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Consider a weight balanced tree such that, the number of nodes in the left sub tree is at least half and at most twice the number of nodes in the right sub tree. The maximum possible height (number of nodes on the path from the root to the farthest leaf) of such a tree on k nodes can be described as

Correct Answer: log3/2 n
Total number of nodes can be described by the recurrence T(n) = T((n-1)/3)) + T(2(n-1)/3) + 1 T(1) = 1. height of the tree will be H(n) = H(2/3(n-1)) + 1, H(1). drawing a recurrence tree and the cost at each level is 1 and the height will be log(3/2)n.

The rate expression for a heterogenous catalytic reaction is given by, - rA = K.KA PA(1 + KA.PA + Kr.PR), where K is surface reaction rate constant and KA and KR are absorption equilibrium constants of A and R respectively. If KR PR >> (1 + KA PA), the apparent activation energy EA is equal to (given E is the activation energy for the reaction and ΔHR and ΔHA are the activation energies of adsorption of R and A)

Letx(t) be the input and y(t) be the output of a continuous time system. Match the system properties P1, P2 and P3 with system relations R1, R2, P3, P4.
Properties
P1 : Linear but NOT time-invariant
P2 : Time-invariant but NOT linear
P3 : Linear and time-invariant
Relations
R1 : y(t) = t2x(t)
R2 : y(t) = t |x(t)|
R3 : y(t) = |x(t)|
R4 : y(t) = x(t - 5)

V1, V2, V3 and V4 are the volumes of four cubes of side lengths x cm, 2x cm, 3x cm and 4 cm respectively. Some statements regarding these volumes are given below :
(i) V1 + V2 + 2V3 4
(ii) V1 + 4V2 + V3 4
(iii) 2(V1 + V3) + V2 = V4
Which of these statements area correct ?

Pick out the wrong statement below:
I. For the same conversion, the holding time required in a batch reactor, is always equal to the space time required in a PFR.
II. Two mixed reactors of unequal size are available for producing a specified product, formed by a homogenous second order reaction. To achieve maximum production rate, the smaller reactor should be placed in series before the larger reactor.
III. Arrehenius equation describing the effect of temperature on rate constant is given by, $${\text{K}} = {\text{A}}.{{\text{e}}^{ - \frac{{{\text{Ea}}}}{{{\text{RT}}}}}}$$
IV. The mechanism for the decomposition of CH3CHO into CH4 and CO in presence of I2 is:
CH3CHO + I2 → CH3I + HI + CO; slow
CH3I + HI → CH4 + I2; fast
Then the rate of disappearance CH3CHO is equal to K.CCH3l.CHl and acts as a catalyst.

Let $$\overrightarrow {\bf{L}} $$ = (Lx, Ly, Lz) denotes the orbital angular momentum operators of a particle and let L+ = Lx + i Ly and L- = Lx - i Ly. The particle is in aneigen state of L2 and Lz eigen values $${\hbar ^2}\left( {l + 1} \right)$$   and $$\hbar l$$  respectively. The expectation value of L+L- in this state is

If a + b + c + d = 4, then find the value of $$\frac{1}{{\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)}}$$     + $$\frac{1}{{\left( {1 - b} \right)\left( {1 - c} \right)\left( {1 - d} \right)}}$$     + $$\frac{1}{{\left( {1 - c} \right)\left( {1 - d} \right)\left( {1 - a} \right)}}$$     + $$\frac{1}{{\left( {1 - d} \right)\left( {1 - a} \right)\left( {1 - b} \right)}}$$     is?

If a + b + c + d = 4, then the value of $$\frac{1}{{\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)}}$$     + $$\frac{1}{{\left( {1 - b} \right)\left( {1 - c} \right)\left( {1 - d} \right)}}$$     + $$\frac{1}{{\left( {1 - c} \right)\left( {1 - d} \right)\left( {1 - a} \right)}}$$     + $$\frac{1}{{\left( {1 - d} \right)\left( {1 - a} \right)\left( {1 - b} \right)}}$$     is?

Given,
3H2 + CO = CH4 + H2O, KP = 101.84 and 4H2 + CO2 = CH4 + 2H2O, KP = 101.17
the KP for the reaction CO + H2O = CO2 + H2 is

Let X1, X2 be two independent normal random variables with means μ1, μ2 and standard deviations σ1, σ2 respectively. Consider Y = X1 - X2; μ1 = μ2 = 1, σ1 = 1, σ2 = 2, Then,

Consider the following system of equations in three real variables x1, x2 and x3
2x1 - x2 + 3x3 = 1
3x1 - 2x2 + 5x3 = 2
-x1 - 4x2 + x3 = 3
This system of equations has