Two stones are thrown up simultaneously from the edge of a cliff 240 m high with initial speed of 10 m/s and 40 m/s respectively. Which of the following graph best represents the time variation of relative position of the second stone with respect to the first? (Assume stones do not rebound after hitting the ground and neglect air resistance, take g = 10 m/s2) (The figures are schematic and not drawn to scale)
Till both are in air (From t = 0 to t = 8 sec)
When second stone hits ground and first stone is in air Δx decreases.
The period of oscillation of a simple pendulum isMeasured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. The accuracy in the determination of g is
Given in the figure are two blocks A and B of weight 20 N and 100 N, respectively. These are being pressed against a wall by a force F as shown. If the coefficient of friction between the blocks is 0.1 and between block B and the wall is 0.15, the frictional force applied by the wall on block B is
Clearly fs = 120 N (for vertical equilibrium of the system)
A particle of mass m moving in the x direction with speed 2v is hit by another particle of mass 2m moving in the y direction wth speed v. If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to
Distance of the centre of mass of a solid uniform cone from its vertex is z0. If the radius of its base is R and its height is h then z0 is equal to
From a solid sphere of mass M and radius R a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its faces is
d = 2R= a√3
From a solid sphere of mass M and radius R, a spherical portion of radius R/2 is removed, as shown in the figure. Taking gravitational potential V = 0 at r = ∞, the potential at the centre of the cavity thus formed is (G = gravitational constant)
A pendulum made of a uniform wire of crosssectional area A has time period . When an additional mass M is added to its bob, the time period changes to TM. If the Young's modulus of the material of the wire is Y then 1/Y is equal to (g = gravitational acceleration)
Consider a spherical shell of radius R at temperature T. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume and pressure.If the shell now undergoes an adiabatic expansion the relation between T and R is
PV =μRT ------ (ii)
A solid body of constant heat capacity 1 J/°C is being heated by keeping it in contact with reservoirs in two ways :
(i) Sequentially keeping in contact with 2 reservoirs such that each reservoir supplies same amount of heat.
(ii) Sequentially keeping in contact with 8 reservoirs such that each reservoir supplies same amount of heat.
In both the cases body is brought from initial temperature 100°C to final temperature 200°C.
Entropy change of the body in the two cases respectively is
Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increases as Vq, where V is the volume of the gas. The value of q is
For a simple pendulum, a graph is plotted between its kinetic energy (KE) and potential energy (PE) against its displacement d. Which one of the following represents these correctly?
(Graphs are schematic and not drawn to scale)
A train is moving on a straight track with speed 20 ms–1. It is blowing its whistle at the frequency of 1000 Hz. The percentage change in the frequency heard by a person standing near the track as the train passes him is (speed of sound = 320 ms–1) close to
A long cylindrical shell carries positive surface charge σ in the upper half and negative surface charge -σ in the lower half. The electric field lines around the cylinder will look like figure given in
(figures are schematic and not drawn to scale)
The field line should resemble that of a dipole.
A uniformly charged solid sphere of radius R has potential V0 (measured with respect to ¥) on its surface. For this sphere the equipotential surfaces with potentials have radius R1, R2, R3 and R4 respectively. Then
In the given circuit, charge Q2 on the 2μF capacitor changes as C is varied from 1 μF to 3 μF. Q2 as a function of C is given properly by : (Figures are drawn schematically and are not to scale)
When 5 V potential difference is applied across a wire of length 0.1 m, the drift speed of electrons is 2.5 × 10–4 ms–1. If the electron density in the wire is 8 × 1028 m–3, the resistivity of the material is close to
In the circuit shown, the current in the 1 W resistor is
Two coaxial solenoids of different radii carry current I in the same direction. Letbe the magnetic force on the inner solenoid due to the outer one andbe the magnetic force on the outer solenoid due to the inner one. Then
Net force on each of them would be zero.
Two long current carrying thin wires, both with current I, are held by insulating threads of length L and are in equilibrium as shown in the figure, with threads making an angle θ with the vertical. If wires have mass λ per unit length then the value of I is (g = gravitational acceleration)
Tcosθ = λgl ...........(1)
Tsinθ = ...........(2)
A rectangular loop of sides 10 cm and 5 cm carrying a current I of 12 A is placed in different orientations as shown in the figures below:
If there is a uniform magnetic field of 0.3 T in the positive z direction, in which orientations the loop would be in (i) stable equilibrium and (ii) unstable equilibrium?
An inductor (L = 0.03 H) and a resistor (R = 0.15 kΩ) are connected in series to a battery of 15V EMF in a circuit shown below. The key K1 has been kept closed for a long time. Then at t = 0, K1 is opened and key K2 is closed simultaneously. At t = 1 ms, the current in the circuit will be (e ≌150)
A red LED emits light at 0.1 watt uniformly around it. The amplitude of the electric field of the light at a distance of 1 m from the diode is
Monochromatic light is incident on a glass prism of angle A. If the refractive index of the material of the prism is μ, a ray, incident at an angle θ, on the face AB would get transmitted through the face AC of the prism provided.
sin θ = μ sin r1
sin r1 =
On a hot summer night, the refractive index of air is smallest near the ground and increases with height form the ground. When a light beam is directed horizontally, the Huygen's principle leads us to conclude that as it travels, the light beam
Consider a plane wavefront travelling horizontally.
As it moves, its different parts move with different speeds. So, its shape will change as shown Þ Light bends upward
Assuming human pupil to have a radius of 0.25 cm and a comfortable viewing distance of 25 cm, the minimum separation between two objects that human eye can resolve at 500 nm wavelength is
As an electron makes a transition from an excited state to the ground state of a hydrogen-like atom/ion
Match List-I (Fundamental Experiment) with List-II (its conclusion) and select the correct option from the choices given below the list:
1. Franck-Hertz exp.– Discrete energy level.
2. Photo-electric effect– Particle nature of light
3. Davison-Germer exp.– Diffraction of electron beam.
A signal of 5 kHz frequency is amplitude modulated on a carrier wave of frequency 2 MHz. The frequencies of the resultant signal is/are
Frequencies of resultant signal are
(2000 + 5) kHz, 2000 kHz, (2000 – 5) kHz, 2005 kHz, 2000 kHz, 1995 kHz
An LCR circuit is equivalent to a damped pendulum. In an LCR circuit the capacitor is charged to Q0 and then connected to the L and R as shown below :
If a student plots graphs of the square of maximum charge (Q2Max) on the capacitor with time (t) for two different values L1 and L2 (L1 > L2) of L then which of the following represents this graph correctly? (Plots are schematic and not drawn to scale)
For a damped pendulum, A =
(Since L plays the same role as m)
The molecular formula of a commercial resin used for exchanging ions in water softening is C8H7SO3Na (mol. wt. 206). What would be the maximum uptake of Ca2+ ions by the resin when expressed in mole per gram resin?
The maximum uptake =
Sodium metal crystallizes in a body centred cubic lattice with a unit cell edge of 4.29 Å. The radius of sodium atom is approximately
Edge length of BCC is 4.29 Å.
edge length =
Which of the following is the energy of a possible excited state of hydrogen?
Energy of excited state is negative and correspond to n>1.
The intermolecular interaction that is dependent on the inverse cube of distance between the molecules is
H-bond is one of the dipole-dipole interaction and dependent on inverse cube of distance between the molecules.
The following reaction is performed at 298 K.
The standard free energy of formation of NO(g) is 86.6 kJ/mol at 298 K. What is the standard free energy of formation of NO2(g) at 298 K ? (Kp = 1.6 × 1012)
The vapour pressure of acetone at 20°C is 185 torr. When 1.2 g of a non-volatile substance was dissolved in 100 g of acetone at 20°C, its vapour pressure was 183 torr. The molar mass (g mol-1) of the substance is
Vapour pressure of pure acetone P°A = 185 torr
Vapour pressure of solution, PS = 183 torr
Molar mass of solvent, MA = 58 g/mole as we know
= 63.68 g/mole
The standard Gibbs energy change at 300 K for the reaction At a given time, the composition of the reaction mixture is . The reaction proceeds in the : [R = 8.314 J/K/mol, e = 2.718]
as QC > KC, hence reaction will shift in backward direction.
Two faraday of electricity is passed through a solution of CuSO4. The mass of copper deposited at the cathode is (at. mass of Cu = 63.5 amu)
So, 2 F charge deposite 1 mol of Cu. Mass deposited = 63.5 g.
Higher order (>3) reactions are rare due to
Higher order greater than 3 for reaction is rare because there is low probability of simultaneous collision of all the reacting species.
3 g of activated charcoal was added to 50 mL of acetic acid solution (0.06N) in a flask. After an hour it was filtered and the strength of the filtrate was found to be 0.042 N. The amount of acetic acid adsorbed (per gram of charcoal) is
Number of moles of acetic acid adsorbed
∴ Weight of acetic acid adsorbed
= 0.9 × 60 mg = 54 mg
Hence, the amount of acetic acid adsorbed per g of charcoal =
Hence, option (1) is correct.
The ionic radii (in Å) of N3- , O2- and F- are respectively
Radius of N3–, O2– and F– follow order N3– > O2– > F–
As per inequality only option (3) is correct that is 1.71 Å, 1.40 Å and 1.36 Å
In the context of the Hall-Heroult process for the extraction of Al, which of the following statement is false?
In Hall-Heroult process Al2O3 (molten) is electrolyte.
From the following statement regarding H2O2, choose the incorrect statement
H2O2 can be reduced or oxidised. Hence, it can act as reducing as well as oxidising agent.
Which one of the following alkaline earth metal sulphates has its hydration enthalpy greater than its lattice enthalpy?
BeSO4 has hydration energy greater than its lattice energy.
Which among the following is the most reactive?
Because of polarity and weak bond interhalogen compounds are more reactive.
Match the catalysts to the correct processes :
a. TiCl3 (i) Wacker process
b. PdCl2 (ii) Ziegler-Natta polymerization
c. CuCl2 (iii) Contact process
d. V2O5 (iv) Deacon's process
TiCl3 - Ziegler-Natta polymerisation
V2O5 - Contact process
PdCl2 - Wacker process
CuCl2 - Deacon's process
Which one has the highest boiling point?
Down the group strength of van der Waal's force of attraction increases hence Xe have highest boiling point.
The number of geometric isomers that can exist for square planar [Pt(Cl)(py)(NH3)(NH2OH)]+ is (py = pyridine)
as per question a = Cl, b = py, c = NH3 and d = NH2OH are assumed.
The color of KMnO4 is due to
Charge transfer spectra from ligand (L) to metal (M) is responsible for color of KMnO4.
Assertion : Nitrogen and Oxygen are the main components in the atmosphere but these do not react to form oxides of nitrogen.
Reason : The reaction between nitrogen and oxygen requires high temperature.
N2 + O2 --> 2NO
Required temperature for above reaction is around 3000°C which is a quite high temperature. This reaction is observed during thunderstorm.
In Carius method of estimation of halogens, 250 mg of an organic compound gave 141 mg of AgBr. The percentage of bromine in the compound is (At. mass Ag = 108; Br = 80)
Percentage of Br
Which of the following compounds will exhibit geometrical isomerism?
For geometrical isomerism doubly bonded carbon must be bonded to two different groups which is only satisfied by 1 - Phenyl - 2 - butene.
Which compound would give 5-keto-2-methyl hexanal upon ozonolysis?
The synthesis of alkyl fluorides is best accomplished by
In the following sequence of reactions :
the product C is
In the reaction the product E is
Which polymer is used in the manufacture of paints and lacquers?
Glyptal is used in manufacture of paints and lacquires.
Which of the vitamins given below is water soluble?
Vitamin C is water soluble vitamin.
Which of the following compounds is not an antacid?
Phenelzine is not antacid, it is anti-depressant.
Which of the following compounds is not colored yellow?
(NH4)3[As (Mo3O10)4], BaCrO4 and K3[Co(NO2)6] are yellow colored compounds but Zn2[Fe(CN)6] is not yellow colored compound.
Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A × B, each having at least three elements is
n(A) = 4, n(B) = 2
n(A × B) = 8
Required numbers = 8C3 + 8C4 + ...... + 8C8
= 28 – (8C0 + 8C1 + 8C2)
= 256 – 37
A complex number z is said to be unimodular if |z| = 1. Suppose z1 and z2 are complex numbers such thatis unimodular and z2 is not unimodular. Then the point z1 lies on a
|z| = 2 i.e. z lies on circle of radius 2.
Let a and b be the roots of equation x2 - 6x - 2 = 0. If an = αn - βn, for n ≥ 1, then the value of is equal to
α + β = 6
αβ = - 2
The value of
If A = is a matrix satisfying the equation AAT = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to
a + 4+2b = 0
2a +2- 2b = 0
a + 1-b =0
2a -2b= -2
a + 2b= -4
-2+1 - b = 0
b = -1
a = -2
The set of all values of l for which the system of linear equations
2x1 - 2x2 + x3 = λx1
2x1 - 3x2 + 2x3 = λx2
-x1 + 2x2 = λx3
has a non-trivial solution
x1(2 - λ) - 2x2 + x3 = 0
2x1 +x2(-λ - 3) + 2x3 = 0
(2 - λ)(λ2 + 3λ- 4)+ 2(-2λ + 2)+(4 - λ- 3) = 0
2λ2 + 6λ - 8 - λ3 - 3λ2 + 4λ - 4λ + 4 -λ + 1 =0
=> λ = 1, 1, -3
The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is
4 digit numbers
5 digit numbers
5 × 4 × 3 × 2 × 1 = 120
Total number of integers
= 72 + 120 = 192
The sum of coefficients of integral powers of x in the binomial expansion of is
Sum of coefficient of integral power of x
We know that
If m is the A.M. of two distinct real numbers l and n (l, n > 1) and G1, G2 and G3 are three geometric means between l and n, then G14 + 2G24+ G34 equals.
The sum of first 9 terms of the series
is equal to
If the function.
is differentiable, the value of k + m is
and 3m – 2k + 2 = 0
k/4 = m and 3m – 2k + 2 = 0
m = 2/5 and k = 8/5
k + m = 8/5 + 2/5 + 10/5 = 2
g is constant at x = 3
k√4 =3m+ 2
2k = 3m + 2 .......(i)
k/4 = m
k = 4 m...........(ii)
8 m = 3 m + 2
The normal to the curve, x2 +2xy – 3y2 = 0 at (1,1)
Curve is x2 + 2xy – 3y2 = 0
Differentiate wr.t. x,
So equation of normal at (1, 1) is
y – 1 = – 1 (x – 1) Þ y = 2 – x Solving it with the curve, we get
x2 + 2x(2 – x) – 3(2 – x)2 = 0
–4x2 + 16x – 12 = 0
x2 – 4x + 3 = 0
x = 1, 3
So points of intersections are (1, 1) & (3, –1) i.e. normal cuts the curve again in fourth quadrant.
Let f(x) be a polynomial of degree four having extreme values at x = 1 and x = 2. If = 3, then f(2) is equal to
Let f(x) = a0 + a1x + a2x2 + a3x3 + a4x4
So, a0 = 0, a1 = 0, a2 = 2
i.e., f(x) = 2x2 + a3x3 + a4x4
Now, f'(x)= 4x + 3a3x2 + 4a4x3
= x[4 + 3a3x + 4a4x2]
Given, f'(1) = 0 and f'(2) = 0
=> 3a3 + 4a4 + 4 = 0 …(i)
and 6a3 + 16a4 + 4 = 0 …(ii)
Solving, a4 = 1/2, a3 = –2
i.e., f (2) = 0
The integral equals
So, option (4).
The integral is equal to
2I = 2
I = 1
The area (in sq. units) of the region described by is
After solving y = 4x – 1 and y2 = 2x
2y2 – y – 1 = 0
y = 1, -1/2
Let y(x) be the solution of the differential equation Then y(e) is equal to
It is best option. Theoretically question is wrong, because initial condition is not given.
x log x dy/dx + y = 2x logx
If x = 1 then y = 0
Solution is y.log x =
y log x = 2(x logx- x) + c
x = 1, y = 0
Then, c = 2, y(e) = 2
The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0), is
Total number of integral coordinates as required
= 39 + 38 + 37 + ....... + 1
= = 780
Locus of the image of the point (2, 3) in the line (2x – 3y + 4) + k(x – 2y + 3) = 0, k ∈ R, is a
After solving equation (i) & (ii)
2x –3y + 4= 0 .....(i)
2x –4y + 6= 0 .....(ii)
x = 1 and y = 2
Slope of AB × Slope of MN = – 1
(y – 3)(y – 1) = –(x – 2)x
y2 – 4y + 3 = –x2 + 2x
x2 + y2 – 2x – 4y + 3 = 0
Circle of radius = √2
The number of common tangents to the circles x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0, is
x2 + y2 – 4x – 6y – 12 = 0
C1(center) = (2,3), r =
x2+ y2 + 6x + 18y + 26 = 0
C2(center) (– 3, –9),
C1C2 = 13, C1C2 = r1 + r2
Number of common tangent is 3.
The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse , is
i.e., a2 = 9, b2 = 5
So, e = 2/3
As, required area
Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is
x2 = 8y
Let Q be (4t, 2t2)
Let P be (h, k)
∴ h = t,
∴ 2 k =h2
∴ Locus of (h, k) is x2 = 2y.
The distance of the point (1, 0, 2) from the point of intersection of the line and the
plane x – y + z = 16, is
P(3λ+2 , 4λ -1, 12λ+ 2)
Lies on plane x – y + z = 16
3λ+2 - 4λ +1+12λ+ 2 = 16
11λ+ 5 = 16
λ = 1 P(5, 3, 14)
The equation of the plane containing the line 2x – 5y + z = 3; x + y + 4z = 5, and parallel to the plane, x + 3y + 6z = 1, is
Required plane is
(2x – 5y + z – 3) + λ(x + y + 4z - 5) = 0
It is parallel to x + 3y + 6z = 1
Solving λ =
∴ Required plane is
(2x – 5y + z – 3) – 11/2 (x + y + 4z – 5) = 0
∴ x + 3y + 6z – 7 = 0
Letbe three non-zero vectors such that no two of them are collinear and If θ is the angle between vectors and then a value of sinθ is
∴ cos θ = - 1/3
∴ sin θ =
If 12 identical balls are to be placed in 3 identical boxes, then the probability that one the boxes contains exactly 3 balls is
Question is wrong but the best suitable option is (1).
Required probability =
The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is
Mean = 16
Sum = 16 × 16 = 256
New sum = 256 – 16 + 3 + 4 + 5 = 252
Mean == 14
If the angles of elevation of the top of a tower from three collinear points A, B and C, on a line leading to the foot of the tower, are 30º, 45º and 60º respectively, then the ratio, AB : BC, is
AO = h cot30º
BO = h
CO = h/√3
Let where |x| < 1/√3 x . Then a value of y is
The negation of is equivalent to