Let $\alpha$ be a root of the equation 1 + x² + x⁴ = 0. Then, the value of $\alpha$¹⁰¹¹ + $\alpha$²⁰²² $-$ $\alpha$³⁰³³ is equal to :

The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to :

The number of values of a $\in$ N such that the variance of 3, 7, 12, a, 43 $-$ a is a natural number is :

For real numbers a, b (a > b > 0), let

Area $\left\{ {(x,y):{x^2} + {y^2} \le {a^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \ge 1} \right\} = 30\pi$

and

Area $\left\{ {(x,y):{x^2} + {y^2} \le {b^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \le 1} \right\} = 18\pi$

Then, the value of (a $-$ b)² is equal to ___________.

Let f and g be twice differentiable even functions on ($-$2, 2) such that $f\left( {{1 \over 4}} \right) = 0$, $f\left( {{1 \over 2}} \right) = 0$, $f(1) = 1$ and $g\left( {{3 \over 4}} \right) = 0$, $g(1) = 2$. Then, the minimum number of solutions of $f(x)g''(x) + f'(x)g'(x) = 0$ in $( - 2,2)$ is equal to ________.

Let the coefficients of x^$-$1 and x^$-$3 in the expansion of ${\left( {2{x^{{1 \over 5}}} - {1 \over {{x^{{1 \over 5}}}}}} \right)^{15}},x > 0$, be m and n respectively. If r is a positive integer such that $m{n^2} = {}^{15}{C_r}\,.\,{2^r}$, then the value of r is equal to __________.

Let M = \left[ {\matrix{ 0 & { - \alpha } \cr \alpha & 0 \cr } } \right]\(, where \)\alpha\( is a non-zero real number an \)N = \sum\limits_{k = 1}^{49} {{M^{2k}}} \(. If \)(I - {M^2})N = - 2I\(, then the positive integral value of \)\alpha is ____________.

Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $f(g(x)) = 8{x^2} - 2x$ and $g(f(x)) = 4{x^2} + 6x + 1$, then the value of $f(2) + g(2)$ is _________.

At what temperature a gold ring of diameter 6.230 cm be heated so that it can be fitted on a wooden bangle of diameter 6.241 cm ? Both the diameters have been measured at room temperature (27$^\circ$C).

(Given : coefficient of linear thermal expansion of gold $\alpha$_L = 1.4 $\times$ 10^$-$5 K^$-$1)

Two point charges Q each are placed at a distance d apart. A third point charge q is placed at a distance x from mid-point on the perpendicular bisector. The value of x at which charge q will experience the maximum Coulomb's force is :

A capacitor is discharging through a resistor R. Consider in time t₁, the energy stored in the capacitor reduces to half of its initial value and in time t₂, the charge stored reduces to one eighth of its initial value. The ratio t₁/t₂ will be

The time period of a satellite revolving around earth in a given orbit is 7 hours. If the radius of orbit is increased to three times its previous value, then approximate new time period of the satellite will be

A vessel contains 16g of hydrogen and 128g of oxygen at standard temperature and pressure. The volume of the vessel in cm³ is :

A block of mass M placed inside a box descends vertically with acceleration 'a'. The block exerts a force equal to one-fourth of its weight on the floor of the box. The value of 'a' will be

A person can throw a ball upto a maximum range of 100 m. How high above the ground he can throw the same ball?

The Vernier constant of Vernier callipers is 0.1 mm and it has zero error of ($-$0.05) cm. While measuring diameter of a sphere, the main scale reading is 1.7 cm and coinciding vernier division is 5. The corrected diameter will be _________ $\times$ 10^$-$2 cm.

Two resistors are connected in series across a battery as shown in figure. If a voltmeter of resistance 2000 $\Omega$ is used to measure the potential difference across 500 $\Omega$ resistor, the reading of the voltmeter will be ___________ V.

A potential barrier of 0.4 V exists across a p-n junction. An electron enters the junction from the n-side with a speed of 6.0 $\times$ 10⁵ ms^$-$1. The speed with which electron enters the p side will be ${x \over 3} \times {10^5}$ ms^$-$1 the value of x is _____________.

(Given mass of electron = 9 $\times$ 10^$-$31 kg, charge on electron = 1.6 $\times$ 10^$-$19 C.)

In a double slit experiment with monochromatic light, fringes are obtained on a screen placed at some distance from the plane of slits. If the screen is moved by 5 $\times$ 10^$-$2 m towards the slits, the change in fringe width is 3 $\times$ 10^$-$3 cm. If the distance between the slits is 1 mm, then the wavelength of the light will be ____________ nm.

An inductor of 0.5 mH, a capacitor of 200 $\mu$F and a resistor of 2 $\Omega$ are connected in series with a 220 V ac source. If the current is in phase with the emf, the frequency of ac source will be ____________ $\times$ 10² Hz.

Consider the species CH₄, NH$_4^ +$ and BH$_4^ -$. Choose the correct option with respect to the these species.

4.0 moles of argon and 5.0 moles of PCl₅ are introduced into an evacuated flask of 100 litre capacity at 610 K. The system is allowed to equilibrate. At equilibrium, the total pressure of mixture was found to be 6.0 atm. The K_p for the reaction is :

[Given : R = 0.082 L atm K^$-$1 mol^$-$1]

Given below are two statements.

$\bullet$ Statement I : In CuSO₄ . 5H₂O, Cu-O bonds are present.

$\bullet$ Statement II : In CuSO₄ . 5H₂O, ligands coordinating with Cu(II) ion are O-and S-based ligands.

In the light of the above statements, choose the correct answer from the options given below.

Number of lone pair(s) of electrons on central atom and the shape BrF₃ molecule respectively, are

A white precipitate was formed when BaCl₂ was added to water extract of an inorganic salt. Further, a gas 'X' with characteristic odour was released when the formed white precipitate was dissolved in dilute HCl. The anion present in the inorganic salt is

A box contains 0.90 g of liquid water in equilibrium with water vapour at 27$^\circ$C. The equilibrium vapour pressure of water at 27$^\circ$C is 32.0 Torr. When the volume of the box is increased, some of the liquid water evaporates to maintain the equilibrium pressure. If all the liquid water evaporates, then the volume of the box must be __________ litre. [nearest integer]

(Given : R = 0.082 L atm K^$-$1 mol^$-$1)

(Ignore the volume of the liquid water and assume water vapours behave as an ideal gas.)

The equation

k = (6.5 $\times$ 10¹²s^$-$1)e^$-$26000K/T

is followed for the decomposition of compound A. The activation energy for the reaction is ________ kJ mol^$-$1. [nearest integer]

(Given : R = 8.314 J K^$-$1 mol^$-$1)

Spin only magnetic moment of [MnBr₆]^4$-$ is _________ B.M. (round off to the closest integer)

Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z $-$ 1) $-$ arg(z + 1) = ${\pi \over 4}$ intersect :

The value of $\mathop {\lim }\limits_{x \to 1} {{({x^2} - 1){{\sin }^2}(\pi x)} \over {{x^4} - 2{x^3} + 2x - 1}}$ is equal to:

If $\int\limits_0^2 {\left( {\sqrt {2x} - \sqrt {2x - {x^2}} } \right)dx = \int\limits_0^1 {\left( {1 - \sqrt {1 - {y^2}} - {{{y^2}} \over 2}} \right)dy + \int\limits_1^2 {\left( {2 - {{{y^2}} \over 2}} \right)dy + I} } }$, then I equals

Let a triangle ABC be inscribed in the circle ${x^2} - \sqrt 2 (x + y) + {y^2} = 0$ such that $\angle BAC = {\pi \over 2}$. If the length of side AB is $\sqrt 2$, then the area of the $\Delta$ABC is equal to :

$\overrightarrow a = \widehat i + 4\widehat j + 3\widehat k$

$\overrightarrow b = 2\widehat i + \alpha \widehat j + 4\widehat k,\,\alpha \in R$

$\overrightarrow c = 3\widehat i - 2\widehat j + 5\widehat k$

If $\alpha$ is the smallest positive integer for which $\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c$ are noncollinear, then the length of the median, in $\Delta$ABC, through A is :

Options:

The total number of four digit numbers such that each of first three digits is divisible by the last digit, is equal to ____________.

The speed of light in media 'A' and 'B' are $2.0 \times {10^{10}}$ cm/s and $1.5 \times {10^{10}}$ cm/s respectively. A ray of light enters from the medium B to A at an incident angle '$\theta$'. If the ray suffers total internal reflection, then

Two long current carrying conductors are placed to each other at a distance of 8 cm between them. The magnitude of magnetic field produced at mid-point between the two conductors due to current flowing in them is 300 $\mu$T. The equal current flowing in the two conductors is :

A small spherical ball of radius 0.1 mm and density 10⁴ kg m^$-$3 falls freely under gravity through a distance h before entering a tank of water. If, after entering the water the velocity of ball does not change and it continue to fall with same constant velocity inside water, then the value of h will be ___________ m.

(Given g = 10 ms^$-$2, viscosity of water = 1.0 $\times$ 10^$-$5 N-sm^$-$2).

In an experiment to determine the velocity of sound in air at room temperature using a resonance tube, the first resonance is observed when the air column has a length of 20.0 cm for a tuning fork of frequency 400 Hz is used. The velocity of the sound at room temperature is 336 ms^$-$1. The third resonance is observed when the air column has a length of _____________ cm.

Sulphur dioxide is one of the components of polluted air. SO₂ is also a major contributor to acid rain. The correct and complete reaction to represent acid rain caused by SO₂ is :

Which of the following carbocations is most stable?

The stable carbocation formed in the above reaction is

In Friedel-Crafts alkylation of aniline, one gets

The cell potential for the given cell at 298 K

Pt| H₂ (g, 1 bar) | H⁺ (aq) || Cu²⁺ (aq) | Cu(s)

is 0.31 V. The pH of the acidic solution is found to be 3, whereas the concentration of Cu²⁺ is 10^$-$x M. The value of x is ___________.

(Given : $E_{C{u^{2 + }}/Cu}^\Theta$ = 0.34 V and ${{2.303\,RT} \over F}$ = 0.06 V)

For the reaction given below :

CoCl₃ . xNH₃ + AgNO₃ (aq) $\to$

If two equivalents of AgCl precipitate out, then the value of x will be _____________.

The number of chiral alcohol(s) with molecular formula C₄H₁₀O is ________.

Let f be a real valued continuous function on [0, 1] and $f(x) = x + \int\limits_0^1 {(x - t)f(t)dt}$.

Then, which of the following points (x, y) lies on the curve y = f(x) ?

Let y = y(x), x > 1, be the solution of the differential equation $(x - 1){{dy} \over {dx}} + 2xy = {1 \over {x - 1}}$, with $y(2) = {{1 + {e^4}} \over {2{e^4}}}$. If $y(3) = {{{e^\alpha } + 1} \over {\beta {e^\alpha }}}$, then the value of $\alpha + \beta$ is equal to _________.

Starting with the same initial conditions, an ideal gas expands from volume V₁ to V₂ in three different ways. The work done by the gas is W₁ if the process is purely isothermal, W₂, if the process is purely adiabatic and W₃ if the process is purely isobaric. Then, choose the correct option

The motion of a simple pendulum executing S.H.M. is represented by the following equation.

$y = A\sin (\pi t + \phi )$, where time is measured in second. The length of pendulum is

Given below are two statements :

Statement I : The electric force changes the speed of the charged particle and hence changes its kinetic energy; whereas the magnetic force does not change the kinetic energy of the charged particle.

Statement II : The electric force accelerates the positively charged particle perpendicular to the direction of electric field. The magnetic force accelerates the moving charged particle along the direction of magnetic field.

In the light of the above statements, choose the most appropriate answer from the options given below :

If the electric potential at any point (x, y, z) m in space is given by V = 3x² volt. The electric field at the point (1, 0, 3) m will be :

The moment of inertia of a uniform thin rod about a perpendicular axis passing through one end is I₁. The same rod is bent into a ring and its moment of inertia about a diameter is I₂. If ${{{I_1}} \over {{I_2}}}$ is ${{x{\pi ^2}} \over 3}$, then the value of x will be ____________.

Which of the following is the correct plot for the probability density ${\psi ^2}$ (r) as a function of distance 'r' of the electron from the nucleus for 2s orbital?

Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : The first ionization enthalpy for oxygen is lower than that of nitrogen.

Reason R : The four electrons in 2p orbitals of oxygen experience more electron-electron repulsion.

In the light of the above statements, choose the correct answer from the options given below.

Two isomers (A) and (B) with Molar mass 184 g/mol and elemental composition C, 52.2%; H, 4.9% and Br 42.9% gave benzoic acid and p-bromobenzoic acid, respectively on oxidation with KMnO₄. Isomer 'A' is optically active and gives a pale yellow precipitate when warmed with alcoholic AgNO₃. Isomer 'A' and 'B' are, respectively

The structure of protein that is unaffected by heating is

2.2 g of nitrous oxide (N₂O) gas is cooled at a constant pressure of 1 atm from 310 K to 270 K causing the compression of the gas from 217.1 mL to 167.75 mL. The change in internal energy of the process, $\Delta$U is '$-$x' J. The value of 'x' is ________. [nearest integer]

(Given : atomic mass of N = 14 g mol^$-$1 and of O = 16 g mol^$-$1. Molar heat capacity of N₂O is 100 J K^$-$1 mol^$-$1)

Elevation in boiling point for 1.5 molal solution of glucose in water is 4 K. The depression in freezing point for 4.5 molal solution of glucose in water is 4 K. The ratio of molal elevation constant to molal depression constant (K_b/K_f) is _________.

In the given reaction,

the number of sp² hybridised carbon(s) in compound 'X' is ________.

In the given reaction,

The number of $\pi$ electrons present in the product 'P' is _________.

If y = y(x) is the solution of the differential equation $\left( {1 + {e^{2x}}} \right){{dy} \over {dx}} + 2\left( {1 + {y^2}} \right){e^x} = 0$ and y (0) = 0, then $6\left( {y'(0) + {{\left( {y\left( {{{\log }_e}\sqrt 3 } \right)} \right)}^2}} \right)$ is equal to

The distance of the origin from the centroid of the triangle whose two sides have the equations $x - 2y + 1 = 0$ and $2x - y - 1 = 0$ and whose orthocenter is $\left( {{7 \over 3},{7 \over 3}} \right)$ is :

Let  $\overrightarrow a = \widehat i - 2\widehat j + 3\widehat k$,   $\overrightarrow b = \widehat i + \widehat j + \widehat k$   and   $\overrightarrow c$   be a vector such that   $\overrightarrow a + \left( {\overrightarrow b \times \overrightarrow c } \right) = \overrightarrow 0$   and   $\overrightarrow b \,.\,\overrightarrow c = 5$. Then the value of   $3\left( {\overrightarrow c \,.\,\overrightarrow a } \right)$   is equal to _________.

Let 3, 6, 9, 12, ....... upto 78 terms and 5, 9, 13, 17, ...... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ________.

A small toy starts moving from the position of rest under a constant acceleration. If it travels a distance of 10m in t s, the distance travelled by the toy in the next t s will be :

The electric field at a point associated with a light wave is given by

E = 200 [sin (6 $\times$ 10¹⁵)t + sin (9 $\times$ 10¹⁵)t] Vm^$-$1

Given : h = 4.14 $\times$ 10^$-$15 eVs

If this light falls on a metal surface having a work function of 2.50 eV, the maximum kinetic energy of the photoelectrons will be

A block of mass 40 kg slides over a surface, when a mass of 4 kg is suspended through an inextensible massless string passing over frictionless pulley as shown below.

The coefficient of kinetic friction between the surface and block is 0.02. The acceleration of block is. (Given g = 10 ms^$-$2.)

In the given figure, the block of mass m is dropped from the point 'A'. The expression for kinetic energy of block when it reaches point 'B' is

The combination of two identical cells, whether connected in series or parallel combination provides the same current through an external resistance of 2$\Omega$. The value of internal resistance of each cell is

The displacement current of 4.425 $\mu$A is developed in the space between the plates of parallel plate capacitor when voltage is changing at a rate of 10⁶ Vs^$-$1. The area of each plate of the capacitor is 40 cm². The distance between each plate of the capacitor is x $\times$ 10^$-$3 m. The value of x is __________.

(Permittivity of free space, E₀ = 8.85 $\times$ 10^$-$12 C² N^$-$1 m^$-$2).