A torque meter is calibrated to reference standards of mass, length and time each with $$5\%$$ accuracy. After calibration, the measured torque with this torque meter will have net accuracy of
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A torque meter is calibrated to reference standards of mass, length and time each with $$5\%$$ accuracy. After calibration, the measured torque with this torque meter will have net accuracy of
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A bullet is shot vertically downwards with an initial velocity of $$100 \text{ m s}^{-1}$$ from a certain height. Within $$10 \text{ s}$$, the bullet reaches the ground and instantaneously comes to rest due to the perfectly inelastic collision. The velocity-time curve for total time $$t = 20 \text{ s}$$ will be: (Take $$g = 10 \text{ m s}^{-2}$$)
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A bag is gently dropped on a conveyor belt moving at a speed of $$2 \text{ m s}^{-1}$$. The coefficient of friction between the conveyor belt and bag is $$0.4$$. Initially, the bag slips on the belt before it stops due to friction. The distance travelled by the bag on the belt during slipping motion is: [Take $$g = 10 \text{ m s}^{-2}$$]
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Sand is being dropped from a stationary dropper at a rate of $$0.5 \text{ kg s}^{-1}$$ on a conveyor belt moving with a velocity of $$5 \text{ m s}^{-1}$$. The power needed to keep belt moving with the same velocity will be
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Two satellites $$A$$ and $$B$$ having masses in the ratio $$4:3$$ are revolving in circular orbits of radii $$3r$$ and $$4r$$ respectively around the earth. The ratio of total mechanical energy of $$A$$ to $$B$$ is
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Two cylindrical vessels of equal cross-sectional area $$16 \text{ cm}^2$$ contain water upto heights $$100 \text{ cm}$$ and $$150 \text{ cm}$$ respectively. The vessels are interconnected so that the water levels in them become equal. The work done by the force of gravity during the process, is [Take density of water $$= 10^3 \text{ kg m}^{-3}$$ and $$g = 10 \text{ m s}^{-2}$$]
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If $$K_1$$ and $$K_2$$ are the thermal conductivities, $$L_1$$ and $$L_2$$ are the lengths and $$A_1$$ and $$A_2$$ are the cross sectional areas of steel and copper rods respectively such that $$\dfrac{K_2}{K_1} = 9$$, $$\dfrac{A_1}{A_2} = 2$$, $$\dfrac{L_1}{L_2} = 2$$. Then, for the arrangement as shown in the figure, the value of temperature $$T$$ of the steel-copper junction in the steady state will be
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Read the following statements:
A. When small temperature difference between a liquid and its surrounding is doubled the rate of loss of heat of the liquid becomes twice.
B. Two bodies P and Q having equal surface areas are maintained at temperature $$10°C$$ and $$20°C$$. The thermal radiation emitted in a given time by P and Q are in the ratio $$1:1.15$$.
C. A Carnot Engine working between $$100 \text{ K}$$ and $$400 \text{ K}$$ has an efficiency of $$75\%$$.
D. When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of heat of the liquid becomes twice.
Choose the correct answer from the options given below:
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Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is $$1:4$$, then
A. The r.m.s. velocity of gas molecules in two vessels will be the same.
B. The ratio of pressure in these vessels will be $$1:4$$.
C. The ratio of pressure will be $$1:1$$.
D. The r.m.s. velocity of gas molecules in two vessels will be in the ratio of $$1:4$$.
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Two identical positive charges $$Q$$ each are fixed at a distance of $$2a$$ apart from each other. Another point charge $$q_0$$ with mass $$m$$ is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge $$q_0$$ executes SHM. The time period of oscillation of charge $$q_0$$ will be
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Two sources of equal emfs are connected in series. This combination is connected to an external resistance $$R$$. The internal resistances of the two sources are $$r_1$$ and $$r_2$$ ($$r_1 > r_2$$). If the potential difference across the source of internal resistance $$r_1$$ is zero then the value of $$R$$ will be
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A magnet hung at $$45°$$ with magnetic meridian makes an angle of $$60°$$ with the horizontal. The actual value of the angle of dip is
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Two bar magnets oscillate in a horizontal plane in earth's magnetic field with time periods of $$3 \text{ s}$$ and $$4 \text{ s}$$ respectively. If their moments of inertia are in the ratio of $$3:2$$ then the ratio of their magnetic moments will be
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A direct current of $$4 \text{ A}$$ and an alternating current of peak value $$4 \text{ A}$$ flow through resistance of $$3 \Omega$$ and $$2 \Omega$$ respectively. The ratio of heat produced in the two resistances in same interval of time will be:
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A beam of light travelling along $$X$$-axis is described by the electric field $$E_y = 900 \sin \omega\left(t - \dfrac{x}{c}\right)$$. The ratio of electric force to magnetic force on a charge $$q$$ moving along $$Y$$-axis with a speed of $$3 \times 10^7 \text{ m s}^{-1}$$ will be: [Given speed of light $$= 3 \times 10^8 \text{ m s}^{-1}$$]
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A microscope was initially placed in air (refractive index 1). It is then immersed in oil (refractive index 2). For a light whose wavelength in air is $$\lambda$$, calculate the change of microscope's resolving power due to oil
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An electron (mass $$m$$) with an initial velocity $$\vec{v} = v_0\hat{i}$$ ($$v_0 > 0$$) is moving in an electric field $$\vec{E} = -E_0\hat{i}$$ ($$E_0$$ is constant). If at $$t = 0$$, de-Broglie wavelength is $$\lambda_0 = \dfrac{h}{mv_0}$$, then its de-Broglie wavelength after time $$t$$ is given by
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What is the half-life period of a radioactive material if its activity drops to $$\dfrac{1}{16^{th}}$$ of its initial value of $$30$$ years?
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A logic gate circuit has two inputs $$A$$ and $$B$$ and output $$Y$$. The voltage waveforms of $$A$$, $$B$$ and $$Y$$ are shown. The logic gate circuit is
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At a particular station, the TV transmission tower has a height of $$100 \text{ m}$$. To triple its coverage range, height of the tower should be increased to
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A ball of mass $$m$$ is thrown vertically upward. Another ball of mass $$2 \text{ m}$$ is thrown at an angle $$\theta$$ with the vertical. Both the balls stay in air for the same period of time. The ratio of the heights attained by the two balls respectively is $$\dfrac{1}{x}$$. The value of $$x$$ is ______.
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A pulley of radius $$1.5 \text{ m}$$ is rotated about its axis by a force $$F = (12t - 3t^2) \text{ N}$$ applied tangentially (while $$t$$ is measured in seconds). If moment of inertia of the pulley about its axis of rotation is $$4.5 \text{ kg m}^2$$, the number of rotations made by the pulley before its direction of motion is reversed, will be $$\dfrac{K}{\pi}$$. The value of $$K$$ is ______.
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A square aluminium (shear modulus is $$25 \times 10^9 \text{ N m}^{-2}$$) slab of side $$60 \text{ cm}$$ and thickness $$15 \text{ cm}$$ is subjected to a shearing force (on its narrow face) of $$18.0 \times 10^4 \text{ N}$$. The lower edge is riveted to the floor. The displacement of the upper edge is ______ $$\mu m$$.
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A mass $$0.9 \text{ kg}$$, attached to a horizontal spring, executes SHM with an amplitude $$A_1$$. When this mass passes through its mean position, then a smaller mass of $$124 \text{ g}$$ is placed over it and both masses move together with amplitude $$A_2$$. If the ratio $$\dfrac{A_1}{A_2}$$ is $$\dfrac{\alpha}{\alpha - 1}$$, then the value of $$\alpha$$ will be ______.
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A $$1 \text{ m}$$ long copper wire carries a current of $$1 \text{ A}$$. If the cross section of the wire is $$2.0 \text{ mm}^2$$ and the resistivity of copper is $$1.7 \times 10^{-8} \Omega \text{ m}$$. The force experienced by moving electron in the wire is ______ $$\times 10^{-23} \text{ N}$$. (Charge of electron $$= 1.6 \times 10^{-19} \text{ C}$$)
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A long cylindrical volume contains a uniformly distributed charge of density $$\rho \text{ C m}^{-3}$$. The electric field inside the cylindrical volume at a distance $$x = \dfrac{2\epsilon_0}{\rho} \text{ m}$$ from its axis is ______ $$\text{V m}^{-1}$$.
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In meter bridge experiment for measuring unknown resistance '$$S$$', the null point is obtained at a distance $$30 \text{ cm}$$ from the left side as shown at point $$D$$. If $$R$$ is $$5.6 \text{ k}\Omega$$, then the value of unknown resistance '$$S$$' will be ______ $$\Omega$$.
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To light, a $$50 \text{ W}$$, $$100 \text{ V}$$ lamp is connected, in series with a capacitor of capacitance $$\dfrac{50}{\pi\sqrt{x}} \mu F$$, with $$200 \text{ V}$$, $$50 \text{ Hz}$$ AC source. The value of $$x$$ will be ______.
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Two beams of light having intensities $$I$$ and $$4I$$ interfere to produce a fringe pattern on a screen. The phase difference between the two beams are $$\dfrac{\pi}{2}$$ and $$\dfrac{\pi}{3}$$ at points $$A$$ and $$B$$ respectively. The difference between the resultant intensities at the two points is $$xI$$. The value of $$x$$ will be ______.
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The one division of main scale of Vernier callipers reads $$1 \text{ mm}$$ and $$10$$ divisions of Vernier scale is equal to the $$9$$ divisions on main scale. When the two jaws of the instrument touch each other the zero of the Vernier lies to the right of zero of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the zero of the Vernier scale lies in between $$4.1 \text{ cm}$$ and $$4.2 \text{ cm}$$ and $$6^{th}$$ Vernier division coincides with a main scale division. The diameter of the bob will be ______ $$ 10^{-2} cm$$.
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$$250 \text{ g}$$ solution of D-glucose in water contains $$10.8\%$$ of carbon by weight. The molality of the solution is nearest to (Given: Atomic Weights are $$H = 1u; C = 12u; O = 16u$$)
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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: Energy of $$2s$$ orbital of hydrogen atom is greater than that of $$2s$$ orbital of lithium.
Reason R: Energies of the orbitals in the same subshell decrease with increase in the atomic number.
In the light of the above statements, choose the correct answer from the options given below
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The incorrect statement is
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Given below are two statements.
Statement I: $$O_2$$, $$Cu^{2+}$$ and $$Fe^{3+}$$ are weakly attracted by magnetic field and are magnetized in the same direction as magnetic field.
Statement II: $$NaCl$$ and $$H_2O$$ are weakly magnetized in opposite direction to magnetic field.
In the light of the above statements, choose the most appropriate answer form 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: Activated charcoal adsorbs $$SO_2$$ more efficiently than $$CH_4$$.
Reason R: Gases with lower critical temperatures are readily adsorbed by activated charcoal.
In the light of the above statements, choose the correct answer from the options given below.
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Given below are two statements:
Statement I: Hydrogen peroxide can act as an oxidizing agent in both acidic and basic conditions.
Statement II: Density of hydrogen peroxide at $$298 \text{ K}$$ is lower than that of $$D_2O$$.
In the light of the above statements. Choose the correct answer from the options
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Given below are two statements
Statement I: The chlorides of Be and Al have Cl-bridged structure. Both are soluble in organic solvents and act as Lewis bases.
Statement II: Hydroxides of Be and Al dissolve in excess alkali to give beryllate and aluminate ions.
In the light of the above statements. Choose the correct answer from the options
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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: [6] Annulene, [8] Annulene and cis-[10] Annulene are respectively aromatic, not-aromatic and aromatic.
Reason R: Planarity is one of the requirements of aromatic systems.
In the light of the above statements, choose the most appropriate answer from the options given below.
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In Carius method of estimation of halogen, $$0.45 \text{ g}$$ of an organic compound gave $$0.36 \text{ g}$$ of $$AgBr$$. Find out the percentage of bromine in the compound. (Molar masses: $$AgBr = 188 \text{ g mol}^{-1}; Br = 80 \text{ g mol}^{-1}$$)
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Match List I with List II
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Boiling point of a $$2\%$$ aqueous solution of a nonvolatile solute A is equal to the boiling point of $$8\%$$ aqueous solution of a non-volatile solute B. The relation between molecular weights of A and B is
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Which of the following methods are not used to refine any metal?
(A) Liquation
(B) Calcination
(C) Electrolysis
(D) Leaching
(E) Distillation
Choose the correct answer from the options given below
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Which oxoacid of phosphorous has the highest number of oxygen atoms present in its chemical formula?
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Given below are two statements:
Statement I: Iron (III) catalyst, acidified $$K_2Cr_2O_7$$ and neutral $$KMnO_4$$ have the ability to oxidise $$I^-$$ to $$I_2$$ independently.
Statement II: Manganate ion is paramagnetic in nature and involves $$p\pi - p\pi$$ bonding.
In the light of the above statements, choose the correct answer from the options.
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The total number of $$Mn = O$$ bonds in $$Mn_2O_7$$ is
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In the above reaction, product B is
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Match List-I with List-II
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Match List I with List II
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Match List-I with List-II.
Choose the correct answer from the options given below
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A sugar 'X' dehydrates very slowly under acidic condition to give furfural which on further reaction with resorcinol gives the coloured product after sometime. Sugar 'X' is
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Amongst the following the number of oxide(s) which are paramagnetic in nature is ______
$$Na_2O, KO_2, NO_2, N_2O, ClO_2, NO, SO_2, Cl_2O$$
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According to MO theory, number of species/ions from the following having identical bond order is ______
$$CN^-, NO^+, O_2, O_2^+, O_2^{2+}$$
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The molar heat capacity for an ideal gas at constant pressure is $$20.785 \text{ J K}^{-1} \text{ mol}^{-1}$$. The change in internal energy is $$5000 \text{ J}$$ upon heating it from $$300 \text{ K}$$ to $$500 \text{ K}$$. The number of moles of the gas at constant volume is ______ (Given: $$R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}$$)
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At $$310 \text{ K}$$, the solubility of $$CaF_2$$ in water is $$2.34 \times 10^{-3} \text{ g/100 mL}$$. The solubility product of $$CaF_2$$ is ______ $$\times 10^{-8} (\text{mol/L})^3$$ (nearest integer). (Given molar mass: $$CaF_2 = 78 \text{ g mol}^{-1}$$)
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$$20 \text{ mL}$$ of $$0.02M K_2Cr_2O_7$$ solution is used for the titration of $$10 \text{ mL}$$ of $$Fe^{2+}$$ solution in the acidic medium. The molarity of $$Fe^{2+}$$ solution is ______ $$\times 10^{-2} M$$
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In the following reaction,
the $$\%$$ yield for reaction I is $$60\%$$ and that of reaction II is $$50\%$$. The overall yield of the complete reaction is ______ $$\%$$.
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$$2NO + 2H_2 \rightarrow N_2 + 2H_2O$$. The above reaction has been studied at $$800°C$$.The related data are given in the table below
The order of the reaction with respect to NO is ______.
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In the titration of $$KMnO_4$$ and oxalic acid in acidic medium, the change in oxidation number of carbon at the end point is ______.
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The conductivity of a solution of complex with formula $$CoCl_3(NH_3)_4$$ corresponds to $$1:1$$ electrolyte, then the primary valency of central metal ion is ______.
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Optical activity of an enantiomeric mixture is $$+12.6°$$ and the specific rotation of $$(+)$$ isomer is $$+30°$$. The optical purity is ______ $$\%$$.
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Let the minimum value $$v_0$$ of $$v=|z|^{2} + |z-3|^{2} + |z-6i|^{2}, z ∈ \mathbb{C} $$ s attained at $$z=z_{0}$$. Then $$|2z_0^2- \overline{z}_0^3+3|^{2}+v_0^2 $$ is equal to
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Suppose $$a_1, a_2, \ldots, a_n$$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is $$5:12$$and $$110 < a_{15} < 120$$, then the sum of the first ten terms of the progression is equal to
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The remainder when $$(2021)^{2022} + (2022)^{2021}$$ is divided by $$7$$ is
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Let $$A(1, 1)$$, $$B(-4, 3)$$, $$C(-2, -5)$$ be vertices of a triangle $$ABC$$, $$P$$ be a point on side $$BC$$, and $$\Delta_1$$ and $$\Delta_2$$ be the areas of triangle $$APB$$ and $$ABC$$ respectively. If $$\Delta_1 : \Delta_2 = 4 : 7$$, then the area enclosed by the lines $$AP$$, $$AC$$ and the $$x$$-axis is
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If the circle $$x^2 + y^2 - 2gx + 6y - 19c = 0$$, $$g, c \in \mathbb{R}$$ passes through the point $$(6, 1)$$ and its centre lies on the line $$x - 2cy = 8$$, then the length of intercept made by the circle on $$x$$-axis is
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Let $$P(a, b)$$ be a point on the parabola $$y^2 = 8x$$ such that the tangent at $$P$$ passes through the centre of the circle $$x^2 + y^2 - 10x - 14y + 65 = 0$$. Let $$A$$ be the product of all possible values of $$a$$ and $$B$$ be the product of all possible values of $$b$$. Then the value of $$A + B$$ is equal to
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined as $$f(x) = a \sin\left(\frac{\pi[x]}{2}\right) + [2 - x]$$, $$a \in \mathbb{R}$$, where $$[t]$$ is the greatest integer less than or equal to $$t$$. If $$\lim_{x \to -1} f(x)$$ exists, then the value of $$\int_0^4 f(x) \, dx$$ is equal to
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$$(p \wedge r) \Leftrightarrow (p \wedge (\sim q))$$ is equivalent to $$(\sim p)$$ when $$r$$ is
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Let a vertical tower $$AB$$ of height $$2h$$ stands on a horizontal ground. Let from a point $$P$$ on the ground a man can see upto height $$h$$ of the tower with an angle of elevation $$2\alpha$$. When from $$P$$, he moves a distance $$d$$ in the direction of $$\overrightarrow{AP}$$, he can see the top of the tower with an angle of elevation $$\alpha$$. If $$d = \sqrt{7}h$$, then $$\tan \alpha$$ is equal to
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Let $$R_1$$ and $$R_2$$ be two relations defined on $$\mathbb{R}$$ by $$a R_1 b \Leftrightarrow ab \geq 0$$ and $$a R_2 b \Leftrightarrow a \geq b$$, then
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Let $$A = \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix}$$. Let $$\alpha, \beta \in \mathbb{R}$$ be such that $$\alpha A^2 + \beta A = 2I$$. Then $$\alpha + \beta$$ is equal to
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Let $$f, g : \mathbb{N} - \{1\} \to \mathbb{N}$$ be functions defined by $$f(a) = \alpha$$, where $$\alpha$$ is the maximum of the powers of those primes $$p$$ such that $$p^\alpha$$ divides $$a$$, and $$g(a) = a + 1$$, for all $$a \in \mathbb{N} - \{1\}$$. Then, the function $$f + g$$ is
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Let a function $$f : \mathbb{R} \to \mathbb{R}$$ be defined as:
$$f(x) = \begin{cases} \int_0^x (5 - |t - 3|) \, dt, & x > 4 \\ x^2 + bx, & x \leq 4 \end{cases}$$
where $$b \in \mathbb{R}$$. If $$f$$ is continuous at $$x = 4$$, then which of the following statements is NOT true?
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$$I = \int_{\pi/4}^{\pi/3} \left(\frac{8 \sin x - \sin 2x}{x}\right) dx$$. Then
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The area of the smaller region enclosed by the curves $$y^2 = 8x + 4$$ and $$x^2 + y^2 + 4\sqrt{3}x - 4 = 0$$ is equal to
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Let $$y = y_1(x)$$ and $$y = y_2(x)$$ be two distinct solutions of the differential equation $$\frac{dy}{dx} = x + y$$, with $$y_1(0) = 0$$ and $$y_2(0) = 1$$ respectively. Then, the number of points of intersection of $$y = y_1(x)$$ and $$y = y_2(x)$$ is
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Let $$\vec{a} = \alpha \hat{i} + \hat{j} + \beta \hat{k}$$ and $$\vec{b} = 3\hat{i} - 5\hat{j} + 4\hat{k}$$ be two vectors, such that $$\vec{a} \times \vec{b} = -\hat{i} + 9\hat{j} + 12\hat{k}$$. Then the projection of $$\vec{b} - 2\vec{a}$$ on $$\vec{b} + \vec{a}$$ is equal to
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Let $$\vec{a} = 2\hat{i} - \hat{j} + 5\hat{k}$$ and $$\vec{b} = \alpha \hat{i} + \beta \hat{j} + 2\hat{k}$$. If $$\left(\left(\vec{a} \times \vec{b}\right) \times \hat{i}\right) \cdot \hat{k} = \frac{23}{2}$$, then $$\left|\vec{b} \times 2\hat{j}\right|$$ is equal to
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If the plane $$P$$ passes through the intersection of two mutually perpendicular planes $$2x + ky - 5z = 1$$ and $$3kx - ky + z = 5$$, $$k < 3$$ and intercepts a unit length on positive $$x$$-axis, then the intercept made by the plane $$P$$ on the $$y$$-axis is
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Let $$S$$ be the sample space of all five digit numbers. If $$p$$ is the probability that a randomly selected number from $$S$$, is a multiple of $$7$$ but not divisible by $$5$$, then $$9p$$ is equal to
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Let $$S = \{z \in \mathbb{C} : \bar{z}^2 + \bar{z} = 0\}$$. Then $$\sum_{z \in S} (\text{Re}(z) + \text{Im}(z))$$ is equal to______.
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Let $$f(x) = 2x^2 - x - 1$$ and $$S = \{n \in \mathbb{Z} : |f(n)| \leq 800\}$$. Then, the value of $$\sum_{n \in S} f(n)$$ is equal to
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If the length of the latus rectum of the ellipse $$x^2 + 4y^2 + 2x + 8y - \lambda = 0$$ is $$4$$, and $$l$$ is the length of its major axis, then $$\lambda + l$$ is equal to_______.
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An ellipse $$E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ passes through the vertices of the hyperbola $$H : \frac{x^2}{49} - \frac{y^2}{64} = -1$$. Let the major and minor axes of the ellipse $$E$$ coincide with the transverse and conjugate axes of the hyperbola $$H$$. Let the product of the eccentricities of $$E$$ and $$H$$ be $$\frac{1}{2}$$. If $$l$$ is the length of the latus rectum of the ellipse $$E$$, then the value of $$113l$$ is equal to______.
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The mean and variance of $$10$$ observations were calculated as $$15$$ and $$15$$ respectively by a student who took by mistake $$25$$ instead of $$15$$ for one observation. Then, the correct standard deviation is______.
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Let $$S$$ be the set containing all $$3 \times 3$$ matrices with entries from $$\{-1, 0, 1\}$$. The total number of matrices $$A \in S$$ such that the sum of all the diagonal elements of $$A^T A$$ is $$6$$ is
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For $$k \in \mathbb{R}$$, let the solutions of the equation $$\cos\left(\sin^{-1}\left(x \cot\left(\tan^{-1}\left(\cos(\sin^{-1} x)\right)\right)\right)\right) = k$$, $$0 < |x| < \frac{1}{\sqrt{2}}$$ be $$\alpha$$ and $$\beta$$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $$x^2 - bx - 5 = 0$$ are $$\frac{1}{\alpha^2} + \frac{1}{\beta^2}$$ and $$\frac{\alpha}{\beta}$$, then $$\frac{b}{k^2}$$ is equal to
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Let $$M$$ and $$N$$ be the number of points on the curve $$y^5 - 9xy + 2x = 0$$, where the tangents to the curve are parallel to $$x$$-axis and $$y$$-axis, respectively. Then the value of $$M + N$$ equals
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Let $$y=y(x)$$ be the solution curve of the differential equation $$\sin(2x^2)\log_e(\tan x^2)\,dy+\left(4xy-4\sqrt2\,x\sin x^2-\frac{\pi}{4}\right)dx=0,$$ $$0<x<\sqrt{\frac{\pi}{2}},$$ which passes through the point $$\left(\sqrt{\frac{\pi}{6}},1\right).$$ Then $$y\left(\sqrt{\frac{\pi}{3}}\right)$$ is equal to ______.
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Let the line $$\dfrac{x-3}{7} = \dfrac{y-2}{-1} = \dfrac{z-3}{-4}$$ intersect the plane containing the lines $$\dfrac{x-4}{1} = \dfrac{y+1}{-2} = \dfrac{z}{1}$$ and $$4ax - y + 5z - 7a = 0 = 2x - 5y - z - 3$$, $$a \in \mathbb{R}$$ at the point $$P(\alpha, \beta, \gamma)$$. Then the value of $$\alpha + \beta + \gamma$$ equals ______.
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