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One mole of a monoatomic real gas satisfies the equation p(V − b) = RT where b is a constant. The relationship of interatomic potential V(r) and interatomic distance r for the gas is given by |
(A) If ∣w∣ = 2, then the set of points z = w − (1/w) is contained in or equal to (B) If ∣w∣ = 1, then the set of points z = w + (1/w) is contained in or equal to Options for both A and B: (p) An ellipse with eccentricity (4/5) (q) The set of points z satisfying Im z = 0 (r) The set of points z satisfying ∣Im z∣ ≤ 1 (s) The set of points z satisfying ∣Re z∣ ≤ 2 (t) The set of points z satisfying ∣z∣ ≤ 3 |
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. lim_(x→2^+ ) {(([x]^3 )/3) −[ (x/3)]^3 } is equal to ... |
lim_(x→∞) (√(16x^2 + 4x)) − (√x^2 ) − (√(9x^2 + 3x)) |
The factors of determinant ((x,a,b),(a,x,b),(a,b,x))are |
Let f(x) = ∣x − 1∣ + ∣x − 2∣ + ∣x − 3∣, then find the value of k for which f(x) = k has 1. no solution 2. only one solution 3. two solutions of same sign 4. two solutions of opposite sign |
prove: ∀n∈N^∗ ,∀(a,b)∈C^2 , a^(2n+1) +b^(2n+1) = Σ_(k=o) ^(2n) (−1)^k a^k b^(2n−k) |
Two particles of mass m each are tied at the ends of a light string of length 2a. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance ′a′ from the center P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force F. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes 2x, is |
Find the compression in the spring if the system shown below is in equilibrium. |
Figure shows an arrangement of blocks, pulley and strings. Strings and pulley are massless and frictionless. The relation between acceleration of the blocks as shown in the figure is |
if A+B=(π/4) so proof (1+tan A)(1+tan B)=2 |
if sin αsin β−cos αcos β+1=0 so proof that 1+cot αtan β=0 |
If the minimum value of ∣z+1+i∣ + ∣z−1−i∣ + ∣2 − z∣ + ∣3 − z∣ is k then (k − 8) equals |
Figure shows a small bob of mass m suspended from a point on a thin rod by a light inextensible string of length l. The rod is rigidly fixed on a circular platform. The platform is set into rotation. The minimum angular speed ω, for which the bob loses contact with the vertical rod, is (1) (√(g/l)) (2) (√((2g)/l)) (3) (√(g/(2l))) (4) (√(g/(4l))) |
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A ball is bouncing elastically with a speed 1 m/s between walls of a railway compartment of size 10 m in a direction perpendicular to walls. The train is moving at a constant velocity of 10 m/s parallel to the direction of motion of the ball. As seen from the ground (a) the direction of motion of the ball changes every 10 seconds. (b) speed of ball changes every 10 seconds. (c) average speed of ball over any 20 second interval is fixed. (d) the acceleration of ball is the same as from the train. |
STATEMENT-1 : z_1 ^2 + z_2 ^2 + z_3 ^2 + z_4 ^2 = 0 where z_1 , z_2 , z_3 and z_4 are the fourth roots of unity. and STATEMENT-2 : (1)^(1/4) = (cos0° + i sin0°)^(1/4) . |
STATEMENT-1 : The locus of z, if arg(((z − 1)/(z + 1))) = (π/2) is a circle. and STATEMENT-2 : ∣((z − 2)/(z + 2))∣ = (π/2), then the locus of z is a circle. |
Let A, B, C be three sets of complex numbers as defined below A = {z : Im z ≥ 1} B = {z : ∣z − 2 − i∣ = 3} C = {z : Re((1 − i)z) = (√2)}. Let z be any point in A ∩ B ∩ C and let w be any point satisfying ∣w − 2 − i∣ < 3. Then, ∣z∣ − ∣w∣ + 3 lies between (1) −6 and 3 (2) −3 and 6 (3) −6 and 6 (4) −3 and 9 |
Find out the value of nth derivative of y=e^(msin^(−1) x ) at x=0 |
Suppose an integer x, a natural number n and a prime number p satisfy the equation 7x^2 − 44x + 12 = p^n . Find the largest value of p. |
if 3^(2n+3) =m,find 3^(−n) |
Pg 1814 Pg 1815 Pg 1816 Pg 1817 Pg 1818 Pg 1819 Pg 1820 Pg 1821 Pg 1822 Pg 1823 |