The dimension of magnetic field in M, L, T and C (Coulomb) is given as
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The dimension of magnetic field in M, L, T and C (Coulomb) is given as
A body is at rest at $$x = 0$$. At $$t = 0$$, it starts moving in the positive $$x$$-direction with a constant acceleration. At the same instant another body passes through $$x = 0$$ moving in the positive $$x$$ direction with a constant speed. The position of the first body is given by $$x_1(t)$$ after time '$$t$$' and that of the second body by $$x_2(t)$$ after the same time interval. Which of the following graphs correctly describes $$(x_1 - x_2)$$ as a function of time '$$t$$'?
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An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range
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A thin rod of length '$$L$$' is lying along the $$x$$-axis with its ends at $$x = 0$$ and $$x = L$$. Its linear density (mass/length) varies with $$x$$ as $$k\left(\frac{x}{L}\right)^n$$, where $$n$$ can be zero or any positive number. If the position $$x_{CM}$$ of the centre of mass of the rod is plotted against '$$n$$', which of the following graphs best approximates the dependence of $$x_{CM}$$ on $$n$$?
A body of mass $$m = 3.513$$ kg is moving along the $$x$$-axis with a speed of $$5.00\ ms^{-1}$$. The magnitude of its momentum is recorded as
A block of mass 0.50 kg is moving with a speed of 2.00 m/s on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is
Consider a uniform square plate of side '$$a$$' and mass '$$m$$'. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is
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A planet in a distant solar system is 10 times more massive than the earth and its radius is 10 times smaller. Given that the escape velocity from the earth is $$11\ kms^{-1}$$, the escape velocity from the surface of the planet would be
This question contains Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement - I: For a mass $$M$$ kept at the centre of a cube of side '$$a$$', the flux of gravitational field passing through its sides is $$4\pi GM$$. Statement - II: If the direction of a field due to a point source is radial and its dependence on the distance '$$r$$' for the source is given as $$1/r^2$$, its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface.
A spherical solid ball of volume $$V$$ is made of a material of density $$\rho_1$$. It is falling through a liquid of density $$\rho_2\ (\rho_2 < \rho_1)$$. Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed v, i.e., $$F_{viscous} = -kv^2 (k > 0)$$. The terminal speed of the ball is
A jar filled with two non mixing liquids 1 and 2 having densities $$\rho_1$$ and $$\rho_2$$ respectively. A solid ball, made of a material of density $$\rho_3$$, is dropped in the jar. It comes to equilibrium in the position shown in the figure. Which of the following is true for $$\rho_1, \rho_2$$ and $$\rho_3$$?
A capillary tube (A) is dropped in water. Another identical tube (B) is dipped in a soap water solution. Which of the following shows the relative nature of the liquid columns in the two tubes?
An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has volume $$V_1$$ and contains ideal gas at pressure $$P_1$$ and temperature $$T_1$$. The other chamber has volume $$V_2$$ and contains ideal gas at pressure $$P_2$$ and temperature $$T_2$$. If the partition is removed without doing any work on the gas, the final equilibrium temperature of the gas in the container will be
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While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at a column length of 18 cm during winter. Repeating the same experiment during summer, she measures the column length to be $$x$$ cm for the second resonance. Then
The speed of sound in oxygen ($$O_2$$) at a certain temperature is $$460\ ms^{-1}$$. The speed of sound in helium (He) at the same temperature will be (assumed both gases to be ideal)
A wave travelling along the $$x$$-axis is described by the equation $$y(x, t) = 0.005 \cos(\alpha x - \beta t)$$. If the wavelength and the time period of the wave are 0.08 m and 2.0 s, respectively, then $$\alpha$$ and $$\beta$$ in appropriate units are
A thin spherical shell of radius $$R$$ has charge $$Q$$ spread uniformly over its surface. Which of the following graphs most closely represents the electric field $$E(r)$$ produced by the shell in the range $$0 \le r < \infty$$, where $$r$$ is the distance from the centre of the shell?
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A parallel plate capacitor with air between the plates has a capacitance of 9 pF. The separation between its plates is '$$d$$'. The space between the plates is now filled with two dielectrics. One of the dielectrics has dielectric constant $$k_1 = 3$$ and thickness $$\frac{d}{3}$$ while the other one has dielectric constant $$k_2 = 6$$ and thickness $$\frac{2d}{3}$$. Capacitance of the capacitor is now
Paragraph: Consider a block of conducting material of resistivity '$$\rho$$' shown in the figure. Current '$$I$$' enters at '$$A$$' and leaves from '$$D$$'. We apply superposition principle to find voltage '$$\Delta V$$' developed between '$$B$$' and '$$C$$'. The calculation is done in the following steps: (i) Take current '$$I$$' entering from '$$A$$' and assume it to spread over a hemispherical surface in the block. (ii) Calculate field $$E(r)$$ at distance '$$r$$' from $$A$$ by using Ohm's law $$E = \rho j$$, where $$j$$ is the current per unit area at '$$r$$'. (iii) From the '$$r$$' dependence of $$E(r)$$, obtain the potential $$V(r)$$ at $$r$$. (iv) Repeat (i), (ii) and (iii) for current '$$I$$' leaving '$$D$$' and superpose results for '$$A$$' and '$$D$$'.
Question: $$\Delta V$$ measured between $$B$$ and $$C$$ is
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Paragraph: Consider a block of conducting material of resistivity '$$\rho$$' shown in the figure. Current '$$I$$' enters at '$$A$$' and leaves from '$$D$$'. We apply superposition principle to find voltage '$$\Delta V$$' developed between '$$B$$' and '$$C$$'. The calculation is done in the following steps: (i) Take current '$$I$$' entering from '$$A$$' and assume it to spread over a hemispherical surface in the block. (ii) Calculate field $$E(r)$$ at distance '$$r$$' from $$A$$ by using Ohm's law $$E = \rho j$$, where $$j$$ is the current per unit area at '$$r$$'. (iii) From the '$$r$$' dependence of $$E(r)$$, obtain the potential $$V(r)$$ at $$r$$. (iv) Repeat (i), (ii) and (iii) for current '$$I$$' leaving '$$D$$' and superpose results for '$$A$$' and '$$D$$'.
Question: For current entering at $$A$$, the electric field at a distance '$$r$$' from $$A$$ is
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A 5 V battery with internal resistance $$2\Omega$$ and a 2 V battery with internal resistance $$1\Omega$$ are connected to a $$10\Omega$$ resistor as shown in the figure. The current in the $$10\Omega$$ resistor is
Relative permittivity and permeability of a material are $$\varepsilon_r$$ and $$\mu_r$$, respectively. Which of the following values of these quantities are allowed for a diamagnetic material?
A horizontal overhead power line is at a height of 4 m from the ground and carries a current of 100 A from east to west. The magnetic field directly below it on the ground is $$(\mu_0 = 4\pi \times 10^{-7}\ T\ m\ A^{-1})$$
Two coaxial solenoids are made by winding thin insulated wire over a pipe of cross sectional area $$A = 10\ cm^2$$ and length = 20 cm. If one of the solenoids has 300 turns and the other 400 turns, their mutual inductance is $$(\mu_0 = 4\pi \times 10^{-7}\ T\ m\ A^{-1})$$
A student measures the focal length of convex lens by putting an object pin at a distance '$$u$$' from the lens and measuring the distance '$$v$$' of the image pin. The graph between '$$u$$' and '$$v$$' plotted by the student should look like
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An experiment is performed to find the refractive index of glass using a travelling microscope. In this experiment distance are measured by
Paragraph: Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere constructively (see in figure).
Question: Electrons accelerated by potential $$V$$ are diffracted from a crystal. If $$d = 1\ \AA$$ and $$i = 30^\circ$$, $$V$$ should be about $$(h = 6.6 \times 10^{-34}\ Js,\ m_e = 9.1 \times 10^{-31}\ kg,\ e = 1.6 \times 10^{-19}\ C)$$
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Paragraph: Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere constructively (see in figure).
Question: If a strong diffraction peak is observed when electrons are incident at an angle '$$i$$' from the normal to the crystal planes with distance '$$d$$' between them (see figure), de Broglie wavelength $$\lambda_{dB}$$ of electrons can be calculated by the relationship ($$n$$ is an integer)
Paragraph: Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere constructively (see in figure).
Question: In an experiment, electrons are made to pass through a narrow slit of width '$$d$$' comparable to their de Broglie wavelength. They are detected on a screen at a distance '$$D$$' from the slit (see figure).
Which of the following graph can be expected to represent the number of electrons '$$N$$' detected as a function of the detector position '$$y$$' ($$y = 0$$ corresponds to the middle of the slit)?
Suppose an electron is attracted towards the origin by a force $$k/r$$ where '$$k$$' is a constant and '$$r$$' is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the $$n^{th}$$ orbital of the electron is found to be '$$r_n$$' and the kinetic energy of the electron to be $$T_n$$. Then which of the following is true?
This question contains Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement - I: Energy is released when heavy nuclei undergo fission or light nuclei undergo fusion. Statement - II: For heavy nuclei, binding energy per nucleon increases with increasing $$Z$$ while for light nuclei it decrease with increasing $$Z$$.
A working transistor with its three legs marked $$P, Q$$ and $$R$$ is tested using a multimeter. No conduction is found between $$P$$ and $$Q$$. By connecting the common (negative) terminal of the multimeter to $$R$$ and the other (positive) terminal to $$P$$ or $$Q$$, some resistance is seen on the multimeter. Which of the following is true for the transistor?
In the circuit below, $$A$$ and $$B$$ represent two inputs and $$C$$ represents the output. The circuit represents
Shown in the figure below is a meter-bridge set up with null deflection in the galvanometer.
The value of the unknown resistor $$R$$ is
Two full turns of the circular scale of a screw gauge cover a distance of 1 mm on its main scale. The total number of divisions on the circular scale is 50. Further, it is found that the screw gauge has a zero error of $$-0.03$$ mm while measuring the diameter of a thin wire, a student notes the main scale reading of 3 mm and the number of circular scale divisions in line with the main scale as 35. The diameter of the wire is
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Which one of the following constitutes a group of the isoelectronic species?
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The ionization enthalpy of hydrogen atom is $$1.312 \times 10^6\ J\ mol^{-1}$$. The energy required to excite the electron in the atom from $$n = 1$$ to $$n = 2$$ is
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Which one of the following pairs of species have the same bond order?
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Oxidising power of chlorine in aqueous solution can be determined by the parameters indicated below: $$$\frac{1}{2}Cl_2(g) \xrightarrow{\frac{1}{2}\Delta_{diss}H^{\ominus}} Cl(g) \xrightarrow{\Delta_{eg}H^{\ominus}} Cl^{-}(g) \xrightarrow{\Delta_{hyd}H^{\ominus}} Cl^{-}(aq).$$$ The energy involved in the conversion of $$\frac{1}{2}Cl_2(g)$$ to $$Cl^{-}(g)$$ (using the data, $$\Delta_{diss}H^{\ominus}_{Cl_2} = 240\ kJ\ mol^{-1},\ \Delta_{eg}H^{\ominus}_{Cl} = -349\ kJ\ mol^{-1},\ \Delta_{hyd}H^{\ominus}_{Cl^{-}} = -381\ kJ\ mol^{-1}$$) will be
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Standard entropy of $$X_2, Y_2$$ and $$XY_3$$ are 60, 40 and 50 $$JK^{-1}\ mol^{-1}$$, respectively. For the reaction, $$\frac{1}{2}X_2 + \frac{3}{2}Y_2 \to XY_3,\ \Delta H = -30\ kJ$$, to be at equilibrium, the temperature will be
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The equilibrium constants $$K_{P_1}$$ and $$K_{P_2}$$ for the reactions $$X \rightleftharpoons 2Y$$ and $$Z \rightleftharpoons P + Q$$, respectively are in the ratio of 1 : 9. If the degree of dissociation of $$X$$ and $$Z$$ be equal then the ratio of total pressure at these equilibria is
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For the following three reactions $$a, b$$ and $$c$$, equilibrium constants are given: (a) $$CO(g) + H_2O(g) \rightleftharpoons CO_2(g) + H_2(g);\ K_1$$ (b) $$CH_4(g) + H_2O(g) \rightleftharpoons CO(g) + 3H_2(g);\ K_2$$ (c) $$CH_4(g) + 2H_2O(g) \rightleftharpoons CO_2(g) + 4H_2(g);\ K_3$$ Which of the following relations is correct?
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Four species are listed below: i. $$HCO_3^{-}$$ ii. $$H_3O^{+}$$ iii. $$HSO_4^{-}$$ iv. $$HSO_3F$$. Which one of the following is the correct sequence of their acid strength?
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The $$pK_a$$ of a weak acid, HA, is 4.80. The $$pK_b$$ of a weak base, BOH, is 4.78. The pH of an aqueous solution of the corresponding salt, BA, will be
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In context with the industrial preparation of hydrogen from water gas $$(CO + H_2)$$, which of the following is the correct statement?
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Which one of the following is the correct statement?
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Among the following substituted silanes the one which will give rise to cross linked silicone polymer on hydrolysis is
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In Kjeldahl's method for the estimation of nitrogen, $$1.4\text{ g}$$ of an organic compound was digested with concentrated $$H_2SO_4$$ and the mixture was then distilled with an excess of $$NaOH$$. The ammonia gas evolved was completely absorbed in $$50\text{ mL}$$ of $$0.5\text{ M } H_2SO_4$$.
The residual unreacted acid required $$60\text{ mL}$$ of $$0.5\text{ M } NaOH$$ for complete neutralization.
Calculate the percentage by mass of nitrogen in the organic compound.
(Round off to the nearest integer)
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The correct decreasing order of priority for the functional groups of organic compounds in the IUPAC system of nomenclature is
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Toluene is nitrated and the resulting product is reduced with tin and hydrochloric acid. The product so obtained is diazotised and then heated with cuprous bromide. The reaction mixture so formed contains
In the following sequence of reactions, the alkene affords the compound '$$B$$' $$$CH_3 CH = CHCH_3 \xrightarrow{O_3} A \xrightarrow{H_2 O,\ Zn} B.$$$ The compound B is
The electrophile, $$E^{\oplus}$$ attacks the benzene ring to generate the intermediate $$\sigma$$-complex. Of the following, which $$\sigma$$-complex is of lowest energy?
The treatment of $$CH_3 MgX$$ with $$CH_3 C \equiv C - H$$ produces
Identify the wrong statements in the following:
In a compound atoms of element $$Y$$ from ccp lattice and those of element $$X$$ occupy $$2/3^{rd}$$ of tetrahedral voids. The formula of the compound will be
At $$80^\circ C$$, the vapour pressure of pure liquid '$$A$$' is 520 mmHg and that of pure liquid '$$B$$' is 1000 mmHg. If a mixture solution of '$$A$$' and '$$B$$' boils at $$80^\circ C$$ and 1 atm pressure, the amount of '$$A$$' in the mixture is (1 atm = 760 mmHg)
The vapour pressure of water at $$20^\circ C$$ is 17.5 mmHg. If 18 g of glucose $$(C_6H_{12}O_6)$$ is added to 178.2 g of water at $$20^\circ C$$, the vapour pressure of the resulting solution will be
Given $$E^\circ_{Cr^{3+}/Cr} = -0.72\ V,\ E^\circ_{Fe^{2+}/Fe} = -0.42\ V$$. The potential for the cell $$Cr|Cr^{3+}(0.1M)\ ||\ Fe^{2+}(0.01M)|Fe$$ is
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For a reaction $$\frac{1}{2}A \to 2B$$, rate of disappearance of '$$A$$' is related to the rate of appearance of '$$B$$' by the expression
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Gold numbers of protective colloids $$A, B, C$$ and $$D$$ are 0.50, 0.01, 0.10 and 0.005, respectively. The correct order of their protective powers is
The hydrocarbon which can react with sodium in liquid ammonia is
Which of the following factors is of no significance for roasting sulphide ores to the oxides and not subjecting the sulphide ores to carbon reduction directly?
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Larger number of oxidation states are exhibited by the actinoids than those by the lanthanoids, the main reason being
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Amount of oxalic acid present in a solution can be determined by its titration with $$KMnO_4$$ solution in the presence of $$H_2SO_4$$. The titration gives unsatisfactory result when carried out in the presence of HCl, because HCl
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The coordination number and the oxidation state of the element '$$E$$' in the complex $$[E(en)_2 (C_2 O_4)] NO_2$$ (where (en) is ethylene diamine) are, respectively,
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In which of the following octahedral complexes of Co (at. no. 27), will the magnitude of $$\Delta_o$$ be the highest?
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The organic chloro compound, which shows complete stereochemical inversion during a $$S_N 2$$ reaction, is
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Phenol, when it first reacts with concentrated sulphuric acid and then with concentrated nitric acid, gives
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Bakelite is obtained from phenol by reacting with
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$$\alpha - D - (+)$$-glucose and $$\beta - D - (+) - $$glucose are
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Statement-1: For every natural number $$n \ge 2$$, $$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \ldots + \frac{1}{\sqrt{n}} > \sqrt{n}$$. Statement-2: For every natural number $$n \ge 2$$, $$\sqrt{n(n+1)} < n + 1$$.
The quadratic equations $$x^2 - 6x + a = 0$$ and $$x^2 - cx + 6 = 0$$ have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is
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The conjugate of a complex number is $$\frac{1}{i-1}$$. Then the complex number is
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In a shop there are five types of ice-creams available. A child buys six ice-creams. Statement-1: The number of different ways the child can buy the six ice-creams is $${}^{10}C_5$$. Statement-2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging $$6\ A's$$ and $$4\ B's$$ in a row.
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How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two $$S$$ are adjacent?
The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is
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Statement-1: $$\sum_{r=0}^{n}(r+1) {}^n C_r = (n+2) 2^{n-1}$$. Statement-2: $$\sum_{r=0}^{n}(r+1) {}^n C_r x^r = (1+x)^n + nx(1+x)^{n-1}$$.
The perpendicular bisector of the line segment joining $$P(1, 4)$$ and $$Q(k, 3)$$ has y-intercept -4. Then a possible value of $$k$$ is
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The point diametrically opposite to the point $$P(1, 0)$$ on the circle $$x^2 + y^2 + 2x + 4y - 3 = 0$$ is
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A parabola has the origin as its focus and the line $$x = 2$$ as the directrix. Then the vertex of the parabola is at
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A focus of an ellipse is at the origin. The directrix is the line $$x = 4$$ and the eccentricity is $$1/2$$. Then the length of the semi-major axis is
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Let $$p$$ be the statement "$$x$$ is an irrational number", $$q$$ be the statement "$$y$$ is a transcendental number", and $$r$$ be the statement "$$x$$ is a rational number iff $$y$$ is a transcendental number". Statement-1: $$r$$ is equivalent to either $$q$$ or $$p$$. Statement-2: $$r$$ is equivalent to $$\sim(p \leftrightarrow \sim q)$$.
The statement $$p \to (q \to p)$$ is equivalent to
The mean of the numbers $$a, b, 8, 5, 10$$ is 6 and the variance is 6.80. Then which one of the following gives possible values of $$a$$ and $$b$$?
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$$AB$$ is a vertical pole with $$B$$ at the ground level and $$A$$ at the top. A man finds that the angle of elevation of the point $$A$$ from a certain point $$C$$ on the ground is $$60^\circ$$. He moves away from the pole along the line $$BC$$ to a point $$D$$ such that $$CD = 7$$ m. From $$D$$ the angle of elevation of the point $$A$$ is $$45^\circ$$. Then the height of the pole is
Let $$R$$ be the real line. Consider the following subsets of the plane $$R \times R$$: $$S = \{(x, y) : y = x + 1\ \text{and}\ 0 < x < 2\},\ T = \{(x, y) : x - y\ \text{is an integer}\}$$. Which one of the following is true?
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Let A be a $$2 \times 2$$ matrix with real entries. Let I be the $$2 \times 2$$ identity matrix. Denote by $$tr(A)$$, the sum of diagonal entries of $$A$$. Assume that $$A^2 = I$$. Statement-1: If $$A \ne I$$ and $$A \ne -I$$, then $$\det A = -1$$. Statement-2: If $$A \ne I$$ and $$A \ne -I$$, then $$tr(A) \ne 0$$.
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Let $$A$$ be a square matrix all of whose entries are integers. Then which one of the following is true?
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Let $$a, b, c$$ be any real numbers. Suppose that there are real numbers $$x, y, z$$ not all zero such that $$x = cy + bz,\ y = az + cx$$ and $$z = bx + ay$$. Then $$a^2 + b^2 + c^2 + 2abc$$ is equal to
The value of $$\cot\left(\operatorname{cosec}^{-1} \frac{5}{3} + \tan^{-1} \frac{2}{3}\right)$$ is
Let $$f: N \to Y$$ be a function defined as $$f(x) = 4x + 3$$, where $$Y = \{y \in N : y = 4x + 3$$ for some $$x \in N\}$$. Show that f is invertible and its inverse is
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Let $$f(x) = \begin{cases} (x - 1)\sin\left(\frac{1}{x - 1}\right), & \text{if } x \ne 1 \\ 0, & \text{if } x = 1 \end{cases}$$. Then which one of the following is true?
Suppose the cube $$x^3 - px + q$$ has three distinct real roots where $$p > 0$$ and $$q > 0$$. Then which one of the following holds?
How many real solutions does the equation $$x^7 + 14 x^5 + 16 x^3 + 30 x - 560 = 0$$ have?
The value of $$\sqrt{2} \int \frac{\sin x\, dx}{\sin\left(x - \frac{\pi}{4}\right)}$$ is
Let $$I = \int_0^1 \frac{\sin x}{\sqrt{x}} dx$$ and $$J = \int_0^1 \frac{\cos x}{\sqrt{x}} dx$$. Then which one of the following is true?
The area of the plane region bounded by the curves $$x + 2y^2 = 0$$ and $$x + 3y^2 = 1$$ is equal to
The solution of the differential equation $$\frac{dy}{dx} = \frac{x + y}{x}$$ satisfying the condition $$y(1) = 1$$ is
The differential equation of the family of circles with fixed radius 5 units and centre on the line $$y = 2$$ is
The non-zero vectors $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ are related by $$\vec{a} = 8\vec{b}$$ and $$\vec{c} = -7\vec{b}$$. Then the angle between $$\vec{a}$$ and $$\vec{c}$$ is
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The vector $$\vec{a} = \alpha \hat{i} + 2\hat{j} + \beta \hat{k}$$ lies in the plane of the vectors $$\vec{b} = \hat{i} + \hat{j}$$ and $$\vec{c} = \hat{j} + \hat{k}$$ and bisects the angle between $$\vec{b}$$ and $$\vec{c}$$. Then which one of the following gives possible values of $$\alpha$$ and $$\beta$$?
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The line passing through the points $$(5, 1, a)$$ and $$(3, b, 1)$$ crosses the $$yz$$-plane at the point $$\left(0, \frac{17}{2}, \frac{-13}{2}\right)$$. Then
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If the straight lines $$\frac{x - 1}{k} = \frac{y - 2}{2} = \frac{z - 3}{3}$$ and $$\frac{x - 2}{3} = \frac{y - 3}{k} = \frac{z - 1}{2}$$ intersect at a point, then the integer $$k$$ is equal to
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It is given that the events $$A$$ and $$B$$ are such that $$P(A) = \frac{1}{4},\ P\left(\frac{A}{B}\right) = \frac{1}{2}$$ and $$P\left(\frac{B}{A}\right) = \frac{2}{3}$$. Then $$P(B)$$ is
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A die is thrown. Let $$A$$ be the event that the number obtained is greater than 3. Let $$B$$ be the event that the number obtained is less than 5. Then $$P(A \cup B)$$ is
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