If $$e$$ is the electronic charge, $$c$$ is the speed of light in free space and $$h$$ is Planck's constant, the quantity $$\frac{1}{4\pi\varepsilon_0} \frac{|e|^2}{hc}$$ has dimensions of:
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If $$e$$ is the electronic charge, $$c$$ is the speed of light in free space and $$h$$ is Planck's constant, the quantity $$\frac{1}{4\pi\varepsilon_0} \frac{|e|^2}{hc}$$ has dimensions of:
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A stone is dropped from the top of a building. When it crosses a point 5 m below the top, another stone starts to fall from a point 25 m below the top. Both stones reach the bottom of building simultaneously. The height of the building is:
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A sphere of radius $$a$$ and mass $$m$$ rolls along a horizontal plane with constant speed $$v_0$$. It encounters an inclined plane at angle $$\theta$$ and climbs upward. Assuming that it rolls without slipping, how far up the sphere will travel?
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Thermodynamic process is shown below on a $$P-V$$ diagram for one mole of an ideal gas. If $$V_2 = 2V_1$$, then the ratio of temperature $$\frac{T_2}{T_1}$$ is:
Given below are two statements:
Statement I: In a diatomic molecule, the rotational energy at a given temperature obeys Maxwell's distribution.
Statement II: In a diatomic molecule, the rotational energy at a given temperature equals the translational kinetic energy for each molecule.
In the light of the above statements, choose the correct answer from the options given below:
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Two identical springs of spring constant $$2k$$ are attached to a block of mass $$m$$ and to fixed support (see figure). When the mass is displaced from equilibrium position on either side, it executes simple harmonic motion. The time period of oscillations of this system is:
The point $$A$$ moves with a uniform speed along the circumference of a circle of radius 0.36 m and covers 30° in 0.1 s. The perpendicular projection $$P$$ from $$A$$ on the diameter $$MN$$ represents the simple harmonic motion of $$P$$. The restoration force per unit mass when $$P$$ touches $$M$$ will be:
$$Y = A\sin(\omega t + \phi_0)$$ is the time-displacement equation of a SHM. At $$t = 0$$ the displacement of the particle is $$Y = \frac{A}{2}$$ and it is moving along negative $$x$$-direction. Then the initial phase angle $$\phi_0$$ will be:
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A charge $$q$$ is placed at one corner of a cube as shown in figure. The flux of electrostatic field $$\vec{E}$$ through the shaded area is:
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In a ferromagnetic material, below the curie temperature, a domain is defined as:
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An electron with kinetic energy $$K_1$$ enters between parallel plates of a capacitor at an angle $$\alpha$$ with the plates. It leaves the plates at angle $$\beta$$ with kinetic energy $$K_2$$. Then the ratio of kinetic energies $$K_1 : K_2$$ will be:
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Match List I with List II.
| List I | List II |
|---|---|
| (a) Rectifier | (i) Used either for stepping up or stepping down the A.C. voltage |
| (b) Stabilizer | (ii) Used to convert A.C. voltage into D.C. voltage |
| (c) Transformer | (iii) Used to remove any ripple in the rectified output voltage |
| (d) Filter | (iv) Used for constant output voltage even when the input voltage or load current change |
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An L.C.R. circuit contains resistance of 110 $$\Omega$$ and a supply of 220 V at 300 rad s$$^{-1}$$ angular frequency. If only capacitance is removed from the circuit, current lags behind the voltage by 45°. If on the other hand, only the inductor is removed the current leads by 45° with the applied voltage. The R.M.S. current flowing in the circuit will be:
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Consider the diffraction pattern obtained from the sunlight incident on a pinhole of diameter 0.1 $$\mu$$m. If the diameter of the pinhole is slightly increased, it will affect the diffraction pattern such that
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An electron of mass $$m_e$$ and a proton of mass $$m_p = 1836 m_e$$ are moving with the same speed. The ratio of their de Broglie wavelength $$\frac{\lambda_{electron}}{\lambda_{proton}}$$ will be:
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The stopping potential for electrons emitted from a photosensitive surface illuminated by light of wavelength 491 nm is 0.710 V. When the incident wavelength is changed to a new value, the stopping potential is 1.43 V. The new wavelength is:
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The wavelength of the photon emitted by a hydrogen atom when an electron makes a transition from $$n = 2$$ to $$n = 1$$ state is:
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The truth table for the following logic circuit is:
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For extrinsic semiconductors; when doping level is increased;
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If a message signal of frequency $$f_m$$ is amplitude modulated with a carrier signal of frequency $$f_c$$ and radiated through an antenna, the wavelength of the corresponding signal in air is
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If $$\vec{P} \times \vec{Q} = \vec{Q} \times \vec{P}$$, the angle between $$\vec{P}$$ and $$\vec{Q}$$ is $$\theta$$ ($$0° < \theta < 360°$$). The value of $$\theta$$ will be ______ °.
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Two particles having masses 4 g and 16 g respectively are moving with equal kinetic energies. The ratio of the magnitudes of their linear momentum is $$n : 2$$. The value of $$n$$ will be ______.
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The initial velocity $$v_i$$ required to project a body vertically upward from the surface of the earth to reach a height of $$10R$$, where $$R$$ is the radius of the earth, may be described in terms of escape velocity $$v_e$$ such that $$v_i = \sqrt{\frac{x}{y}} \times v_e$$. The value of $$x$$ will be
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A reversible heat engine converts one-fourth of the heat input into work. When the temperature of the sink is reduced by 52 K, its efficiency is doubled. The temperature in Kelvin of the source will be ______.
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The percentage increase in the speed of transverse waves produced in a stretched string if the tension is increased by 4%, will be ______ %.
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The peak electric field produced by the radiation coming from the 8 W bulb at a distance of 10 m is $$\frac{x}{10}\sqrt{\frac{\mu_0 c}{\pi}}$$ V m$$^{-1}$$. The efficiency of the bulb is 10% and it is a point source. The value of $$x$$ is ______,
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Two small spheres each of mass 10 mg are suspended from a point by threads 0.5 m long. They are equally charged and repel each other to a distance of 0.20 m. The charge on each of the sphere is $$\frac{a}{21} \times 10^{-8}$$ C. The value of $$a$$ will be ______. [Given $$g = 10$$ m s$$^{-2}$$]
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Two identical conducting spheres with negligible volume have 2.1 nC and $$-0.1$$ nC charges, respectively. They are brought into contact and then separated by a distance of 0.5 m. The electrostatic force acting between the spheres is ______ $$\times 10^{-9}$$ N.
[Given : $$4\pi \epsilon_{\circ}$$ = $$ \frac{1}{9\times 10^{9}}$$ SI Units}]
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A current of 6 A enters one corner $$P$$ of an equilateral triangle $$PQR$$ having 3 wires of resistance 2 $$\Omega$$ each and leaves by the corner $$R$$. The currents $$i_1$$ in ampere is ______
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The wavelength of an X-ray beam is 10 Å. The mass of a fictitious particle having the same energy as that of the X-ray photons is $$\frac{x}{3}h$$ kg. The value of $$x$$ is ______. ($$h$$ = Planck's constant)
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Which among the following species has unequal bond lengths?
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The solubility of $$Ca(OH)_2$$ in water is:
[Given: The solubility product of $$Ca(OH)_2$$ in water $$= 5.5 \times 10^{-6}$$]
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Water does not produce CO on reacting with:
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Which of the following compound is added to the sodium extract before addition of silver nitrate for testing of halogens?
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The correct order of acid character of the following compounds is:
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Correct statement about the given chemical reaction is:
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The major product of the following reaction is:
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The correct sequence of reagents used in the preparation of 4-bromo-2-nitroethylbenzene from benzene is:
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Given below are two statements:
Statement I: The pH of rain water is normally ~5.6.
Statement II: If the pH of rain water drops below 5.6, it is called acid rain.
In the light of the above statements, choose the correct answer from the options given below:
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Which one of the following statements is FALSE for hydrophilic sols?
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The method used for the purification of Indium is:
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The correct order of bond dissociation enthalpy of halogens is:
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Given below are two statements:
Statement I: $$\alpha$$ and $$\beta$$ forms of sulphur can change reversibly between themselves with slow heating or slow cooling.
Statement II: At room temperature the stable crystalline form of sulphur is monoclinic sulphur.
In the light of the above statements, choose the correct answer from the options given below:
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The major components of German Silver are:
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In which of the following order the given complex ions are arranged correctly with respect to their decreasing spin only magnetic moment?
(i) $$[FeF_6]^{3-}$$
(ii) $$[Co(NH_3)_6]^{3+}$$
(iii) $$[NiCl_4]^{2-}$$
(iv) $$[Cu(NH_3)_4]^{2+}$$
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Given below are two statements:
Statement I: The identification of $$Ni^{2+}$$ is carried out by dimethyl glyoxime in the presence of $$NH_4OH$$.
Statement II: The dimethyl glyoxime is a bidentate neutral ligand.
In the light of the above statements, choose the correct answer from the options given below:
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The major product of the following reaction is:
$$CH_3CH_2CH = CH_2 \xrightarrow[ Rh \text{ catalyst}]{H_2/CO}$$
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What is 'X' in the given reaction?
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Carbylamine test is used to detect the presence of primary amino group in an organic compound. Which of the following compound is formed when this test is performed with aniline?
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Which of the following is correct structure of $$\alpha$$-anomer of maltose?
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Electromagnetic radiation of wavelength 663 nm is just sufficient to ionise the atom of metal A. The ionization energy of metal A in kJ mol$$^{-1}$$ is ______. (Rounded-off to the nearest integer)
[$$h = 6.63 \times 10^{-34}$$ Js, $$c = 3.00 \times 10^8$$ ms$$^{-1}$$, $$N_A = 6.02 \times 10^{23}$$ mol$$^{-1}$$]
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Five moles of an ideal gas at 293 K is expanded isothermally from an initial pressure of 2.1 MPa to 1.3 MPa against at constant external pressure 4.3 MPa. The heat transferred in this process is ______ kJ mol$$^{-1}$$.
(Rounded-off to the nearest integer)
[Use $$R = 8.314$$ J mol$$^{-1}$$ K$$^{-1}$$]
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Consider titration of NaOH solution versus 1.25 M oxalic acid solution. At the end point following burette readings were obtained.
(i) 4.5 mL (ii) 4.5 mL (iii) 4.4 mL (iv) 4.4 mL (v) 4.4 mL
If the volume of oxalic acid taken was 10.0 mL then the molarity of the NaOH solution is ______ M. (Rounded-off to the nearest integer)
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Among the following, number of metal/s which can be used as electrodes in the photoelectric cell is ______. (Integer answer)
(A) Li (B) Na (C) Rb (D) Cs
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The unit cell of copper corresponds to a face centered cube of edge length 3.596 $$\mathring{A}$$ with one copper atom at each lattice point. The calculated density of copper in kg/m$$^3$$ is ______. [Molar mass of Cu: 63.54 g; Avogadro Number $$= 6.022 \times 10^{23}$$]
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If a compound AB dissociates to the extent of 75% in an aqueous solution, the molality of the solution which shows a 2.5 K rise in the boiling point of the solution is ______ molal. (Rounded-off to the nearest integer)
[$$K_b = 0.52$$ K kg mol$$^{-1}$$].
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Copper reduces $$NO_3^-$$ into NO and $$NO_2$$ depending upon the concentration of $$HNO_3$$ in solution. (Assuming fixed $$[Cu^{2+}]$$ and $$P_{NO} = P_{NO_2}$$), the $$HNO_3$$ concentration at which the thermodynamic tendency for reduction of $$NO_3^-$$ into NO and $$NO_2$$ by copper is same is $$10^x$$ M. The value of $$2x$$ is ______. (Rounded-off to the nearest integer)
[Given, $$E^\circ_{Cu^{2+}/Cu} = 0.34$$ V, $$E^\circ_{NO_3^-/NO} = 0.96$$ V, $$E^\circ_{NO_3^-/NO_2} = 0.79$$ V and at 298 K, $$\frac{RT}{F}(2.303) = 0.059$$]
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The rate constant of a reaction increases by five times on increase in temperature from 27°C to 52°C. The value of activation energy in kJ mol$$^{-1}$$ is ______. (Rounded-off to the nearest integer) [$$R = 8.314$$ J K$$^{-1}$$ mol$$^{-1}$$]
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The spin only magnetic moment of a divalent ion in aqueous solution (atomic number 29) is ______ BM.
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The number of compound/s given below which contain/s $$-COOH$$ group is ______. (Integer answer)
(A) Sulphanilic acid (B) Picric acid (C) Aspirin (D) Ascorbic acid
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Let $$\alpha$$ and $$\beta$$ be the roots of $$x^2 - 6x - 2 = 0$$. If $$a_n = \alpha^n - \beta^n$$ for $$n \geq 1$$, then the value of $$\dfrac{a_{10} - 2a_8}{3a_9}$$ is:
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If $$\alpha, \beta \in R$$ are such that $$1 - 2i$$ (here $$i^2 = -1$$) is a root of $$z^2 + \alpha z + \beta = 0$$, then $$(\alpha - \beta)$$ is equal to:
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The minimum value of $$f(x) = a^{a^x} + a^{1 - a^x}$$, where $$a, x \in R$$ and $$a > 0$$, is equal to:
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If $$0 < x, y < \pi$$ and $$\cos x + \cos y - \cos(x + y) = \frac{3}{2}$$, then $$\sin x + \cos y$$ is equal to:
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If the curve $$x^2 + 2y^2 = 2$$ intersects the line $$x + y = 1$$ at two points $$P$$ and $$Q$$, then the angle subtended by the line segment $$PQ$$ at the origin is
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A hyperbola passes through the foci of the ellipse $$\frac{x^2}{25} + \frac{y^2}{16} = 1$$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:
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The contrapositive of the statement "If you will work, you will earn money" is:
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If for the matrix, $$A = \begin{bmatrix} 1 & -\alpha \\ \alpha & \beta \end{bmatrix}$$, $$AA^T = I_2$$, then the value of $$\alpha^4 + \beta^4$$ is:
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Let $$A$$ be a $$3 \times 3$$ matrix with det$$(A) = 4$$. Let $$R_i$$ denote the $$i^{th}$$ row of $$A$$. If a matrix $$B$$ is obtained by performing the operation $$R_2 \to 2R_2 + 5R_3$$ on $$2A$$, then det$$(B)$$ is equal to:
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The following system of linear equations
$$2x + 3y + 2z = 9$$
$$3x + 2y + 2z = 9$$
$$x - y + 4z = 8$$
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$$\operatorname{cosec}\left[2\cot^{-1}(5) + \cos^{-1}\left(\frac{4}{5}\right)\right]$$ is equal to:
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A function $$f(x)$$ is given by $$f(x) = \frac{5^x}{5^x + 5}$$, then the sum of the series $$f\left(\frac{1}{20}\right) + f\left(\frac{2}{20}\right) + f\left(\frac{3}{20}\right) + \ldots + f\left(\frac{39}{20}\right)$$ is equal to:
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Let $$x$$ denote the total number of one-one functions from a set $$A$$ with 3 elements to a set $$B$$ with 5 elements and $$y$$ denote the total number of one-one functions from the set $$A$$ to the set $$A \times B$$. Then:
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The shortest distance between the line $$x - y = 1$$ and the curve $$x^2 = 2y$$ is:
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The integral $$\int \frac{e^{3\log_e 2x} + 5e^{2\log_e 2x}}{e^{4\log_e x} + 5e^{3\log_e x} - 7e^{2\log_e x}} dx$$, $$x > 0$$, is equal to (where $$c$$ is a constant of integration)
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If $$I_n = \int_{\pi/4}^{\pi/2} \cot^n x \, dx$$, then
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$$\lim_{n \to \infty} \left[\frac{1}{n} + \frac{n}{(n+1)^2} + \frac{n}{(n+2)^2} + \ldots + \frac{n}{(2n-1)^2}\right]$$ is equal to
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A plane passes through the points $$A(1, 2, 3)$$, $$B(2, 3, 1)$$ and $$C(2, 4, 2)$$. If $$O$$ is the origin and $$P$$ is $$(2, -1, 1)$$, then the projection of $$\vec{OP}$$ on this plane is of length:
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In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:
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Let $$A$$ be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of $$A$$ leaves remainder 2 when divided by 5 is:
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The total number of two digit numbers $$'n'$$, such that $$3^n + 7^n$$ is a multiple of 10, is ______.
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If the remainder when $$x$$ is divided by 4 is 3, then the remainder when $$(2020 + x)^{2022}$$ is divided by 8 is ______.
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A line is a common tangent to the circle $$(x - 3)^2 + y^2 = 9$$ and the parabola $$y^2 = 4x$$. If the two points of contact $$(a, b)$$ and $$(c, d)$$ are distinct and lie in the first quadrant, then $$2(a + c)$$ is equal to ______.
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If $$\lim_{x \to 0} \frac{ax - (e^{4x} - 1)}{ax(e^{4x} - 1)}$$ exists and is equal to $$b$$, then the value of $$a - 2b$$ is ______.
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A function $$f$$ is defined on $$[-3, 3]$$ as
$$f(x) = \begin{cases} \min\{|x|, 2 - x^2\}, & -2 \leq x \leq 2 \\ [|x|], & 2 < |x| \leq 3 \end{cases}$$
where $$[x]$$ denotes the greatest integer $$\leq x$$. The number of points, where $$f$$ is not differentiable in $$(-3, 3)$$ is ______.
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If the curves $$x = y^4$$ and $$xy = k$$ cut at right angles, then $$(4k)^6$$ is equal to ______.
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The value of $$\int_{-2}^{2} |3x^2 - 3x - 6| dx$$ is ______
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If the curve, $$y = y(x)$$ represented by the solution of the differential equation $$(2xy^2 - y)dx + x \, dy = 0$$, passes through the intersection of the lines, $$2x - 3y = 1$$ and $$3x + 2y = 8$$, then $$|y(1)|$$ is equal to ______.
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Let $$\vec{a} = \hat{i} + \alpha\hat{j} + 3\hat{k}$$ and $$\vec{b} = 3\hat{i} - \alpha\hat{j} + \hat{k}$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\vec{a}$$ and $$\vec{b}$$ is $$8\sqrt{3}$$ square units, then $$\vec{a} \cdot \vec{b}$$ is equal to ______.
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A line $$l$$ passing through origin is perpendicular to the lines
$$l_1: \vec{r} = (3 + t)\hat{i} + (-1 + 2t)\hat{j} + (4 + 2t)\hat{k}$$
$$l_2: \vec{r} = (3 + 2s)\hat{i} + (3 + 2s)\hat{j} + (2 + s)\hat{k}$$
If the co-ordinates of the point in the first octant on $$l_2$$ at a distance of $$\sqrt{17}$$ from the point of intersection of $$l$$ and $$l_1$$ are $$(a, b, c)$$, then $$18(a + b + c)$$ is equal to ______.
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