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An engine whistling at a constant frequency n_{0} and moving with a constant velocity goes past a stationary observer. As the engine crosses him, the frequency of the sound heard by him changes by a factor f. The actual difference in the frequency of the sound heard by him before and after the engine across him is
Simply use the above relation
A ball of density ρ_{0} falls from rest from point P onto the surface of a liquid of density ρ in time T. It enters the liquid stops moves up and returns to P in a total time 3T. Neglect viscosity, surface tension and splashing. The ratio p/p_{0} is equal to
When an object is placed in front of a concave mirror of focal length f, a virtual image is produced with a magnification of 2. To obtain a real image with a magnification of 2. The object has to be moved by a distance equal to
Calculate the initial and final distance of object from the concave mirror.
A solid cube is placed on a horizontal surface. The coefficient of friction between them is μ, where μ < 1/2 . A variable horizontal force is applied on the cube’s upper face, perpendicular to one edge and passing through the midpoint of that edge. The maximum acceleration with which it can move without toppling is
To avoid t toppling
A spherical body of mass m, radius r and moment of inertia I about its centre moves along the xaxis. Its centre of mass moves with velocity = v_{0} and it rotates about its centre of mass with angular velocity ω_{0}.L_{O} = lω_{0} + mv_{o}r . The angular momentum of the body about the origin O is
As we know
L_{0} = L_{1} + L_{2}
L_{1} = due to rotational motion
L_{2} = due to linear motion.
DIRECTION ; This section contains 2 multiple choice questions from 8 to 9. Each question has 4 choices (A), (B), (C) and (D) for its answer, out which ONE OR MORE is/are correct
The temperature of a solid object is observed to be constant during a period. In this period
If temperature of a body is constant that means amount of hear energy received and amount of hear energy emitted are equal.
The electric potential decreases uniformly from 120 V to 80 V as one moves on the axis from x = –1 cm to x = 1 cm. The electric field at the origin
If direction of field is along the xaxis then its magnitude will be 20 V/cm other wise greatest then hat
This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions and based upon the second paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE IS CORRECT.
paragraph for question no.12 and 13
The base of a hollow right cone of semi vertical angle 30^{o} is fixed to a horizontal plane. Two particle each of mass m are connected by a light inextensible string which passes through a small hole in the vertex of the cone. One particle A hangs at rest inside the cone. The other particle B moves on the outer smooth surface of the cone at a distance ℓ from vertex in a horizontal circle with centre at A. Neglecting friction, now answer the following.
The tention in the string is
Now apply newton’s law of motion for the particle B.
The base of a hollow right cone of semi vertical angle 30^{o} is fixed to a horizontal plane. Two particle each of mass m are connected by a light inextensible string which passes through a small hole in the vertex of the cone. One particle A hangs at rest inside the cone. The other particle B moves on the outer smooth surface of the cone at a distance ℓ from vertex in a horizontal circle with centre at A. Neglecting friction, now answer the following.
The angular velocity of B is
Now apply newton’s law of motion for the particle B.
Paragraph For Question No.14 and 16
The switch S has been closed for long time and the electric circuit shown carries a steady current. Let C_{1} = 3μF, C_{2 }= 6μF, R1 = 4 kΩ and R_{2} = 7.0 kΩ. The power dissipated in R_{2} is 2.8 W.
The power dissipated to the resistor R_{1} is
Initially current will pass through the resistance only.
The switch S has been closed for long time and the electric circuit shown carries a steady current. Let C_{1} = 3μF, C_{2} = 6μF, R1 = 4 kΩ and R_{2} = 7.0 kΩ. The power dissipated in R_{2} is 2.8 W.
The power dissipated to the resistor R_{1} is
The charge on capacitance C_{1} and C_{2} are respectively
Initially current will pass through the resistance only.
The switch S has been closed for long time and the electric circuit shown carries a steady current. Let C_{1} = 3μF, C_{2} = 6μF, R1 = 4 kΩ and R_{2} = 7.0 kΩ. The power dissipated in R_{2} is 2.8 W.
The power dissipated to the resistor R_{1} is
Long time after switch is opened the charge on C_{1} is
Initially current will pass through the resistance only.
Column – I gives certain situation involving two thin conducting shells connected by a conducting wire via a key K. In all situation one sphere has net charge +q and other sphere has no net charge. After the key K is pressed
column – II gives some resulting effect. Match the figures in column with the statements in column – II.
In such type of situation, we have to use these basic facts.
(a) charge will flow between the two body if there is potential difference between the two.
(b) during this process energy will lose due to sparking
A particle of mass 2 kg is moving on straight line under the action of force F = (8 – 2x)N. The
particle is released at rest from x = 6m. For the subsequent motion match the following (All the
values in the right column are in there SI units).
At equilibrium F_{net} will be zero and at the extreme point, particle will be in state of rest.
This section contains 6 questions. Each question, when worked out will result in one integer from 0 to 9 (both inclusive).
In the arrangement shown in figure, pulley are light and frictionless, threads are inextensible and mass of blocks A, B and C are m1 = 5 kg, m2 = 4 kg and m3 = 2.5 kg respectively and coefficient of friction for both the planes is μ = 0.50. Calculate the acceleration of block A (in m/sec), when the system is released from rest. (g = 10 m/sec)
In this situation block B will not move at all.
Now apply newton's laws of motion.
The figure shows the velocity and acceleration of a point like body at the initial moment of its motion. The acceleration vector of the body remains constant. The minimum radius of curvature of trajectory of the body is
Where R is radius of curvature.
A disc ‘A’ of mass M is placed at rest on the smooth inclined surface of inclination θ. A ball B of mass m is suspended vertically from the centre of the disc A by a light inextensible string of light ℓ as shown in the figure. If the acceleration of the disc B immediately after the system is released from rest
apply newton’s laws of motion for the body A and body B
A uniform rod AB of mass M and length R √2 is moving in a vertical plane inside a hollow sphere of radius R. The sphere is rolling on a fixed horizontal surface without slipping with velocity of its centre of mass 2v, when the end B is at the lowest position, its speed is found to be v as shown in the figure. If the kinetic energy of rod at this instant is 4/k mv^{2}. Find k.
length of rod is R √2
Velocity of point P
2v  ωR = v
KE of rod = rotational KE t translatory KE
Sulphur reacts with chlorine in 1 : 2 ratio and forms X. Hydrolysis of X gives a sulphur compoundY. What is the hybridization state of central atom in the compound Y?
Hybridisation of S in H_{2}SO_{3} = 1/2 (6+2+0) = 4 = sp^{3} (3σ + 1π)
Find out the correct representation of trans  decaline
50 ml of a solution containing 10^{3} mol of Ag^{+} is mixed with 50 ml of a 0.1 M HCl solution. Howmuch [Ag^{+}] remains in solution? Given: K_{sp} of AgCl = 10^{10}
Where X is a compound that forms azodye with benzene diazonium chloride in faintly basic medium.
Hence the products P, X and Y are respectively.
Which of the following solutions show lowering of vopour pressure on mixing?
In B and D compound after mixing undergoes H – bonding and hence boiling point will be raised and hence vapour pressure lowered.
Which of the following will produce aromatic compound as a product?
This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions and based upon the second paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE IS CORRECT.
Paragraph For Question No.12 and 13
A white compound (A) on strong heating decomposes to produce two products (B) and (C). (B) on reaction with white phosphorus produces (D), which is a strong dehydrating agent. (D) on reaction with perchloric acid converts it to its anhydride.
The compound (A) is
Paragraph For Question No.12 and 13
A white compound (A) on strong heating decomposes to produce two products (B) and (C). (B) on reaction with white phosphorus produces (D), which is a strong dehydrating agent. (D) on reaction with perchloric acid converts it to its anhydride.
The product/s obtained on hydrolysis of (D) is
A’ is a substance that converts into B, C and D by three first order parallel paths simultaneously according to the following stoichiometry
The partial t_{1/2} of A along path I and path II are 173.25 min and 346.5 min respectively. The energies of activation of the reaction along path I, path II and path III are 40, 60 and 80 kJ/mol respectively.
The percent distribution of C in the product mixture B, C and D at any time is equal to
If k_{1}, k_{2} and k_{3} be the rate constants of the reaction along path I, II and III respectively, then overall rate constant of consumption of A will be k_{1} + k_{2} + k_{3}. So,
A’ is a substance that converts into B, C and D by three first order parallel paths simultaneously according to the following stoichiometry
The partial t_{1/2} of A along path I and path II are 173.25 min and 346.5 min respectively. The energies of activation of the reaction along path I, path II and path III are 40, 60 and 80 kJ/mol respectively.
The initial rate of consumption of A and the sum of the initial rate of formation of B, C and D arerespectively, taking [A] = 0.25 M, equal to
A’ is a substance that converts into B, C and D by three first order parallel paths simultaneously according to the following stoichiometry
The partial t_{1/2} of A along path I and path II are 173.25 min and 346.5 min respectively. The energies of activation of the reaction along path I, path II and path III are 40, 60 and 80 kJ/mol respectively.
The overall energy of activation of A along all the three parallel path is equal to
Among the following, the molecule with the highest dipole moment is
5.6 L an unknown gas at NTP requires 12.5 calories to rise its temperature by 10^{o}C at constant volume. What is the atomicity of the gas?
The gas is thus diatomic
After electrolysis of a NaCl solution with inert electrodes for a certain period of time, 60 ml of the solution was left which was found to be 1 N in NaOH. During the same time 31.75 g of Cu was deposited in a Cu voltameter in series with the electrolytic cell. Calculate the percentage of the theoretical yield of the sodium hydroxide obtained.
No. of eq. of NaOH that can be produced theoretically for 100% current efficiency = no of equivalents of NaCl decomposed = no of eq. of Cu deposited
No. of eq. of NaOH produced experimentally =
Of the following amines how many can be separated by Hoffmann’s mustard oil reaction?
Hoffmann’s mustard oil reaction is a test of 1^{o} amine.
If the equation x^{4} – 4x^{3} + ax^{2} + bx + 1 = 0 has four positive roots then
Find the point p on the line 2x + 3y + = 0 such that PA  PB is maximum where A is (2,0) and B is (0,2)
Thus the max value of PA – PB is AB
This is possible only when P lies on AB but P lies on AB
∴ P is the point of intersection of x + y = 2 and 2x + 3y + 1 = 0.
Find the coefficient of x^{5} in the expansion of (2 – x + 3x^{2})^{6}
\
θ = nπ,n ∈ z or 2θ = nπ ∵ θ = nπ/2 is not for possible as n is odd tanθ is not define.
Hence θ = nπ, n ∈ z is the only solution.
This section contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct
If x, y, z are respectively perpendiculars from the circumcenter to the sides of ΔABC, (a, b, c are usual meanings) then
The angle between the lines whose direction cosine are connected by the relations ℓ –5m+3n=0 and 7 ℓ^{2} + 5m^{2} – 3n^{2} = 0
If are three mutually perpendicular unit vectors and is a unit vector which makes equal angle with and then the value of
This section contains 2 paragraphs. Based upon one of paragraphs 2 multiple choice questions and based on the other paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct
Paragraph for Question Nos. 12 to 13
PQ is the double ordinate of the parabola y2 = 4x which passes through the focus S. ΔPQA is an isosceles right angle triangle, where A is on the axis of the parabola. Line PA meets the parabola at C and QA meets the parabola at B.
The area of trapezium PBCQ is
end point of latus rectum P(1, 2) & Q(1, 2) ΔPAQ is isosceles right angled
∴ slope of PA is – 1
In equation y – 2 =  (x – 1)
x + y – 3 = 0
similarly equation of line QB
x – y – 3 = 0
solving x + y – 3 = 0 with parabola y^{2} = 4x
(3x)2 = 4x
x = 1,9
∴ coordinate of B & C are (9, 6) & (9, 6) respectively
Area of trapezium
= 1/2 x (12 + 4) x 8
= 64 sq units
PQ is the double ordinate of the parabola y2 = 4x which passes through the focus S. ΔPQA is an isosceles right angle triangle, where A is on the axis of the parabola. Line PA meets the parabola at C and QA meets the parabola at B.
The circumradius of trapezium PBCQ is
let the circumcenter of trapezium be T(n, 0)
Then PT = PB
or n = 7
radius √40 = 2 √10
Let z be a complex number satisfying z^{2} + 2αz + 1 = 0 where α is a parameter which can take any real value.
The roots of this equation lie on a certain circle if
one root lies inside the unit circle and other will outside the unit circle case where α is very large then
PQ is the double ordinate of the parabola y2 = 4x which passes through the focus S. ΔPQA is an isosceles right angle triangle, where A is on the axis of the parabola. Line PA meets the parabola at C and QA meets the parabola at B.
For every large value of α the roots are approximately
one root lies inside the unit circle and other will outside the unit circle
case where α is very large then
This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as
illustrated in the following example:
If the correct matches are A – p, s and t; B – q and r; C – p and q; and D – s and t; then the correct darkening of bubbles will look like the following:
A function is defined as f; { a_{1},a_{2},a_{3},a_{4},a,,a_{6} } → { b_{1},b_{2},b_{3} }
total number of function = 3^{6 }= 729
(a) total number of onto function
(b) since f(a_{i}) ≠ bi
It means that a_{1}, a_{2}, a_{3 }cannot be assigned images b_{1}, b_{2}, b_{3}
Number of function = 2^{3}3^{3} = 216
(c) number of invertible function = 0
as function is not one – one
(d) total many one function = 729 – 0
= 729
This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as
illustrated in the following example:
If the correct matches are A – p, s and t; B – q and r; C – p and q; and D – s and t; then the correct darkening of bubbles will look like the following:
Match the following:
This section contains 6 questions. Each question, when worked out will result in one integer from 0 to 9(both inclusive).
Two friends have equal number of sons. There are 3 tickets for a cricket match which are to be distributed among the sons. The probability that two tickets go to the sons of one and one tickets goes to the sons of other is 6/7 . Then total number of boys is equal to (sum of son of each friend).
let each friend has n sons.
∴ 3 tickets can be distributed among 2n sons in ^{2n}C_{3} ways.
The number of ways distributing 3 tickets such that two tickets go to the sons of one and one tickets goes to sons of the other.
^{n}C_{2} x ^{n}C_{1} + ^{n}C_{1} x ^{n}C_{2} = 2 x ^{n}C_{1} x ^{n}C_{2} probability that two tickets go to the sons of one and one tickets goes the sons of the other
But from the question
hence total number of boys = 8
A line through the origin meets the circle x^{2} + y^{2} = a^{2 }at P and the hyperbola x^{2} – y^{2} = a^{2} at Q.
Then locus of the point of intersections of tangent to the circle at P with the tangent at Q to the
hyperbola is the curve (a^{4} + 4y^{4})x^{2} = a^{K} then K is equal to .
the equation of the tangents to the circle x2 + y2 = a2 at P and the hyperbola x2 – y2 = a2 at Q are
Where y = mx is intersecting line through (2, 0)
Let (h, k) be the point of intersection of these two lines
In a ΔABC, then the value of Δ,where • denote absolute value.
The area of region in first quadrant in which points are nearer to the origin then to the line x = 3.
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