The velocity-displacement graph describing the motion of a bicycle is shown in the figure.
The acceleration-displacement graph of the bicycle's motion is best described by:
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The velocity-displacement graph describing the motion of a bicycle is shown in the figure.
The acceleration-displacement graph of the bicycle's motion is best described by:
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A block of 200 g mass moves with a uniform speed in a horizontal circular groove, with vertical side walls of radius 20 cm. If the block takes 40 s to complete one round, the normal force by the side walls of the groove is:
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A block of mass $$m$$ slides along a floor while a force of magnitude $$F$$ is applied to it at an angle $$\theta$$ as shown in figure. The coefficient of kinetic friction is $$\mu_K$$. Then, the block's acceleration $$a$$ is given by: ($$g$$ is acceleration due to gravity)
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The maximum and minimum distances of a comet from the Sun are $$1.6 \times 10^{12}$$ m and $$8.0 \times 10^{10}$$ m respectively. If the speed of the comet at the nearest point is $$6 \times 10^{4}$$ m s$$^{-1}$$, the speed at the farthest point is:
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Four equal masses, $$m$$ each are placed at the corners of a square of length $$l$$ as shown in the figure. The moment of inertia of the system about an axis passing through $$A$$ and parallel to $$DB$$ would be:
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The pressure acting on a submarine is $$3 \times 10^5$$ Pa at a certain depth. If the depth is doubled, the percentage increase in the pressure acting on the submarine would be: (Assume that atmospheric pressure is $$1 \times 10^5$$ Pa, density of water is $$10^3$$ kg m$$^{-3}$$, g = 10 ms$$^{-2}$$)
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In thermodynamics, heat and work are:
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The volume $$V$$ of an enclosure contains a mixture of three gases, 16 g of oxygen, 28 g of nitrogen and 44 g of carbon dioxide at absolute temperature $$T$$. Consider $$R$$ as universal gas constant. The pressure of the mixture of gases is:
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Time period of a simple pendulum is $$T$$ inside a lift when the lift is stationary. If the lift moves upwards with an acceleration $$\frac{g}{2}$$, the time period of pendulum will be:
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A 25 m long antenna is mounted on an antenna tower. The height of the antenna tower is 75 m. The wavelength (in meter) of the signal transmitted by this antenna would be:
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For changing the capacitance of a given parallel plate capacitor, a dielectric material of dielectric constant $$K$$ is used, which has the same area as the plates of the capacitor. The thickness of the dielectric slab is $$\frac{3}{4}d$$, where $$d$$ is the separation between the plates of parallel plate capacitor. The new capacitance $$C'$$ in terms of original capacitance $$C_0$$ is given by the following relation:
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A conducting wire of length $$l$$, area of cross-section $$A$$ and electric resistivity $$\rho$$ is connected between the terminals of a battery. A potential difference $$V$$ is developed between its ends, causing an electric current. If the length of the wire of the same material is doubled and the area of cross-section is halved, the resultant current would be:
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A bar magnet of length 14 cm is placed in the magnetic meridian with its north pole pointing towards the geographic north pole. A neutral point is obtained at a distance of 18 cm from the center of the magnet. If $$B_H = 0.4$$ G, the magnetic moment of the magnet is (1 G = $$10^{-4}$$ T):
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A conducting bar of length $$L$$ is free to slide on two parallel conducting rails as shown in the figure.
Two resistors $$R_1$$ and $$R_2$$ are connected across the ends of the rails. There is a uniform magnetic field $$\vec{B}$$ pointing into the page. An external agent pulls the bar to the left at a constant speed $$v$$. The correct statement about the directions of induced currents $$I_1$$ and $$I_2$$ flowing through $$R_1$$ and $$R_2$$ respectively is:
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An $$RC$$ circuit as shown in the figure is driven by a $$AC$$ source generating a square wave. The output wave pattern monitored by $$CRO$$ would look close to:
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A plane electromagnetic wave of frequency 500 MHz is traveling in a vacuum along the $$y$$-direction. At a particular point in space and time, $$\vec{B} = 8.0 \times 10^{-8}\hat{z}$$ T. The value of the electric field at this point is: (speed of light = $$3 \times 10^{8}$$ ms$$^{-1}$$; $$\hat{x}, \hat{y}, \hat{z}$$ are unit vectors along $$x, y$$ and $$z$$ direction.)
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For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric ($$U_e$$) and magnetic ($$U_m$$) fields is:
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The angle of deviation through a prism is minimum when
(A) Incident ray and emergent ray are symmetric to the prism
(B) The refracted ray inside the prism becomes parallel to its base
(C) Angle of incidence is equal to that of the angle of emergence
(D) When angle of emergence is double the angle of incidence
Choose the correct answer from the options given below:
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The stopping potential in the context of photoelectric effect depends on the following property of incident electromagnetic radiation:
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One main scale division of a vernier callipers is $$a$$ cm and $$n^{th}$$ division of the vernier scale coincide with $$(n-1)^{th}$$ division of the main scale. The least count of the callipers in mm is:
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The resistance $$R = \frac{V}{I}$$, where $$V = (50 \pm 2)$$ V and $$I = (20 \pm 0.2)$$ A. The percentage error in $$R$$ is $$x$$%. The value of $$x$$ to the nearest integer is ________.
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Consider a frame that is made up of two thin massless rods $$AB$$ and $$AC$$ as shown in the figure. A vertical force $$\vec{P}$$ of magnitude 100 N is applied at point $$A$$ of the frame.
Suppose the force $$\vec{P}$$ is resolved parallel to the arms $$AB$$ and $$AC$$ of the frame. The magnitude of the resolved component along the arm $$AC$$ is $$x$$ N. The value of $$x$$, to the nearest integer, is ________.
[Given: $$\sin(35^\circ) = 0.573, \cos(35^\circ) = 0.819, \sin(110^\circ) = 0.939, \cos(110^\circ) = -0.342$$]
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Consider a 20 kg uniform circular disk of radius 0.2 m. It is pin supported at its center and is at rest initially. The disk is acted upon by a constant force $$F = 20$$ N through a massless string wrapped around its periphery as shown in the figure.
Suppose the disk makes $$n$$ number of revolutions to attain an angular speed of 50 rad s$$^{-1}$$. The value of $$n$$, to the nearest integer, is ________.
[Given: In one complete revolution, the disk rotates by 6.28 rad]
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A ball of mass 10 kg moving with a velocity $$10\sqrt{3}$$ m s$$^{-1}$$ along $$X$$-axis, hits another ball of mass 20 kg which is at rest. After the collision, the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along $$Y$$-axis at a speed of 10 m s$$^{-1}$$. The second piece starts moving at a speed of 20 m s$$^{-1}$$ at an angle $$\theta$$ (degree) with respect to the $$X$$-axis. The configuration of pieces after the collision is shown in the figure. The value of $$\theta$$ to the nearest integer is ________.
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In the figure given, the electric current flowing through the 5 k$$\Omega$$ resistor is $$x$$ mA.
The value of $$x$$ to the nearest integer is ________.
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A sinusoidal voltage of peak value 250 V is applied to a series LCR circuit, in which $$R = 8$$ $$\Omega$$, $$L = 24$$ mH and $$C = 60$$ $$\mu$$F. The value of power dissipated at resonant condition is $$x$$ kW. The value of $$x$$ to the nearest integer is ________.
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A fringe width of 6 mm was produced for two slits separated by 1 mm apart. The screen is placed 10 m away. The wavelength of light used is $$x$$ nm. The value of $$x$$ to the nearest integer is ________.
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The first three spectral lines of H-atom in the Balmer series are given $$\lambda_1, \lambda_2, \lambda_3$$ considering the Bohr atomic model, the wave lengths of first and third spectral lines $$\left(\frac{\lambda_1}{\lambda_3}\right)$$ are related by a factor of approximately $$x \times 10^{-1}$$. The value of $$x$$, to the nearest integer, is ________.
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The value of power dissipated across the zener diode ($$V_z = 15$$ V) connected in the circuit as shown in the figure is $$x \times 10^{-1}$$ W.
The value of $$x$$ to the nearest integer is ________.
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In the logic circuit shown in the figure, if input $$A$$ and $$B$$ are 0 to 1 respectively, the output at $$Y$$ would be $$x$$. The value of $$x$$ is ________.
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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: The H-O-H bond angle in water molecule is 104.5°.
Reason R: The lone pair - lone pair repulsion of electrons is higher than the bond pair - bond pair repulsion.
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Given below are two statements:
Statement I: $$H_2O_2$$ can act as both oxidising and reducing agent in basic medium.
Statement II: In the hydrogen economy, the energy is transmitted in the form of dihydrogen. In the light of the above statements, choose the correct answer from the options given below:
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A group 15 element, which is a metal and forms a hydride with strongest reducing power among group 15 hydrides. The element is:
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Given below are two statements:
Statement I: Both $$CaCl_2 \cdot 6H_2O$$ and $$MgCl_2 \cdot 8H_2O$$ undergo dehydration on heating.
Statement II: BeO is amphoteric, whereas the oxides of other elements in the same group are acidic.
In the light of the above statements, choose the correct answer from the options given below:
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Among the following, the aromatic compounds are:
Choose the correct answer from the following options:
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Assertion A: Enol form of acetone [$$CH_3COCH_3$$] exists in < 0.1% quantity. However, the enol form of acetyl acetone [$$CH_3COCH_2OCCH_3$$] exists in approximately 15% quantity.
Reason R: enol form of acetyl acetone is stabilized by intramolecular hydrogen bonding, which is not possible in enol form of acetone.
Choose the correct statement:
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In chromatography technique, the purification of a compound is independent of:
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The product "A" and "B" formed in below reactions are:
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The type of pollution that gets increased during the daytime and in the presence of $$O_3$$ is:
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Given below are two statements:
Statement I: The E° value of $$Ce^{4+}/Ce^{3+}$$ is +1.74 V.
Statement II: Ce is more stable in $$Ce^{4+}$$ state than $$Ce^{3+}$$ state.
In the light of the above statements, choose the most appropriate answer from the options given below:
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The process that involves the removal of sulphur from the ores is:
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Match List-I with List-II:
Choose the correct answer from the options given below:
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Match List-I with List-II:
Choose the correct 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: Size of $$Bk^{3+}$$ ion is less than $$Np^{3+}$$ ion.
Reason R: The above is a consequence of the lanthanide contraction.
In the light of the above statements, choose the correct answer from the options given below:
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Which of the following is Lindlar catalyst?
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The product "P" in the above reaction is:
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In the above chemical reaction, intermediate "X" and reagent/condition "A" are:
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Which of the following reaction DOES NOT involve Hoffmann Bromamide degradation?
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The functions of antihistamine are:
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Which among the following pairs of Vitamins is stored in our body relatively for longer duration?
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Complete combustion of 750 g of an organic compound provides 420 g of $$CO_2$$ and 210 g of $$H_2O$$. The percentage composition of carbon and hydrogen in organic compound is 15.3 and ________ respectively. (Round off to the Nearest Integer)
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A 6.50 molal solution of KOH (aq.) has a density of 1.89 g cm$$^{-3}$$. The molarity of the solution is ________ mol dm$$^{-3}$$. (Round off to the Nearest Integer).
[Atomic masses: K: 39.0 u; O: 16.0 u; H: 1.0 u]
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When light of wavelength 248 nm falls on a metal of threshold energy 3.0 eV, the de-Broglie wavelength of emitted electrons is ________ $$\mathring{A}$$ (angstrom). (Round off to the Nearest Integer).
[Use: $$\sqrt{3} = 1.73$$, $$h = 6.63 \times 10^{-34}$$ Js; $$m_e = 9.1 \times 10^{-31}$$ kg; $$c = 3.0 \times 10^{8}$$ ms$$^{-1}$$; $$1$$ eV $$= 1.6 \times 10^{-19}$$ J]
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For the reaction $$A(g) \rightleftharpoons B(g)$$ at 495 K, $$\Delta_rG^\circ = -9.478$$ kJ mol$$^{-1}$$. If we start the reaction in a closed container at 495 K with 22 millimoles of A, the amount of B is the equilibrium mixture is ________ millimoles. (Round off to the Nearest Integer).
[$$R = 8.314$$ J mol$$^{-1}$$ K$$^{-1}$$; $$\ln 10 = 2.303$$]
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Two salts $$A_2X$$ and MX have the same value of solubility product of $$4.0 \times 10^{-12}$$. The ratio of their molar solubilities i.e. $$\frac{S(A_2X)}{S(MX)}$$ = ________. (Round off to the Nearest Integer).
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$$2MnO_4^- + bC_2O_4^{2-} + cH^+ \rightarrow xMn^{2+} + yCO_2 + zH_2O$$
If the above equation is balanced with integer coefficients, the value of c is ________. (Round off to the Nearest Integer).
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A certain element crystallises in a bcc lattice of unit cell edge length 27 $$\mathring{A}$$. If the same element under the same conditions crystallises in the fcc lattice, the edge length of the unit cell in $$\mathring{A}$$ will be ________. (Round off to the Nearest Integer).
[Assume each lattice point has a single atom]
[Assume $$\sqrt{3} = 1.73$$, $$\sqrt{2} = 1.41$$]
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$$AB_2$$ is 10% dissociated in water to $$A^{2+}$$ and $$B^-$$. The boiling point of 10.0 molal aqueous solution of $$AB_2$$ is ________ °C. (Round off to the Nearest Integer).
[Given: Molal elevation constant of water $$K_b = 0.5$$ K kg mol$$^{-1}$$, boiling point of pure water = 100°C]
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The decomposition of formic acid on gold surface follows first order kinetics. If the rate constant at 300 K is $$1.0 \times 10^{-3}$$ s$$^{-1}$$ and the activation energy $$E_a = 11.488$$ kJ mol$$^{-1}$$, the rate constant at 200 K is ________ $$\times 10^{-5}$$ s$$^{-1}$$. (Round off to the Nearest Integer).
[Given $$R = 8.314$$ J mol$$^{-1}$$ K$$^{-1}$$]
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The equivalents of ethylene diamine required to replace the neutral ligands from the coordination sphere of the trans-complex of $$CoCl_3 \cdot 4NH_3$$ is ________. (Round off to the Nearest Integer).
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If for $$x \in \left(0, \frac{\pi}{2}\right)$$, $$\log_{10} \sin x + \log_{10} \cos x = -1$$ and $$\log_{10}(\sin x + \cos x) = \frac{1}{2}(\log_{10} n - 1)$$, $$n > 0$$, then the value of $$n$$ is equal to:
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Let a complex number $$z$$, $$|z| \neq 1$$, satisfy $$\log_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^2}\right) \leq 2$$. Then, the largest value of $$|z|$$ is equal to:
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If $$n$$ is the number of irrational terms in the expansion of $$\left(3^{1/4} + 5^{1/8}\right)^{60}$$, then $$(n-1)$$ is divisible by:
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Let $$[x]$$ denote greatest integer less than or equal to $$x$$. If for $$n \in N$$, $$\left(1 - x + x^3\right)^n = \sum_{j=0}^{3n} a_j x^j$$, then $$\sum_{j=0}^{\left[\frac{3n}{2}\right]} a_{2j} + 4\sum_{j=0}^{\left[\frac{3n-1}{2}\right]} a_{2j+1}$$ is equal to:
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The number of roots of the equation, $$(81)^{\sin^2 x} + (81)^{\cos^2 x} = 30$$ in the interval $$[0, \pi]$$ is equal to:
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If the three normals drawn to the parabola, $$y^2 = 2x$$ pass through the point $$(a, 0)$$, $$a \neq 0$$, then $$a$$ must be greater than:
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The locus of the midpoints of the chord of the circle, $$x^2 + y^2 = 25$$ which is tangent to the hyperbola, $$\frac{x^2}{9} - \frac{y^2}{16} = 1$$ is:
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Which of the following Boolean expression is a tautology?
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Consider three observations $$a, b$$ and $$c$$ such that $$b = a + c$$. If the standard deviation of $$a+2, b+2, c+2$$ is $$d$$, then which of the following is true?
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Let $$A = \begin{bmatrix} i & -i \\ -i & i \end{bmatrix}$$, $$i = \sqrt{-1}$$. Then, the system of linear equations $$A^8 \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 64 \end{bmatrix}$$ has:
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Let $$S_k = \sum_{r=1}^{k} \tan^{-1}\left(\frac{6^r}{2^{2r+1} + 3^{2r+1}}\right)$$, then $$\lim_{k \to \infty} S_k$$ is equal to:
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The number of elements in the set $$\{x \in R : (|x| - 3)|x + 4| = 6\}$$ is equal to:
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Let the functions $$f : R \to R$$ and $$g : R \to R$$ be defined as:
$$f(x) = \begin{cases} x+2, & x < 0 \\ x^2, & x \geq 0 \end{cases}$$ and $$g(x) = \begin{cases} x^3, & x < 1 \\ 3x-2, & x \geq 1 \end{cases}$$
Then, the number of points in $$R$$ where $$(f \circ g)(x)$$ is NOT differentiable is equal to:
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The range of $$a \in R$$ for which the function $$f(x) = (4a-3)(x + \log_e 5) + 2(a-7)\cot\left(\frac{x}{2}\right)\sin^2\left(\frac{x}{2}\right)$$, $$x \neq 2n\pi$$, $$n \in N$$, has critical points, is:
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If $$y = y(x)$$ is the solution of the differential equation, $$\frac{dy}{dx} + 2y\tan x = \sin x$$, $$y\left(\frac{\pi}{3}\right) = 0$$, then the maximum value of the function $$y(x)$$ over $$R$$ is equal to:
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Let a vector $$\alpha\hat{i} + \beta\hat{j}$$ be obtained by rotating the vector $$\sqrt{3}\hat{i} + \hat{j}$$ by an angle 45° about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices $$(\alpha, \beta)$$, $$(0, \beta)$$ and $$(0, 0)$$ is equal to:
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If for $$a > 0$$, the feet of perpendiculars from the points $$A(a, -2a, 3)$$ and $$B(0, 4, 5)$$ on the plane $$lx + my + nz = 0$$ are points $$C(0, -a, -1)$$ and $$D$$ respectively, then the length of line segment $$CD$$ is equal to:
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Let the position vectors of two points $$P$$ and $$Q$$ be $$3\hat{i} - \hat{j} + 2\hat{k}$$ and $$\hat{i} + 2\hat{j} - 4\hat{k}$$, respectively. Let $$R$$ and $$S$$ be two points such that the direction ratios of lines $$PR$$ and $$QS$$ are $$(4, -1, 2)$$ and $$(-2, 1, -2)$$, respectively. Let lines $$PR$$ and $$QS$$ intersect at $$T$$. If the vector $$\vec{TA}$$ is perpendicular to both $$\vec{PR}$$ and $$\vec{QS}$$ and the length of vector $$\vec{TA}$$ is $$\sqrt{5}$$ units, then the modulus of a position vector of $$A$$ is:
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Let $$P$$ be a plane $$lx + my + nz = 0$$ containing the line, $$\frac{1-x}{1} = \frac{y+4}{2} = \frac{z+2}{3}$$. If plane $$P$$ divides the line segment $$AB$$ joining points $$A(-3, -6, 1)$$ and $$B(2, 4, -3)$$ in ratio $$k:1$$ then the value of $$k$$ is equal to:
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A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is:
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Let $$z$$ and $$w$$ be two complex numbers such that $$w = z\bar{z} - 2z + 2$$, $$\left|\frac{z+i}{z-3i}\right| = 1$$ and $$\text{Re}(w)$$ has minimum value. Then, the minimum value of $$n \in N$$ for which $$w^n$$ is real, is equal to ________.
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Consider an arithmetic series and a geometric series having four initial terms from the set $$\{11, 8, 21, 16, 26, 32, 4\}$$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to ________.
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Let $$ABCD$$ be a square of side of unit length. Let a circle $$C_1$$ centered at $$A$$ with unit radius is drawn. Another circle $$C_2$$ which touches $$C_1$$ and the lines $$AD$$ and $$AB$$ are tangent to it, is also drawn. Let a tangent line from the point $$C$$ to the circle $$C_2$$ meet the side $$AB$$ at $$E$$. If the length of $$EB$$ is $$\alpha + \sqrt{3}\beta$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ is equal to ________.
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If $$\lim_{x \to 0} \frac{ae^x - b\cos x + ce^{-x}}{x \sin x} = 2$$, then $$a + b + c$$ is equal to ________.
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Let $$P = \begin{bmatrix} -30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{bmatrix}$$ and $$A = \begin{bmatrix} 2 & 7 & \omega^2 \\ -1 & -\omega & 1 \\ 0 & -\omega & -\omega+1 \end{bmatrix}$$ where $$\omega = \frac{-1+i\sqrt{3}}{2}$$, and $$I_3$$ be the identity matrix of order 3. If the determinant of the matrix $$\left(P^{-1}AP - I_3\right)^2$$ is $$\alpha\omega^2$$, then the value of $$\alpha$$ is equal to ________.
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The total number of $$3 \times 3$$ matrices $$A$$ having entries from the set $$\{0, 1, 2, 3\}$$ such that the sum of all the diagonal entries of $$AA^T$$ is 9, is equal to ________.
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Let $$f : (0, 2) \to R$$ be defined as $$f(x) = \log_2\left(1 + \tan\left(\frac{\pi x}{4}\right)\right)$$. Then, $$\lim_{n \to \infty} \frac{2}{n}\left(f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + \ldots + f(1)\right)$$ is equal to ________.
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If the normal to the curve $$y(x) = \int_0^x (2t^2 - 15t + 10)\,dt$$ at a point $$(a, b)$$ is parallel to the line $$x + 3y = -5, a > 1$$, then the value of $$|a + 6b|$$ is equal to ________.
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Let $$f : R \to R$$ be a continuous function such that $$f(x) + f(x+1) = 2$$ for all $$x \in R$$. If $$I_1 = \int_0^8 f(x)\,dx$$ and $$I_2 = \int_{-1}^3 f(x)\,dx$$, then the value of $$I_1 + 2I_2$$ is equal to ________.
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Let the curve $$y = y(x)$$ be the solution of the differential equation, $$\frac{dy}{dx} = 2(x+1)$$. If the numerical value of area bounded by the curve $$y = y(x)$$ and $$x$$-axis is $$\frac{4\sqrt{8}}{3}$$, then the value of $$y(1)$$ is equal to ________.
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