A system in a normalized state $$\left| \psi \right\rangle = {c_1}\left| {{\alpha _1}} \right\rangle + {c_2}\left| {{\alpha _2}} \right\rangle $$ with $$\left| {{\alpha _1}} \right\rangle $$ and $$\left| {{\alpha _2}} \right\rangle $$ representing two different eigen states of the system requires that the constants c<sub>1</sub> and c<sub>2</sub> must satisfy the condition

A particle is in the normalized state $$\left| \psi \right\rangle $$ which is a superposition of the energy eigen states $$\left| {{E_0} = 10\,eV} \right\rangle $$ and $$\left| {{E_1} = 30\,eV} \right\rangle .$$ The average value of energy of the particle in the state $$\left| \psi \right\rangle $$ is 20 eV. The state $$\left| \psi \right\rangle $$ is given by

A

$$\frac{1}{2}\left| {{E_0} = 10\,eV} \right\rangle + \frac{{\sqrt 3 }}{4}\left| {{E_1} = 30\,eV} \right\rangle $$

B

$$\frac{1}{{\sqrt 3 }}\left| {{E_0} = 10\,eV} \right\rangle + \sqrt {\frac{2}{3}} \left| {{E_1} = 30\,eV} \right\rangle $$

C

$$\frac{1}{2}\left| {{E_0} = 10\,eV} \right\rangle - \frac{{\sqrt 3 }}{4}\left| {{E_1} = 30\,eV} \right\rangle $$

D

$$\frac{1}{{\sqrt 2 }}\left| {{E_0} = 10\,eV} \right\rangle - \frac{1}{{\sqrt 2 }}\left| {{E_1} = 30\,eV} \right\rangle $$

A particle of mass m is confined in the potential
Quantum Mechanics mcq question image

\[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{2}m{\omega ^2}{x^2},}&{{\text{for }}x Let the wave function of the particle be given by $$\psi \left( x \right) = - \frac{1}{{\sqrt 5 }}{\psi _0} + \frac{2}{{\sqrt 5 }}{\psi _1}$$
where $${\psi _0}$$ and $${\psi _1}$$ are the eigen functions of the ground state and the first excited slate respectively. The expectation value of the energy is

A

$$\frac{{31}}{{10}}\hbar \omega $$

B

$$\frac{{25}}{{10}}\hbar \omega $$

C

$$\frac{{13}}{{10}}\hbar \omega $$

D

$$\frac{{11}}{{10}}\hbar \omega $$

Two boys are playing on a ground. Both the boys are less than 10 years old. Age of the younger boy is equal to the cube root of the product of the age of the two boys. If we place the digit representing the age of the younger boy to the left of the digit representing the age of the elder boy, we get the age of the father of the younger boy. Similarly, we place the digit representing the age of the elder boy to the left of the digit representing the age of the younger boy and divide the figure by 2, we get the age of the mother of the younger boy. The mother of the younger boy is younger than his father by 3 years. Then, what are the ages of elder and younger boys ?

A

E = 15 & Y = 3

B

E = 14 & Y = 12

C

E = 40 & Y = 22

D

E = 4 & Y = 2

Let the eigen values of a 2 × 2 matrix A be 1, -2 with eigen vectors x₁ and x₂ respectively. Then the eigen values and eigen vectors of the matrix A² - 3A + 4$$I$$ would, respectively, be

A

2, 14; x₁, x₂

B

2, 14; x₁ + x₂, x₁ - x₂

C

2, 0; x₁, x₂

D

2, 0; x₁ + x₂, x₁ - x₂

If $${\text{sin}}\left( {{{60}^ \circ } - \theta } \right)$$ = $${\text{cos}}\left( {\psi - {{30}^ \circ }} \right),$$ then the value of $${\text{tan}}\left( {\psi - \theta } \right)$$ is (assume that $$\theta $$ and $$\psi $$ are both positive acute angles with$$\theta {30^ \circ }$$ ) ?

A

$$\frac{1}{{\sqrt 3 }}$$

B

0

C

$$\sqrt 3 $$

D

1

To constitute a matter of res judicata which of the following conditions must concur?
1. The matter directly and substantially in issue in the subsequent suit or issue must be the same matter which was directly and substantially in issue either actually (section 11, explanation III) or constructively (section 11, explanation IV) in the former suit
2. The former suit must have been a suit between the same parties under whom they or any of them claim. Explanation VI of Section 11 must be read with this condition
3. The parties as aforesaid must have litigated under the same title in the former suit
4. The court which decided the former suit must have been a court competent to try the subsequent suit of the suit in which such issue has been subsequently raised. Explanation II of section 11 is to be read with condition
5. The matter directly and substantially in issue in the subsequent suit must have been heard and finally decided by the court in the first suit. Explanation V of section 11 is to be read with this condition

A

1, 2

B

3, 4

C

2, 4, 5

D

All of these

Consider the system of equations A_{(n × n)} X_{(n × 1)} = λ_{(n × 1)} where, λ is a scalar. Let (λ_i, x_i) be an eigen-pair of an eigen value and its corresponding eigen vector for real matrix A. Let $$I$$ be a(n × n) unit matrix. Which one of the following statement is NOT correct?

A

For a homogeneous n × n system of linear equations, (A - λ$$I$$)x = 0 having a nontrivial solution, the rank of (A - λ$$I$$) is less than n

B

For matrix A^m, m being a positive integer, (λ_i^m, x_i^m) will be the eigen-pair for all i

C

If A^T = A^-1, then |λ_i| = 1 for all i

D

If A^T = A, then λ_i is real for all i

According to morphological details, pSi can be divided as sponge-like pSi and pSi featuring cylindrical pores.

A

True

B

False

The table below presents the data of seats won by leading political parties in 6 states. Study the given information and answer the questions that follow. Seats won by different political parties in different states Political Parties Stats State A State B State C State D State E State F Cock 12 21 3 2 4 19 Eagle 4 3 23 2 1 13 Bat 1 21 6 1 18 5 OwI 2 1 11 .0 9 14 Parrot 15 7 3 0 6 0 Swan 18 2 6 4 2 0 In which state were elections held for the maximum number of seats compared to the other states?

A

State B

B

State C

C

State D

D

State A

$\psi$MO = $\psi$A + $\psi$B.

A

True

B

False

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