Given that $$K$$ = energy, $$V$$ = velocity, $$T$$ = time. If they are chosen as the fundamental units, then what is dimensional formula for surface tension?
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Given that $$K$$ = energy, $$V$$ = velocity, $$T$$ = time. If they are chosen as the fundamental units, then what is dimensional formula for surface tension?
The graph of an object's motion (along the $$x$$-axis) is shown in the figure. The instantaneous velocity of the object at points $$A$$ and $$B$$ are $$v_A$$ and $$v_B$$ respectively. Then
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A car of mass 1000 kg is moving at a speed of $$30\ \text{m/s}$$. Brakes are applied to bring the car to rest. If the net retarding force is $$5000\ \text{N}$$, the car comes to stop after travelling $$d$$ m in $$t$$ s. Then
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An engine pumps water continuously through a hose. Water leaves the hose with velocity $$v$$ and $$m$$ is mass per unit length of the water jet. If this jet hits a surface and came to rest instantaneously, the force on the surface is
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A particle gets displaced by $$\Delta \vec{r} = (2\hat{i} + 3\hat{j} + 4\hat{k})\ \text{m}$$ under the action of a force $$\vec{F} = (7\hat{i} + 4\hat{j} + 3\hat{k})$$. The change in its kinetic energy is
A circular hole of diameter $$R$$ is cut from a disc of mass $$M$$ and radius $$R$$; the circumference of the cut passes through the centre of the disc. The moment of inertia of the remaining portion of the disc about an axis perpendicular to the disc and passing through its centre is
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A solid sphere having mass $$m$$ and radius $$r$$ rolls down an inclined plane. Then its kinetic energy is
Which graph correctly presents the variation of acceleration due to gravity with the distance from the centre of the earth (radius of the earth = $$R_E$$)?
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A structural steel rod has a radius of 10 mm and length of 1.0 m. A 100 kN force stretches it along its length. Young's modulus of structural steel is $$2 \times 10^{11}\ \text{Nm}^{-2}$$. The percentage strain is about
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The load versus elongation graphs for four wires of same length and made of the same material are shown in the figure. The thinnest wire is represented by the line
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A square hole of side length $$\ell$$ is made at a depth of $$h$$ and a circular hole of radius $$r$$ is made at a depth of $$4h$$ from the surface of water in a water tank kept on a horizontal surface. If $$\ell << h, r << h$$ and the rate of water flow from the holes is the same, then $$r$$ is equal to
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The heat radiated per unit area in 1 hour by a furnace whose temperature is 3000 K is ($$\sigma = 5.7 \times 10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$$)
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A perfect gas at $$27^\circ$$C is heated at constant pressure so as to double its volume. The final temperature of the gas will be, close to
This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement 1: In an adiabatic process, change in internal energy of a gas is equal to work done on/by the gas in the process. Statement 2: The temperature of a gas remains constant in an adiabatic process.
Following are expressions for four plane simple harmonic waves (i) $$y_1 = A \cos 2\pi \left(n_1 t + \dfrac{x}{\lambda_1}\right)$$ (ii) $$y_2 = A \cos 2\pi \left(n_1 t + \dfrac{x}{\lambda_1} + 0.5\right)$$ (iii) $$y_3 = A \cos 2\pi \left(n_2 t + \dfrac{x}{\lambda_2}\right)$$ (iv) $$y_4 = A \cos 2\pi \left(n_2 t - \dfrac{x}{\lambda_2}\right)$$. The pairs of waves which will produce destructive interference and stationary waves respectively in a medium, are
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This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement 1: In the resonance tube experiment, if the tuning fork is replaced by another identical turning fork but with its arm having been filled, the length of the air column should be increased to obtain resonance again. Statement 2: On filling the arms, the frequency of a tuning fork increases.
The electric potential $$V(x)$$ in a region around the origin is given by $$V(x) = 4x^2$$ volts. The electric charge enclosed in a cube of 1 m side with its centre at the origin is (in coulomb)
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Two circuits (a) and (b) have charged capacitors of capacitance C, 2C and 3C with open switches. Charges on each of the capacitor are as shown in the figures. On closing the switches
Circuit (a) Circuit (b)
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This question has 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 possibility of an electric bulb fusing is higher at the time of switching ON. Statement 2: Resistance of an electric bulb when it is not lit up is much smaller than when it is lit up.
A bar magnet of length 6 cm has a magnetic moment of $$4\ \text{J T}^{-1}$$. Find the strength of magnetic field at a distance of 200 cm from the centre of the magnet along its equatorial line.
The velocity of certain ions that pass undeflected through crossed electric field $$E = 7.7\ \text{kV/m}$$ and magnetic field $$B = 0.14\ \text{T}$$ is
In an $$LCR$$ circuit shown in the following figure, what will be the readings of the voltmeter across the resistor and ammeter if an a.c. source of 220 V and 100 Hz is connected to it as shown?
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Which of the following processes play a part in the formation of a rainbow? (i) Refraction (ii) Total internal reflection (iii) Dispersion (iv) Interference
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In a Young's double slit experiment with light of wavelength $$\lambda$$, fringe pattern on the screen has fringe width $$\beta$$. When two thin transparent glass (refractive index $$\mu$$) plates of thickness $$t_1$$ and $$t_2$$ ($$t_1 > t_2$$) are placed in the path of the two beams respectively, the fringe pattern will shift by a distance
Two polaroids have their polarizing directions parallel so that the intensity of a transmitted light is maximum. The angle through which either polaroid must be turned if the intensity is to drop by one-half is
The electron of a hydrogen atom makes a transition from the $$(n+1)^{\text{th}}$$ orbit to the $$n^{\text{th}}$$ orbit. For large $$n$$ the wavelength of the emitted radiation is proportional to
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In the Rutherford experiment, $$\alpha$$-particles are scattered from a nucleus as shown. Out of the four paths, which path is not possible?
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A sample originally contained $$10^{20}$$ radioactive atoms, which emit $$\alpha$$-particles. The ratio of $$\alpha$$ particles emitted in the third year to that emitted during the second year is 0.3. How many $$\alpha$$ particles were emitted in the first year?
Which logic gate with inputs $$A$$ and $$B$$ performs the same operation as that performed by the following circuit?
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Broadcasting antennas are generally
The concentrated sulphuric acid that is peddled commercial is 95% $$\text{H}_2\text{SO}_4$$ by weight. If the density of this commercial acid is $$1.834\ \text{g cm}^{-3}$$, the molarity of this solution is
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The ratio of number of oxygen atoms (O) in 16.0 g ozone ($$\text{O}_3$$), 28.0 g carbon monoxide (CO) and 16.0 g oxygen ($$\text{O}_2$$) is (Atomic mass : C = 12, O = 16 and Avogadro's constant $$N_A = 6.0 \times 10^{23}\ \text{mol}^{-1}$$)
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The limiting line in Balmer series will have a frequency of (Rydberg constant, $$R_\infty = 3.29 \times 10^{15}\ \text{cycles/s}$$)
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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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Among the following, the species having the smallest bond is
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Based on lattice energy and other considerations, which one of the following alkali metal chloride is expected to have the highest melting point?
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For 1 mol of an ideal gas at a constant temperature $$T$$, the plot of $$(\log P)$$ against $$(\log V)$$ is a ($$P$$ : Pressure, $$V$$ : Volume)
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The entropy of a sample of a certain substance increases by $$0.836\ \text{J K}^{-1}$$ on adding reversibly $$0.3344\ \text{J}$$ of heat at constant temperature. The temperature of the sample is:
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The electron affinity of chlorine is 3.7 eV. 1 gram of chlorine is completely converted to $$\text{Cl}^-$$ ion in a gaseous state. ($$1\ \text{eV} = 23.06\ \text{kcal mol}^{-1}$$). Energy released in the process is
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The solubility (in mol $$\text{L}^{-1}$$) of AgCl ($$K_{sp} = 1.0 \times 10^{-10}$$) in a 0.1 M KCl solution will be
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Which of the oxide groups among the following cannot be reduced by carbon?
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Which of the following cannot be represented by resonance structures?
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Copper wire test for halogens is known as
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The IUPAC name of the compound
is
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Among the following the order of reactivity towards nucleophilic addition is
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Green house gases can be arranged in 'Global Warming Potential' sequence as
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Among the following which is the best description of water in the solid phase?
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A solid has a '$$bcc$$' structure. If the distance of nearest approach between two atoms is $$1.73\ \text{Å}$$, the edge length of the cell is
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A battery is constructed of Cr and $$\text{Na}_2\text{Cr}_2\text{O}_7$$. The unbalanced chemical equation when such a battery discharges is following: $$$\text{Na}_2\text{Cr}_2\text{O}_7 + \text{Cr} + \text{H}^+ \rightarrow \text{Cr}^{3+} + \text{H}_2\text{O} + \text{Na}^+$$$ If one Faraday of electricity is passed through the battery during the charging, the number of moles of $$\text{Cr}^{3+}$$ removed from the solution is
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$$K_1, K_2$$ and $$K_3$$ are the equilibrium constants of the following reactions (I), (II) and (III) respectively: (I) $$\text{N}_2 + 2\text{O}_2 \rightleftharpoons 2\text{NO}_2$$ (II) $$2\text{NO}_2 \rightleftharpoons \text{N}_2 + 2\text{O}_2$$ (III) $$\text{NO}_2 \rightleftharpoons \tfrac{1}{2}\text{N}_2 + \text{O}_2$$. The correct relation from the following is
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Reaction rate between two substance $$A$$ and $$B$$ is expressed as following: $$\text{rate} = k[A]^n[B]^m$$. If the concentration of $$A$$ is doubled and concentration of $$B$$ is made half of initial concentration, the ratio of the new rate to the earlier rate will be:
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If $$x$$ is the mass of the gas adsorbed on mass $$m$$ of the adsorbent at pressure $$p$$, Freundlich adsorption isotherm gives a straight line on plotting
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The $$d$$-electron configurations of $$\text{Cr}^{2+}, \text{Mn}^{2+}, \text{Fe}^{2+}$$ and $$\text{Co}^{2+}$$ are $$d^4, d^5, d^6$$ and $$d^7$$ respectively. Which one of the following will exhibit the lowest paramagnetic behaviour? (Atomic no. Cr = 24, Mn = 25, Fe = 26, Co = 27).
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Which is not the correct Statement? (At. nos. Ce = 58, Lu = 71, La = 57, Yb = 70)
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How many cyclic structures are possible for $$\text{C}_4\text{H}_6$$?
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The most basic compound among the following is
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$$\text{C}_2\text{H}_5\text{Br} \xrightarrow{\text{AgCN}} X \xrightarrow[\text{Zn-Hg/HCl}]{\text{Reduction}} Y$$. Here Y is
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Which one of the following is a chain growth polymerisation?
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Which one of the following compounds is an antifertility drug?
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All of the following statements apply to proteins except
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The value of $$k$$ for which the equation $$(K-2)x^2 + 8x + K + 4 = 0$$ has both roots real, distinct and negative can be:
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Let $$Z_1$$ and $$Z_2$$ be any two complex number. Statement 1: $$|Z_1 - Z_2| \geq |Z_1| - |Z_2|$$ Statement 2: $$|Z_1 + Z_2| \leq |Z_1| + |Z_2|$$
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If the number of 5-element subsets of the set $$A = \{a_1, a_2, \ldots, a_{20}\}$$ of 20 distinct elements is $$k$$ times the number of 5-element subsets containing $$a_4$$, then $$k$$ is
The difference between the fourth term and the first term of a Geometrical Progression is 52. If the sum of its first three terms is 26, then the sum of the first six terms of the progression is
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The sum of the series $$1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + 2 \cdot 6^2 + \ldots + 2(2m)^2$$ is
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If $$f(y) = 1 - (y-1) + (y-1)^2 - (y-1)^3 + \ldots - (y-1)^{17}$$ then the coefficient of $$y^2$$ in it is
If the straight lines $$x + 3y = 4, 3x + y = 4$$ and $$x + y = 0$$ form a triangle, then the triangle is
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The point of intersection of the lines $$(a^3 + 3)x + ay + a - 3 = 0$$ and $$(a^5 + 2)x + (a+2)y + 2a + 3 = 0$$ ($$a$$ real) lies on the $$y$$-axis for
The equation of the circle passing through the point $$(1, 2)$$ and through the points of intersection of $$x^2 + y^2 - 4x - 6y - 21 = 0$$ and $$3x + 4y + 5 = 0$$ is given by
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Statement 1: $$y = mx - \dfrac{1}{m}$$ is always a tangent to the parabola, $$y^2 = -4x$$ for all non-zero values of $$m$$. Statement 2: Every tangent to the parabola, $$y^2 = -4x$$ will meet its axis at a point whose abscissa is non-negative.
If the eccentricity of a hyperbola $$\dfrac{x^2}{9} - \dfrac{y^2}{b^2} = 1$$, which passes through $$(K, 2)$$, is $$\dfrac{\sqrt{13}}{3}$$, then the value of $$K^2$$ is
$$\lim_{x \to 0} \left(\dfrac{x - \sin x}{x}\right) \sin\left(\dfrac{1}{x}\right)$$
The Statement that is TRUE among the following is
The frequency distribution of daily working expenditure of families in a locality is as follows:
If the mode of the distribution is Rs. 140, then the value of $$b$$ is
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If two vertical poles 20 m and 80 m high stand apart on a horizontal plane, then the height (in m) of the point of intersection of the lines joining the top of each pole to the foot of other is
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Let $$X$$ and $$Y$$ are two events such that $$P(X \cup Y) = P(X \cap Y)$$. Statement 1: $$P(X \cap Y') = P(X' \cap Y) = 0$$ Statement 2: $$P(X) + P(Y) = 2 P(X \cap Y)$$
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If $$A = \begin{pmatrix} \alpha - 1 \\ 0 \\ 0 \end{pmatrix}, B = \begin{pmatrix} \alpha + 1 \\ 0 \\ 0 \end{pmatrix}$$ be two matrices, then $$AB^T$$ is a non-zero matrix for $$|\alpha|$$ not equal to
If the system of equations $$\begin{aligned} x + y + z &= 6 \\ x + 2y + 3z &= 10 \\ x + 2y + \lambda z &= 0 \end{aligned}$$ has a unique solution, then $$\lambda$$ is not equal to
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The range of the function $$f(x) = \dfrac{x}{1+|x|}, x \in R$$, is
Let $$f(x) = \sin x, g(x) = x$$. Statement 1: $$f(x) \leq g(x)$$ for $$x$$ in $$(0, \infty)$$ Statement 2: $$f(x) \leq 1$$ for $$x$$ in $$(0, \infty)$$ but $$g(x) \to \infty$$ as $$x \to \infty$$.
If $$x + |y| = 2y$$, then $$y$$ as a function of $$x$$, at $$x = 0$$ is
If a circular iron sheet of radius 30 cm is heated such that its area increases at the uniform rate of $$6\pi\ \text{cm}^2/\text{hr}$$, then the rate (in mm/hr) at which the radius of the circular sheet increases is
Let $$f(x)$$ be an indefinite integral of $$\cos^3 x$$. Statement 1: $$f(x)$$ is a periodic function of period $$\pi$$. Statement 2: $$\cos^3 x$$ is a periodic function.
If $$\int_e^x t f(t)\, dt = \sin x - x \cos x - \dfrac{x^2}{2}$$, for all $$x \in R - \{0\}$$, then the value of $$f\left(\dfrac{\pi}{6}\right)$$ is
The parabola $$y^2 = x$$ divides the circle $$x^2 + y^2 = 2$$ into two parts whose areas are in the ratio
Let $$y(x)$$ be a solution of $$\dfrac{(2+\sin x)}{(1+y)} \dfrac{dy}{dx} = \cos x$$. If $$y(0) = 2$$, then $$y\left(\dfrac{\pi}{2}\right)$$ equals
$$ABCD$$ is parallelogram. The position vectors of $$A$$ and $$C$$ are respectively, $$3\hat{i} + 3\hat{j} + 5\hat{k}$$ and $$\hat{i} - 5\hat{j} - 5\hat{k}$$. If $$M$$ is the midpoint of the diagonal $$DB$$, then the magnitude of the projection of $$\overrightarrow{OM}$$ on $$\overrightarrow{OC}$$, where $$O$$ is the origin, is
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If $$\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}, \vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}$$ and $$\vec{c} = \lambda\hat{i} + \hat{j} + (2\lambda - 1)\hat{k}$$ are coplanar vectors, then $$\lambda$$ is equal to
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The values of $$a$$ for which the two points $$(1, a, 1)$$ and $$(-3, 0, a)$$ lie on the opposite sides of the plane $$3x + 4y - 12z + 13 = 0$$, satisfy
A line with positive direction cosines passes through the point $$P(2, -1, 2)$$ and makes equal angles with the coordinate axes. If the line meets the plane $$2x + y + z = 9$$ at point $$Q$$, then the length $$PQ$$ equals
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