Chemistry
Match the entries in Column I with the correctly related quantum number(s) in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 $$\times$$ 4 matrix given in the ORS.
Column I | Column II | ||
---|---|---|---|
(A) | Orbital angular momentum of the electron in a hydrogen-like atomic orbital. | (P) | Principal quantum number |
(B) | A hydrogen-like one-electron wave function obeying Pauli's principle. | (Q) | Azimuthal quantum number |
(C) | Shape, size and orientation of hydrogen like atomic orbitals. | (R) | Magnetic quantum number |
(D) | Probability density of electron at the nucleus in hydrogen-like atom. | (S) | Electron spin quantum number |
The correct stability order for the following species is :
Cellulose upon acetylation with excess acetic anhydride/H$$_2$$SO$$_4$$ (catalytic) gives cellulose triacetate whose structure is :
In the following reaction sequence, the correct structure of E, F and G are :
Among the following, the coloured compound is :
Both [Ni(CO)$$_4$$] and [Ni(CN)$$_4$$]$$^{2-}$$ are diamagnetic. They hybridisations of nickel in these complexes, respectively, are :
The IUPAC name of [Ni(NH$$_3$$)$$_4$$] [NiCl$$_4$$] is :
Electrolysis of dilute aqueous NaCl solution was carried out by passing 10 milli ampere current. The time required to liberate 0.01 mol of H$$_2$$ gas at the cathode is (1 Faraday = 96500 C mol$$^{-1}$$].
Among the following, the surfactant that will form micelles in aqueous solution at the lowest molar concentration at ambient condition is:
Solubility product constants (K$$_{sp}$$) of salts of types MX, MX$$_2$$ and M$$_3$$X at temperature T are 4.0 $$\times$$ 10$$^{-8}$$, 3.2 $$\times$$ 10$$^{-14}$$ and 2.7 $$\times$$ 10$$^{-15}$$, respectively. Solubilities (mol dm$$^{-3}$$) of the salts at temperature 'T' are in the order:
Statement 1 : Aniline on reaction with NaNO$$_2$$/HCl at 0$$^\circ$$C followed by coupling with $$\beta$$-naphthol gives a dark blue coloured precipitate.
Statement 2 : The colour of the compound formed in the reaction of aniline with NaNO$$_2$$/HCl at 0$$^\circ$$C followed by coupling with $$\beta$$-naphthol is due to the extended conjugation.
Statement 1 : [Fe(H$$_2$$O)$$_5$$NO]SO$$_4$$ is paramagnetic.
Statement 2 : The Fe in [Fe(H$$_2$$O)$$_5$$NO]SO$$_4$$ has three unpaired electrons.
Statement 1 : The geometrical isomers of the complex [M(NH$$_3$$)$$_4$$Cl$$_2$$] are optically inactive.
Statement 2 : Both geometrical isomers of the complex [M(NH$$_3$$)$$_4$$Cl$$_2$$] possess axis of symmetry.
Statement 1 : There is a natural asymmetry between converting work to heat and converting heat to work.
Statement 2 : No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.
Compound H is formed by the reaction of
The structure of compound I is :
The structures of compound J, K and L, respectively, are:
The number of atoms in this HCP unit cell is :
The volume of this HCP unit cell is :
The empty space in this HCP unit cell is :
Match the compounds in Column I with their characteristic test(s)/reaction(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 $$\times$$ 4 matrix given in the ORS.
Column I | Column II | ||
---|---|---|---|
(A) | ![]() |
(P) | sodium fusion extract of the compound gives Prussian blue colour with FeSO$$_4$$. |
(B) | ![]() |
(Q) | gives positive FeCl$$_3$$ test. |
(C) | ![]() |
(R) | gives white precipitate with AgNO$$_3$$. |
(D) | ![]() |
(S) | reacts with aldehydes to form the corresponding hydrazone derivative. |
Match the conversions in Column I with the type(s) of reaction(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 $$\times$$ 4 matrix given in the ORS.
Column I | Column II | ||
---|---|---|---|
(A) | PbS $$\to$$ PbO | (P) | roasting |
(B) | CaCO$$_3$$ $$\to$$ CaO | (Q) | Calcination |
(C) | ZnS $$\to$$ Zn | (R) | carbon reduction |
(D) | Cu$$_2$$S $$\to$$ Cu | (S) | self reduction |
Mathematics
Which of the following is true?
The shortest distance between $${L_1}$$ and $${L_2}$$ is :
The unit vector perpendicular to both $${L_1}$$ and $${L_2}$$ is :
$$x\sqrt {{x^2} - 1} \,\,dy - y\sqrt {{y^2} - 1} \,dx = 0$$ satify $$y\left( 2 \right) = {2 \over {\sqrt 3 }}.$$
STATEMENT-1 : $$y\left( x \right) = \sec \left( {{{\sec }^{ - 1}}x - {\pi \over 6}} \right)$$ and
STATEMENT-2 : $$y\left( x \right)$$ given by $${1 \over y} = {{2\sqrt 3 } \over x} - \sqrt {1 - {1 \over {{x^2}}}} $$
Let $$g\left( x \right) = \int\limits_0^{{e^x}} {{{f'\left( t \right)} \over {1 + {t^2}}}} \,dt.$$
Which of the following is true?
Which of the following is true?
and $$y = \sqrt {{{1 - \sin x} \over {\cos x}}} $$ bounded by the lines $$x=0$$ and $$x = {\pi \over 4}$$ is
for an arbitrary constant $$C$$, the value of $$J -I$$ equals :
$$g\left( u \right) = 2{\tan ^{ - 1}}\left( {{e^u}} \right) - {\pi \over 2}.$$ Then, $$g$$ is
Let $$g(x) = \log f(x)$$, where $$f(x)$$ is a twice differentiable positive function on (0, $$\infty$$) such that $$f(x + 1) = xf(x)$$. Then for N = 1, 2, 3, ..., $$g''\left( {N + {1 \over 2}} \right) - g''\left( {{1 \over 2}} \right) = $$
with vertex at the point $$A$$. Let $$B$$ be one of the end points of its latus rectum. If $$C$$ is the focus of the hyperbola nearest to the point $$A$$, then the area of the triangle $$ABC$$ is
$$\,{L_1}:\,\,2x\,\, + \,\,3y\, + \,p\,\, - \,\,3 = 0$$
$$\,{L_2}:\,\,2x\,\, + \,\,3y\, + \,p\,\, + \,\,3 = 0$$
where p is a real number, and $$\,C:\,{x^2}\, + \,{y^2}\, + \,6x\, - 10y\, + \,30 = 0$$
STATEMENT-1 : If line $${L_1}$$ is a chord of circle C, then line $${L_2}$$ is not always a diameter of circle C
and
STATEMENT-2 : If line $${L_1}$$ is a diameter of circle C, then line $${L_2}$$ is not a chord of circle C.
STATEMENT-1: The numbers $${b_1},\,{b_{2\,}},\,{b_3},\,{b_4}\,$$ are neither in A.P. nor in G.P. and
STATEMENT-2 The numbers $${b_1},\,{b_{2\,}},\,{b_3},\,{b_4}\,$$ are in H.P.
Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.
Column I | Column II | ||
---|---|---|---|
(A) | The number of permutations containing the word ENDEA is | (P) | 5! |
(B) | The number of permutations in which the letter E occurs in the first and the last position is | (Q) | 2 $$\times$$ 5! |
(C) | The number of permutations in which none of the letters D, L, N occurs in the last five positions is | (R) | 7 $$\times$$ 5! |
(D) | The number of permutations in which the letters A, E, O occur only in odd positions is | (S) | 21 $$\times$$ 5! |
STATEMENT - 1 : $$\left( {{p^2} - q} \right)\left( {{b^2} - ac} \right) \ge 0$$
and
STATEMENT - 2 : $$b \ne pa$$ or $$c \ne qa$$
Consider three points $$P = ( - \sin (\beta - \alpha ), - cos\beta ),Q = (cos(\beta - \alpha ),\sin \beta )$$ and $$R = (\cos (\beta - \alpha + \theta ),\sin (\beta - \theta ))$$ where $$0 < \alpha ,\beta ,\theta < {\pi \over 4}$$. Then :
An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent is :
Consider the lines given by:
$${L_1}:x + 3y - 5 = 0$$
$${L_2}:3x - ky - 1 = 0$$
$${L_3}:5x + 2y - 12 = 0$$
Match the Statement/Expressions in Column I with the Statements/Expressions in Column II.
Column I | Column II | ||
---|---|---|---|
(A) | L$$_1$$, L$$_2$$, L$$_3$$ are concurrent, if | (P) | $$K = - 9$$ |
(B) | One of L$$_1$$, L$$_2$$, L$$_3$$ is parallel to atleast one of the other two, if | (Q) | $$K = - {6 \over 5}$$ |
(C) | L$$_1$$, L$$_2$$, L$$_3$$ form a triangle, if | (R) | $$K = {5 \over 6}$$ |
(D) | L$$_1$$, L$$_2$$, L$$_3$$ do not form a triangle, if | (S) | $$K = 5$$ |
Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.
Column I | Column II | ||
---|---|---|---|
(A) | The minimum value of $${{{x^2} + 2x + 4} \over {x + 2}}$$ is | (P) | 0 |
(B) | Let A and B be 3 $$\times$$ 3 matrices of real numbers, where A is symmetric, B is skew-symmetric and (A + B) (A $$-$$ B) = (A $$-$$ B) (A + B). If (AB)$$^t$$ = ($$-1$$)$$^k$$ AB, where (AB)$$^t$$ is the transpose of the matrix AB, then the possible values of k are | (Q) | 1 |
(C) | Let $$a=\log_3\log_3 2$$. An integer k satisfying $$1 < {2^{( - k + 3 - a)}} < 2$$, must be less than | (R) | 2 |
(D) | If $$\sin \theta = \cos \varphi $$, then the possible values of $${1 \over \pi }\left( {\theta + \varphi - {\pi \over 2}} \right)$$ are | (S) | 3 |
Physics
STATEMENT-2: If the observer and the object are moving at velocities $${\overrightarrow v _1}$$ and $${\overrightarrow v _2}$$ respectively with reference to a laboratory frame, the velocity of the object with respect to the observer is $${\overrightarrow v _2}$$ - $${\overrightarrow v _1}$$.
Consider a system of three charges $${q \over 3},{q \over 3}$$ and $$ - {{2q} \over 3}$$ placed at points A, B and C, respectively, as shown in the figure. Take O to be the centre of the circle of radius R and angle CAB = 60$$^\circ$$
A radioactive sample S1 having activity of 5 $$\mu$$Ci has twice the number of nuclei as another sample S2 which has an activity of 10 $$\mu$$Ci. The half lives of S1 and S2 can be :
A transverse sinusoidal wave moves along a string in the positive x-direction at a speed of 10 cm/s. The wavelength of the waves is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap-shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is :
A block (B) is attached to two unstretched springs S1 and S2 with spring constants k and 4k respectively (see figure I). The other ends are attached to identical supports M1 and M2 not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block displaced towards wall 1 by a small distance x (figure II) and released. The block returns and moves a maximum distance y towards wall 2. Displacements x and y are measured with respect to the equilibrium position of the block B. The ratio $$\frac{y}{x}$$ is :
A bob of mass M is suspended by a massless string of length L. The horizontal velocity V at position A is just sufficient to make it reach the point B. The angle $$\theta$$ at which the speed of the bob is half of that at A, satisfies,
A glass tube of uniform internal radius (r) has a valve separating the two identical ends. Initially, the valve is in a tightly closed position. End 1 has a hemispherical soap bubble of radius r. End 2 has sub-hemispherical soap bubble as shown in figure. Just after opening the valve,
A vibrating string of certain length 1 under a tension T resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length 75 cm inside a tube closed at one end. The string also generates 4 beats per second when excited along with a tuning fork of frequency n. Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be 340 m/s, the frequency n of the tuning fork in Hz is:
A parallel plate capacitor C with plates of unit area and separation d is filled with a liquid of dielectric constant K = 2. The level of liquid is $$\frac{d}{3}$$ initially. Suppose the liquid level decreases at a constant speed V, the time constant as a function of time t is:
A light beam is travelling from Region I to Region IV (Refer figure). The refractive index in Regions I, II, III and IV are $${n_0},{{{n_0}} \over 2},{{{n_0}} \over 6}$$ and $${{{n_0}} \over 8}$$, respectively. The angle of incidence $$\theta$$ for which the beam just misses entering Region IV is
STATEMENT 1 : It is easier to pull a heavy object than to push it on a level ground.
and
STATEMENT 2 : The magnitude of frictional force depends on the nature of the two surfaces in contact.
STATEMENT 1 : For practical purposes, the earth is used as a reference at zero potential in electrical circuits.
and
STATEMENT 2 : The electrical potential of a sphere of radius R with charge Q uniformly distributed on the surface is given by $${Q \over {4\pi {\varepsilon _0}R}}$$
STATEMENT 1 : The sensitivity of a moving coil galvanometer is increased by placing a suitable magnetic material as a core inside the coil.
and
STATEMENT 2 : Soft iron has a high magnetic permeability and cannot be easily magnetized or demagnetized.
The electric field at r = R is :
For a = 0, the value of d (maximum value of $$\rho$$ as shown in the figure) is
The electric field within the nucleus is generally observed to be linearly dependent on r. This implies
The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is :
The centre of mass of the disk undergoes simple harmonic motion with angular frequency $$\omega$$ equal to:
The maximum value of V$$_0$$ for which the disk will roll without slipping is:
Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 $$\times$$ 4 matrix given in the ORS.
Column I | Column II | ||
---|---|---|---|
(A) | Potential energy of a simple pendulum (y-axis) as a function of displacement (x) axis | (P) | ![]() |
(B) | Displacement (y-axis) as a function of time (x-axis) for a one dimensional motion at zero or constant acceleration when the body is moving along the positive x-direction | (Q) | ![]() |
(C) | Range of a projectile (y-axis) as a function of its velocity (x-axis) when projected at a fixed angle | (R) | ![]() |
(D) | The square of the time period (y-axis) of a simple pendulum as a function of its length (x-axis) | (S) | ![]() |
An optical component and an object S placed along its optic axis are given in Column I. The distance between the object and the component can be varied. The properties of images are given in Column II. Match all the properties of images from Column II with the appropriate components given in Column I. Indicate your answer by darkening the appropriate bubbles of the 4 $$\times$$ 4 matrix given in the ORS.
Column I | Column II | ||
---|---|---|---|
(A) | ![]() |
(P) | Real Image |
(B) | ![]() |
(Q) | Virtual Image |
(C) | ![]() |
(R) | Magnified Image |
(D) | ![]() |
(S) | Image at infinity |
Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the 4 $$\times$$ 4 matrix given in the ORS.
Column I | Column II | ||
---|---|---|---|
(A) | An insulated container has two chambers separated by a valve. Chamber I contains an ideal gas and the Chamber II has vacuum. The valve is opened.![]() |
(P) | The temperature of the gas decreases |
(B) | An ideal monatomic gas expands to twice its original volume such that its pressure P $$\propto$$ $$\frac{1}{\mathrm{V}^2}$$, where V is the volume of the gas | (Q) | The temperature of the gas increase or remains constant. |
(C) | An ideal monoatomic gas expands to twice its original volume such that its pressure P $$\propto$$ $$\frac{1}{\mathrm{V}^{4/3}}$$, where V is its volume | (R) | The gas loses heat |
(D) | An ideal monoatomic gas expands such that its pressure P and volume V follows the behaviour shown in the graph![]() |
(S) | The gas gains heat |