In an expression $$a \times 10^b$$;
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In an expression $$a \times 10^b$$;
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Young's modulus is determined by the equation given by $$Y = 49000 \frac{m}{l} \frac{dyn}{cm^2}$$ where $$M$$ is the mass and $$l$$ is the extension of wire used in the experiment. Now error in Young modulus $$(Y)$$ is estimated by taking data from $$M - l$$ plot in graph paper. The smallest scale divisions are $$5 \text{ g}$$ and $$0.02 \text{ cm}$$ along load axis and extension axis respectively. If the value of $$M$$ and $$l$$ are $$500 \text{ g}$$ and $$2 \text{ cm}$$ respectively then percentage error of $$Y$$ is :
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A clock has $$75 \text{ cm}$$ long second hand and $$60 \text{ cm}$$ minute hand respectively. In 30 minutes duration the tip of second hand will travel $$x$$ distance more than the tip of minute hand. The value of $$x$$ in meter is nearly (Take $$\pi = 3.14$$) :
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A stationary particle breaks into two parts of masses $$m_A$$ and $$m_B$$ which move with velocities $$v_A$$ and $$v_B$$ respectively. The ratio of their kinetic energies $$(K_B : K_A)$$ is :
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Three bodies A, B and C have equal kinetic energies and their masses are $$400 \text{ g}$$, $$1.2 \text{ kg}$$ and $$1.6 \text{ kg}$$ respectively. The ratio of their linear momenta is :
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A player caught a cricket ball of mass $$150 \text{ g}$$ moving at a speed of $$20 \text{ m/s}$$. If the catching process is completed in $$0.1 \text{ s}$$, the magnitude of force exerted by the ball on the hand of the player is:
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Two planets $$A$$ and $$B$$ having masses $$m_1$$ and $$m_2$$ move around the sun in circular orbits of $$r_1$$ and $$r_2$$ radii respectively. If angular momentum of $$A$$ is $$L$$ and that of $$B$$ is $$3L$$, the ratio of time period $$\left(\frac{T_A}{T_B}\right)$$ is:
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Correct Bernoulli's equation is (symbols have their usual meaning) :
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Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P-V diagram. The relation between the ratio $$\frac{V_a}{V_d}$$ and the ratio $$\frac{V_b}{V_c}$$ is:
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A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature $$(27°C)$$. The ratio of specific heat of gases at constant volume respectively is:
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Two charged conducting spheres of radii $$a$$ and $$b$$ are connected to each other by a conducting wire. The ratio of charges of the two spheres respectively is:
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In the given circuit, the terminal potential difference of the cell is :
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Paramagnetic substances:
A. align themselves along the directions of external magnetic field.
B. attract strongly towards external magnetic field.
C. has susceptibility little more than zero.
D. move from a region of strong magnetic field to weak magnetic field.
Choose the most appropriate answer from the options given below:
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A LCR circuit is at resonance for a capacitor $$C$$, inductance $$L$$ and resistance $$R$$. Now the value of resistance is halved keeping all other parameters same. The current amplitude at resonance will be now:
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Critical angle of incidence for a pair of optical media is $$45°$$. The refractive indices of first and second media are in the ratio:
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A proton and an electron are associated with same de-Broglie wavelength. The ratio of their kinetic energies is: (Assume $$h = 6.63 \times 10^{-34} \text{ J s}$$, $$m_e = 9.0 \times 10^{-31} \text{ kg}$$ and $$m_p = 1836 \; m_e$$)
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Average force exerted on a non-reflecting surface at normal incidence is $$2.4 \times 10^{-4} \text{ N}$$. If $$360 \text{ W/cm}^2$$ is the light energy flux during span of 1 hour 30 minutes, Then the area of the surface is:
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Binding energy of a certain nucleus is $$18 \times 10^8 \text{ J}$$. How much is the difference between total mass of all the nucleons and nuclear mass of the given nucleus:
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The output Y of following circuit for given inputs is :
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The diameter of a sphere is measured using a vernier caliper whose 9 divisions of main scale are equal to 10 divisions of vernier scale. The shortest division on the main scale is equal to $$1 \text{ mm}$$. The main scale reading is $$2 \text{ cm}$$ and second division of vernier scale coincides with a division on main scale. If mass of the sphere is $$8.635 \text{ g}$$, the density of the sphere is:
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Three vectors $$\vec{OP}$$, $$\vec{OQ}$$ and $$\vec{OR}$$ each of magnitude $$A$$ are acting as shown in figure. The resultant of the three vectors is $$A\sqrt{x}$$. The value of $$x$$ is ________.
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A uniform thin metal plate of mass $$10 \text{ kg}$$ with dimensions is shown in the figure below. The ratio of $$x$$ and $$y$$ coordinates of center of mass of the plate is $$\frac{n}{9}$$. The value of $$n$$ is ________.
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A liquid column of height $$0.04 \text{ cm}$$ balances excess pressure of a soap bubble of certain radius. If density of liquid is $$8 \times 10^3 \text{ kg m}^{-3}$$ and surface tension of soap solution is $$0.28 \text{ Nm}^{-1}$$, then diameter of the soap bubble is ______ cm. (if $$g = 10 \text{ m s}^{-2}$$)
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A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is $$\left(\frac{a-1}{a}\right)$$ then the value of $$a$$ is ________.
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An electric field, $$\vec{E} = \frac{2\hat{i} + 6\hat{j} + 8\hat{k}}{\sqrt{6}}$$ passes through the surface of $$4 \text{ m}^2$$ area having unit vector $$\hat{n} = \left(\frac{2\hat{i} + \hat{j} + \hat{k}}{\sqrt{6}}\right)$$. The electric flux for that surface is ______ Vm.
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Resistance of a wire at $$0°C$$, $$100°C$$ and $$t°C$$ is found to be $$10\Omega$$, $$10.2\Omega$$ and $$10.95\Omega$$ respectively. The temperature $$t$$ in Kelvin scale is ________.
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An electron with kinetic energy $$5 \text{ eV}$$ enters a region of uniform magnetic field of $$3 \; \mu T$$ perpendicular to its direction. An electric field $$E$$ is applied perpendicular to the direction of velocity and magnetic field. The value of $$E$$, so that electron moves along the same path, is ______ $$\text{NC}^{-1}$$. (Given, mass of electron $$= 9 \times 10^{-31} \text{ kg}$$, electric charge $$= 1.6 \times 10^{-19} \text{ C}$$)
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A square loop PQRS having 10 turns, area $$3.6 \times 10^{-3} \text{ m}^2$$ and resistance $$100\Omega$$ is slowly and uniformly being pulled out of a uniform magnetic field of magnitude $$B = 0.5 \text{ T}$$ as shown. Work done in pulling the loop out of the field in $$1.0 \text{ s}$$ is ______ $$\times 10^{-8} \text{ J}$$.
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A parallel beam of monochromatic light of wavelength $$600 \text{ nm}$$ passes through single slit of $$0.4 \text{ mm}$$ width. Angular divergence corresponding to second order minima would be ______ $$\times 10^{-3} \text{ rad}$$.
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In an alpha particle scattering experiment distance of closest approach for the $$\alpha$$ particle is $$4.5 \times 10^{-14} \text{ m}$$. If target nucleus has atomic number 80, then maximum velocity of $$\alpha$$-particle is ______ $$\times 10^5 \text{ m/s}$$ approximately. $$\left(\frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ SI unit, mass of } \alpha \text{ particle} = 6.72 \times 10^{-27} \text{ kg}\right)$$
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Combustion of glucose $$(C_6H_{12}O_6)$$ produces $$CO_2$$ and water. The amount of oxygen (in g) required for the complete combustion of $$900 \text{ g}$$ of glucose is : [Molar mass of glucose in $$\text{gmol}^{-1} = 180$$]
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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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Match List I with List II
Choose the correct answer from the options given below:
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Given below are two statements: Statement I: $$N(CH_3)_3$$ and $$P(CH_3)_3$$ can act as ligands to form transition metal complexes. Statement II: As N and P are from same group, the nature of bonding of $$N(CH_3)_3$$ and $$P(CH_3)_3$$ is always same with transition metals. In the light of the above statements, choose the most appropriate answer from the options given below:
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For the given hypothetical reactions, the equilibrium constants are as follows: $$X \rightleftharpoons Y; K_1 = 1.0$$, $$Y \rightleftharpoons Z; K_2 = 2.0$$, $$Z \rightleftharpoons W; K_3 = 4.0$$. The equilibrium constant for the reaction $$X \rightleftharpoons W$$ is
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Among the following halogens $$F_2, Cl_2, Br_2$$ and $$I_2$$. Which can undergo disproportionation reactions?
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Thiosulphate reacts differently with iodine and bromine in the reactions given below: $$2S_2O_3^{2-} + I_2 \rightarrow S_4O_6^{2-} + 2I^-$$, $$S_2O_3^{2-} + 5Br_2 + 5H_2O \rightarrow 2SO_4^{2-} + 4Br^- + 10H^+$$. Which of the following statement justifies the above dual behaviour of thiosulphate?
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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 stability order of +1 oxidation state of Ga, In and Tl is $$Ga < In < Tl$$. Reason R: The inert pair effect stabilizes the lower oxidation state down the group. In the light of the above statements, choose the correct answer from the options given below:
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Which of the following are aromatic?
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Given below are two statements:
Statement I: IUPAC name of Compound A is 4-chloro-1,3-dinitrobenzene.
Statement II: IUPAC name of Compound B is 4-ethyl-2-methylaniline. In the light of the above statements, choose the most appropriate answer from the options given below:
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In the given compound, the number of $$2°$$ carbon atom/s is _______.
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Iron (III) catalyses the reaction between iodide and persulphate ions, in which
A. $$Fe^{3+}$$ oxidises the iodide ion
B. $$Fe^{3+}$$ oxidises the persulphate ion
C. $$Fe^{2+}$$ reduces the iodide ion
D. $$Fe^{2+}$$ reduces the persulphate ion.
Choose the most appropriate answer from the options given below:
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Number of Complexes with even number of electrons in $$t_{2g}$$ orbitals is - $$[Fe(H_2O)_6]^{2+}$$, $$[Co(H_2O)_6]^{2+}$$, $$[Co(H_2O)_6]^{3+}$$, $$[Cu(H_2O)_6]^{2+}$$, $$[Cr(H_2O)_6]^{2+}$$
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An octahedral complex with the formula $$CoCl_3 \cdot nNH_3$$ upon reaction with excess of $$AgNO_3$$ solution gives 2 moles of $$AgCl$$. Consider the oxidation state of $$Co$$ in the complex is '$$x$$'. The value of "$$x + n$$" is ______
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Which among the following compounds will undergo fastest $$S_N2$$ reaction.
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Identify the major products A and B respectively in the following set of reactions.
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Identify the product (P) in the following reaction:
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Match List I with List II
Choose the correct answer from the options given below:
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The incorrect statement regarding the given structure is
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A hypothetical electromagnetic wave is shown below. The frequency of the wave is $$x \times 10^{19}$$ Hz. $$x =$$ ______ (nearest integer)
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Number of molecules from the following which are exceptions to octet rule is ______ $$CO_2$$, $$NO_2$$, $$H_2SO_4$$, $$BF_3$$, $$CH_4$$, $$SiF_4$$, $$ClO_2$$, $$PCl_5$$, $$BeF_2$$, $$C_2H_6$$, $$CHCl_3$$, $$CBr_4$$
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Consider the figure provided. 1 mol of an ideal gas is kept in a cylinder, fitted with a piston, at the position A, at $$18°C$$. If the piston is moved to position B, keeping the temperature unchanged, then '$$x$$' L atm work is done in this reversible process. $$x =$$ ______ L atm. (nearest integer) [Given: Absolute temperature $$= °C + 273.15$$, $$R = 0.08206 \text{ L atm mol}^{-1} \text{K}^{-1}$$]
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The number of optical isomers in following compound is: ________
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A solution containing $$10 \text{ g}$$ of an electrolyte $$AB_2$$ in $$100 \text{ g}$$ of water boils at $$100.52°C$$. The degree of ionization of the electrolyte $$(\alpha)$$ is ______ $$\times 10^{-1}$$. (nearest integer) [Given: Molar mass of $$AB_2 = 200 \text{ g mol}^{-1}$$, $$K_b$$ (molal boiling point elevation const. of water) $$= 0.52 \text{ K kg mol}^{-1}$$, boiling point of water $$= 100°C$$; $$AB_2$$ ionises as $$AB_2 \rightarrow A^{2+} + 2B^-$$]
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Consider the following reaction $$A + B \rightarrow C$$. The time taken for A to become $$1/4^{th}$$ of its initial concentration is twice the time taken to become $$1/2$$ of the same. Also, when the change of concentration of $$B$$ is plotted against time, the resulting graph gives a straight line with a negative slope and a positive intercept on the concentration axis. The overall order of the reaction is ________
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The 'spin only' magnetic moment value of $$MO_4^{2-}$$ is ______ BM. (Where M is a metal having least metallic radii among $$Sc, Ti, V, Cr, Mn$$ and $$Zn$$). (Given atomic number: $$Sc = 21, Ti = 22, V = 23, Cr = 24, Mn = 25$$ and $$Zn = 30$$)
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Major product $$B$$ of the following reaction has ______ $$\pi$$-bond.
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If $$279 \text{ g}$$ of aniline is reacted with one equivalent of benzenediazonium chloride, the maximum amount of aniline yellow formed will be ______ g. (nearest integer) (consider complete conversion).
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Number of amine compounds from the following giving solids which are soluble in NaOH upon reaction with Hinsberg's reagent is ________
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The sum of all the solutions of the equation $$(8)^{2x} - 16 \cdot (8)^x + 48 = 0$$ is :
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Let $$z$$ be a complex number such that $$|z + 2| = 1$$ and $$\text{Im}\left(\frac{z+1}{z+2}\right) = \frac{1}{5}$$. Then the value of $$|\text{Re}(z + 2)|$$ is
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If the set $$R = \{(a, b) : a + 5b = 42, a, b \in \mathbb{N}\}$$ has $$m$$ elements and $$\sum_{n=1}^{m}(1 - i^{n!}) = x + iy$$, where $$i = \sqrt{-1}$$, then the value of $$m + x + y$$ is
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If $$\sin x = -\frac{3}{5}$$, where $$\pi < x < \frac{3\pi}{2}$$, then $$80(\tan^2 x - \cos x)$$ is equal to
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The equations of two sides AB and AC of a triangle ABC are $$4x + y = 14$$ and $$3x - 2y = 5$$, respectively. The point $$\left(2, -\frac{4}{3}\right)$$ divides the third side BC internally in the ratio $$2 : 1$$. The equation of the side BC is
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Let the circles $$C_1 : (x - \alpha)^2 + (y - \beta)^2 = r_1^2$$ and $$C_2 : (x - 8)^2 + \left(y - \frac{15}{2}\right)^2 = r_2^2$$ touch each other externally at the point $$(6, 6)$$. If the point $$(6, 6)$$ divides the line segment joining the centres of the circles $$C_1$$ and $$C_2$$ internally in the ratio $$2 : 1$$, then $$(\alpha + \beta) + 4(r_1^2 + r_2^2)$$ equals
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Let $$H : \frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1$$ be the hyperbola, whose eccentricity is $$\sqrt{3}$$ and the length of the latus rectum is $$4\sqrt{3}$$. Suppose the point $$(\alpha, 6), \alpha > 0$$ lies on $$H$$. If $$\beta$$ is the product of the focal distances of the point $$(\alpha, 6)$$, then $$\alpha^2 + \beta$$ is equal to
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Let $$A = \begin{bmatrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{bmatrix}$$. If $$A^3 = 4A^2 - A - 21I$$, where $$I$$ is the identity matrix of order $$3 \times 3$$, then $$2a + 3b$$ is equal to
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Let $$[t]$$ be the greatest integer less than or equal to $$t$$. Let $$A$$ be the set of all prime factors of 2310 and $$f : A \rightarrow \mathbb{Z}$$ be the function $$f(x) = \left[\log_2\left(x^2 + \left[\frac{x^3}{5}\right]\right)\right]$$. The number of one-to-one functions from $$A$$ to the range of $$f$$ is
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For the function $$f(x) = (\cos x) - x + 1, x \in \mathbb{R}$$, between the following two statements (S1) $$f(x) = 0$$ for only one value of $$x$$ in $$[0, \pi]$$. (S2) $$f(x)$$ is decreasing in $$\left[0, \frac{\pi}{2}\right]$$ and increasing in $$\left[\frac{\pi}{2}, \pi\right]$$.
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Let $$f(x) = 4\cos^3 x + 3\sqrt{3}\cos^2 x - 10$$. The number of points of local maxima of $$f$$ in interval $$(0, 2\pi)$$ is
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The number of critical points of the function $$f(x) = (x - 2)^{2/3}(2x + 1)$$ is
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Let $$I(x) = \int \frac{6}{\sin^2 x (1 - \cot x)^2} dx$$. If $$I(0) = 3$$, then $$I\left(\frac{\pi}{12}\right)$$ is equal to
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The value of $$k \in \mathbb{N}$$ for which the integral $$I_n = \int_0^1 (1 - x^k)^n dx, n \in \mathbb{N}$$, satisfies $$147I_{20} = 148I_{21}$$ is
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Let $$f(x)$$ be a positive function such that the area bounded by $$y = f(x), y = 0$$ from $$x = 0$$ to $$x = a > 0$$ is $$e^{-a} + 4a^2 + a - 1$$. Then the differential equation, whose general solution is $$y = c_1 f(x) + c_2$$, where $$c_1$$ and $$c_2$$ are arbitrary constants, is
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Let $$y = y(x)$$ be the solution of the differential equation $$(1 + y^2)e^{\tan x} dx + \cos^2 x(1 + e^{2\tan x}) dy = 0, y(0) = 1$$. Then $$y\left(\frac{\pi}{4}\right)$$ is equal to
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The set of all $$\alpha$$, for which the vectors $$\vec{a} = \alpha t\hat{i} + 6\hat{j} - 3\hat{k}$$ and $$\vec{b} = t\hat{i} - 2\hat{j} - 2\alpha t\hat{k}$$ are inclined at an obtuse angle for all $$t \in \mathbb{R}$$, is
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If the shortest distance between the lines $$L_1 : \vec{r} = (2 + \lambda)\hat{i} + (1 - 3\lambda)\hat{j} + (3 + 4\lambda)\hat{k}, \lambda \in \mathbb{R}$$ and $$L_2 : \vec{r} = 2(1 + \mu)\hat{i} + 3(1 + \mu)\hat{j} + (5 + \mu)\hat{k}, \mu \in \mathbb{R}$$ is $$\frac{m}{\sqrt{n}}$$, where $$\gcd(m, n) = 1$$, then the value of $$m + n$$ equals
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Let $$P(x, y, z)$$ be a point in the first octant, whose projection in the $$xy$$-plane is the point $$Q$$. Let $$OP = \gamma$$; the angle between $$OQ$$ and the positive $$x$$-axis be $$\theta$$; and the angle between $$OP$$ and the positive $$z$$-axis be $$\phi$$, where $$O$$ is the origin. Then the distance of $$P$$ from the $$x$$-axis is
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Let the sum of two positive integers be 24. If the probability, that their product is not less than $$\frac{3}{4}$$ times their greatest possible product, is $$\frac{m}{n}$$, where $$\gcd(m, n) = 1$$, then $$n - m$$ equals
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The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to ________
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Let the positive integers be written in the form:
If the $$k^{th}$$ row contains exactly $$k$$ numbers for every natural number $$k$$, then the row in which the number 5310 will be, is ________
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Let $$\alpha = \sum_{r=0}^{n}(4r^2 + 2r + 1)^nC_r$$ and $$\beta = \left(\sum_{r=0}^{n}\frac{^nC_r}{r+1}\right) + \frac{1}{n+1}$$. If $$140 < \frac{2\alpha}{\beta} < 281$$, then the value of $$n$$ is ________
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If the orthocentre of the triangle formed by the lines $$2x + 3y - 1 = 0$$, $$x + 2y - 1 = 0$$ and $$ax + by - 1 = 0$$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $$(3, 4)$$ and $$(-6, -8)$$, then the value of $$|a - b|$$ is ________
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The value of $$\lim_{x \to 0} 2\left(\frac{1 - \cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x} \cdots \sqrt[10]{\cos 10x}}{x^2}\right)$$ is
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Let $$A = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}$$. If the sum of the diagonal elements of $$A^{13}$$ is $$3^n$$, then $$n$$ is equal to ________
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If the range of $$f(\theta) = \frac{\sin^4\theta + 3\cos^2\theta}{\sin^4\theta + \cos^2\theta}, \theta \in \mathbb{R}$$ is $$[\alpha, \beta]$$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $$\frac{\alpha}{\beta}$$, is equal to ________
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Let the area of the region enclosed by the curve $$y = \min\{\sin x, \cos x\}$$ and the $$x$$ axis between $$x = -\pi$$ to $$x = \pi$$ be $$A$$. Then $$A^2$$ is equal to ________
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Let $$\vec{a} = 9\hat{i} - 13\hat{j} + 25\hat{k}$$, $$\vec{b} = 3\hat{i} + 7\hat{j} - 13\hat{k}$$ and $$\vec{c} = 17\hat{i} - 2\hat{j} + \hat{k}$$ be three given vectors. If $$\vec{r}$$ is a vector such that $$\vec{r} \times \vec{a} = (\vec{b} + \vec{c}) \times \vec{a}$$ and $$\vec{r} \cdot (\vec{b} - \vec{c}) = 0$$, then $$\frac{|593\vec{r} + 67\vec{a}|^2}{(593)^2}$$ is equal to ________
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Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $$X$$ and $$Y$$ respectively denote the number of blue and yellow balls. If $$\bar{X}$$ and $$\bar{Y}$$ are the means of $$X$$ and $$Y$$ respectively, then $$7\bar{X} + 4\bar{Y}$$ is equal to ________
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