A particle has an initial velocity $$3\hat{i} + 4\hat{j}$$ and an acceleration of $$0.4\hat{i} + 0.3\hat{j}$$. Its speed after $$10$$ s is
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A particle has an initial velocity $$3\hat{i} + 4\hat{j}$$ and an acceleration of $$0.4\hat{i} + 0.3\hat{j}$$. Its speed after $$10$$ s is
Consider a rubber ball freely falling from a height $$h = 4.9$$ m onto a horizontal elastic plate. Assume that the duration of collision is negligible and the collision with the plate is totally elastic. Then the velocity as a function of time the height as function of time will be
A thin uniform rod of length $$\ell$$ and mass $$m$$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $$\omega$$. Its centre of mass rises to a maximum height of
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The height at which the acceleration due to gravity becomes $$\frac{g}{9}$$ (where g $$=$$ the acceleration due to gravity on the surface of the earth) in terms of $$R$$, the radius of the earth is
Two wires are made of the same material and have the same volume. However wire 1 has crosssectional area $$A$$ and wire-2 has cross-sectional area $$3A$$. If the length of wire 1 increases by $$\Delta x$$ on applying force $$F$$, how much force is needed to stretch wire 2 by the same amount?
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A long metallic bar is carrying heat from one of its ends to the other end under steady-state. The variation of temperature $$\theta$$ along the length $$x$$ of the bar from its hot end is best described by which of the following figure.
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Directions: Questions 7 to 9 are based on the following figure showing a thermodynamic cycle ABCDA on a P-V diagram. (Figure not available; the cycle ABCDA involves gas state changes with given pressure-volume values from the original paper.) Assuming the gas to be ideal the work done on the gas in taking it from $$A$$ to $$B$$ is
The work done on the gas in taking it from $$D$$ to $$A$$ is
The net work done on the gas in the cycle ABCDA is
One kg of a diatomic gas is at a pressure of $$8 \times 10^4$$ N/m$$^2$$. The density of the gas is $$4$$ kg/m$$^{-3}$$. What is the energy of the gas due to its thermal motion?
If $$x, v$$ and $$a$$ denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period $$T$$, then, which of the following does not change with time?
A motor cycle starts from rest and accelerates along a straight path at $$2$$ m/s$$^2$$. At the starting point of the motor cycle there is a stationary electric siren. How far has the motor cycle gone when the driver hears the frequency of the siren at $$94\%$$ of its value when the motor cycle was at rest? (speed of sound $$= 330$$ ms$$^{-1}$$)
Three sound waves of equal amplitudes have frequencies $$(v - 1), v, (v + 1)$$. They superpose to give beats. The number of beats produced per second will 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 - 1: For a charged particle moving from point $$P$$ to point $$Q$$, the net work done by an electrostatic field on the particle is independent of the path connecting point $$P$$ to point $$Q$$. Statement-2: The net work done by a conservative force on an object moving along a closed loop is zero
Let $$P(r) = \frac{Q}{\pi R^4}r$$ be the charge density distribution for a solid sphere of radius $$R$$ and total charge $$Q$$. For a point '$$p$$' inside the sphere at distance $$r_1$$ from the centre of the sphere, the magnitude of electric field is
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Two points $$P$$ and $$Q$$ are maintained at the potentials of $$10$$ V and $$-4$$ V respectively. The work done in moving $$100$$ electrons from $$P$$ to $$Q$$ is
A charge $$Q$$ is placed at each of the opposite corners of a square. A charge $$q$$ is placed at each of the other two corners. If the net electrical force on $$Q$$ is zero, then the $$Q/q$$ equals
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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-1: The temperature dependence of resistance is usually given as $$R = R_0(1 + \alpha\Delta t)$$. The resistance of a wire changes from $$100\Omega$$ to $$150\Omega$$ when its temperature is increased from $$27°C$$ to $$227°C$$. This implies that $$\alpha = 2.5 \times 10^{-3}/°C$$. Statement 2: $$R = R_t(1 + \alpha\Delta T)$$ is valid only when the change in the temperature $$\Delta T$$ is small and $$\Delta R = (R - R_0) \ll R_0$$.
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The magnitude of the magnetic field $$(B)$$ due to loop $$ABCD$$ at the origin $$(O)$$ is
Due to the presence of the current $$I_1$$ at the origin
An inductor of inductance $$L = 400$$ mH and resistors of resistances $$R_1 = 2\Omega$$ and $$R_2 = 2\Omega$$ are connected to a battery of emf $$12$$ V as shown in the figure. The internal resistance of the battery is negligible. The switch $$S$$ is 
closed at $$t = 0$$. The potential drop across $$L$$ as a function of time is
In an optics experiment, with the position of the object fixed, a student varies the position of a convex lens and for each position, the screen is adjusted to get a clear image of the object. A graph between the object distance $$u$$ and the image distance $$v$$, from the lens, is plotted using the same scale for the two axes. A straight line passing through the origin and making an angle of $$45°$$ with the x-axis meets the experimental curve at $$P$$. The coordinates of $$P$$ will be
A transparent solid cylindrical rod has a refractive index of $$\frac{2}{\sqrt{3}}$$. It is surrounded by air. A light ray is incident at the mid point of one end of the rod as shown in the figure. The incident angle $$\theta$$ for which the light ray 
grazes along the wall of the rod is
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A mixture of light, consisting of wavelength $$590$$ nm and an unknown wavelength, illuminates Young's double slit and gives rise to two overlapping interference patterns on the screen. The central maximum of both lights coincide. Further, it is observed that the third bright fringe of known light coincides with the $$4^{th}$$ bright fringe of the unknown light. From this data, the wavelength of the unknown light is
The surface of a metal is illuminated with the light of $$400$$ nm. The kinetic energy of the ejected photoelectrons was found to be $$1.68$$ eV. The work function of the metal is (hc $$= 1240$$ eVnm)
The transition from the state $$n = 4$$ to $$n = 3$$ in a hydrogen like atom results in ultraviolet radiation. Infrared radiation will be obtained in the transition from
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The above is a plot of binding energy per nucleon $$E_b$$, against the nuclear mass $$M$$; $$A, B, C, D, E, F$$ correspond to different nuclei. Consider four reactions: (i) $$A + B \to C + \varepsilon$$ (ii) $$C \to A + B + \varepsilon$$ (iii) $$D + E \to F + \varepsilon$$ and (iv) $$F \to D + E + \varepsilon$$ where $$\varepsilon$$ is the energy released? In which reactions is $$\varepsilon$$ positive? 
A p-n junction (D) shown in the figure can act as a rectifier. An alternating current source $$(V)$$ is connected in the circuit. 
The logic circuit shown below has the input waveforms 'A' and 'B' as shown. Pick out the correct output waveform. 
In an experiment the angles are required to be measured using an instrument. $$29$$ divisions of the main scale exactly coincide with the $$30$$ divisions of the vernier scale. If the smallest division of the main scale is half-a-degree $$(= 0.5°)$$, then the least count of the instrument is
In an atom, an electron is moving with a speed of $$600$$ m/s with an accuracy of $$0.005\%$$. Certainity with which the position of the electron can be located is ($$h = 6.6 \times 10^{-34}$$ kg m$$^2$$ s$$^{-1}$$, mass of electron, $$e_m = 9.1 \times 10^{-31}$$ kg)
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Calculate the wavelength (in nanometer) associated with a proton moving at $$1.0 \times 10^3$$ ms$$^{-1}$$ (Mass of proton $$= 1.67 \times 10^{-27}$$ kg and h $$= 6.63 \times 10^{-34}$$ Js):
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In which of the following arrangements, the sequence is not strictly according to the property written against it?
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The set representing the correct order of ionic radius is :
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Using MO theory predict which of the following species has the shortest bond length?
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On the basis of the following thermochemical data: $$\left(\Delta_f G° H^+_{(aq)} = 0\right)$$ $$$H_2O(\ell) \to H^+(aq) + OH^-(aq); \Delta H = 57.32 \text{ kJ}$$$ $$$H_2(g) + \frac{1}{2}O_2(g) \to H_2O(\ell); \Delta H = -286.20 \text{ kJ}$$$ The value of enthalpy of formation of $$OH^-$$ ion at $$25°C$$ is:
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Solid $$Ba(NO_3)_2$$ is gradually dissolved in a $$1.0 \times 10^{-4}$$ M $$Na_2CO_3$$ solution. At what concentration of $$Ba^{2+}$$ will a precipitate begin to form ? ($$K_{sp}$$ for $$BaCO_3 = 5.1 \times 10^{-9}$$).
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The bond dissociation energy of B$$-$$F in BF$$_3$$ is $$646$$ kJ mol$$^{-1}$$ whereas that of C$$-$$F in CF$$_4$$ is $$515$$ kJ mol$$^{-1}$$. The correct reason for higher B-F bond dissociation energy as compared to that of $$C - F$$ is :
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Arrange the carbanions, $$(CH_3)_3\bar{C}, \bar{C}Cl_3, (CH_3)_2\bar{C}H, C_6H_5\bar{C}H_2$$, in order of their decreasing stability :
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The alkene that exhibits geometrical isomerism is:
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The number of stereoisomers possible for a compound of the molecular formula $$CH_3 - CH = CH - CH(OH) - Me$$ is :
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The IUPAC name of neopentane is
Copper crystallizes in fcc with a unit cell length of $$361$$ pm. What is the radius of copper atom?
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Two liquids $$X$$ and $$Y$$ form an ideal solution. At $$300$$ K, vapour pressure of the solution containing $$1$$ mol of $$X$$ and $$3$$ mol of $$Y$$ is $$550$$ mmHg. At the same temperature, if $$1$$ mol of $$Y$$ is further added to this solution, vapour pressure of the solution increases by $$10$$ mmHg. Vapour pressure (in mmHg) of $$X$$ and $$Y$$ in their pure states will be, respectively :
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A binary liquid solution is prepared by mixing $$n$$-heptane and ethanol. Which one of the following statements is correct regarding the behaviour of the solution?
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In a fuel cell methanol is used as fuel and oxygen gas is used as an oxidizer. The reaction is $$CH_3OH(\ell) + \frac{3}{2}O_2(g) \to CO_2(g) + 2H_2O(\ell)$$ At $$298$$ K standard Gibb's energies of formation for $$CH_3OH(\ell), H_2O(\ell)$$ and $$CO_2(g)$$ are $$-166.2, -237.2$$ and $$-394.4$$ kJ mol$$^{-1}$$ respectively. If standard enthalpy of combustion of methanol is $$-726$$ kJ mol$$^{-1}$$, efficiency of the fuel cell will be
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Given : $$E°_{Fe^{3+}/Fe} = -0.036$$ V, $$E°_{Fe^{2+}/Fe} = -0.439$$ V. The value of standard electrode potential for the change, $$Fe^{3+}_{(aq)} + e^- \to Fe^{2+}(aq)$$ will be :
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The half life period of a first order chemical reaction is $$6.93$$ minutes. The time required for the completion of $$99\%$$ of the chemical reaction will be ($$\log 2 = 0.301$$) :
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Which of the following statements is incorrect regarding physissorptions?
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Which one of the following reactions of Xenon compounds is not feasible?
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Knowing that the Chemistry of lanthanoids (Ln) is dominated by its $$+3$$ oxidation state, which of the following statements in incorrect?
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In context with the transition elements, which of the following statements is incorrect?
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Which of the following has an optical isomer?
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Which of the following pairs represents linkage isomers?
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Which of the following on heating with aqueous KOH, produces acetaldehyde?
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The major product obtained on interaction of phenol with sodium hydroxide and carbon dioxide is :
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In Cannizzaro reaction given below: 
the slowest step is :
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A liquid was mixed with ethanol and a drop of concentrated $$H_2SO_4$$ was added. A compound with a fruity smell was formed. The liquid was:
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Buna-N synthetic rubber is a copolymer of :
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The two functional groups present in a typical carbohydrate are :
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If the roots of the equation $$bx^2 + cx + a = 0$$ be imaginary, then for all real values of $$x$$, the expression $$3b^2x^2 + 6bcx + 2c^2$$ is
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If $$\left|z - \frac{4}{z}\right| = 2$$, then the maximum value of $$|z|$$ is equal to
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From $$6$$ different novels and $$3$$ different dictionaries, $$4$$ novels and $$1$$ dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is
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The sum to the infinity of the series $$1 + \frac{2}{3} + \frac{6}{3^2} + \frac{10}{3^3} + \frac{14}{3^4} + \ldots$$ is
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The remainder left out when $$8^{2n} - (62)^{2n+1}$$ is divided by $$9$$ is
The lines $$p(p^2 + 1)x - y + q = 0$$ and $$(p^2 + 1)^2 x + (p^2 + 1)y + 2q = 0$$ are perpendicular to a common line for
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Three distinct points A, B and C are given in the 2-dimensional coordinate plane such that the ratio of the distance of any one of them from the point $$(1, 0)$$ to the distance from the point $$(-1, 0)$$ is equal to $$\frac{1}{3}$$. Then the circumcentre of the triangle $$ABC$$ is at the point
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If $$P$$ and $$Q$$ are the points of intersection of the circles $$x^2 + y^2 + 3x + 7y + 2p - 5 = 0$$ and $$x^2 + y^2 + 2x + 2y - p^2 = 0$$, then there is a circle passing through $$P, Q$$ and $$(1, 1)$$ for
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The ellipse $$x^2 + 4y^2 = 4$$ is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point $$(4, 0)$$. Then the equation of the ellipse is
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Statement-1: $$\sim (p \leftrightarrow \sim q)$$ is equivalent to $$p \leftrightarrow q$$. Statement-2: $$\sim (p \leftrightarrow \sim q)$$ is a tautology.
If the mean deviation of number $$1, 1+d, 1+2d, \ldots, 1+100d$$ from their mean is $$255$$, then the $$d$$ is equal to
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Statement-1 : The variance of first $$n$$ even natural numbers is $$\frac{n^2 - 1}{4}$$. Statement-2 : The sum of first $$n$$ natural numbers is $$\frac{n(n+1)}{2}$$ and the sum of squares of first $$n$$ natural numbers is $$\frac{n(n+1)(2n+1)}{6}$$.
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If $$A, B$$ and $$C$$ are three sets such that $$A \cap B = A \cap C$$ and $$A \cup B = A \cup C$$, then
Let A be a $$2 \times 2$$ matrix Statement-1 : $$\text{adj}(\text{adj } A) = A$$ Statement-2 : $$|\text{adj } A| = |A|$$
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Let $$a, b, c$$ be such that $$b(a + c) \neq 0$$. If $$$\begin{vmatrix} a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1 \end{vmatrix} + \begin{vmatrix} a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2}a & (-1)^{n+1}b & (-1)^n c \end{vmatrix} = 0,$$$ then the value of '$$n$$' is
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Let $$A$$ and $$B$$ denote the statements A: $$\cos\alpha + \cos\beta + \cos\gamma = 0$$ B: $$\sin\alpha + \sin\beta + \sin\gamma = 0$$ If $$\cos(\beta - \gamma) + \cos(\gamma - \alpha) + \cos(\alpha - \beta) = -\frac{3}{2}$$, then
For real $$x$$, let $$f(x) = x^3 + 5x + 1$$, then
Let $$f(x) = (x + 1)^2 - 1, x \geq -1$$. Statement-1: The set $$\{x : f(x) = f^{-1}(x)\} = \{0, -1\}$$ Statement-2 : f is a bijection.
Let $$f(x) = x|x|$$ and $$g(x) = \sin x$$. Statement-1 : $$g \circ f$$ is differentiable at $$x = 0$$ and its derivative is continuous at that point. Statement-2 : $$g \circ f$$ is twice differentiable at $$x = 0$$.
Let $$y$$ be an implicit function of $$x$$ defined by $$x^{2x} - 2x^x \cot y - 1 = 0$$. Then $$y'(1)$$ equals
Given $$P(x) = x^4 + ax^3 + bx^2 + cx + d$$ such that $$x = 0$$ is the only real root of $$P'(x) = 0$$. If $$P(-1) < P(1)$$, then in the interval $$[-1, 1]$$
The shortest distance between the line $$y - x = 1$$ and the curve $$x = y^2$$ is
$$\int_0^\pi [\cot x] dx$$, $$[\cdot]$$ denotes the greatest integer function, is equal to
The area of the region bounded by the parabola $$(y - 2)^2 = x - 1$$, the tangent to the parabola at the point $$(2, 3)$$ and the $$x$$-axis is
The differential equation which represents the family of curves $$y = c_1 e^{c_2 x}$$, where $$c_1$$ and $$c_2$$ are arbitrary constants is
If $$\vec{u}, \vec{v}, \vec{w}$$ are non-coplanar vectors and $$p, q$$ are real numbers, then the equality $$[3\vec{u} \;\; p\vec{v} \;\; p\vec{w}] - [p\vec{v} \;\; \vec{w} \;\; q\vec{u}] - [2\vec{w} \;\; q\vec{v} \;\; q\vec{u}] = 0$$ holds for
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Let the line $$\frac{x-2}{3} = \frac{y-1}{-5} = \frac{z-2}{2}$$ lies in the plane $$x + 3y - \alpha z + \beta = 0$$. Then $$(\alpha, \beta)$$ equals
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The projections of a vector on the three coordinate axis are $$6, -3, 2$$ respectively. The direction cosines of the vector are
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In a binomial distribution $$B\left(n, p = \frac{1}{4}\right)$$, if the probability of at least one success is greater than or equal to $$\frac{9}{10}$$, then $$n$$ is greater than
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One ticket is selected at random from $$50$$ tickets numbered $$00, 01, 02, \ldots, 49$$. Then the probability that the sum of the digits on the selected ticket is $$8$$, given that the product of these digits is zero, equals
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