Calculate the amount of aluminium present in $\text{Al}_2(\text{SO}_4)_3$ if the compound contains 4 g of sulfur.
[Molar mass of sulfur = 32 g/mol , Molar mass of Al =27 g/mol ]

What is the observed electronic configuration of Cu ?

What is the energy of an electron in a hydrogen atom in a stationary state corresponding to n = 2 ?

Identify increasing order of acidity for following diprotic acids in aqueous solutions.

Which of the following molecules is paramagnetic?

Which of the following statements is true about ideal gas ?

What is the change in entropy of surrounding for the reaction,
$\text{H}_{2(g)} + 1/2\ \text{O}_{2(g)} \rightarrow \text{H}_2\text{O}_{(l)}$
at 298 K if standard enthalpy of formation of water is $-286$ kJ ?

In a particular reaction, 4 kJ heat is released by the system and 12 kJ work done on the system. Calculate the $\Delta H$ and $\Delta U$.

Which of the following relations is correct for a reaction $\text{S}_{(s)} + \text{O}_{2(g)} \rightarrow \text{SO}_{2(g)}$ ?

The solubility product of a sparingly soluble salt BA is $6.4 \times 10^{-13}$. Calculate it's solubility in $\text{g dm}^{-3}$ .
Molar mass of salt is $190\ \text{g mol}^{-1}$ .

The solubility of AgCl in 0.1 M NaCl is S mol/L. If the solubility product of AgCl is $1.8 \times 10^{-10}$, then S is approximately:

Identify from following a weak electrolyte based on their dissociation nature in aqueous medium.

What is the oxidation number of oxygen in $\text{KO}_2$ ?

The most abundant element in the universe is

Which of the following compounds is formed when chlorine reacts with dry slake lime $\text{Ca(OH)}_2$ ?

Which of the following is an aromatic alcohol?

What type of following reactions, the oxidative rancidity is?

Find the number of hyperconjugation structures in the isopropyl carbocation.

Which from following alkenes is highly stabilized ?

Which of the following alkanes is NOT obtained when a mixture of ethyl iodide and n-propyl iodide is subjected to Wurtz reaction?

A compound forms hcp structure. What is the number of tetrahedral voids formed in 0.8 mol of it.

Calculate the number of atoms present in 90 g of metal if it forms bcc structure.
[ $\rho \times a^3 = 1.8 \times 10^{-22}$ g ]

In the fcc structure, the number of tetrahedral holes per sphere is

For a binary solution, when a nonvolatile nonelectrolyte solute is dissolved in solvent, the vapour pressure of solvent decreases by 20 % . what is mole fraction of solvent in solution ?

Calculate the boiling point of an aqueous solution containing 18 g glucose in 100 g water if the molal elevation constant of water is $0.5\ \text{K kg mol}^{-1}$.
[Molar mass of glucose $= 180\ \text{g mol}^{-1}$ and boiling point of water $= 100\ ^\circ\text{C}$]

Which of the following is the colligative property of a solution?

Identify the reaction from following so that the standard potential of $\text{Cu}^{+2}/\text{Cu}$ electrode is 0.34 V. with respective to SHE ?

The cells A and B contain aqueous solutions of ferrous chloride and ferric chloride respectively. The mass of iron deposited by the same quantity of electricity in cells A and B is in the ratio.

What happens during the discharge of a lead storage battery?

The rate constant of reaction is $1.5 \times 10^7\ \text{s}^{-1}$ at 300 K and $3.0 \times 10^7\ \text{s}^{-1}$ at 330 K. What is the activation energy for the reaction? [ $R \times 2.303 = 19.15\ \text{JK}^{-1}\text{mol}^{-1}$]

The rate constant for the reaction $2\text{N}_2\text{O}_5 \rightarrow 4\text{NO}_2 + \text{O}_2$ is $3.0 \times 10^{-5}\ \text{s}^{-1}$. If the rate is $2.4 \times 10^{-5}\ \text{mole L}^{-1}\ \text{s}^{-1}$ then the concentration of $\text{N}_2\text{O}_5\,(\text{mol L}^{-1})$ is

What is order of radioactive decay of a substance ?

Nano materials not applied in

Which of the following properties is responsible for the development of colour to the colloids?

Identify the factor responsible for colour of transition metal compounds.

Identify the number of unpaired electrons present and geometry respectively of $[\text{Co(NH}_3)_6]^{3+}$ complex.

Identify oxidation state of cobalt ion in the complex, $[\text{Co(NH}_3)_5\text{Br}]\text{SO}_4$ .

Which of the following alkyl halides is hydrolyzed most rapidly by the $\text{SN}^2$ mechanism ?

Which of the following alkyl halides undergoes $\text{SN}^1$ reaction most readily?

What is the nature of alcohols in aqueous medium?

Which of following observations is found in Tollens test for aldehydes?

An organic compound $\text{CH}_3\text{-CH=CH-CH}_2\text{-CHO}$ is taken in two different containers A and B. Sample in A is treated with $\text{H}_2$ / Ni forming new compound P. Sample in B is treated with $\text{LiAlH}_4$ and hydrolyzed further forming new compound Q. Identify P and Q.

What is the common name for the simplest carboxylic acid, HCOOH ?

Which of the following reactions uses diazonium salt ?

Which of the following statements is NOT correct?

Identify the linkage in backbone of single strand of DNA ?

Which of the following ionic species is present in hemoglobin and responsible for transformation of oxygen ?

Which of the following processes develops cross-links in elastomer?

Which of the following is not a basis of classification of polymers ?

Which of the following detergents is used as germicide?

A ball is thrown in the air. Its height at any time $t$ is given by $h = 3 + 14t - 5t^2$, then the maximum height it can reach

If $(1 + i) \cdot (1 + 2i) \ldots\ldots\ldots (1 + ni) = x + iy$ (Where $i = \sqrt{-1}$ ), then the value of $(2) \cdot (5) \cdot (10)\ldots\ldots\ldots(1 + n^2)$

The number of arrangements of the numbers strictly between 10 and 1000 formed with the digits 0, 1, 2, 3, 4, 5, 6 without repetition is

The value of $\dfrac{\sin^2 3A}{\sin^2 A} - \dfrac{\cos^2 3A}{\cos^2 A}$ is

If $0 \leq x \leq \dfrac{\pi}{2}$ , then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$ is

The number of lines passing through A (3, 4) and the sum of whose non-zero intercepts is zero is (are)

If $4ab = 3h^2$, then the ratio of the slopes of the lines represented by $ax^2 + 2hxy + by^2 = 0$ is...

The equations of the tangent to the curve $x^2 + y^2 = 10$, where the tangent is parallel to the line $2x + y - 1 = 0$, are

If the line $y = 2x + \lambda$ is a tangent to the hyperbola $36x^2 - 25y^2 = 3600$, then $\lambda =$

$\displaystyle\lim_{x \to 0}\dfrac{(5^x - 1)^4\,\text{cosec}\,(x\log 5)}{\tan(x\log 5) \cdot \log(1 + x^2\log 25)} = \ldots\ldots$

Simplest form of the following switching circuit is

With usual notations in $\triangle$ABC, if $b\cos^2\dfrac{C}{2} + c\cos^2\dfrac{B}{2} = \dfrac{3a}{2}$ then

In a triangle ABC, with usual notations $a = \sqrt{3} + 1$, $b = \sqrt{3} - 1$ and $\angle C = 60^\circ$ then the values of $\angle A$ and $\angle B$ respectively are

Inverse of $\begin{bmatrix} 3 & -2 \\ 1 & 4 \end{bmatrix}$ is

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A\,(\text{adj }A) = AA^T$, Then $5a + b =$

If $y = \tan^{-1}\left[\dfrac{x - \sqrt{1 - x^2}}{x + \sqrt{1 - x^2}}\right]$ , then $\dfrac{dy}{dx} =$

$\cos\left(\cos^{-1}\left(-\dfrac{1}{2}\right) + \dfrac{\pi}{3}\right) =$

$\sin^{-1}\left(\dfrac{12}{13}\right) + \cos^{-1}\left(\dfrac{4}{5}\right) + \tan^{-1}\left(\dfrac{63}{16}\right) =$

If $f(x) = \dfrac{4x + 3}{6x - 4}$, $x \neq \dfrac{2}{3}$ and $(\text{fof})(x) = g(x)$ where $g: \mathbb{R} - \left\{\dfrac{2}{3}\right\} \rightarrow \mathbb{R} - \left\{\dfrac{2}{3}\right\}$, then $(g\,o\,g\,o\,g\,o\,g\,o\,g)\,(3) =$

If the derivative of the function $f(x) = \begin{cases} ax^2 + b & \text{if } x < -1 \\ bx^2 + ax + 4 & \text{if } x \geq -1 \end{cases}$ is continuous everywhere then

The derivative of $\log_8(\log_5 x)$ w. r. t. $x$ is

If $y^{\frac{1}{m}} + y^{\frac{-1}{m}} = 2x$ , then $(x^2 - 1)y_1^{\ 2} =$

A triangle has two fixed vertices A $(a, 0)$ and B$(0, b)$ . Let its third vertex C is moving along the line $x = y$. If $s$ is the area of triangle ABC, then $\dfrac{ds}{dx} =$

The function $f(x) = \tan^{-1}(\sin x + \cos x)$ is an increasing function in the interval.....

The minimum value of $\dfrac{\log x}{x}$ in the interval $(2, \infty)$ is

$\displaystyle\int \dfrac{(x + 1)(x + \log x)^2}{x}\,dx =$

$\displaystyle\int \cot^4 x\,dx$ is equal to

$\displaystyle\int \dfrac{\cos^3 x}{\sin^2 x + \sin x}\,dx =$

$\displaystyle\int_{\pi/6}^{\pi/3} \dfrac{\sin x - \cos x}{1 + \sin x\cos x}\,dx =$

$\displaystyle\int_0^{\log 5} \dfrac{e^x\sqrt{e^x - 1}}{e^x + 3}\,dx =$

The integrating factor of the differential equation $(1 + t^2) + \left(x - e^{\tan^{-1}t}\right)\dfrac{dt}{dx} = 0$ is

The degree of the differential equation obtained from the equation $(y - a)^2 = 4(x - b)$ [where $a$ and b are arbitrary constants] is

The solution of the differential equation $\left(x + 2y^3\right)\dfrac{dy}{dx} - y = 0$ is

The rate of reduction is proportional to the square root of a persons existing assets at that moment. If his assets at the beginning are 10000 and they dwindle down to 5625 in 2 years, then the person will be bankrupt in

If a vector $3\hat{i} + 4\hat{j} - 5\hat{k}$ is rotated through a certain angle about the origin in the anti-clockwise direction, then the components of the new vector are $a + 1, -3, 5$ . The possible values of $a$ is

Let A, B, C, D be the points in the plane with position vectors $-2\hat{i} - \hat{j}$, $4\hat{i}$, $3\hat{i} + 3\hat{j}$ and $-3\hat{i} + 2\hat{j}$ respectively, then $\square$ABCD is

Let $\bar{a} = \hat{i} + \hat{j}$, $\bar{c} = \hat{i} - \hat{j}$ and a vector $\bar{b}$ be such that $\bar{a} \times \bar{b} = \bar{c}$ and $\bar{a} \cdot \bar{b} = 3$ then $|\bar{b}| =$

The acute angle between the vector $2\hat{i} + \hat{j} - 3\hat{k}$ and the plane containing the vectors $2\hat{i} + 3\hat{j} - \hat{k}$ and $\hat{i} - \hat{j} + 2\hat{k}$ is

A parallelogram is constructed on $5\bar{a} + 2\bar{b}$ and $\bar{a} - 3\bar{b}$ as its adjacent sides, with $|\bar{a}| = 2\sqrt{2}, |\bar{b}| = 3$ . The angle between $\bar{a}$ and $\bar{b}$ is $\dfrac{\pi}{4}$ . Then the length of the diagonals of the parallelogram are

The lines $\dfrac{x - 2}{1} = \dfrac{y - 3}{1} = \dfrac{z - 4}{-k}$ and $\dfrac{x - 1}{k} = \dfrac{y - 4}{2} = \dfrac{z - 5}{1}$ are coplanar if

The direction ratios of the normal to the plane passing through (1, 0, 0), (0, 1, 0) which makes an angle of measure $45^\circ$ with the plane $2x + 3y = 7$ are....

The lines $\dfrac{x - 1}{-1} = \dfrac{y + 2}{1} = \dfrac{z - 3}{-2}$ and $\dfrac{x - 1}{1} = \dfrac{y + 2}{1} = \dfrac{z + 1}{-2}$ are

If the product of the distances of the point (1, 2, 3) from the origin and the plane $2x - 3y + z + k = 0$ is 7, then the value of k is

The vector equation of the line whose cartesian equations are $x = 2, 2y - 3z + 7 = 0$

The feasible region represented by the constraints $y - 2x \leq 4, x + y \geq 5, x \leq 4, y \geq 2, x, y \geq 0$ is ...........

A fair die is thrown at random. Let A be the event that the number obtained on the die is a non-even prime number and B be the event that the number obtained on the die is an odd number.
Let $p : P(A) = \dfrac{1}{3}$, $q$ : A and B are independent events.
Then the truth values of statements $p$ and $q$ respectively are.....

Three ships A, B, C sail from England to India. If the odds in favour of their safe arrival are 2:5, 3:7 and 6:11 respectively, then the probability that the exactly two ships arrive safely is

A random variable X has the probability distribution

$X = x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$P(X = x)$	$0.23$	$0.15$	$0.12$	$0.10$	$0.20$	$0.07$	$0.08$	$0.05$

for the events E = {X is a prime number} and F = {X $\leq$ 3}, then P (E$\cup$F)=

The probability distribution of a random variable X is given by

$X = x$	$1$	$2$	$3$	$4$
$P(X = x)$	$k$	$2k$	$3k$	$4k$

Then the c.d.f. of X is given by

Two dice are thrown successively 4 times and getting a doublet is called a success. The probability of getting at least 1 success is

A wire has mass $(0.3 \pm 0.003)$ gram, radius $(0.5 \pm 0.005)$ cm and length $(6 \pm 0.06)$ cm. The maximum percentage error in the measurement of density is

Let $\vec{P} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{Q} = -(\hat{i} + \hat{j} + \hat{k})$. The angle between $(\vec{P} - \vec{Q})$ and $\vec{P}$ is

A ball is released from height 'h' which makes perfectly elastic collision with ground. The frequency of periodic vibratory motion is (g=acceleration due to gravity)

Two boys are standing at points A and B on ground where distance AB = a. The boy at point B starts running perpendicular to line AB with velocity '$V_1$'. The boy at point A starts running simultaneously with velocity 'V' and catches the other boy in time 't'. The value of 't' is

A particle is moving with constant angular acceleration $4\,\text{rad/s}^2$ in circular path. At what time the magnitudes of its tangential acceleration and centripetal acceleration will be equal ?

A particle describes a horizontal circle of radius 'r' in conical funnel with smooth inner surface with a speed of $0.5$ m/s. The height of the plane of the circle from the vertex of the funnel is (acceleration due to gravity, $g = 10\,\text{m/s}^2$)

A block of mass 'm' moving on a frictionless horizontal surface collides with a spring of spring constant 'K' and compresses it through a distance 'x'. The maximum momentum of the block after collision is

A Solid sphere has mass M and radius R. Its moment of inertia about a parallel axis passing through a point at a distance R/3 from its centre is

The masses and radii of the earth and moon are $M_1$, $R_1$ and $M_2$, $R_2$ and respectively, Their centres are at a distance 'd' apart. The minimum speed with which body of mass 'm' should be projected from a distance $2d/3$ from the centre of $M_1$ so as to escape to infinity is

A rectangular block of mass 'm' and cross-sectional area 'A' floats on a liquid of density '$\varrho$'. It is given a small vertical displacement from equilibrium, it starts oscillating with frequency (g=acceleration due to gravity)

A closed pipe containing liquid showed a pressure $P_1$ by gauge. When the valve was opened, pressure was reduced to $P_2$. The speed of water flowing out of the pipe is ($\varrho$=density of water)

Water flows through a horizontal pipe at a speed 'V'. Internal diameter of the pipe is 'd'. If the water is emerging at a speed '$V_1$' then the diameter of the nozzle is

A liquid is at rest in a container. In a sphere of influence, the liquid molecule at its centre is

Rate of flow of heat through a cylindrical rod is '$H_1$'. The temperature at the ends of the rod are $T_1$ and $T_2$. If all the dimensions of the rod become double and the temperature difference remains the same and if the rate of flow of heat becomes '$H_2$' then $H_2 =$

The energy spectrum of a black body exhibits a maximum around a wavelength '$\lambda$'. The temperature of the black body is now changed such that the energy is maximum around a wavelength $2\lambda/3$. The power radiated by the black body will now increase by a factor

A black rectangular surface of area 'A' emits energy 'E' per second at $27^\circ$C. If length and breadth is reduced to half of its initial value and temperature is raised to $327^\circ$C then energy emitted per second becomes

A monoatomic gas at pressure P having volume V expands isothermally to a volume 3V and then adiabatically to a volume 81 V. The final pressure of the gas is ($\gamma = 5/3$).

An ideal gas at $27^\circ$C is compressed adiabatically to $\left(\dfrac{8}{27}\right)$ of its original volume. If $\gamma = 5/3$, then the rise in temperature of a gas is

The translational kinetic energy of the molecules of a gas at absolute temperature (T) can be doubled by

For a particle P performing S.H.M., when displacement is 'x', potential energy and restoring force acting on it is denoted by 'E' and 'F' respectively. The relation between x, E and F is

A progressive wave of frequency 50 Hz is travelling with velocity 350 m/s through a medium. The change in phase at given time interval of 0.01s is

When a sonometer wire vibrates in the third overtone, the number of antinodes and nodes formed on the wire are respectively

When the observer moves towards a stationary source with velocity '$V_1$', the apparent frequency of the emitted note is '$F_1$'. When the observer moves away from the source with velocity '$V_1$', the apparent frequency is '$F_2$'. If 'V' is the speed of sound in air and $F_1/F_2 = 2$, then $V/V_1 =$

The equation of wave motion is $Y = 6\sin\left(12\pi t - 0.02\pi x + \dfrac{\pi}{3}\right)$ where x is in metre and time in second. The velocity of the wave is

The electric field in the region is $\vec{E} = a\hat{i} + b\hat{j}$ where 'a' and 'b' are constants. The net electric flux passing through a square area of side 'l' parallel to Y-Z plane is

Two point charges $q_1 = 6\ \mu\text{C}$ and $q_2 = 4\ \mu\text{C}$ are kept at points A and B in air where distance $AB = 10$ cm. What is the increase in potential energy of the system when $q_2$ is moved towards $q_1$, by 2 cm ? $\left(\dfrac{1}{4\pi\epsilon_0} = 9 \times 10^9\ \text{SI units}\right)$

The electric field between the plates of a parallel plate capacitor is 'E'. If the charge on the plates is Q then the force on each plate is

Two condensers of capacities, '2C' and 'C' are joined in parallel and charged upto potential 'V'. The battery is then disconnected and condenser of capacity 'C' is filled completely with a medium of dielectric constant 'K'. The potential difference across the capacitors in the second case is

A potentiometer wire is 4m long and potential difference of 3V is maintained between the ends. The e.m.f. of the cell which balances against a length of 100 cm of the potentiometer wire is

A metre bridge experiment, the resistance in the left gap is $20\ \Omega$ and in the right gap is $60\ \Omega$. The bridge is balanced. The distance of the null point from the centre of the wire is

The magnetic moment produced in a sample of 2 gram is $8 \times 10^{-7}\,\text{A/m}^2$. If its density is $4\ \text{g/cm}^3$ then the magnetization of the sample is

A wire carrying current 'I' along x axis has length 'L' and it is kept in a magnetic field $B(\hat{i}+2\hat{j}-2\hat{k})$ T. The magnitude of magnetic force acting on the wire is

A charged particle is moving in a uniform magnetic field in a circular path of radius 'R'. When the energy of the particle becomes 3 times the original, the new radius will be

Torque acting on a coil carrying current 'I' situated parallel to magnetic field of induction 'B' having of turns 'n' and area 'A' is

A metal rod of length 'L' completes the circuit as shown. The area of the circuit is perpendicular to magnetic field 'B'. Total resistance of the circuit is 'R'. The force needed to move the rod in the direction as shown with constant speed 'V' is

When a capacitor is connected in series LR circuit, the alternating current flowing in the circuit

With gradual increase in frequency of an a.c. supply, the impedance of an LCR series circuit

An alternating e.m.f. is given by $e = e_0 \sin\omega t$. In what time the e.m.f. will have half its maximum value, if 'e' Starts from zero?

A step down transformer has turns ratio 20:1. If 8V are applied across $0.4\ \Omega$ secondary then the primary current will be

A simple microscope is used to see the object first in blue light and then in red light. Due to the change from blue to red light, what is the effect on its magnifying power?

In Young's double slit experiment, the fringe width is 0.4 mm. What is the distance between $4^{\text{th}}$ dark band and $6^{\text{th}}$ bright band on the same side of the interference pattern?

The ratio of intensities at two points on the screen in Young's double slit experiment when waves from the two slits have a path difference of zero and $\lambda/4$ is ($\lambda$ is the wavelength of light used) ($\cos 0^\circ = 1$, $\cos\pi/2 = 0$)

In Young's double slit experiment, the distance between the slits is 3 mm and the slits are 2m away from the screen. Two interference patterns can be obtained on the screen due to light of wavelength 480 nm and 600 nm respectively. The separation on the screen between the $5^{\text{th}}$ order bright fringes on two interference patterns is

For a photosensitive material, work function is '$W_0$' and stopping potential is 'V'. The wavelength of the incident radiation is (h=Planck's constant, c = velocity of light, e = electronic charge)

Using Einstein's photoelectric equation, the graphical representation between the kinetic energy (E) of emitted Photoelectrons and the frequency of incident radiation ($\nu$) is show correctly in figure

The force acting on the electron in hydrogen atom (Bohr's theory) is related to the principal quantum number n as

Using Bohr's atomic model, the orbital period of electron in hydrogen atom in the $n^{\text{th}}$ orbit is ($\epsilon_0$=permittivity of free space, h=Planck's constant, m = mass of electron, e = electronic charge)

In a transistor amplifier, a change of 0.2 mA in the base current causes a change of 5 mA in the collector current. If input resistance is $2\ \text{k}\Omega$ and voltage gain is 75, the load resistance used in the circuit is

The resultant gate and its Boolean expression in the given circuit is

When a small amount of impurity atoms are added to a semiconductor then generally its resistivity