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Geometrical shapes of the complexes formed by the reaction of $$Ni^{2+}$$ with $$Cl^-, CN^-$$ ans $$H_2O$$, respectively, are
$$AgNO_3$$ (aq.) was added to an aqueous KCl solution gradually and the conductivity of the solution was measured. The plot of conductance $$(\bigwedge)$$ versus the volume of $$AgNO_3$$ is-
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Bombardment of aluminum by $$\alpha$$- particle leads to its artificial disintegration in two ways, (i) and (ii) as shown. Products X, Y and Z respectively are-
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Extra pure $$N_2$$ can be obtained by heating-
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Among the following compounds, the most acidic is -
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The major product of the following reaction is -
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Dissolving 120 g of urea (mol. wt. 60) in 1000 g of water gave a solution of density 1.15 g/mL. The molarity of the solution is -
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Extraction of metal from the ore cassiterite involves -
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Amongst the given options, the compound(s) in which all the atoms are in one plane in all the possible conformations (if any), is (are)
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The correct statement(s) pertaining to the adsorption of a gas on a solid surface is (are)
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According to kinetic theory of gases -
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When a metal rod M is dipped into an aqueous colourless concentrated solution of compound N, the solution turns light blue. Addition of aqueous NaCl to the blue solution gives a white precipitate O. Addition of aq. $$NH_3$$ dissolves O and gives an intense blue solution.
The metal rod M is -
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The compounds N is -
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The final solution contains -
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An acyclic hydrocarbon P, having molecular formula $$C_6H_{10}$$, gave acetone as the only organic product through the following sequence of reactions, in which Q is an intermediate organic compound.
The structure of compound P is -
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The structure of compound Q is -
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Reaction of $$Br_2$$ with $$Na_2CO_3$$ in aq. solution gives sodium bromide and sodium bromate with evolution of $$CO_2$$ gas. The number of sodium bromide molecules involved in the balanced chemical equation is.
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The difference in the oxidation numbers of the two types of sulphur atoms in $$Na_2S_4O_6$$ is.
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The maximum number of electrons that can have principal quantum number, n = 3 and spin quantum number, ms = - 1/2, is.
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A decapeptide (mol. wt. 796) on complete hydrolysis gives glycine (mol. wt. 75), alanine and phenylalanine. Glycine contributes 47.0% to the total weight of the hydrolysed products. The number of glycine units present in the decapeptide is.
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To an evacuated vessel with movable piston under external pressure of 1 atm., 0.1 mol of He and 1.0 mol of an unknown compound (vapour pressure 0.68 atm, at $$0^\circ C$$) are introduced. Considering the ideal gas behaviour, the total volume (in litre) of the gases at 0ºC is close to.
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The total number of alkenes possible by dehydrobromination of 3-bromo-3-cyclopentylhexane using alcoholic KOH is.
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The work function $$(\phi)$$ of some metals is listed below. The number of metals which will show photoelectric effect when light of 300 nm wavelength falls on the metal is.
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5.6 liter of helium gas at STP is adiabatically compressed to 0.7 liter. Taking the initial temperature to be $$T_1$$, the work done in the process is-
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A ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m. The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of angular velocity of ball (in radian/s) is-
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Consider an electric field $$\overrightarrow{E} = E_0\hat{x}$$, where $$E_0$$ is a constant. The flux through the shaded area (as shown in the figure) due to this field is-
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A police car with a siren of frequency 8 kHz is moving with uniform velocity 36 km/hr towards a tall building which reflects the sound waves. The speed of sound in air is 320 m/s. The frequency of the siren heard by the car driver is-
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A meter bridge is set-up as shown, to determine an unknown resistance 'X' using a standard 10 ohm resistor. The galvanometer show null point when tapping-key is at 52 cm mark. The end-corrections are 1 cm and 2 cm respectively for the ends A and B. The determine value of 'X' is-
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A 2 $$\mu F$$ capacitor is charged as shown in figure. The percentage of its stored energy dissipated after the switch S is turned to position 2 is-
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The wavelength of the first spectral line in the Balmer series of hydrogen atom is 6561 Å. The wave length of the second spectral line in the Balmer series of singly-ionized helium atom is-
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A spherical metal shall A of radius $$R_A$$ and a solid metal sphere B of radius $$R_B(< R_A)$$ are kept far apart and each is given charge '+Q'. Now they are connected by a thin metal wire. Then-
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A metal rod of length 'L' and mass 'm' is pivoted at one end. A thin disk of mass 'M' and radius 'R' (< L) is attached at its centre to the free end of the rod. Consider two ways the disc is attached :(case A) The disc is not free to rotate about its center and (case B) the disc is free to rotate about its center. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true ?
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An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true ?
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A composite book is made of slabs A, B, C, D and E of different thermal conductivities (given in terms of a constant K) and sizes (given in terms of length, L) as shown in the figure. All slabs are of same width. Heat 'Q' flow only from left to right through the blocks. Then in steady state-
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Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.
The phase space diagram for a ball thrown vertically up from ground is-
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The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $$E_1$$ and $$E_2$$ are the total mechanical energies respectively. Then-
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Consider the spring-mass system, with the mass submerged in water, as shown in figure. The phase space diagram for one cycle of this system is-
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A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let 'N' be the number density of free electrons, each of mass 'm'. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency 'wp', which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency w, where a part of the energy is absorbed and a part of it is reflected. As w approaches wp, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Taking the electronic charge as 'e' and the permittivity as $$'ε_0'$$, use dimensional analysis to determine the correct expression for $$\omega_p$$.
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Estimate the wavelength at which plasma reflection will occur for metal having the density of electrons $$N \approx 4 \times 10^{27} m^{-3}$$. Take $$ε_0 = 10^{-11}$$ and $$m \approx 10^{-30}$$, where these quantities are in proper SI unit-
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A block is moving on an inclined plane making an angle $$45^\circ$$ with the horizontal and the coefficient of friction is $$\mu$$. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down. If we define N = 10 $$\mu$$, then N is.
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A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of 0.3 $$m/s^2$$. The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is (P/10). The value of P is.
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Four point charge, each of +q, are rigidly fixed at the four corners of a square planar soap film of side 'a'. The surface tension of the soap film is $$\gamma$$. The system of charges and planar film are in equilibrium, and $$a = k\left[\frac{q^2}{\gamma}\right]^{\frac{1}{N}}$$, where 'k' is a constant. Then N is
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Four solid spheres each of diameter $$\sqrt{5}$$ cm and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm. The momentum of inertia of the system about the diagonal of the square is $$N \times 10^{-4} kg -m^2$$, then N is.
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The activity of a freshly prepared radioactive sample is $$10^{10}$$ disintegrations per second, whose mean life is $$10^{9}$$ s. The mass of an atom of this radioisotope is $$10^{-25}$$ kg. The mass (in mg) of the radioactive sample is.
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A long circular tube of length 10 m and radius 0.3 carries a current I along its curved surface as shown. A wire-loop of resistance 0.005 ohm and of radius 0.1 m is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as $$I = I_0 \cos(300 t)$$ where $$I_0$$ is constant. If the magnetic moment of the loop is $$N \mu_0 I_0 \sin(300 t)$$. then 'N' is.
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Steel wire of length 'L' at $$40^\circ C$$ is suspended from the ceiling and then a mass 'm' is hung from its free end. The wire is cooled down from $$40^\circ C$$ to $$30^\circ C$$ to region its original length 'L'. The coefficient of linear thermal expansion of the steel is $$10^{-5}/^\circ C$$, Young's modulus of steel is $$10^{11} N/m^2$$ and radius of the wire is 1 mm. Assume that L >> diameter of the wire. Then the value of 'm' in kg is nearly.
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The value of $$\int_{\sqrt{\ln 2}}^{\sqrt{\ln 3}} \frac{x \sin x^2}{\sin x^2 + \sin(\ln 6 - x^2)} dx $$ is
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Let the straight line x = b divide the area enclosed by $$y = (1 - x^2), y = 0$$ and x = 0 into two parts $$R_1(0 \leq x \leq b)$$ and $$R_2(b \leq x leq 1)$$ such that $$R_1 - R_2 = \frac{1}{4}$$. Then b equals
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Let $$\overrightarrow{a} = \hat{i} + \hat{j} + \hat{k}, \overrightarrow{b} = \hat{i} - \hat{j} + \hat{k}$$ and $$\overrightarrow{c} = \hat{i} - \hat{j} - \hat{k}$$ be three vectors. A vector $$\overrightarrow{v}$$ in the place of $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$, whose projection on $$\overrightarrow{c}$$ is $$\frac{1}{\sqrt{3}}$$, is given by
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Let $$(x_0, y_0)$$ be the solution of the following equations
$$(2x)^{\ln 2} = (3y)^{\ln 3}$$
$$3^{\ln x} = 2^{\ln y}$$
Then $$x_0$$ is
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Let $$\alpha$$ and $$\beta$$ be the roots of $$x^2 - 6x - 2 = 0$$, with $$\alpha > \beta$$. If $$a_n = \alpha^n - \beta^n$$ for $$n \geq 1$$, then the value of $$\frac{a_{10} - 2a_8}{2a_9}$$ is
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A straight line L through the point (3, -2) is inclined at an angle $$60^\circ$$ to the line $$\sqrt{3} x + y = 1$$. If L also intersects the x-axis, then the equation of L is
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Let $$P = \left\{\theta : \sin \theta - \cos \theta = \sqrt{2} \cos \theta\right\}$$ and $$Q = \left\{\theta : \sin \theta + \cos \theta = \sqrt{2} \sin \theta\right\}$$ be two sets. Then
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The vector(s) which is/are coplanar with vectors $$\hat{i} + \hat{j} + 2\hat{k}$$ and $$\hat{i} + 2\hat{j} + \hat{k}$$, and perpendicular to the vector $$\hat{i} + \hat{j} + \hat{k}$$ is/are
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Let $$f : R \rightarrow R$$ be a function such that
$$f(x + y) = f(x) + f(y), \forall x, y \in R$$
If f(x) is differentiable at x = 0, Then
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Let M and N be two $$3 \times 3$$ non-singular skew-symmetric matrices such that $$MN = NM$$. If $$P^T$$ denotes the transpose of P, then $$M^2N^2(M^TN)^{-1}(MN^{-1})^T$$ is equal to
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Let the eccentricity of the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be reciprocal to that of the ellipse $$x^2 + 4y^2 = 4$$. If the hyperbola passes through a focus of the ellipse, then
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Let a, b and c be three real numbers satisfying
$$\begin{bmatrix}a & b & c \end{bmatrix}\begin{bmatrix}1 & 9 & 7 \\8 & 2 & 7 \\7 & 3 & 7\end{bmatrix} = \begin{bmatrix}0 & 0 & 0 \end{bmatrix} .............(E)$$
If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is
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Let $$\omega$$ be a solution of $$x^3 - 1 = 0$$ with $$Im(\omega) > 0$$. If a = 2 with b and c satisfying (E), then the value of $$\frac{3}{\omega^a} + \frac{1}{\omega^b} + \frac{3}{\omega^c}$$ is equal to
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Let b = 6, with a and c satisfying (E). If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation $$ax^2 + bx + c = 0$$, then $$\sum_{n=0}^{\infty}\left(\frac{1}{\alpha} + \frac{1}{\beta}\right)^n$$ is
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Let $$U_1$$ and $$U_2$$ be two urns such that $$U_1$$ contains 3 white and 2 red balls, and $$U_2$$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $$U_1$$ and put into $$U_2$$. However, if tail appears then 2 balls are drawn at random from $$U_1$$ and put into $$U_2$$. Now 1 ball is drawn at random from $$U_2$$.
The probability of the drawn ball from $$U_2$$ being white is
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Given that the drawn ball from $$U_2$$ is white, the probability that head appeared on the coin is
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Let $$a_1, a_2, a_3, ….., a_{100}$$ be an arithmetic progression with $$a_1 = 3$$ and $$S_p = \sum_{i=1}^{p}a_i, 1 \leq p \leq 100$$. For any integer n with $$1 \leq n \leq 20$$, let $$m = 5n$$. If $$\frac{S_m}{S_n}$$ does not depend on n, then $$a_2$$ is
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Consider the parabola $$y^2 = 8x$$. Let $$\triangle _1$$ be the area of the triangle formed by the end points of its latus rectum and the point $$P\left(\frac{1}{2}, 2\right)$$ on the parabola, and $$\triangle_2$$ be the area of the triangle formed by drawing tangents at P and at the end points of the latus rectum. Then $$\frac{\triangle_1}{\triangle_2}$$ is
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The positive integer value of $$n > 3$$ satisfying the equation $$\frac{1}{\sin\left(\frac{\pi}{n}\right)} = \frac{1}{\sin\left(\frac{2 \pi}{n}\right)} + \frac{1}{\sin\left(\frac{3 \pi}{n}\right)}$$ is
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Let $$f(\theta) = \sin\left(\tan^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2\theta}}\right)\right)$$, where $$-\frac{\pi}{4} < \theta < \frac{\pi}{4}$$. Then the value of $$\frac{d}{d(\tan \theta)}(f(\theta))$$ is
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If z is any complex number satisfying $$\mid z - 3 - 2i \mid \leq 2$$, then the minimum value of $$\mid 2z - 6 + 5i \mid$$ is
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The minimum value of the sum of real numbers $$a^{-5}, a^{-4}, 3a^{-3}, 1, a^{8}$$ and $$a^{10}$$ with a > 0 is
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Let $$f : [1, \infty) \rightarrow [2, \infty)$$ be a differentiable function such that $$f(1) = 2$$. If $$6 \int_{1}^{x} f(t) dt = 3xf(x) - x^3$$ forall $$x \geq 1$$, then the value of f(2) is
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