point depression constant of water = 1.86 K kg mol$$^{-1}$$
Freezing point of water = 273 K
Atomic masses: C : 12.0u, O : 16.0u, H : 1.0u]
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For the galvanic cell,
Zn(s) + Cu$$^{2+}$$(0.02M) $$\rightarrow$$ Zn$$^{2+}$$(0.04M) + Cu(s)
E$$_{cell}$$ = _________ $$\times 10^{-2}$$ V (Nearest integer)
[Use: E$$^0$$ Cu/Cu$$^{2+}$$ = $$-0.34$$ V, E$$_{Zn/Zn^{2+}}$$ = +0.76 V, $$\frac{2.303RT}{F}$$ = 0.059 V]
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The overall stability constant of the complex ion $$[Cu(NH_3)_4]^{2+}$$ is $$2.1 \times 10^{13}$$. The overall dissociation constant is $$y \times 10^{-14}$$. Then $$y$$ is _________ (Nearest integer)
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A metal surface is exposed to 500 nm radiation. The threshold frequency of the metal for photoelectric current is $$4.3 \times 10^{14}$$ Hz. The velocity of ejected electron is _________ $$\times 10^5$$ ms$$^{-1}$$ (Nearest integer)
[Use: h = $$6.63 \times 10^{-34}$$ Js, m$$_e$$ = $$9.0 \times 10^{-31}$$ kg]
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If $$(\sqrt{3} + i)^{100} = 2^{99}(p + iq)$$, then $$p$$ and $$q$$ are roots of the equation:
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A 10 inches long pencil $$AB$$ with mid point $$C$$ and a small eraser $$P$$ are placed on the horizontal top of a table such that $$PC = \sqrt{5}$$ inches and $$\angle PCB = \tan^{-1}(2)$$. The acute angle through which the pencil must be rotated about $$C$$ so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is:
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The value of $$$2\sin\left(\frac{\pi}{8}\right)\sin\left(\frac{2\pi}{8}\right)\sin\left(\frac{3\pi}{8}\right)\sin\left(\frac{5\pi}{8}\right)\sin\left(\frac{6\pi}{8}\right)\sin\left(\frac{7\pi}{8}\right)$$$ is:
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A circle $$C$$ touches the line $$x = 2y$$ at the point $$(2, 1)$$ and intersects the circle $$C_1 : x^2 + y^2 + 2y - 5 = 0$$ at two points $$P$$ and $$Q$$ such that $$PQ$$ is a diameter of $$C_1$$. Then the diameter of $$C$$ is:
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The point $$P\left(-2\sqrt{6}, \sqrt{3}\right)$$ lies on the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ having eccentricity $$\frac{\sqrt{5}}{2}$$. If the tangent and normal at $$P$$ to the hyperbola intersect its conjugate axis at the points $$Q$$ and $$R$$ respectively, then $$QR$$ is equal to:
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The locus of the mid points of the chords of the hyperbola $$x^2 - y^2 = 4$$, which touch the parabola $$y^2 = 8x$$, is:
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$$\lim_{x \to 2}\left(\sum_{n=1}^{9} \frac{x}{n(n+1)x^2 + 2(2n+1)x + 4}\right)$$ is equal to:
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Consider the two statements:
$$(S_1) : (p \rightarrow q) \vee (\sim q \rightarrow p)$$ is a tautology.
$$(S_2) : (p \wedge \sim q) \wedge (\sim p \vee q)$$ is a fallacy.
Then:
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Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{bmatrix}$$. Then $$A^{2025} - A^{2020}$$ is equal to:
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Two fair dice are thrown. The numbers on them are taken as $$\lambda$$ and $$\mu$$, and a system of linear equations
$$x + y + z = 5$$
$$x + 2y + 3z = \mu$$
$$x + 3y + \lambda z = 1$$
is constructed. If $$p$$ is the probability that the system has a unique solution and $$q$$ is the probability that the system has no solution, then:
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If $$\sum_{r=1}^{50} \tan^{-1} \frac{1}{2r^2} = p$$, then the value of $$\tan p$$ is:
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The domain of the function $$\operatorname{cosec}^{-1}\left(\frac{1+x}{x}\right)$$ is:
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Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Let $$f(x) = x - [x]$$, $$g(x) = 1 - x + [x]$$, and $$h(x) = \min\{f(x), g(x)\}$$, $$x \in [-2, 2]$$. Then $$h$$ is:
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The local maximum value of the function, $$f(x) = \left(\frac{2}{x}\right)^{x^2}$$, $$x \gt 0$$, is:
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The value of $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{1 + \sin^2 x}{1 + \pi^{\sin x}}\right) dx$$ is:
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If the value of the integral $$\int_0^5 \frac{x + [x]}{e^{x-[x]}} dx = \alpha e^{-1} + \beta$$, where $$\alpha, \beta \in R$$, $$5\alpha + 6\beta = 0$$, and $$[x]$$ denotes the greatest integer less than or equal to $$x$$; then the value of $$(\alpha + \beta)^2$$ is equal to:
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Let $$y(x)$$ be the solution of the differential equation $$2x^2 dy + (e^y - 2x)dx = 0$$, $$x > 0$$. If $$y(e) = 1$$, then $$y(1)$$ is equal to:
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A hall has a square floor of dimension 10 m $$\times$$ 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is $$\cos^{-1}\frac{1}{5}$$, then the height of the hall (in meters) is:
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Let $$P$$ be the plane passing through the point $$(1, 2, 3)$$ and the line of intersection of the planes $$\vec{r} \cdot (\hat{i} + \hat{j} + 4\hat{k}) = 16$$ and $$\vec{r} \cdot (-\hat{i} + \hat{j} + \hat{k}) = 6$$. Then which of the following points does NOT lie on $$P$$?
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A fair die is tossed until six is obtained on it. Let $$X$$ be the number of required tosses, then the conditional probability $$P(X \geq 5 \mid X \gt 2)$$ is:
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Let $$\lambda \neq 0$$ be in $$R$$. If $$\alpha$$ and $$\beta$$ are the roots of the equation $$x^2 - x + 2\lambda = 0$$, and $$\alpha$$ and $$\gamma$$ are the roots of the equation $$3x^2 - 10x + 27\lambda = 0$$, then $$\frac{\beta\gamma}{\lambda}$$ is equal to _________
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The least positive integer $$n$$ such that $$\frac{(2i)^n}{(1-i)^{n-2}}$$, $$i = \sqrt{-1}$$, is a positive integer, is _________
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The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit 1 and they all are multiple of 11, is _________
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Let $$a_1, a_2, \ldots, a_{10}$$ be an A.P. with common difference $$-3$$ and $$b_1, b_2, \ldots, b_{10}$$ be a G.P. with common ratio 2. Let $$c_k = a_k + b_k$$, $$k = 1, 2, \ldots, 10$$. If $$c_2 = 12$$ and $$c_3 = 13$$, then $$\sum_{k=1}^{10} c_k$$ is equal to _________
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Let $$\binom{n}{k}$$ denote $$^nC_k$$ and $$\left[\frac{n}{k}\right] = \begin{cases} \binom{n}{k}, & \text{if } 0 \leq k \leq n \\ 0, & \text{otherwise} \end{cases}$$. If $$A_k = \sum_{i=0}^{9} \binom{9}{i} \left[\binom{12}{12-k+i}\right] + \sum_{i=0}^{8} \binom{8}{i} \left[\binom{13}{13-k+i}\right]$$ and $$A_4 - A_3 = 190p$$, then $$p$$ is equal to _________
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Let the mean and variance of four numbers 3, 7, $$x$$ and $$y$$ ($$x > y$$) be 5 and 10 respectively. Then the mean of four numbers 3 + 2x, 7 + 2y, $$x + y$$ and $$x - y$$ is _________
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Let $$A$$ be a $$3 \times 3$$ real matrix. If $$\det(2\text{Adj}(2\text{Adj}(\text{Adj}(2A)))) = 2^{41}$$, then the value of $$\det(A^2)$$ equals _________
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Let $$a$$ and $$b$$ respectively be the points of local maximum and local minimum of the function $$f(x) = 2x^3 - 3x^2 - 12x$$. If $$A$$ is the total area of the region bounded by $$y = f(x)$$, the $$x$$-axis and the lines $$x = a$$ and $$x = b$$, then 4A is equal to _________
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If the projection of the vector $$\hat{i} + 2\hat{j} + \hat{k}$$ on the sum of the two vectors $$2\hat{i} + 4\hat{j} - 5\hat{k}$$ and $$-\lambda\hat{i} + 2\hat{j} + 3\hat{k}$$ is 1, then $$\lambda$$ is equal to _________
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Let $$Q$$ be the foot of the perpendicular from the point $$P(7, -2, 13)$$ on the plane containing the lines $$\frac{x+1}{6} = \frac{y-1}{7} = \frac{z-3}{8}$$ and $$\frac{x-1}{3} = \frac{y-2}{5} = \frac{z-3}{7}$$. Then $$(PQ)^2$$ is equal to _________
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