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A particle moving in the xy-plane experiences a velocity dependent force $$\vec{F} = k(v_y\hat{i} + v_x\hat{j})$$, where $$v_x$$ and $$v_y$$ are the x and y components of its velocity $$\vec{v}$$. If $$\vec{a}$$ is the acceleration of the particle, then which of the following statements is true for the particle?
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When a car is at rest, its driver sees rain drops falling on it vertically. When driving the car with speed $$v$$, he sees that rain drops coming at an angle $$60^\circ$$ from the horizontal. On further increasing the speed of the car to $$(1+\beta)v$$, this angle changes to $$45^\circ$$. The value of $$\beta$$ is close to:
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Particle A of mass $$m_1$$ moving with velocity $$(\sqrt{3}\hat{i} + \hat{j})\,\text{ms}^{-1}$$ collides with another particle B of mass $$m_2$$ which is at rest initially. Let $$\vec{v}_1$$ and $$\vec{v}_2$$ be the velocities of particles A and B after collision respectively. If $$m_1 = 2m_2$$ and after collision $$\vec{v}_1 - (\hat{i} + \sqrt{3}\hat{j})\,\text{ms}^{-1}$$, the angle between $$\vec{v}_1$$ and $$\vec{v}_2$$ is:
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The linear mass density of a thin rod AB of length L varies from A to B as $$\lambda(x) = \lambda_0\left(1 + \frac{x}{L}\right)$$, where $$x$$ is the distance from A. If M is the mass of the rod then its moment of inertia about an axis passing through A and perpendicular to the rod is:
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Two planets have masses M and 16M and their radii are $$a$$ and $$2a$$, respectively. The separation between the centres of the planets is $$10a$$. A body of mass $$m$$ is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of smaller planet, the minimum firing speed needed is:
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A fluid is flowing through a horizontal pipe of varying cross-section, with speed $$v\,\text{ms}^{-1}$$ at a point where the pressure is P Pascal. At another point where pressure is $$\frac{P}{2}$$ Pascal its speed is $$V\,\text{ms}^{-1}$$. If the density of the fluid is $$\rho\,\text{kg m}^{-3}$$ and the flow is streamline, then $$V$$ is equal to:
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Three rods of identical cross-section and length are made of three different materials of thermal conductivity $$K_1$$, $$K_2$$ and $$K_3$$, respectively. They are joined together at their ends to make a long rod (see figure). One end of the long rod is maintained at $$100^\circ\text{C}$$ and the other at $$0^\circ\text{C}$$ (see figure). If the joints of the rod are at $$70^\circ\text{C}$$ and $$20^\circ\text{C}$$ in steady state and there is no loss of energy from the surface of the rod, the correct relationship between $$K_1$$, $$K_2$$ and $$K_3$$ is:
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In a dilute gas at pressure P and temperature 't', the time between successive collision of a molecule varies with T as:
Assuming the nitrogen molecule is moving with r.m.s. velocity at 400 K, the de-Broglie wave length of nitrogen molecule is close to: (Given: nitrogen molecule weight: $$4.64 \times 10^{-26}\,\text{kg}$$, Boltzman constant: $$1.38 \times 10^{-23}\,\text{J K}^{-1}$$, Planck constant: $$6.63 \times 10^{-34}\,\text{J s}$$)
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When a particle of mass $$m$$ is attached to a vertical spring of spring constant $$k$$ and released, its motion is described by $$y(t) = y_0\sin^2\omega t$$, where 'y' is measured from the lower end of unstretched spring. Then $$\omega$$ is:
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Two identical electric point dipoles have dipole moments $$\vec{p}_1 = p\hat{i}$$ and $$\vec{p}_2 = -p\hat{i}$$ and are held on the x-axis at distance 'a' from each other. When released, they move along the x-axis with the direction of their dipole moments remaining unchanged. If the mass of each dipole is 'm', their speed when they are infinitely far apart is:
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Consider the force F on a charge 'q' due to a uniformly charged spherical shell of radius R carrying charge Q distributed uniformly over it. Which one of the following statements is true for F, if 'q' is placed at distance r from the centre of the shell?
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In the figure shown, the current in the $$10\,\text{V}$$ battery is close to:
A circuit to verify Ohm's law uses ammeter and voltmeter in series or parallel connected correctly to the resistor. In the circuit:
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A charged particle going around in a circle can be considered to be a current loop. A particle of a mass $$m$$ carrying charge $$q$$ is moving in a plane with speed $$v$$ under the influence of magnetic field $$\vec{B}$$. The magnetic moment of this moving particle is:
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A square loop of side $$2a$$ and carrying current I is kept in xz plane with its centre at origin. A long wire carrying the same current I is placed parallel to z-axis and passing through point $$(0, b, 0)$$, $$(b >> a)$$. The magnitude of torque on the loop about z-axis will be:
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For a plane electromagnetic wave, the magnetic field at a point x and time t is:
$$\vec{B}(x, t) = [1.2 \times 10^{-7}\sin(0.5 \times 10^3 x + 1.5 \times 10^{11}t)\hat{k}]\,\text{T}$$.
The instantaneous electric field $$\vec{E}$$ corresponding to $$\vec{B}$$ is:
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A double convex lens has power P and same radii of curvature R of both the surfaces. The radius of curvature of a surface of a plano-convex lens made of the same material with power $$1.5\,P$$ is:
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Given the masses of various atomic particles $$m_P = 1.0072\,\text{u}$$, $$m_n = 1.0087\,\text{u}$$, $$m_e = 0.000548\,\text{u}$$, $$m_{\bar{v}} = 0$$, $$m_d = 2.0141\,\text{u}$$, where p = proton, n = neutron, e = electron, $$\bar{v}$$ = antineutrino and d = deuteron. Which of the following process is allowed by momentum and energy conservation:
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A student measuring the diameter of a pencil of circular cross-section with the help of a vernier scale records the following four readings 5.50 mm, 5.55 mm, 5.34 mm, 5.65 mm. The average of these four readings is 5.5375 mm and the standard deviation of the data is 0.07395 mm. The average diameter of the pencil should therefore be recorded as:
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The centre of mass of a solid hemisphere of radius $$8\,\text{cm}$$ is $$x\,\text{cm}$$ from the centre of the flat surface. Then value of $$x$$ is___
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An engine operates by taking a monatomic ideal gas through the cycle shown in the figure. The percentage efficiency of the engine is close to___
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In a series LR circuit, power of $$400\,\text{W}$$ is dissipated from a source of $$250\,\text{V}$$, $$50\,\text{Hz}$$. The power factor of the circuit is $$0.8$$. In order to bring the power factor to unity, a capacitor of value C is added in series to the L and R. Taking the value of C as $$\left(\frac{n}{3\pi}\right)\,\mu F$$, then value of n is___
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A Young's double-slit experiment is performed using monochromatic light of wavelength $$\lambda$$. The intensity of light at a point on the screen, where the path difference is $$\lambda$$, is $$K$$ units. The intensity of light at a point where the path difference is $$\dfrac{\lambda}{6}$$ is given by $$\dfrac{nK}{12}$$, where n is an integer. The value of n is___
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The output characteristics of a transistor is shown in the figure. When $$V_{CE}$$ is $$10\,\text{V}$$ and $$I_C = 4.0\,\text{mA}$$, then value of $$\beta_{ac}$$ is___
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The average molar mass of chlorine is $$35.5\,\text{g mol}^{-1}$$. The ratio of $$^{35}\text{Cl}$$ to $$^{37}\text{Cl}$$ in naturally occurring chlorine is close to:
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For a reaction $$4M(s) + nO_2(g) \to 2M_2O_n(s)$$. The free energy change is plotted as a function of temperature. The temperature below which the oxide is stable could be inferred from the plot as the point at which:
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The value of $$K_c$$ is 64 at 800 K for the reaction
$$\text{N}_2(g) + 3\text{H}_2(g) \to 2\text{NH}_3(g)$$
The value of $$K_c$$ for the following reaction is:
$$\text{NH}_3(g) \to \frac{1}{2}\text{N}_2(g) + \frac{3}{2}\text{H}_2(g)$$
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Dihydrogen of high purity (> 99.95%) is obtained through:
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Match the following compounds (Column-I) with their uses (Column-II):
S. No. Column-I S.No. Column-II
(I) $$\text{Ca(OH)}_2$$ - (A) casts of statues
(II) NaCl - (B) white wash
(III) $$\text{CaSO}_4 \cdot \frac{1}{2}\text{H}_2\text{O}$$ - (C) antacid
(IV) $$\text{CaCO}_3$$ - (D) washing soda preparation
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Match the following:
Test/Method Reagent
(i) Lucas Test (a) $$\text{C}_6\text{H}_5\text{SO}_2\text{Cl}$$ / aq. KOH
(ii) Dumas method (b) $$\text{HNO}_3$$ / $$\text{AgNO}_3$$
(iii) Kjeldahl's method (c) CuO / $$\text{CO}_3$$
(iv) Hinsberg Test (d) Conc. HCl and $$\text{ZnCl}_2$$
(e) $$\text{H}_2\text{SO}_4$$
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The IUPAC name of the following compound is:
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A crystal is made up of metal ions 'M1' and 'M2' and oxide ions. Oxide ions form a ccp lattice structure. The cation 'M1' occupies 50% of octahedral voids and the cation 'M2' occupies 12.5% of tetrahedral voids of oxide lattice. The oxidation numbers of 'M1' and 'M2' are respectively:
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A set of solutions is prepared using 180 g of water as a solvent and 10 g of different non-volatile solutes A, B and C. The relative lowering of vapour pressure in the presence of these solutes are in the order [Given, molar mass of $$A = 100\,\text{g mol}^{-1}$$; $$B = 200\,\text{g mol}^{-1}$$; $$C = 10,000\,\text{g mol}^{-1}$$]
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For the given cell; $$\text{Cu}(s)|\text{Cu}^{2+}(C_1\,M)||\text{Cu}^{2+}(C_2\,M)|\text{Cu}(s)$$
change in Gibbs energy $$(\Delta G)$$ is negative, it:
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The element that can be refined by distillation is:
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The reaction of NO with $$\text{N}_2\text{O}_4$$ at 250 K gives:
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Reaction of an inorganic sulphite X with dilute $$\text{H}_2\text{SO}_4$$ generated compound Y. Reaction of Y with NaOH gives X. Further, the reaction of X with Y and water affords compound Z. Y and X, respectively, are:
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Mischmetal is an alloy consisting mainly of:
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For a $$d^4$$ metal ion in an octahedral field, the correct electronic configuration is:
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The increasing order of the boiling points of the major products A, B and C of the following reactions will be:
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The correct match between Item-I (starting material) and Item-II (reagent) for the preparation of benzaldehyde is:
Item-I Item-II
(I) Benzene (P) HCl and $$\text{SnCl}_2$$, $$\text{H}_3\text{O}^+$$
(II) Benzonitrile (Q) $$\text{H}_2$$, $$\text{Pd-BaSO}_4$$, S and quinoline
(III) Benzoyl Chloride (R) Co, HCl and $$\text{AlCl}_3$$
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Which of the following compounds can be prepared in good yield by Gabriel phthalimide synthesis?
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The correct match between item-I and item-II is:
Item-I Item-II
(a) Natural rubber (I) 1,3-butadiene + styrene
(b) Neoprene (II) 1,3-butadiene
(c) Buna-N (III) chloroprene
(d) Buna-S (IV) Isoprene
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Which one of the following statement is not true?
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The atomic number of Unnilunium is______.
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If the solubility product of $$\text{AB}_2$$ is $$3.20 \times 10^{-11}\,\text{M}^3$$, then the solubility of $$\text{AB}_2$$ in pure water is ______ $$\times 10^{-4}\,\text{mol L}^{-1}$$ [Assuming that neither kind of ion reacts with water]
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The rate of a reaction decreased by 3.555 times when the temperature was changed from $$40\,^\circ\text{C}$$ to $$30\,^\circ\text{C}$$. The activation energy (in $$\text{kJ mol}^{-1}$$) of the reaction is______.
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For Freundlich adsorption isotherm, a plot of $$\log(x/m)$$ (y-axis) and $$\log p$$ (x-axis) gives a straight line, the intercept and slope for the line is 0.4771 and 2, respectively. The mass of gas, adsorbed per gram of adsorbent if the initial pressure is 0.04 atm is ______ $$\times 10^{-4}\,\text{g}$$.
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A solution of phenol in chloroform when treated with aqueous NaOH gives compound P as a major product. The mass percentage of carbon in P is______.
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If $$\alpha$$ and $$\beta$$ are the roots of the equation $$2x(2x+1) = 1$$, then $$\beta$$ is equal to:
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Let $$z = x + iy$$ be a non-zero complex number such that $$z^2 = i|z|^2$$, where $$i = \sqrt{-1}$$, then $$z$$ lies on the:
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The common difference of the A.P. $$b_1, b_2, \ldots, b_m$$ is 2 more than common difference of A.P. $$a_1, a_2, \ldots, a_n$$. If $$a_{40} = -159$$, $$a_{100} = -399$$ and $$b_{100} = a_{70}$$, then $$b_1$$ is equal to:
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If the constant term in the binomial expansion of $$\left(\sqrt{x} - \frac{k}{x^2}\right)^{10}$$ is 405, then $$|k|$$ equals:
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Let $$L$$ denote the line in the $$xy$$-plane with $$x$$ and $$y$$ intercepts as 3 and 1 respectively. Then the image of the point $$(-1, -4)$$ in the line is:
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The centre of the circle passing through the point $$(0, 1)$$ and touching the parabola $$y = x^2$$ at the point $$(2, 4)$$ is:
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If the normal at an end of latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity $$e$$ of the ellipse satisfies:
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Consider the statement: "For an integer n, if $$n^3 - 1$$ is even, then $$n$$ is odd". The contrapositive statement of this statement is:
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The angle of elevation of the summit of a mountain from a point on the ground is $$45^\circ$$. After climbing up one km towards the summit at an inclination of $$30^\circ$$ from the ground, the angle of elevation of the summit is found to be $$60^\circ$$. Then the height (in km) of the summit from the ground is:
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Let $$\theta = \frac{\pi}{5}$$ and $$A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$$. If $$B = A + A^4$$, then $$\det(B)$$:
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For a suitably chosen real constant $$a$$, let a function, $$f : \mathbb{R} - \{-a\} \to \mathbb{R}$$ be defined by $$f(x) = \frac{a-x}{a+x}$$. Further suppose that for any real number $$x \neq -a$$, and $$f(x) \neq -a$$, $$(f \circ f)(x) = x$$. Then $$f\left(-\frac{1}{2}\right)$$ is equal to:
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined by $$f(x) = \max\{x, x^2\}$$. Let $$S$$ denote the set of all points in $$\mathbb{R}$$, where $$f$$ is not differentiable. Then:
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The set of all real values of $$\lambda$$ for which the function $$f(x) = (1 - \cos^2 x) \cdot (\lambda + \sin x)$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, has exactly one maxima and exactly one minima, is:
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For all twice differentiable functions $$f : \mathbb{R} \to \mathbb{R}$$, with $$f(0) = f(1) = f'(0) = 0$$,
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If the tangent to the curve $$y = f(x) = x\log_e x$$, $$(x > 0)$$ at a point $$(c, f(c))$$ is parallel to the line-segment joining the points $$(1, 0)$$ and $$(e, e)$$, then $$c$$ is equal to:
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The integral $$\int_1^2 e^x \cdot x^x(2 + \log_e x)\,dx$$ equals:
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The area (in sq. units) of the region enclosed by the curves $$y = x^2 - 1$$ and $$y = 1 - x^2$$ is equal to:
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If $$y = \left(\frac{2}{\pi}x - 1\right) \operatorname{cosec} x$$ is the solution of the differential equation, $$\frac{dy}{dx} + p(x)y = -\frac{2}{\pi} \operatorname{cosec} x$$, $$0 < x < \frac{\pi}{2}$$, then the function $$p(x)$$ is equal to:
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A plane P meets the coordinate axes at A, B and C respectively. The centroid of $$\triangle ABC$$ is given to be $$(1, 1, 2)$$. Then the equation of the line through this centroid and perpendicular to the plane P is:
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The probabilities of three events A, B and C are given $$P(A) = 0.6$$, $$P(B) = 0.4$$ and $$P(C) = 0.5$$. If $$P(A \cup B) = 0.8$$, $$P(A \cap C) = 0.3$$, $$P(A \cap B \cap C) = 0.2$$, $$P(B \cap C) = \beta$$ and $$P(A \cup B \cup C) = \alpha$$, where $$0.85 \leq \alpha \leq 0.95$$, then $$\beta$$ lies in the interval:
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The number of words (with or without meaning) that can be formed from all the letters of the word 'LETTER' in which vowels never come together is_____.
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Consider the data on x taking the values $$0, 2, 4, 8, \ldots, 2^n$$ with frequencies $$^nC_0, ^nC_1, ^nC_2, \ldots, ^nC_n$$ respectively. If the mean of this data is $$\frac{728}{2^n}$$, then n is equal to_______.
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The sum of distinct values of $$\lambda$$ for which the system of equations:
$$(\lambda - 1)x + (3\lambda + 1)y + 2\lambda z = 0$$
$$(\lambda - 1)x + (4\lambda - 2)y + (\lambda + 3)z = 0$$
$$2x + (3\lambda + 1)y + 3(\lambda - 1)z = 0$$
Has non-zero solutions, is_______.
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Suppose that a function $$f : \mathbb{R} \to \mathbb{R}$$ satisfies $$f(x+y) = f(x)f(y)$$ for all $$x, y \in \mathbb{R}$$ and $$f(1) = 3$$. If $$\sum_{i=1}^{n} f(i) = 363$$, then $$n$$ is equal to_____.
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If $$\vec{x}$$ and $$\vec{y}$$ be two non-zero vectors such that $$|\vec{x} + \vec{y}| = |\vec{x}|$$ and $$2\vec{x} + \lambda\vec{y}$$ is perpendicular to $$\vec{y}$$, then the value of $$\lambda$$ is_____.
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