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Q 1.
If \(3x + 4y = 12\sqrt2\) is a tangent to the ellipse \(\frac{x^ 2 }{a^ 2} + \frac{y^ 2 }9 = 1\), for some a ∈ R then the distance between the foci of the ellipse is :

Q 2.
Let A, B, C and D be four non-empty sets. The Contrapositive statement of “If A ⊆ B and B ⊆ D then A ⊆ C ” is :

Q 3.
The coefficient of \(x^ 7\) in the expression \((1 + x)^ {10} + x(1 + x) ^9 + x^ 2 (1 + x) ^8 + ⋯ +x ^{10}\) is :

Q 4.
In a workshop, there are five machines and the probability of any one of them to be out of service on a day is \(\frac1 4 \). If the probability that at most two machines will be out of service on the same day is \((\frac3 4 )^{ 3}\) k, then k is equal to

Q 5.
The locus of mid points of the perpendiculars drawn from points on the line x = 2y to the line x = y is :

Q 6.
If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ….. is (102)m, then m is equal to :

Q 7.
If \(\frac{ 3+isin \theta }{4- icos \theta}, \theta ∈ [0,2\pi]\) is a real number, then the argument of sin

Q 8.
The value of α for which \(4α \int_{-1 }^2 e^ {−α|x|} dx = \)5, is :

Q 9.

Q 10.
Let f(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If \(\lim\limits_{x \to 0} (2 + \frac{f(x) }{x^ 3 }) = 4\), then which one of the following is not true?

Q 11.
The number of ordered pairs (r, k) for which \(6 ⋅ ^{ 35 }C_r = (k ^2 − 3) ⋅ ^{ 36}C_ {r+1}\) , where k is an integer, is

Q 12.
Let \(a_1, a_2, a_3, … \)be a G.P. such that \(a_1 < 0, a_1 + a_2 = 4\: and \: a_3 + a_4 = 16\). If \(\sum_{i=1}^{9}ai=4λ\), then λ is equal to :

Q 13.
Let a, b and c be three unit vectors such that \(\vec{a}+ \vec{b}+ \vec{c} = 0\) . If \(\lambda =\vec{a}. \vec{b}+ \vec{b}. \vec{c}+ \vec{c}. \vec{a}\) and \( \vec{d}= \vec{a} * \vec{b}+ \vec{b}* \vec{c}+ \vec{c}* \vec{a}\) , then the ordered pair (λ, \( \vec{d}\)) is equal to :

Q 14.
Let y = y(x) be the solution curve of the differential equation,\( (y^ 2 − x) \frac{dy}{ dx} = 1\), satisfying y(0) = 1 This curve intersects the x − axis at a point whose abscissa is :

Q 15.
If \( θ_1 \)and \(θ_2\) be respectively the smallest and the largest values of θ in (0,2π) − {π} which satisfy the equation, \(2 cot^2 θ −\frac{ 5} {sin θ }+4 = 0, then \int_{\theta_1}^{\theta_2} cos^2 3θ\: dθ \) is equal to :

Q 16.
Let α and β are the roots of the equation\( x^ 2 − x − 1 = 0. If pk = (α)^ k + (β)^ k , k ≥ 1\) then which one of the following statements is not true?

Q 17.
The area (in sq. units) of the region {(x, y)ϵR|4\(x^2\) ≤ y ≤ 8x + 12} is :

Q 18.
The value of c in Lagrange’s mean value theorem for the function \(f(x) = x^3 − 4x^2 + 8x + 11\), where x ∈ [0,1] is :

Q 19.
Let y = y(x) be a function of x satisfying \(y\sqrt{1 − x^2} = k − x\sqrt{1 − y^2}\) where k is a constant and \(y (\frac1 2 ) = −\frac 1 4\) . Then \(\frac{dy}{ dx }\:at\: x = \frac1 2\) , is equal to :

Q 20.
Let the tangents drawn from the origin to the circle,

Q 21.

Q 22.
If the foot of perpendicular drawn from the point (1, 0, 3) on a line passing through (
Q 23.
Q 24.
If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20,
Q 25.
Let \(
Q 26.
The balancing length for a cell is 560 cm in a potentiometer experiment. When an external resistance of 10 Ω is connected in parallel to the cell, the balancing length changes by 60 cm. If the internal resistance of the cell is N/10 Ω, where N is an integer then value of N is ______.
Q 27.
M grams of steam at 100°C is mixed with 200g of ice at its melting point in a thermally insulated container. If it produces liquid water at 40°C [heat of vaporization of water is 540 cal/g and heat of fusion of ice is 80 cal/g], the value of M is______.
Q 28.
Consider uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is μ = 0.4, the maximum possible value of 100 × b/a for box not to topple before moving is
Q 29.
A 60 pF capacitor is fully charged by a 20V supply. It is then disconnected from the supply and is connected to another uncharged 60 pF capacitor in parallel. The electrostatic energy that is lost in this process by the time the charge is redistributed between them is (in nJ)________.
Q 30.
The sum of two forces \(\vec P\) and \(\vec Q\) is \(\vec R\) such that \(|\vec R|=|\vec P|\) The angle θ (in degrees) that the resultant of \(2\vec P\) and \(\vec Q\) will make with \(\vec Q\) is
Q 31.
Which of the following statements is correct?

Q 32.
Two open beakers one containing a solvent and the other containing a mixture of that solvent with a non-volatile solute are together sealed in a container. Over time

Q 33.
For the reaction, \(2H_2(g) + 2NO(g) → N_2(g) + 2H_2O(g)\); the observed rate expression is, rate = \(k_f[NO]^2 [H_2]\).

Q 34.

Consider the following reactions

Q 35.
The bond order and the magnetic characteristics of CN– are

Q 36.
The number of possible optical isomers for the complexes \(MA_2B_2\) with \(sp^3 \)and \(dsp^2\) hybridized metal atom, respectively, is

Q 37.

In the following reaction sequence

the major product B is :

Q 38.

In the following reactions, products (A) and (B), respectively, are

\(NaOH + Cl_2 \)\((A)\) + side products       (hot and conc.)

\(Ca(OH)_2 + Cl_2 (B) \)+ side products     (dry)

Q 39.

In the following reaction sequence, structures of A and B, respectively will be

Q 40.
The refining method used when the metal and the impurities have low and high melting temperatures, respectively, is

Q 41.

The correct order of stability for the following alkoxides is

Q 42.
Within each pair of element F & Cl, S & Se, and Li & Na, respectively, the elements that release more energy upon an electron gain are

Q 43.
The ammonia (\(NH_3\)) released on quantitative reaction of 0.6 g urea (\(NH_2CONH_2\)) with sodium hydroxide (NaOH) can be neutralized by

Q 44.
A chromatography column, packed with silica gel as stationary phase, was used to separate a mixture of compounds consisting of (A) benzanilide (B) aniline and (C) acetophenone. When the column is eluted with a mixture of solvents, hexane: ethyl acetate (20 : 80), the sequence of obtained compounds is

Q 45.
The equation that is incorrect is

Q 46.

Among the statements(a)-(d), the incorrect ones are :

(a) Octahedral Co(III) complexes with strong field ligands have very high magnetic moments

(b) When \( Δ_0 < P\), the d-electron configuration of Co(III) in an octahedral complex is \(t^4 _{eg}e_g ^2\)

(c) Wavelength of light absorbed by\( [Co(en)_3] ^3+\) is lower than that of \([CoF_6]^ {3–}\)

(d) If the \(Δ_0 \) for an octahedral complex of Co(III) is \(18,000 cm^{–1}\) , the Δt for its tetrahedral complex with the same ligand will be \(16,000 cm^{–1}\)

Q 47.
The redox reaction among the following is

Q 48.

Identify the correct labels of A, B and C in the following graph from the options given below :

Root mean square speed \((V_{rms}) \); most probable speed \((V_{mp})\); Average speed \((V_{av})\)

Q 49.

For the following reactions

\(k_s\) and \(k_e\), are, respectively, the rate constants for substitution and elimination, and μ = \(k_s\)/\(k_e\), the correct option is ________.

Q 50.

Among statements (a)-(d), the correct ones are :

(a) Decomposition of hydrogen peroxide gives dioxygen

(b) Like hydrogen peroxide, compounds, such as \(KClO_3, Pb(NO_3)_2\) and \(NaNO_3 \)when heated liberate dioxygen.

(c) 2-Ethylanthraquinone is useful for the industrial preparation of hydrogen peroxide.

(d) Hydrogen peroxide is used for the manufacture of sodium perborate.


Q 51.
The number of sp2 hybridised carbons present in ''Aspartame'' is ______
Q 52.
3g of acetic acid is added to 250 mL of 0.1 M HCl and the solution made up to 500 mL. To 20 mL of this solution ½ mL of 5 M NaOH is added. The pH of the solution is
Q 53.
The standard heat of formation \((Δ_fH_{298^{ -}} )\) of ethane (in kJ/mol), if the heat of combustion of ethane, hydrogen and graphite are –1560, – 393.5 and –286kJ/mol, respectively is _______
Q 54.
The flocculation value of HCl for arsenic sulphide sol. is 30m mol \(L^{–1}\)L–1 . if \(H_2SO_4 \)is used for the flocculation of arsenic sulphide, the amount, in grams, of \(H_2SO_4 \) in 250 ml required for the above purpose is _______ (Molecular mass of\(H_2SO_4 \) = 98g/mol)
Q 55.

Consider the following reactions:

Q 56.
The electric field of a plane electromagnetic wave is given by \(\vec E= E_o\frac {\hat i+ \hat j}{\sqrt2} cos(kz+\omega t)\) At t = 0, a positively charged particle is at the point \((x,y,z)=(0,0,\frac{\pi}{k})\) If its instantaneous velocity at (t=0) is  \(v_o\hat k\) the force acting on it due to the wave is

Q 57.
In a Young's double slit experiment, the separation between the slits is 0.15 mm. In the experiment, a source of light of wavelength 589 nm is used and the interference pattern is observed on a screen kept 1.5 m away. The separation between the successive bright fringes on the screen is

Q 58.
A stationary observer receives sound from two identical tuning forks, one of which approaches and the other one recedes with the same speed (much less than the speed of sound). The observer hears 2 beats/sec. The oscillation frequency of each tuning fork is 0 = 1400 Hz and the velocity of sound in air is 350 m/s. The speed of each tuning fork is close to

Q 59.
An elevator in a building can carry a maximum of 10 persons, with the average mass of each person being 68 kg. The mass of the elevator itself is 920 kg and it moves with a constant speed of 3 m/s. The frictional force opposing the motion is 6000 N. If the elevator is moving up with its full capacity, the power delivered by the motor to the elevator (\(g = 10 m/s^2\) ) must be at least

Q 60.
Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures,\( T_1\) and\( T_2\). The temperature of the hot reservoir of the first engine is \(T_1\) and the temperature of the cold reservoir of the second engine is \(T_2\). T is temperature of the sink of first engine which is also the source for the second engine. How is T related to\( T_1\) and \(T_2\), if both the engines perform equal amount of work ?

Q 61.
An ideal fluid flows (laminar flow) through a pipe of non-uniform diameter. The maximum and minimum diameters of the pipes are 6.4 cm and 4.8 cm, respectively. The ratio of the minimum and the maximum velocities of fluid in this pipe is

Q 62.
A planar loop of wire rotates in a uniform magnetic field. Initially at t = 0, the plane of the loop is perpendicular to the magnetic field. If it rotates with a period of 10 s about an axis in its plane then the magnitude of induced emf will be maximum and minimum, respectively at 5.0 s and 7.5 s

Q 63.
In a building there are 15 bulbs of 45 W, 15 bulbs of 100 W, 15 small fans o 10 W and 2 heaters of 1 kW. The voltage of electric main is 220 V. The minimum fuse capacity (rated value) of building will be

Q 64.
A thin lens made of glass (refractive index = 1.5) of focal length f = 16 cm is immersed in a liquid of refractive index 1.42. If its focal length in liquid is f, then the ratio\( f_ℓ/f \)is closest to the integer

Q 65.
An emf of 20 V is applied at time t = 0 to a circuit containing in series 10 mH inductor and 5 Ω resistor. The ratio of the currents at time \(t = \infty\) and at t = 40 s is close to : (Take \(e^2 = 7.389\))

Q 66.

The figure gives experimentally measured B vs. H variation in a ferromagnetic material. The retentivity, co-ercivity and saturation, respectively, of the material are

Q 67.
A mass of 10 kg is suspended by a rope of length 4 m, from the ceiling. A force F is applied horizontally at the mid-point of the rope such that the top half of the rope makes an angle of 45° with the vertical. Then F equals : (Take \( g = 10 ms{^–2}\) and the rope to be massless)

Q 68.
The dimensions of \(\frac{B^2}{2\mu_o}\) where B is magnetic field and \(μ_0\) is the magnetic permeability of vacuum, is

Q 69.

In the figure, potential difference between A and B is

Q 70.
Mass per unit area of a circular disc of radius a depends on the distance r from its centre as σ(r) = A + Br. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is

Q 71.
An electron (of mass m) and a photon have the same energy E in the range of a few eV. The ratio of the de-Broglie wavelength associated with the electron and the wavelength of the photon is (c = speed of light in vacuum)

Q 72.
The activity of a radioactive sample falls from\( 700 s^{–1} \;to \;500 s^{–1}\) in 30 minutes. Its half life is close to

Q 73.
Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecules changes from \(τ_1\)to \(τ_2\). If \(C_P/C_V = γ\) for this gas then a good estimate for \(τ_1/τ_2\) is given by

Q 74.
A box weighs 196 N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to \((Take\ g = 10 ms^{–2}\) at the north pole and the radius of the earth = 6400 km)

Q 75.
A particle of mass m and charge q has an initial velocity \(\vec v=v_o\hat j\) If an electric field \(\vec E=E_o\hat i\) and magnetic field \(\vec B=B_o\hat i\) act on the particle, its speed will double after a time

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