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A particle strikes a horizontal frictionless floor with a speed u at an angle φ with the vertical andrebounds with a speed v at an angle φ with the vertical. The coefficient of restitution betweenthe particle and floor is e. The angle φ is equal to
C
∵ vertical component after collision = eucos θ
& Horizontal component = usin θ
In the given situation disc and ring (2kg) are connected with a string. Both are placed on the rough surface. A force F = 10N is applied on the centre of disc horizontally. Assume initially both bodies were in rest. Then frictional force acting onthe ring will be
C
Zero
A flat coil of area A and n turns is placed at the centre of a ring of radius r (r^{2} >> A) and resistance R. The two are coplanar when current in the coil increases from zero to i, the total charge circulating in the ring is
In the arrangement shown, all surfaces are frictionless. The rod R is constrained to move vertically. The vertical acceleration of R is a_{1} and the horizontal acceleration of the edge w is a_{2}. The ratio a_{1}/a_{2}is equal to
Relative acceleration of rod with respect to wedge is parallel to the inclined plane.
Which of the following is the most accurate instrument for measuring length?
A block of mass m is placed on a wedge of mass 2m which rests on a rough horizontal surface. There is no friction between the block and the wedge. The minimum coefficient of friction between the wedge and the ground so that thewedge does not move is
A spherical object of mass 1 kg and radius 1 m is falling vertically downward inside a viscous liquid in a gravity free space. At a certain instant the velocity of the sphere is 2 m/s. If the coefficient of viscosity of the liquid is 1/18π ns/m^{2}, then velocity of ball will become 0.5 m/s after a time
Two identical simple pendulums A and B are fixed at same point. They are displaced by an angle α and β (α and β are very small and β > α) and released from rest. Find the time after which B reaches its initial position for the first time. Collisions are elastic and length of the strings is l.
A photon collides with a stationary H – atom in ground state inelastically. Energy of the colliding photon is 10.2 eV. Almost instantaneously, another photon collides with same H – atom inelastically with an energy of 15 eV. What will be observed by the detector?
An electron in H – atom is excited from ground state level to first excited level. Select the correct statements:
A,B,C,D
In the given circuit, current in resistance R is 1A, then
A bag of mass M hangs by a long thread and a bullet (mass m) comes horizontally with velocity v and gets caught in the bag. Then for the combined system (bag + bullet)
Apply conservation of linear momentum KE = P^{2}/2 X mass
A particle starts SHM at time t = 0. Its amplitude is A and angular frequency isω. At time t = 0 its kinetic energy is 4/E , where E is total energy. Assuming potential energy to be zero at mean
position, the displacementtime equation of the particle can be written as
A disc having radius R is rolling without slipping on a horizontal plane as shown. Centre of the disc has a velocity v and acceleration a as shown.
Q. Speed of point P having coordinates (x, y) is
A
A disc having radius R is rolling without slipping on a horizontal plane as shown. Centre of the disc has a velocity v and acceleration a as shown.
Q. If V = , the angle θ between acceleration of the top most point and the horizontal is
B
A certain substance has a mass of 50g/mol. When 300J of heat is added to 25 gm of sample of this material, its temperature rises from 25 to 45^{o}C.
Q.. The thermal heat capacity of the substance is
C
Thermal heat capacity,
Two thin symmetrical lenses of different nature have equal radii of curvature of all faces R = 20 cm. The lenses are put close together and immersed in water. The focal length of the system is 24 cm. The difference between refractive indices of the lenses is Find n. The refractive index of water is
Which one of the following is not true about diborane?
D
D
The wave function of atomic orbital of Hlike atoms is given as under:
Given that the radius is in Å, then which of the following is radius for nodal surface for He+ ion
Reimer – Tiemann’s reaction
I_{2}(s)  I^{} (0.1 M) half cell is connected to a H+ (aq.)  H_{2}(1 bar)  Pt half cell and emf is found to be 0.7714 V. If = 0.535 V, find the pH of H^{+} H half cell.
In the graph between √v and Z for the Mosley’s equation, theintercept OX is 1 on √v axis. What is the frequency whenatomic number (Z) is 51?
Which of the following equations must be used for the exact calculation of [H^{+}] of an aqueous HCl solution at any concentration C_{HC}l.[k_{w} = 10^{14} M^{2}]
Which of the following salts give different result by the action of heat?
(A) Both carbonates give oxides and CO_{2} gas.
(B) Both bicarbonates give carbonates, CO_{2} and H_{2}O
(C)
(D) Both the sulphates give oxides and SO_{3}
Which of the following is/are correct for the reaction with equilibrium constant K?
A, B, C
are eq. constant
are rate cons tant
Which of the following reaction is incorrect regarding the formation of major product (alkene)?
B
(Major product)
Due to greater probability of losing βH (9 : 3 ratio).
This section contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out which ONE OR MORE is/are correct.
Select the correct statement(s) about the above compounds from the following:
All reducing sugars are muatrotating. Although (IV) is an α  hydroxy ketone and hence reducing but it can’t mutarotate as it not a carbohydrate, can’t form ring. In (II) the glycosidic linkage is in between two anomeric carbons and hence ring opening can’t occur, thus non – reducing as well as non – mutarotating.
HI cannot be prepared by the action of conc. H_{2}SO_{4} on KI because
fact
Which of the following will not undergo aldol condensation?
The number of bicarbonates that do not exist in solid form among the following is....................
No bicarbonates exist in solid due to inefficient packing except Ammonium and Na+ to Cs+. Only NaHCO3 , KHCO3 , RbHCO3 , CsHCO3 and NH4HCO3 exist in solid.
So the correct answer is 4.
In Borax
(i) Number of B – O – B bond is x
(ii) Number of B – B bond is y
(iii) Number of sp^{2} hybridized B atom is z
Calculate the value of x + y + z
Examine the structural formulas of following compounds and find out number of compounds
which show higher rate of nucleophilic addition than
This section contains 6 questions. Each question, when worked out will result in one integer from 0 to 9 (both inclusive).
Number of nuclear particles (Projectiles + ejectiles) involved in the conversion of the nuclide _{92}U^{238} into _{94}Pu^{239} by neutron – capture method is ..............
If then is equal to (where [•] and {•} denotes GIF and fraction parts of x),
n ∈ N
The solution of the differential equation y´y´´´=3(y´´)^{2} is
A point on the hypotenuse of a triangle is at distance a and b from the side of the triangle then minimum length of the hypotenuse is
This section contains 6 multiple choice questions numbered 1 to 6. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct
A(2, 0), B(0, 4) . The point M on line y = x for which perimeter of ΔAMB is least is
Use optimization technique
Let where [.] denotes the greatest integer function. The value of S is
Usemaxima minima concept
It is the minimum distance between 2 curve
Let 0 ≤ α < β < γ ≤ 2π and if cos(x + α) + cos (x + β) + cos (x + γ) = 0 ∀x∈R , then value of γ  α is equal to
Putting x = −α, x = −β, x = −γ in the equation
Let the circles S_{1} ≡ x^{2} + y^{2} – 4x – 8y + 4 = 0 and S_{2} be its image in the line y = x, the equation of the circle touching y = x at (1, 1) and orthogonal to S_{2} is
Centre of circle S_{1} = (2, 4)
Centre of circle S_{2} = (4, 2)
Radius of circle S_{1} = radius of circle S_{2} = 4
∴ equation of circle S_{2}
(x – 4)^{2} + (y – 2)^{2} = 16
⇒ x^{2} + y^{2} – 8x – 4y + 4 = 0 . . . (i)
Equation of circle touching y = x at (1, 1) can be taken as
(x – 1)^{2} + (y – 1)^{2} + λ(x – y) = 0
or, x^{2} + y^{2} + x (λ – 2) + y(– λ – 2) + 2 = 0 . . . (ii)
As this is orthogonal to S_{2}
If f is a periodic function and g is a non–periodic function, then which of the following is not always a non–periodic function?
Suppose period of f is T. Now fog may or may not be periodics. For example if f(x) = sin x and
g(x) = x + sin x, then fog is periodic of period 2π. On the other hand if f(x) = sin x and
g(x) = x^{2}, then fog is a non–periodic function.
gof is always periodic of periodic T. Similarly fof is always periodic of period T.
A is 3 × 3 orthogonal matrix and if B = 5A^{5} , then det (B) must be equal to
A is orthogonal matrix (3 × 3)
⇒ AA^{T} = I_{3}
det (AA^{T}) = det (I_{3})
det A. det A^{T} = 1
(det A)^{2} = 1
det (A) = ±1 (1)
Q B = 5A^{5}
∴ det (B) = 5^{3}. (det A)^{5}
⇒ det (B) = 125 (±1)^{5} = ±125
Which of the following is/are true for the equation e(k − x log x) = 1 ?
Consider the eq uati onwhich represents point of intersection of and y = x ln x. Clearly, no point of intersection if k < 0. One point of intersection if k = 0 and
Two points of intersection if
In triangle ABC, a = 4 and b = c = A point P moves inside the triangle such that the square of its distance from BC is half of the area of rectangle contained by its distances from the other two sides. If D be the centre of locus of P, then
PM = k
Equation of AB = −x + y = 2
Equation of AC = x + y = 2
According to equation
Consider a triangle ABC with BC = 3. Choose a point D on BC such that BD = 2. Find the value of AB^{2} + 2AC^{2}− 3AD^{2} .
Drop the perpendicular from A to BC and let F be its foot. Further more, suppose BF = x and AF = y.
Then, by Pyt hag orean theore m,
The equation of four circles are Then the radius of a circle touching all the four circles internally is equal to
Let radius of new circle = r
From symmetry centre of the required circle is (0, 0)
(Q C_{1}O is the distance from origin to centre of any of four circles).
If the coordinates (x, y, z) of the point S which is equidistant from the points O(0, 0, 0), A(n^{5}, 0, 0), B(0, n^{4}, 0), C(0, 0, n) obey the relation 2(x + y + z) + 1 = 0. If n ∈ Z , then n = (  is the modulus function)
Minimum value of z_{1} + 1 + z_{2} + 1 + z_{1}z_{2} + 1, if z_{1} = 1, z_{2} = 1 is
If x^{2} + Px – 444P = 0 has integral roots (where P is a prime number) then the value of
The equatio n x^{2} + Px – 444P = 0 has integral roots.
Since P = 2 does not give the integral roots.
⇒ D must be perfect square of an odd integer. i.e.
D^{2} = P(P + 1776) since perfect square.
⇒ P + 1776 must be multiple of P
⇒ 1776 must be a multiple of P
Now, 1776 = 2^{4}.3.37 when; P = 2 or 3 or 37.
(i) P = 2 then P(P + 1776) = 2(2 + 1776) = 3556, not a perfect square.
(ii) P = 3 then P(P + 1776) = 3(3 + 1776) = 5337, not a perfect square.
(iii) P = 37 then P(P + 1776) = 37(37 + 1776) = 37^{2}.7^{2} which is odd.
∴ P = 37
The number of integral values of ‘a’ so that the point of local minima of f(x) = x^{3} – 3ax^{2} + 3(a^{2} – 1)x + 1 is less than 4 and point of local maximum is greater than −2, is/are
f'(x) = 3(x^{2}− 2ax + a^{2} −1)
f'(x) = 0 ⇒ (x − a)2 = 1, x = a + 1, a – 1
−2<a − 1, a + 1 < 4
⇒ −1<a , a < 3
⇒ a ∈ (−1, 3)
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