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The following observations were taken for determining surface tension $$T$$ of water by capillary method:
diameter of capillary, $$D = 1.25 \times 10^{-2}$$ m
rise of water, $$h = 1.45 \times 10^{-2}$$ m
Using $$g = 9.80$$ m s$$^{-2}$$ and the simplified relation $$T = \frac{rhg}{2} \times 10^{3}$$ N m$$^{-1}$$, the possible error in surface tension is closest to:
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A body is thrown vertically upwards. Which one of the following graphs correctly represents the velocity $$v$$ vs time $$t$$?
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A time dependent force $$F = 6t$$ acts on a particle of mass 1 kg. If the particle starts from the rest, the work done by the force during the first 1 sec will be:
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A body of mass $$m = 10^{-2}$$ kg is moving in a medium and experiences a frictional force $$F = -kv^{2}$$. Its initial speed is $$v_{0} = 10$$ m s$$^{-1}$$. After 10 s its kinetic energy is $$\frac{1}{8}mv_{0}^{2}$$, then the value of $$k$$ will be:
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The moment of inertia of a uniform cylinder of length $$l$$ and radius $$R$$ about its perpendicular bisector is $$I$$. What is the ratio $$l/R$$ such that the moment of inertia is minimum?
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A slender uniform rod of mass $$M$$ and length $$l$$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $$\theta$$ with the vertical is:
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The variation of acceleration due to gravity $$g$$ with distance $$d$$ from the centre of the earth is best represented by ($$R$$ = Earth's radius):
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A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of:
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A copper ball of mass 100 g is at a temperature $$T$$. It is dropped in a copper calorimeter of mass 100 g, filled with 170 g of water at room temperature. Subsequently, the temperature of the system is found to be 75°C. $$T$$ is given by:
(Given: room temperature = 30°C, specific heat of copper = 0.1 cal g$$^{-1}$$ °C$$^{-1}$$)
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An external pressure $$P$$ is applied on a cube at 0°C so that it is equally compressed from all sides. $$K$$ is the bulk modulus of the material of the cube and $$\alpha$$ is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating. The temperature should be raised by:
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$$C_{p} - C_{v} = \frac{R}{M}$$ and $$C_{v}$$ are specific heats at constant pressure and constant volume respectively. It is observed that, $$C_{p} - C_{v} = a$$ for hydrogen gas and $$C_{p} - C_{v} = b$$ for nitrogen gas. The correct relation between $$a$$ and $$b$$ is:
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The temperature of an open room of volume 30 m$$^{3}$$ increases from 17°C to 27°C due to the sunshine. The atmospheric pressure in the room remains $$1 \times 10^{5}$$ Pa. If $$n_{i}$$ and $$n_{f}$$ are the number of molecules in the room before and after heating, then $$n_{f} - n_{i}$$ will be:
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A particle is executing simple harmonic motion with a time period $$T$$. At time $$t = 0$$, it is at its position of equilibrium. The kinetic energy - time graph of the particle will look like:
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An observer is moving with half the speed of light towards a stationary microwave source emitting waves at frequency 10 GHz. What is the frequency of the microwave measured by the observer? (speed of light = $$3 \times 10^{8}$$ m s$$^{-1}$$)
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An electric dipole has fixed dipole moment $$\vec{p}$$, which makes angle $$\theta$$ with respect to $$x$$-axis. When subjected to an electric field $$\vec{E}_{1} = E\hat{i}$$, it experiences a torque $$\vec{T}_{1} = \tau\hat{k}$$. When subjected to another electric field $$\vec{E}_{2} = \sqrt{3}E_{1}\hat{j}$$, it experiences a torque $$\vec{T}_{2} = -\vec{T}_{1}$$. The angle $$\theta$$ is:
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A capacitance of 2 μF is required in an electrical circuit across a potential difference of 1.0 kV. A large number of 1 μF capacitors are available which can withstand a potential difference of not more than 300 V. The minimum number of capacitors required to achieve this is:
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In the given circuit diagram, when the current reaches a steady-state in the circuit, the charge on the capacitor of capacitance $$C$$ will be:
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In the above circuit the current in each resistance is:
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Which of the following statements is false?
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When a current of 5 mA is passed through a galvanometer having a coil of resistance 15 Ω, it shows full-scale deflection. The value of the resistance to be put in series with the galvanometer to convert it into a voltmeter of range 0 - 10 V is:
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A magnetic needle of magnetic moment $$6.7 \times 10^{-2}$$ A m$$^{2}$$ and moment of inertia $$7.5 \times 10^{-6}$$ kg m$$^{2}$$ is performing simple harmonic oscillations in a magnetic field of 0.01 T. Time taken for 10 complete oscillations is:
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In a coil of resistance 100 Ω, a current is induced by changing the magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is:
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An electron beam is accelerated by a potential difference $$V$$ to hit a metallic target to produce X-rays. It produces continuous as well as characteristic X-rays. If $$\lambda_{min}$$ is the smallest possible wavelength of X-ray in the spectrum, the variation of $$\log\lambda_{min}$$ with $$\log V$$ is correctly represented in:
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A diverging lens with magnitude of focal length 25 cm is placed at a distance of 15 cm from a converging lens of magnitude of focal length 20 cm. A beam of parallel light falls on the diverging lens. The final image formed is:
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In a Young's double slit experiment, slits are separated by 0.5 mm, and the screen is placed 150 cm away. A beam of light consisting of two wavelengths, 650 nm and 520 nm, is used to obtain interference fringes on the screen. The least distance from the common central maximum to the point where the bright fringes due to both the wavelengths coincide is:
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A particle $$A$$ of mass $$m$$ and initial velocity $$v$$ collides with a particle $$B$$ of mass $$\frac{m}{2}$$ which is at rest. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths $$\lambda_{A}$$ to $$\lambda_{B}$$ after the collision is:
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Some energy levels of a molecule are shown in the figure. The ratio of the wavelengths $$r = \frac{\lambda_{1}}{\lambda_{2}}$$, is given by:
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A radioactive nucleus $$A$$ with a half-life $$T$$, decays into a nucleus $$B$$. At $$t = 0$$, there is no nucleus $$B$$. At some time $$t$$, the ratio of the number of $$B$$ to that of $$A$$ is 0.3. Then, $$t$$ is given by: (Consider $$\log_{e} x = \log x$$)
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In a common emitter amplifier circuit using an $$n$$-$$p$$-$$n$$ transistor, the phase difference between the input and the output voltages will be:
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In amplitude modulation, the sinusoidal carrier frequency used is denoted by $$\omega_{c}$$ and the signal frequency is denoted by $$\omega_{m}$$. The bandwidth $$\Delta\omega_{m}$$ of the signal is such that $$\Delta\omega_{m} \ll \omega_{c}$$. Which of the following frequencies is not contained in the modulated wave?
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1 gram of a carbonate M$$_{2}$$CO$$_{3}$$ on treatment with excess HCl produces 0.01186 moles of CO$$_{2}$$. The molar mass of M$$_{2}$$CO$$_{3}$$ in g mol$$^{-1}$$ is:
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The most abundant elements by mass in the body of a healthy human adult are: Oxygen (61.4%); Carbon (22.9%), Hydrogen (10.0%); and Nitrogen (2.6%). The weight which a 75 kg person would gain if all $$^{1}$$H atoms are replaced by $$^{2}$$H atoms is:
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Among the following species which option contains all isoelectronic species?
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The radius of the second Bohr orbit for hydrogen atom is
(Planck's constant, (h) = $$6.6262 \times 10^{-34}$$ Js; mass of electron = $$9.1091 \times 10^{-31}$$ kg; charge of electron = $$1.60210 \times 10^{-19}$$ C; permittivity of vacuum, ($$\epsilon_{0}$$) = $$8.854185 \times 10^{-12}$$ kg$$^{-1}$$ m$$^{-3}$$ A$$^{2}$$)
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Which of the following species is not paramagnetic?
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Given:
C(graphite) + O$$_{2}$$(g) → CO$$_{2}$$(g); $$\Delta_{r}$$H° = -393.5 kJ mol$$^{-1}$$
H$$_{2}$$(g) + $$\frac{1}{2}$$O$$_{2}$$(g) → H$$_{2}$$O(l); $$\Delta_{r}$$H° = -285.8 kJ mol$$^{-1}$$
CO$$_{2}$$(g) + 2H$$_{2}$$O(l) → CH$$_{4}$$(g) + 2O$$_{2}$$(g); $$\Delta_{r}$$H° = +890.3 kJ mol$$^{-1}$$
Based on the above thermochemical equations, the value of $$\Delta_{r}$$H° at 298 K for the reaction
C(graphite) + 2H$$_{2}$$(g) → CH$$_{4}$$(g) will be:
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$$\Delta U$$ is equal to:
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pK$$_{a}$$ of a weak acid HA and pK$$_{b}$$ of a weak base BOH are 3.2 and 3.4 respectively. The pH of their salt AB solution at 25°C is:
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Both lithium and magnesium display several similar properties due to the diagonal relationship; however, the one which is incorrect is:
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Which of the following molecules is least resonance stabilized?
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Which of the following compounds will form significant amount of meta product during mono-nitration reaction?
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A water sample has ppm level concentration of following anions
F$$^{-}$$ = 10; SO$$_{4}^{2-}$$ = 100; NO$$_{3}^{-}$$ = 50
The anion/anions that make/makes the water sample unsuitable for drinking is/are
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A metal crystallizes in a face centred cubic structure. If the edge length of its unit cell is 'a', the closest approach between two atoms in the metallic crystal will be
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The freezing point of benzene decreases by 0.45°C on adding 0.2 g of acetic acid to 20 g of benzene. If acetic acid associates to form a dimer in benzene, then what is the percentage association of acetic acid in benzene?
K$$_{f}$$ for benzene = 5.12 K kg mol$$^{-1}$$
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Given
E°$$_{Cl_{2}/Cl^{-}}$$ = 1.36 V, E°$$_{Cr^{3+}/Cr}$$ = -0.74 V
E°$$_{Cr_{2}O_{7}^{2-}/Cr^{3+}}$$ = 1.33 V, E°$$_{MnO_{4}^{-}/Mn^{2+}}$$ = 1.51 V
Among the following, the strongest reducing agent is:
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Two reactions A$$_{1}$$ and A$$_{2}$$ have identical pre-exponential factors. The activation energy of A$$_{1}$$ is more than A$$_{2}$$ by 10 kJ mol$$^{-1}$$. If k$$_{1}$$ and k$$_{2}$$ are the rate constants for reactions A$$_{1}$$ and A$$_{2}$$, respectively at 300 K, then $$\ln\frac{k_{2}}{k_{1}}$$ is equal to
(R = 8.314 J mol$$^{-1}$$ K$$^{-1}$$)
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The Tyndall effect is observed only when the following conditions are satisfied,
i. The diameter of the dispersed particles is much smaller than the wavelength of the light used.
ii. The diameter of the dispersed particle is not much smaller than the wavelength of the light used.
iii. The refractive indices of the dispersed phase and dispersion medium are almost similar in magnitude.
iv. The refractive indices of the dispersed phase and dispersion medium differ greatly in magnitude.
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The products obtained when chlorine gas reacts with cold and dilute aqueous NaOH are
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Which of the following reactions is an example of a redox reaction?
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In the following reactions, ZnO is respectively acting as a/an,
(i) ZnO + Na$$_{2}$$O → Na$$_{2}$$ZnO$$_{2}$$
(ii) ZnO + CO$$_{2}$$ → ZnCO$$_{3}$$
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The sodium salt of an organic acid X produces effervescence with concentrated H$$_{2}$$SO$$_{4}$$. X reacts with the acidified aqueous CaCl$$_{2}$$ solution to give a white precipitate which decolourises acidic solution of KMnO$$_{4}$$. X is
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On treatment of 100 mL of 0.1 M solution of CoCl$$_{3}$$.6H$$_{2}$$O with excess AgNO$$_{3}$$; $$1.2 \times 10^{22}$$ ions are precipitated. The complex is:
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The major product obtained in the following reaction is,
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The increasing order of the reactivity of the following halides for the S$$_{N}$$1 reaction is
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3-Methyl-pent-2-ene on reaction with HBr in presence of peroxide forms an addition product. The number of possible stereoisomers for the product is
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Which of the following, upon treatment with tert-BuONa followed by addition of bromine water, fails to decolourize the colour of bromine?
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The correct sequence of reagents for the following conversion will be
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The major product obtained in the following reaction is
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The formation of which of the following polymers involves hydrolysis reaction?
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Which of the following compounds will behave as reducing sugar in an aqueous KOH solution?
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If, for a positive integer $$n$$, the quadratic equation,
$$x(x+1) + (x+1)(x+2) + \ldots + (x+\overline{n-1})(x+n) = 10n$$
has two consecutive integral solutions, then $$n$$ is equal to:
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Let $$\omega$$ be a complex number such that $$2\omega + 1 = z$$ where $$z = \sqrt{-3}$$. If
$$\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^{2}-1 & \omega^{2} \\ 1 & \omega^{2} & \omega^{7} \end{vmatrix} = 3k$$,
then $$k$$ can be equal to:
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A man $$X$$ has 7 friends, 4 of them are ladies and 3 are men. His wife $$Y$$ also has 7 friends, 3 of them are ladies and 4 are men. Assume $$X$$ and $$Y$$ have no common friends. Then the total number of ways in which $$X$$ and $$Y$$ together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of $$X$$ and $$Y$$ are in this party is:
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For any three positive real numbers $$a$$, $$b$$ and $$c$$. If $$9(25a^{2} + b^{2}) + 25(c^{2} - 3ac) = 15b(3a + c)$$. Then
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The value of $$(^{21}C_{1} - ^{10}C_{1}) + (^{21}C_{2} - ^{10}C_{2}) + \dots + (^{21}C_{10} - ^{10}C_{10})$$ is:
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If $$5\tan^{2}x - \cos^{2}x = 2\cos 2x + 9$$, then the value of $$\cos 4x$$ is:
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Let $$k$$ be an integer such that the triangle with vertices $$(k, -3k)$$, $$(5, k)$$ and $$(-k, 2)$$ has area 28 sq. units. Then the orthocenter of this triangle is at the point:
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The radius of a circle, having minimum area, which touches the curve $$y = 4 - x^{2}$$ and the lines, $$y = |x|$$ is:
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The eccentricity of an ellipse whose centre is at the origin is $$\frac{1}{2}$$. If one of its directrices is $$x = -4$$, then the equation of the normal to it at $$\left(1, \frac{3}{2}\right)$$ is:
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A hyperbola passes through the point $$P(\sqrt{2}, \sqrt{3})$$ and has foci at $$( \pm 2, 0)$$. Then the tangent to this hyperbola at $$P$$ also passes through the point
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$$\lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{(\pi - 2x)^{3}}$$ equals
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The statement $$p \to q \to (\sim p \to q \to q)$$ is
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A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:
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Let a vertical tower $$AB$$ have its end $$A$$ on the level ground. Let $$C$$ be the mid-point of $$AB$$ and $$P$$ be a point on the ground such that $$AP = 2AB$$. If $$\angle BPC = \beta$$, then $$\tan\beta$$ is equal to:
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If $$A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}$$, then Adj$$(3A^{2} + 12A)$$ is equal to:
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If $$S$$ is the set of distinct values of $$b$$ for which the following system of linear equations
$$x + y + z = 1$$
$$x + ay + z = 1$$
$$ax + by + z = 0$$
has no solution, then $$S$$ is:
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The function $$f : R \to \left(-\frac{1}{2}, \frac{1}{2}\right)$$ defined as $$f(x) = \frac{x}{1+x^{2}}$$, is:
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Let $$a, b, c \in R$$. If $$f(x) = ax^{2} + bx + c$$ is such that $$a + b + c = 3$$ and $$f(x + y) = f(x) + f(y) + xy$$, $$\forall$$ $$x, y \in R$$, then $$\sum_{n=1}^{10} f(n)$$ is equal to:
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If for $$x \in \left(0, \frac{1}{4}\right)$$, the derivative of $$\tan^{-1}\left(\frac{6x\sqrt{x}}{1-9x^{3}}\right)$$ is $$\sqrt{x} \cdot g(x)$$, then $$g(x)$$ equals:
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Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:
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The normal to the curve $$y(x-2)(x-3) = x + 6$$ at the point where the curve intersects the $$y$$-axis passes through the point:
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Let $$I_{n} = \int \tan^{n}x \, dx$$ ($$n > 1$$). If $$I_{4} + I_{6} = a\tan^{5}x + bx^{5} + c$$, then the ordered pair $$(a, b)$$, is equal to
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The integral $$\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}$$ is equal to
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The area (in sq. units) of the region $$\{(x, y) : x \geq 0, x + y \leq 3, x^{2} \leq 4y \text{ and } y \leq 1 + \sqrt{x}\}$$ is
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If $$(2 + \sin x)\frac{dy}{dx} + (y+1)\cos x = 0$$ and $$y(0) = 1$$, then $$y\left(\frac{\pi}{2}\right)$$ is equal to
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Given, $$\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j}$$. Let $$\vec{c}$$ be a vector such that $$|\vec{c} - \vec{a}| = 3$$, $$|\vec{a} \times \vec{b} \times \vec{c}| = 3$$ and the angle between $$\vec{c}$$ and $$\vec{a} \times \vec{b}$$ be 30°. Then $$\vec{a} \cdot \vec{c}$$ is equal to:
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If the image of the point $$P(1, -2, 3)$$ in the plane, $$2x + 3y - 4z + 22 = 0$$ measured parallel to the line, $$\frac{x}{1} = \frac{y}{4} = \frac{z}{5}$$ is $$Q$$, then $$PQ$$ is equal to:
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The distance of the point $$(1, 3, -7)$$ from the plane passing through the point $$(1, -1, -1)$$, having normal perpendicular to both the lines $$\frac{x-1}{1} = \frac{y+2}{-2} = \frac{z-4}{3}$$ and $$\frac{x-2}{2} = \frac{y+1}{-1} = \frac{z+7}{-1}$$, is:
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For three events, $$A$$, $$B$$ and $$C$$, $$P$$(Exactly one of $$A$$ or $$B$$ occurs) = $$P$$(Exactly one of $$B$$ or $$C$$ occurs) = $$P$$(Exactly one of $$C$$ or $$A$$ occurs) = $$\frac{1}{4}$$ and $$P$$(All the three events occur simultaneously) = $$\frac{1}{16}$$. Then the probability that at least one of the events occurs, is:
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If two different numbers are taken from the set $$\{0, 1, 2, 3, \ldots, 10\}$$; then the probability that their sum as well as absolute difference are both multiple of 4, is:
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