Consider the following statements :

(A) The principal quantum number 'n' is a positive integer with values of 'n' = 1, 2, 3, ...

(B) The azimuthal quantum number 'l' for a given 'n' (principal quantum number) can have values as 'l' = 0, 1, 2, ...... n

(C) Magnetic orbital quantum number 'm_l' for a particular 'l' (azimuthal quantum number) has (2l + 1) values.

(D) $\pm$ 1/2 are the two possible orientations of electron spin.

(E) For l = 5, there will be a total of 9 orbital

Which of the above statements are correct?

Compound A contains 8.7% Hydrogen, 74% Carbon and 17.3% Nitrogen. The molecular formula of the compound is,

Given : Atomic masses of C, H and N are 12, 1 and 14 amu respectively.

The molar mass of the compound A is 162 g mol^$-$1.

The correct IUPAC name of the following compound is :

The major product (P) of the given reaction is

(where, Me is $-$CH₃)

In the above reaction 'A' is

When sugar 'X' is boiled with dilute H₂SO₄ in alcoholic solution, two isomers 'A' and 'B' are formed. 'A' on oxidation with HNO₃ yields saccharic acid where as 'B' is laevorotatory. The compound 'X' is :

(a) CoCl₃.4NH₃,    (b) CoCl₃.5NH₃,      (c) CoCl₃.6NH₃    and    (d) CoCl(NO₃)₂.5NH₃.

Number of complex(es) which will exist in cis-trans form is/are _______________.

Let f(x) be a quadratic polynomial such that f($-$2) + f(3) = 0. If one of the roots of f(x) = 0 is $-$1, then the sum of the roots of f(x) = 0 is equal to :

Let a triangle be bounded by the lines L₁ : 2x + 5y = 10; L₂ : $-$4x + 3y = 12 and the line L₃, which passes through the point P(2, 3), intersects L₂ at A and L₁ at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to :

Let $\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k$ and $\overrightarrow b = - 2\widehat i + \alpha \widehat j + \widehat k$, where $\alpha \in R$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\overrightarrow a$ and $\overrightarrow b$ is $\sqrt {15({\alpha ^2} + 4)}$, then the value of $2{\left| {\overrightarrow a } \right|^2} + \left( {\overrightarrow a \,.\,\overrightarrow b } \right){\left| {\overrightarrow b } \right|^2}$ is equal to :

The value of

$\mathop {\lim }\limits_{n \to \infty } 6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {{r^2} + 3r + 3}}} \right)} } \right\}$ is equal to :

Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their variance is 20. A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of students can fail is _________.

If one of the diameters of the circle ${x^2} + {y^2} - 2\sqrt 2 x - 6\sqrt 2 y + 14 = 0$ is a chord of the circle ${(x - 2\sqrt 2 )^2} + {(y - 2\sqrt 2 )^2} = {r^2}$, then the value of r² is equal to ____________.

If the system of linear equations
$2x - 3y = \gamma + 5$,
$\alpha x + 5y = \beta + 1$, where $\alpha$, $\beta$, $\gamma$ $\in$ R has infinitely many solutions then the value
of | 9$\alpha$ + 3$\beta$ + 5$\gamma$ | is equal to ____________.

Let $$A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)\( where \)i = \sqrt { - 1} \(. Then, the number of elements in the set { n \)\in$$ {1, 2, ......, 100} : Aⁿ = A } is ____________.

Let S = {1, 2, 3, 4}. Then the number of elements in the set { f : S $\times$ S $\to$ S : f is onto and f (a, b) = f (b, a) $\ge$ a $\forall$ (a, b) $\in$ S $\times$ S } is ______________.

A $\sqrt {34}$ m long ladder weighing 10 kg leans on a frictionless wall. Its feet rest on the floor 3 m away from the wall as shown in the figure. If E_f and F_w are the reaction forces of the floor and the wall, then ratio of ${F_w}/{F_f}$ will be :

(Use g = 10 m/s².)

Water falls from a 40 m high dam at the rate of 9 $\times$ 10⁴ kg per hour. Fifty percentage of gravitational potential energy can be converted into electrical energy. Using this hydroelectric energy number of 100 W lamps, that can be lit, is :

(Take g = 10 ms^$-$2)

A sample of an ideal gas is taken through the cyclic process ABCA as shown in figure. It absorbs, 40 J of heat during the part AB, no heat during BC and rejects 60 J of heat during CA. A work of 50 J is done on the gas during the part BC. The internal energy of the gas at A is 1560 J. The workdone by the gas during the part CA is :

Resistance of the wire is measured as 2 $\Omega$ and 3 $\Omega$ at 10$^\circ$C and 30$^\circ$C respectively. Temperature co-efficient of resistance of the material of the wire is :

In Young's double slit experiment performed using a monochromatic light of wavelength $\lambda$, when a glass plate ($\mu$ = 1.5) of thickness x$\lambda$ is introduced in the path of the one of the interfering beams, the intensity at the position where the central maximum occurred previously remains unchanged. The value of x will be :

Isobutyraldehyde on reaction with formaldehyde and K₂CO₃ gives compound 'A'. Compound 'A' reacts with KCN and yields compound 'B', which on hydrolysis gives a stable compound 'C'. The compound 'C' is

With respect to the following reaction, consider the given statements :

(A) o-Nitroaniline and p-nitroaniline are the predominant products.

(B) p-Nitroaniline and m-nitroaniline are the predominant products.

(C) HNO₃ acts as an acid.

(D) H₂SO₄ acts as an acid.

Choose the correct option.

For combustion of one mole of magnesium in an open container at 300 K and 1 bar pressure, $\Delta$_CH^$\Theta$ = $-$601.70 kJ mol^$-$1, the magnitude of change in internal energy for the reaction is __________ kJ. (Nearest integer)

(Given : R = 8.3 J K^$-$1 mol^$-$1)

A radioactive element has a half life of 200 days. The percentage of original activity remaining after 83 days is ___________. (Nearest integer)

(Given : antilog 0.125 = 1.333, antilog 0.693 = 4.93)

Let R₁ = {(a, b) $\in$ N $\times$ N : |a $-$ b| $\le$ 13} and

R₂ = {(a, b) $\in$ N $\times$ N : |a $-$ b| $\ne$ 13}. Then on N :

The number of ways to distribute 30 identical candies among four children C₁, C₂, C₃ and C₄ so that C₂ receives at least 4 and at most 7 candies, C₃ receives at least 2 and at most 6 candies, is equal to :

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

Let f : R $\to$ R be a continuous function satisfying f(x) + f(x + k) = n, for all x $\in$ R where k > 0 and n is a positive integer. If ${I_1} = \int\limits_0^{4nk} {f(x)dx}$ and ${I_2} = \int\limits_{ - k}^{3k} {f(x)dx}$, then :

Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 $\tan x(\cos x - y)$. If the curve passes through the point $\left( {{\pi \over 4},0} \right)$, then the value of $\int\limits_0^{\pi /2} {y\,dx}$ is equal to :

If vertex of a parabola is (2, $-$1) and the equation of its directrix is 4x $-$ 3y = 21, then the length of its latus rectum is :

The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfies f(a) + 2f(b) $-$ f(c) = f(d) is :

Let $\overrightarrow a$ be a vector which is perpendicular to the vector $3\widehat i + {1 \over 2}\widehat j + 2\widehat k$. If $\overrightarrow a \times \left( {2\widehat i + \widehat k} \right) = 2\widehat i - 13\widehat j - 4\widehat k$, then the projection of the vector $\overrightarrow a$ on the vector $2\widehat i + 2\widehat j + \widehat k$ is :

If $\mathop {\lim }\limits_{x \to 1} {{\sin (3{x^2} - 4x + 1) - {x^2} + 1} \over {2{x^3} - 7{x^2} + ax + b}} = - 2$, then the value of (a $-$ b) is equal to ___________.

Let for n = 1, 2, ......, 50, S_n be the sum of the infinite geometric progression whose first term is n² and whose common ratio is ${1 \over {{{(n + 1)}^2}}}$. Then the value of

${1 \over {26}} + \sum\limits_{n = 1}^{50} {\left( {{S_n} + {2 \over {n + 1}} - n - 1} \right)}$ is equal to ___________.

Velocity (v) and acceleration (a) in two systems of units 1 and 2 are related as ${v_2} = {n \over {{m^2}}}{v_1}$ and ${a_2} = {{{a_1}} \over {mn}}$ respectively. Here m and n are constants. The relations for distance and time in two systems respectively are :

A ball is spun with angular acceleration $\alpha$ = 6t² $-$ 2t where t is in second and $\alpha$ is in rads^$-$2. At t = 0, the ball has angular velocity of 10 rads^$-$1 and angular position of 4 rad. The most appropriate expression for the angular position of the ball is :

A coil is placed in a time varying magnetic field. If the number of turns in the coil were to be halved and the radius of wire doubled, the electrical power dissipated due to the current induced in the coil would be :

(Assume the coil to be short circuited.)

Among the following, basic oxide is :

In the structure of SF₄, the lone pair of electrons on S is in.

2.5 g of protein containing only glycine (C₂H₅NO₂) is dissolved in water to make 500 mL of solution. The osmotic pressure of this solution at 300 K is found to be 5.03 $\times$ 10^$-$3 bar. The total number of glycine units present in the protein is ____________.

(Given : R = 0.083 L bar K^$-$1 mol^$-$1)

For the given reactions

Sn²⁺ + 2e^$-$ $\to$ Sn

Sn⁴⁺ + 4e^$-$ $\to$ Sn

the electrode potentials are ; $E_{S{n^{2 + }}/Sn}^o = - 0.140$ V and $E_{S{n^{4 + }}/Sn}^o = + 0.010$ V. The magnitude of standard electrode potential for $S{n^{4 + }}/S{n^{2 + }}$ i.e. $E_{S{n^{4 + }}/S{n^{2 + }}}^o$ is _____________ $\times$ 10^$-$2 V. (Nearest integer)

The complete combustion of 0.492 g of an organic compound containing 'C', 'H' and 'O' gives 0.793 g of CO₂ and 0.442 g of H₂O. The percentage of oxygen composition in the organic compound is ______________. (nearest integer)

The major product of the following reaction contains ____________ bromine atom(s).

0.01 M KMnO₄ solution was added to 20.0 mL of 0.05 M Mohr's salt solution through a burette. The initial reading of 50 mL burette is zero. The volume of KMnO₄ solution left in the burette after the end point is _____________ mL. (nearest integer)

The term independent of x in the expansion of

$(1 - {x^2} + 3{x^3}){\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \right)^{11}},\,x \ne 0$ is :

Let f : R $\to$ R be a differentiable function such that $f\left( {{\pi \over 4}} \right) = \sqrt 2 ,\,f\left( {{\pi \over 2}} \right) = 0$ and $f'\left( {{\pi \over 2}} \right) = 1$ and

let $g(x) = \int_x^{\pi /4} {(f'(t)\sec t + \tan t\sec t\,f(t))\,dt}$ for $x \in \left[ {{\pi \over 4},{\pi \over 2}} \right)$. Then $\mathop {\lim }\limits_{x \to {{\left( {{\pi \over 2}} \right)}^ - }} g(x)$ is equal to :

The area of the bounded region enclosed by the curve

$y = 3 - \left| {x - {1 \over 2}} \right| - |x + 1|$ and the x-axis is :

Let x = x(y) be the solution of the differential equation

$2y\,{e^{x/{y^2}}}dx + \left( {{y^2} - 4x{e^{x/{y^2}}}} \right)dy = 0$ such that x(1) = 0. Then, x(e) is equal to :

If cot$\alpha$ = 1 and sec$\beta$ = $- {5 \over 3}$, where $\pi < \alpha < {{3\pi } \over 2}$ and ${\pi \over 2} < \beta < \pi$, then the value of $\tan (\alpha + \beta )$ and the quadrant in which $\alpha$ + $\beta$ lies, respectively are :

Two objects of equal masses placed at certain distance from each other attracts each other with a force of F. If one-third mass of one object is transferred to the other object, then the new force will be :

A water drop of radius 1 $\mu$m falls in a situation where the effect of buoyant force is negligible. Co-efficient of viscosity of air is 1.8 $\times$ 10^$-$5 Nsm^$-$2 and its density is negligible as compared to that of water 10⁶ gm^$-$3. Terminal velocity of the water drop is :

(Take acceleration due to gravity = 10 ms^$-$2)

What will be the effect on the root mean square velocity of oxygen molecules if the temperature is doubled and oxygen molecule dissociates into atomic oxygen?

Two point charges A and B of magnitude +8 $\times$ 10^$-$6 C and $-$8 $\times$ 10^$-$6 C respectively are placed at a distance d apart. The electric field at the middle point O between the charges is 6.4 $\times$ 10⁴ NC^$-$1. The distance 'd' between the point charges A and B is :

The space inside a straight current carrying solenoid is filled with a magnetic material having magnetic susceptibility equal to 1.2 $\times$ 10^$-$5. What is fractional increase in the magnetic field inside solenoid with respect to air as medium inside the solenoid?

A student needs to prepare a buffer solution of propanoic acid and its sodium salt with pH 4. The ratio of ${{[C{H_3}C{H_2}CO{O^ - }]} \over {[C{H_3}C{H_2}COOH]}}$ required to make buffer is ___________.

Given : ${K_a}(C{H_3}C{H_2}COOH) = 1.3 \times {10^{ - 5}}$

Match List - I with List - II :

Choose the correct answer from the options given below :

${[Fe{(CN)_6}]^{4 - }}$

${[Fe{(CN)_6}]^{3 - }}$

${[Ti{(CN)_6}]^{3 - }}$

${[Ni{(CN)_4}]^{2 - }}$

${[Co{(CN)_6}]^{3 - }}$

Among the given complexes, number of paramagnetic complexes is ____________.

Let f, g : R $\to$ R be functions defined by

f(x) = \left\{ {\matrix{ {[x]} & , & {x < 0} \cr {|1 - x|} & , & {x \ge 0} \cr } } \right.\( and \)g(x) = \left\{ {\matrix{ {{e^x} - x} & , & {x < 0} \cr {{{(x - 1)}^2} - 1} & , & {x \ge 0} \cr } } \right. where [x] denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly :

Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$. Let e' and l' respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If ${e^2} = {{11} \over {14}}l$ and ${\left( {e'} \right)^2} = {{11} \over 8}l'$, then the value of $77a + 44b$ is equal to :

Let the image of the point P(1, 2, 3) in the line $L:{{x - 6} \over 3} = {{y - 1} \over 2} = {{z - 2} \over 3}$ be Q. Let R ($\alpha$, $\beta$, $\gamma$) be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the value of 22 ($\alpha$ + $\beta$ + $\gamma$) is equal to __________.

Sum of squares of modulus of all the complex numbers z satisfying $\overline z = i{z^2} + {z^2} - z$ is equal to ___________.

A block of mass 2 kg moving on a horizontal surface with speed of 4 ms^$-$1 enters a rough surface ranging from x = 0.5 m to x = 1.5 m. The retarding force in this range of rough surface is related to distance by F = $-$kx where k = 12 Nm^$-$1. The speed of the block as it just crosses the rough surface will be :

Two parallel, long wires are kept 0.20 m apart in vacuum, each carrying current of x A in the same direction. If the force of attraction per meter of each wire is 2 $\times$ 10^$-$6 N, then the value of x is approximately :

An EM wave propagating in x-direction has a wavelength of 8 mm. The electric field vibrating y-direction has maximum magnitude of 60 Vm^$-$1. Choose the correct equations for electric and magnetic fields if the EM wave is propagating in vacuum :

Let K₁ and K₂ be the maximum kinetic energies of photo-electrons emitted when two monochromatic beams of wavelength $\lambda$₁ and $\lambda$₂, respectively are incident on a metallic surface. If $\lambda$₁ = 3$\lambda$₂ then :

In a Young's double slit experiment, an angular width of the fringe is 0.35$^\circ$ on a screen placed at 2 m away for particular wavelength of 450 nm. The angular width of the fringe, when whole system is immersed in a medium of refractive index 7/5, is ${1 \over \alpha }$. The value of $\alpha$ is ___________.

All resistances in figure are 1 $\Omega$ each. The value of current 'I' is ${a \over 5}$ A. The value of a is _________.

A capacitor C₁ of capacitance 5 $\mu$F is charged to a potential of 30 V using a battery. The battery is then removed and the charged capacitor is connected to an uncharged capacitor C₂ of capacitance 10 $\mu$F as shown in figure. When the switch is closed charge flows between the capacitors. At equilibrium, the charge on the capacitor C₂ is __________ $\mu$C.

A liquid of density 750 kgm^$-$3 flows smoothly through a horizontal pipe that tapers in cross-sectional area from A₁ = 1.2 $\times$ 10^$-$2 m² to A₂ = ${{{A_1}} \over 2}$. The pressure difference between the wide and narrow sections of the pipe is 4500 Pa. The rate of flow of liquid is ___________ $\times$ 10^$-$3 m³s^$-$1.

A uniform disc with mass M = 4 kg and radius R = 10 cm is mounted on a fixed horizontal axle as shown in figure. A block with mass m = 2 kg hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is ____________ N.

(Take g = 10 ms^$-$2)

A student in the laboratory measures thickness of a wire using screw gauge. The readings are 1.22 mm, 1.23 mm, 1.19 mm and 1.20 mm. The percentage error is ${x \over {121}}\%$. The value of x is ____________.

In the given circuit the input voltage V_in is shown in figure. The cut-in voltage of p-n junction diode (D₁ or D₂) is 0.6 V. Which of the following output voltage (V₀) waveform across the diode is correct?

In the given circuit, the magnitude of V_L and V_C are twice that of V_R. Given that f = 50 Hz, the inductance of the coil is ${1 \over {K\pi }}$ mH. The value of K is ____________.

A zener of breakdown voltage V_z = 8 V and maximum zener current, I_ZM = 10 mA is subjectd to an input voltage V_i = 10 V with series resistance R = 100 $\Omega$. In the given circuit R_L represents the variable load resistance. The ratio of maximum and minimum value of R_L is _____________.

A tunning fork of frequency 340 Hz resonates in the fundamental mode with an air column of length 125 cm in a cylindrical tube closed at one end. When water is slowly poured in it, the minimum height of water required for observing resonance once again is ___________ cm.

(Velocity of sound in air is 340 ms^$-$1)

A car covers AB distance with first one-third at velocity v₁ ms^$-$1, second one-third at v₂ ms^$-$1 and last one-third at v₃ ms^$-$1. If v₃ = 3v₁, v₂ = 2v₁ and v₁ = 11 ms^$-$1 then the average velocity of the car is _____________ ms^$-$1.