The angle between vector $$\vec{Q}$$ and the resultant of $$(2\vec{Q} + 2\vec{P})$$ and $$(2\vec{Q} - 2\vec{P})$$ is :
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The angle between vector $$\vec{Q}$$ and the resultant of $$(2\vec{Q} + 2\vec{P})$$ and $$(2\vec{Q} - 2\vec{P})$$ is :
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Time periods of oscillation of the same simple pendulum measured using four different measuring clocks were recorded as $$4.62 \text{ s}, 4.632 \text{ s}, 4.6 \text{ s}$$ and $$4.64 \text{ s}$$. The arithmetic mean of these readings in correct significant figure is :
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If $$G$$ be the gravitational constant and $$u$$ be the energy density then which of the following quantity have the dimensions as that of the $$\sqrt{uG}$$ :
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A wooden block of mass $$5 \text{ kg}$$ rests on a soft horizontal floor. When an iron cylinder of mass $$25 \text{ kg}$$ is placed on the top of the block, the floor yields and the block and the cylinder together go down with an acceleration of $$0.1 \text{ ms}^{-2}$$. The action force of the system on the floor is equal to:
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A body of mass $$50 \text{ kg}$$ is lifted to a height of $$20 \text{ m}$$ from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be :
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Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis AB as shown in figure is $$\sqrt{8/x}$$. The value of $$x$$ is :
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A simple pendulum doing small oscillations at a place R height above earth surface has time period of $$T_1 = 4 \text{ s}$$. $$T_2$$ would be its time period if it is brought to a point which is at a height $$2R$$ from earth surface. Choose the correct relation $$[R = \text{radius of earth}]$$ :
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In hydrogen like system the ratio of coulombian force and gravitational force between an electron and a proton is in the order of :
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Match List I with List II :
(Where $$a$$ = radius of planet orbit, $$r$$ = radius of planet, $$M$$ = mass of Sun, $$m$$ = mass of planet)
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Given below are two statements :
Statement I : When a capillary tube is dipped into a liquid, the liquid neither rises nor falls in the capillary. The contact angle may be $$0°$$.
Statement II : The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well.
In the light of the above statement, choose the correct answer from the options given below.
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The heat absorbed by a system in going through the given cyclic process is :
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If the collision frequency of hydrogen molecules in a closed chamber at $$27°C$$ is $$Z$$, then the collision frequency of the same system at $$127°C$$ is :
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In the given figure $$R_1 = 10\Omega, R_2 = 8\Omega, R_3 = 4\Omega$$ and $$R_4 = 8\Omega$$. Battery is ideal with emf $$12 \text{ V}$$. Equivalent resistance of the circuit and current supplied by battery are respectively :
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In a co-axial straight cable, the central conductor and the outer conductor carry equal currents in opposite directions. The magnetic field is zero :
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Two conducting circular loops A and B are placed in the same plane with their centers coinciding as shown in figure. The mutual inductance between them is :
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An alternating voltage of amplitude $$40 \text{ V}$$ and frequency $$4 \text{ kHz}$$ is applied directly across the capacitor of $$12\mu\text{F}$$. The maximum displacement current between the plates of the capacitor is nearly :
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Light emerges out of a convex lens when a source of light kept at its focus. The shape of wavefront of the light is :
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Given below are two statements :
Statement I :
Figure shows the variation of stopping potential $$(V_0)$$ with frequency $$(\nu)$$ for the two photosensitive materials $$M_1$$ and $$M_2$$. The slope gives value of $$\frac{h}{e}$$, where $$h$$ is Planck's constant, $$e$$ is the charge of electron.
Statement II : $$M_2$$ will emit photoelectrons of greater kinetic energy for the incident radiation having same frequency.
In the light of the above statements, choose the most appropriate answer from the options given below.
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An electron rotates in a circle around a nucleus having positive charge $$Ze$$. Correct relation between total energy $$(E)$$ of electron to its potential energy $$(U)$$ is :
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Following gates section is connected in a complete suitable circuit.
For which of the following combination, bulb will glow (ON) :
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A body moves on a frictionless plane starting from rest. If $$S_n$$ is distance moved between $$t = n - 1$$ and $$t = n$$ and $$S_{n-1}$$ is distance moved between $$t = n - 2$$ and $$t = n - 1$$, then the ratio $$\frac{S_{n-1}}{S_n}$$ is $$\left(1 - \frac{2}{x}\right)$$ for $$n = 10$$. The value of $$x$$ is ______.
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Three blocks $$M_1, M_2, M_3$$ having masses $$4 \text{ kg}, 6 \text{ kg}$$ and $$10 \text{ kg}$$ respectively are hanging from a smooth pulley using rope 1, 2 and 3 as shown in figure. The tension in the rope 1, $$T_1$$ when they are moving upward with acceleration of $$2 \text{ ms}^{-2}$$ is ______ N (if $$g = 10 \text{ m/s}^2$$).
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The density and breaking stress of a wire are $$6 \times 10^4 \text{ kg/m}^3$$ and $$1.2 \times 10^8 \text{ N/m}^2$$ respectively. The wire is suspended from a rigid support on a planet where acceleration due to gravity is $$\frac{1}{3}^{rd}$$ of the value on the surface of earth. The maximum length of the wire with breaking is ______ m (take, $$g = 10 \text{ m/s}^2$$).
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Three capacitors of capacitances $$25\mu\text{F}, 30\mu\text{F}$$ and $$45\mu\text{F}$$ are connected in parallel to a supply of $$100 \text{ V}$$. Energy stored in the above combination is $$E$$. When these capacitors are connected in series to the same supply, the stored energy is $$\frac{9}{x}E$$. The value of $$x$$ is ______.
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The electric field between the two parallel plates of a capacitor of capacitance $$1.5\mu\text{F}$$ drops to one third of its initial value in $$6.6\mu s$$ when the plates are connected by a thin wire. The resistance of this wire is ______ $$\Omega$$. (Given, $$\log 3 = 1.1$$)
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In the experiment to determine the galvanometer resistance by half-deflection method, the plot of $$1/\theta$$ vs the resistance $$(R)$$ of the resistance box is shown in the figure. The figure of merit of the galvanometer is ______ $$\times 10^{-1} \text{ A/division}$$. [The source has emf 2V]
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A $$2 \text{ A}$$ current carrying straight metal wire of resistance $$1\Omega$$, resistivity $$2 \times 10^{-6}\Omega\text{m}$$, area of cross-section $$10 \text{ mm}^2$$ and mass $$500 \text{ g}$$ is suspended horizontally in mid air by applying a uniform magnetic field $$\vec{B}$$. The magnitude of $$B$$ is ______ $$\times 10^{-1} \text{ T}$$ (given, $$g = 10 \text{ m/s}^2$$).
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An ac source is connected in given series LCR circuit. The rms potential difference across the capacitor of $$20\mu\text{F}$$ is ______ V.
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In Young's double slit experiment, carried out with light of wavelength $$5000 \text{ \AA}$$, the distance between the slits is $$0.3 \text{ mm}$$ and the screen is at $$200 \text{ cm}$$ from the slits. The central maximum is at $$x = 0 \text{ cm}$$. The value of $$x$$ for third maxima is ______ mm.
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If three helium nuclei combine to form a carbon nucleus then the energy released in this reaction is ______ $$\times 10^{-2} \text{ MeV}$$. (Given $$1\text{u} = 931 \text{ MeV/c}^2$$, atomic mass of helium $$= 4.002603 \text{ u}$$)
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An organic compound has $$42.1\%$$ carbon, $$6.4\%$$ hydrogen and remainder is oxygen. If its molecular weight is 342, then its molecular formula is :
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The incorrect postulates of the Dalton's atomic theory are : (A) Atoms of different elements differ in mass. (B) Matter consists of divisible atoms. (C) Compounds are formed when atoms of different element combine in a fixed ratio. (D) All the atoms of given element have different properties including mass. (E) Chemical reactions involve reorganisation of atoms. Choose the correct answer from the options given below :
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Given below are two statements :
Statement I : In group 13, the stability of +1 oxidation state increases down the group.
Statement II : The atomic size of gallium is greater than that of aluminium.
In the light of the above statements, choose the most appropriate answer from the options given below :
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The statement(s) that are correct about the species $$O^{2-}, F^-, Na^+$$ and $$Mg^{2+}$$. (A) All are isoelectronic (B) All have the same nuclear charge (C) $$O^{2-}$$ has the largest ionic radii (D) $$Mg^{2+}$$ has the smallest ionic radii. Choose the most appropriate answer from the options given below :
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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) : Enthalpy of neutralisation of strong monobasic acid with strong monoacidic base is always $$-57 \text{ kJ mol}^{-1}$$.
Reason (R) : Enthalpy of neutralisation is the amount of heat liberated when one mole of $$H^+$$ ions furnished by acid combine with one mole of $$OH^-$$ ions furnished by base to form one mole of water.
In the light of the above statements, choose the correct answer from the options given below.
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The following reaction occurs in the Blast furnace where iron ore is reduced to iron metal $$Fe_2O_{3(s)} + 3CO_{(g)} \rightleftharpoons Fe_{(l)} + 3CO_{2(g)}$$. Using the Le-Chatelier's principle, predict which one of the following will not disturb the equilibrium.
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The number of neutrons present in the more abundant isotope of boron is '$$x$$'. Amorphous boron upon heating with air forms a product, in which the oxidation state of boron is '$$y$$'. The value of $$x + y$$ is ______
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Number of $$\sigma$$ and $$\pi$$ bonds present in ethylene molecule is respectively :
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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) : Cis form of alkene is found to be more polar than the trans form. Reason (R) : Dipole moment of trans isomer of 2-butene is zero. In the light of the above statements, choose the correct answer from the options given below :
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For the Compounds : (A) $$H_3C-CH_2-O-CH_2-CH_2-CH_3$$
(B) $$H_3C-CH_2-CH_2-CH_2-CH_2-CH_3$$ (
C)
(D)
The increasing order of boiling point is :
Given below are two statements: Statement I : Nitration of benzene involves the following step
Statement II : Use of Lewis base promotes the electrophilic substitution of benzene. In the light of the above statements, choose the most appropriate answer from the options given below :
The reaction at cathode in the cells commonly used in clocks involves.
Molar ionic conductivities of divalent cation and anion are $$57 \text{ S cm}^2 \text{ mol}^{-1}$$ and $$73 \text{ S cm}^2 \text{ mol}^{-1}$$ respectively. The molar conductivity of solution of an electrolyte with the above cation and anion will be :
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The metal that shows highest and maximum number of oxidation state is :
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Which one of the following complexes will exhibit the least paramagnetic behaviour? [Atomic number, $$Cr = 24, Mn = 25, Fe = 26, Co = 27$$]
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The correct order of ligands arranged in increasing field strength.
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Given below are two statements: Statement I : Bromination of phenol in solvent with low polarity such as $$CHCl_3$$ or $$CS_2$$ requires Lewis acid catalyst. Statement II : The Lewis acid catalyst polarises the bromine to generate $$Br^+$$. In the light of the above statements, choose the correct answer from the options given below :
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Identify compound (Z) in the following reaction sequence.
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Identify '$$A$$' in the following reaction:
Which of the following gives a positive test with ninhydrin?
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$$9.3 \text{ g}$$ of pure aniline is treated with bromine water at room temperature to give a white precipitate of the product '$$P$$'. The mass of product '$$P$$' obtained is $$26.4 \text{ g}$$. The percentage yield is ______ %.
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The value of Rydberg constant $$(R_H)$$ is $$2.18 \times 10^{-18} \text{ J}$$. The velocity of electron having mass $$9.1 \times 10^{-31} \text{ kg}$$ in Bohr's first orbit of hydrogen atom $$=$$ ______ $$\times 10^5 \text{ ms}^{-1}$$ (nearest integer).
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In the Lewis dot structure for $$NO_2^-$$, total number of valence electrons around nitrogen is ______
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The heat of combustion of solid benzoic acid at constant volume is $$-321.30 \text{ kJ}$$ at $$27°C$$. The heat of combustion at constant pressure is $$(-321.30 - xR) \text{ kJ}$$, the value of $$x$$ is ______.
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An artificial cell is made by encapsulating $$0.2 \text{ M}$$ glucose solution within a semipermeable membrane. The osmotic pressure developed when the artificial cell is placed within a $$0.05 \text{ M NaCl}$$ solution at $$300 \text{ K}$$ is ______ $$\times 10^{-1} \text{ bar}$$. (nearest integer). [Given : $$R = 0.083 \text{ L bar mol}^{-1} \text{ K}^{-1}$$] Assume complete dissociation of $$NaCl$$
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During Kinetic study of reaction $$2A + B \to C + D$$, the following results were obtained :
I: A = 0.1 M, B = 0.1 M, initial rate of formation of D = $$6.0 \times 10^{-3}$$
II: A = 0.3 M, B = 0.2 M, initial rate = $$7.20 \times 10^{-2}$$
III: A = 0.3 M, B = 0.4 M, initial rate = $$2.88 \times 10^{-1}$$
IV: A = 0.4 M, B = 0.1 M, initial rate = $$2.40 \times 10^{-2}$$
Based on above data, overall order of the reaction is ______
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The spin-only magnetic moment value of the ion among $$Ti^{2+}, V^{2+}, Co^{3+}$$ and $$Cr^{2+}$$, that acts as strong oxidising agent in aqueous solution is ______ BM (Near integer). (Given atomic numbers : $$Ti : 22, V : 23, Cr : 24, Co : 27$$)
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The number of halobenzenes from the following that can be prepared by Sandmeyer's reaction is ______
Consider the given chemical reaction sequence :
Total sum of oxygen atoms in Product A and Product B are ______
In a borax bead test under hot condition, a metal salt (one from the given) is heated at point B of the flame, resulted in green colour salt bead. The spin-only magnetic moment value of the salt is ______ BM (Nearest integer) [Given atomic number of $$Cu = 29, Ni = 28, Mn = 25, Fe = 26$$]
Consider the following two statements : Statement I : For any two non-zero complex numbers $$z_1, z_2$$, $$(|z_1| + |z_2|)\left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \leq 2(|z_1| + |z_2|)$$. Statement II : If $$x, y, z$$ are three distinct complex numbers and $$a, b, c$$ are three positive real numbers such that $$\frac{a}{|y-z|} = \frac{b}{|z-x|} = \frac{c}{|x-y|}$$, then $$\frac{a^2}{y-z} + \frac{b^2}{z-x} + \frac{c^2}{x-y} = 1$$. Between the above two statements,
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If $$\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \ldots + \frac{1}{\sqrt{99}+\sqrt{100}} = m$$ and $$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{99 \cdot 100} = n$$, then the point $$(m, n)$$ lies on the line
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Suppose $$\theta \in [0, \frac{\pi}{4}]$$ is a solution of $$4\cos\theta - 3\sin\theta = 1$$. Then $$\cos\theta$$ is equal to :
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Let two straight lines drawn from the origin $$O$$ intersect the line $$3x + 4y = 12$$ at the points $$P$$ and $$Q$$ such that $$\triangle OPQ$$ is an isosceles triangle and $$\angle POQ = 90°$$. If $$l = OP^2 + PQ^2 + QO^2$$, then the greatest integer less than or equal to $$l$$ is :
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If $$A(1, -1, 2), B(5, 7, -6), C(3, 4, -10)$$ and $$D(-1, -4, -2)$$ are the vertices of a quadrilateral $$ABCD$$, then its area is :
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Let a circle $$C$$ of radius 1 and closer to the origin be such that the lines passing through the point $$(3, 2)$$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle $$C$$ from the point $$(5, 5)$$ is :
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Let the line $$2x + 3y - k = 0, k > 0$$, intersect the $$x$$-axis and $$y$$-axis at the points $$A$$ and $$B$$, respectively. If the equation of the circle having the line segment $$AB$$ as a diameter is $$x^2 + y^2 - 3x - 2y = 0$$ and the length of the latus rectum of the ellipse $$x^2 + 9y^2 = k^2$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, then $$2m + n$$ is equal to
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Let $$A$$ and $$B$$ be two square matrices of order 3 such that $$|A| = 3$$ and $$|B| = 2$$. Then $$|A^T A(\text{adj}(2A))^{-1}(\text{adj}(4B))(\text{adj}(AB))^{-1}AA^T|$$ is equal to :
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If the system of equations $$11x + y + \lambda z = -5$$, $$2x + 3y + 5z = 3$$, $$8x - 19y - 39z = \mu$$ has infinitely many solutions, then $$\lambda^4 - \mu$$ is equal to :
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Let $$A = \{1, 3, 7, 9, 11\}$$ and $$B = \{2, 4, 5, 7, 8, 10, 12\}$$. Then the total number of one-one maps $$f : A \to B$$, such that $$f(1) + f(3) = 14$$, is :
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Let $$f(x) = x^5 + 2x^3 + 3x + 1, x \in \mathbb{R}$$, and $$g(x)$$ be a function such that $$g(f(x)) = x$$ for all $$x \in \mathbb{R}$$. Then $$\frac{g(7)}{g'(7)}$$ is equal to :
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If the function $$f(x) = \frac{\sin 3x + \alpha \sin x - \beta \cos 3x}{x^3}, x \in \mathbb{R}$$, is continuous at $$x = 0$$, then $$f(0)$$ is equal to :
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Let a rectangle $$ABCD$$ of sides 2 and 4 be inscribed in another rectangle $$PQRS$$ such that the vertices of the rectangle $$ABCD$$ lie on the sides of the rectangle $$PQRS$$. Let $$a$$ and $$b$$ be the sides of the rectangle $$PQRS$$ when its area is maximum. Then $$(a + b)^2$$ is equal to :
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For the function $$f(x) = \sin x + 3x - \frac{2}{\pi}(x^2 + x)$$, where $$x \in [0, \frac{\pi}{2}]$$, consider the following two statements : (I) $$f$$ is increasing in $$(0, \frac{\pi}{2})$$. (II) $$f'$$ is decreasing in $$(0, \frac{\pi}{2})$$. Between the above two statements,
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The value of $$\int_{-\pi}^{\pi} \frac{2y(1+\sin y)}{1+\cos^2 y} dy$$ is :
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The integral $$\int_0^{\pi/4} \frac{136\sin x}{3\sin x + 5\cos x} dx$$ is equal to :
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If $$y = y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} + 2y = \sin(2x), y(0) = \frac{3}{4}$$, then $$y\left(\frac{\pi}{8}\right)$$ is equal to:
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If the line $$\frac{2-x}{3} = \frac{3y-2}{4\lambda+1} = 4-z$$ makes a right angle with the line $$\frac{x+3}{3\mu} = \frac{1-2y}{6} = \frac{5-z}{7}$$, then $$4\lambda + 9\mu$$ is equal to :
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Let $$d$$ be the distance of the point of intersection of the lines $$\frac{x+6}{3} = \frac{y}{2} = \frac{z+1}{1}$$ and $$\frac{x-7}{4} = \frac{y-9}{3} = \frac{z-4}{2}$$ from the point $$(7, 8, 9)$$. Then $$d^2 + 6$$ is equal to :
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The coefficients $$a, b, c$$ in the quadratic equation $$ax^2 + bx + c = 0$$ are chosen from the set $$\{1, 2, 3, 4, 5, 6, 7, 8\}$$. The probability of this equation having repeated roots is :
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The number of ways of getting a sum 16 on throwing a dice four times is ______
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Let $$a_1, a_2, a_3, \ldots$$ be in an arithmetic progression of positive terms. Let $$A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \ldots + a_{2k-1}^2 - a_{2k}^2$$. If $$A_3 = -153, A_5 = -435$$ and $$a_1^2 + a_2^2 + a_3^2 = 66$$, then $$a_{17} - A_7$$ is equal to ______
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If the constant term in the expansion of $$\left(1 + 2x - 3x^3\right)\left(\frac{3}{2}x^2 - \frac{1}{3x}\right)^9$$ is $$p$$, then $$108p$$ is equal to ______
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Suppose $$AB$$ is a focal chord of the parabola $$y^2 = 12x$$ of length $$l$$ and slope $$m < \sqrt{3}$$. If the distance of the chord $$AB$$ from the origin is $$d$$, then $$ld^2$$ is equal to ______
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Let $$f$$ be a differentiable function in the interval $$(0, \infty)$$ such that $$f(1) = 1$$ and $$\lim_{t \to x} \frac{t^2 f(x) - x^2 f(t)}{t - x} = 1$$ for each $$x > 0$$. Then $$2f(2) + 3f(3)$$ is equal to ______
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From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $$X$$ denote the number of defective items in the sample. If the variance of $$X$$ is $$\sigma^2$$, then $$96\sigma^2$$ is equal to ______
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The number of distinct real roots of the equation $$|x||x + 2| - 5|x + 1| - 1 = 0$$ is ______
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If $$S = \{a \in \mathbb{R} : |2a - 1| = 3[a] + 2\{a\}\}$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$ and $$\{t\}$$ represents the fractional part of $$t$$, then $$72\sum_{a \in S} a$$ is equal to ______
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The area of the region enclosed by the parabolas $$y = x^2 - 5x$$ and $$y = 7x - x^2$$ is ______
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Let $$\vec{a} = \hat{i} - 3\hat{j} + 7\hat{k}, \vec{b} = 2\hat{i} - \hat{j} + \hat{k}$$ and $$\vec{c}$$ be a vector such that $$(\vec{a} + 2\vec{b}) \times \vec{c} = 3(\vec{c} \times \vec{a})$$. If $$\vec{a} \cdot \vec{c} = 130$$, then $$\vec{b} \cdot \vec{c}$$ is equal to ______
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