TN TRB PG Assistant Mock Test- 4 (Mathematics) - TN TET MCQ

Concept:
Particular integration of the given 3rd order differential equation
P.I. Calculation:
P.I. = (1/f(D)) e^ax = (1/f(a)) e^ax, f(a) ≠ 0
If f(a) = 0 then go f'(a)
P.I. = (1/f'(a)) e^ax
D = dy/dx
Calculation:
(d³y/dx³) - 3(d²y/dx²) + 4y = e³ˣ
Partial integration of the given 3rd order differential equation
P.I. = (1/f(D)) e^ax = (1/f(a)) e^ax
Where f(D) = D³ - 3D² + 4 , a = 3
f(a) = (3)² - 3(3)² + 4 = 4
Put the value f(a) and a in P.I. equation
P.I. = (1/4) e³ˣ

Concept:
Equation of the form y = x × g(p) + f(p) is called Lagrange's form.
When g(p) = p, then the equation, y = px + f(p) is called Clairaut's equation and the solution of such type of equation is given by: y = Cx + f(C).
Calculation:
Given:
y = px + √(4 + p²) where f(p) = √(4 + p²)
The above equation represents Clairaut's form so the solution is y = Cx + f(C).
∴ y = Cx + √(4 + C²)
∴ (y - Cx)² = 4 + C²
∴ (y - Cx)² - C² = 4 is the required solution.

Concept:
For a n x n matrix say A, |kA| = kⁿ|A|
For a n x n matrix say A, |A^m| = |A|^m
Calculation:
Given:
A = 
and B = 4A2
|B| = |4A²| = 4³ . |A²|
|A| = 3(6 - 4) = 6
|A²| = |A|² = 36
|B| = 64 × 36 = 2304

Let,
I = ∫ eˣ {f(x) + f'(x)} dx
= ∫ eˣ f(x) dx + ∫ eˣ f'(x) dx + C
By solving through integration by parts, we get
= {eˣ f(x) - ∫ f'(x) eˣ dx} + ∫ eˣ f'(x) dx + C
= f(x) · eˣ + C\
Where C is constant

Given :-
The function f(z) is continuous at z = z₀ 
Concept used :-
If the function f(z) is analytic at z = z₀ then so it is continuous.
Solution :-
If the function f(z) is analytic at z = z₀ then so it is continuous. But converse is not always true
Which means if f(z) is continuous at is analytic at z = z₀

Concept:
M dx + N dy = 0, will be exact differential equation if:

Where M and N are functions of x and y.
Calculation:
M = α xy³ + y cos x
N = x²y² + β sin x


The differential equation to be exact,
3 α xy² + cos x = 2xy² + β cos x
⇒ 3α = 2 ⇒ α = 2/3.
And β = 1

Concept:
The volume of a solid Revolution: Disks and waster.
If a region in the plane is revolved about a line in the same plane, the resulting objects is shown as 'solid of revolution'.
The line about which we rotate the shape is called the "axis of revolution".
(1) Disk method
This is used when we rotate a single curve, y = f(x) around x (or) y-axis.
Assume y = f(x) is a continuous non-negative function in the interval [a, b]


The volume of solid formed by revolving the region bounded by curve and the x axis between x = a and x = b about x-axis is

The cross-section perpendicular to the axis of revolution has the form of a disk of radius R = f(x).
Similarly, we fund the volume of solid when region bounded by x = f(y) and y-axis between y = c and y = d, and rotated about y-axis.



Calculation:
Given function and a = 0, b =
Volume



= 76π/3

Analysis:

Let, n = 3

The matrix for n = 3 from the given conditions will be:

Applying the following transformation:

∴ The rank if n = 3 will be:

ρ (A) = 1

We are given the set of vectors in :

Since S contains 5 vectors in , and the maximum number of linearly independent vectors in is 3, the set must be linearly dependent.
Thus, (A) is correct.
Choosing the Vectors v₁ = (1, 0, 0), v₂ = (0, 1, 0), v₃ = (1, 1, 0):
a(1,0,0) + b(0,1,0) + c(1,1,0) = (0,0,0)
a + c = 0 ⇒ c = -a
b + c = 0 ⇒ c = -b
Third component : 0 = 0 (always true)
Since we found c = -a and c = -b , we get a = b ,
and we can choose nonzero values for a, b, c such that this equation holds, meaning these three vectors are linearly dependent.
Thus, (B) is incorrect
Since S is a set of 5 vectors in  , any 4 vectors must be linearly dependent because the rank of the matrix formed by these vectors is at most 3
Thus, (C) is correct
Hence Option (3) is the correct answer.

Explanation:
Given:
Matrix
A.A^T =

a + 2b = 0 ____ (1)
a² + b² = 1 ____ (2)
∴ a = ± 2/√5, b = ± 1/√5

Concept-
The Dihedral group of an n-gon (which has order 2n) is generated by the set
where r represents rotation by 2π/n and s is any reflection across a line of symmetry.
Explanation-
In Dihedral group the generating set are set of rotation say r ,
set of reflection say s
so cardinality of generating set in Dihedral group is {r,s}
Therefore, Correct Option is Option 3).

Concept:

• A square matrix such that A2 = I is called the involuntary matrix.
• A square matrix such that A2 = A is called the Idempotent matrix.
• A square matrix such that AAT = ATA = In is called the Orthogonal matrix.

• Any square matrix of order n is said to be nilpotent matrix if there exists least positive integer m such that A^m = O, where O is the null matrix of order n.

Calculation:

Given:

So, lets calculate A²

As we know that if A is a square matrix such that A2 = I is called then A is an involuntary matrix.

The Gamma function is defined as:

t = x²
dt = 2x dx
dx = dt / (2√t)
Our integral becomes:


This integral now matches the form of the Gamma function with z = 1/2:

Γ(1/2) = √π
Therefore,
Hence, option (3) is the correct answer.

Concept:
Generalized Cauchy Integral Formula:
For an analytic function f(z) inside and on a simple closed contour C , and for any point a inside C :

Explanation:
Applying this Formula:
Here, we are given the integral:
Comparing with the formula, we see that n = 2 , so we use:

Since 1! = 1 , we get:

Rewriting the Integral in Terms of f'(z) :
We now use Cauchy's Integral Formula for f'(a) :

Multiplying both sides by 2πi, we get:

Now, comparing this with our given integral result:
,
we see that this exactly matches:

Hence Option (2) is the correct answer.

(A)
This indicates that f(x, y) is integrated first with respect to y , then with respect to x
The outer integral is over x from 0 to 2, and the inner integral is over y from 0 to x
⇒ x = 0 , x = 2 and y = 0 , y = x
By Changing Order of integration:
y = 0 , y = 2 and x = y , x = 2
Now, Integral becomes

⇒ A → III
(B)
The outer integral is over y from 0 to 1, and the inner integral is over x from y to 1
This indicates that f(x, y) is integrated first with respect to x , and the integration limit depends on y
By Changing Order of integration:
⇒ x = 0 to 1 and y = 0 to x
Now Integrals becomes :

⇒ B → II
(C)
here , x = 0 to 2 and y = x to 2
Changing order of integration:
y = 0 to 2 and x = 0 to y
Now, Integrals becomes

⇒ C → IV
(D)
Here y = 0 to 1 and x = 0 to y
Changing order of integration:
here x = 0 to 1 and y = x to 1
Now, Integrals becomes

⇒ D → I
Matching List-I with List-II
(A) matches (III)
(B) matches (II)
(C) matches (IV)
(D) matches (I)
Hence Option (2) is the correct answer.

The symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group S_n defined over a finite set of n symbols consists of the permutations that can be performed on the ${\displaystyle }$
n symbols. Since there are ${\displaystyle }$
n! such permutation operations, the order (number of elements) of the symmetric group ${\displaystyle {\mathrm{}}_{}}$
S_n is ${\displaystyle }$
n!.
Multiplication of Symmetric Group :
1 2 3)
Images of the first Permutation element
1 --> 2, 2 --> 3, 3 --> 1, 4 --> 4, 5 --> 5, 6 --> 6
(5 6 4 1)
Images of the Second Permutation element
1 --> 5, 2 --> 2, 3 --> 3, 4 --> 1, 5 --> 6, 6 --> 4
For Multiplying the Element of the Group
we have to process from backward to forward
1 --> 5, 5 --> 5 so, The Image of 1 under Permutation is 5
2 --> 2, 2 --> 3 so, The image of 2 under Permutation is 3
3 --> 3, 3 --> 1 so, The image of 3 under Permutation is 1
4 --> 1, 1 --> 2 so, The image of 4 under Permutation is 2
5 --> 6, 6 --> 6 so, The image of 5 under Permutation is 6
6 --> 4, 4 --> 4 so, The image of 6 under Permutation is 4
So, The multiplication of (1 2 3)(5 6 4 1) = (1 5 6 4 2 3)

Concept:

Transpose of a matrix: It is defined as "Matrix 'A' which is formed by replacing all the rows of a given matrix into columns and vice-versa". It is denoted by A^T or A'

Properties of transpose:

1) (A + B)^T = A^T + B^T

2) (A^T)^T = A

3) (ABC)^T = CTB^TA^T

Symmetric matrix: If A^T = A then, A is called symmetric matrix.

Skew symmetric matrix: If A^T = - A then, A is called skew-symmetric matrix.

Null matrix: matrix which has all its elements equal to 0 is called null matrix.

Calculation:

Let

(ABT - BA^T) = P -----(1)

⇒ P^T = (ABT - BAT)^T

⇒ P^T = (AB^T)^T - (BA^T)^T (Using property-1)

⇒ P^T = (B^T)^T AT- (A^T)^T BT (Using property-3)

⇒ PT = BAT - A BT = - ( ABT - BAT) (Using property-2)

⇒ (ABT - BAT)T = - ( ABT - BAT)

Hence, the given matrix is a skew-symmetric matrix.

Concept:
Steps for obtaining adjoint of the matrix
We need to calculate cofactors of all elements
⇒ Cij or Aij=(-1)(i+j) Mij
Here, Mij = Minor of matrix A
Transpose of the cofactor is called adjoint of the matrix
Adj (A) = CT
Calculation:
Given that,

The cofactor α = A₁₁ = δ
The cofactor β = A₁₂ = -γ
The cofactor γ = A₂₁ = -β
The cofactor δ = A₂₂ = -α

⇒ adj Shortcut Trick

Statement (A):
If P^-1 XP  is a diagonal matrix for some invertible matrix P, then X has a basis of eigenvectors 
P^-1 XP being diagonal means X can be written in diagonal form
A matrix is diagonalizable if and only if there exists a basis of eigenvectors
Since P is invertible, the columns of P are eigenvectors of X
So (A) is correct
Statement (B):
If X is a diagonal matrix with different diagonal values and XY = YX , then Y must also be diagonal
Let's say X is:

where (diagonal values are different)
If XY = YX , it means Y must be forced to have zeroes outside the diagonal, otherwise multiplication would not work properly
So Y must also be diagonal
So (B) is correct
Statement (C):
If X² is diagonal, then X must be diagonal
Let's take an example:

Here, X² = 0, which is diagonal
But X itself is not diagonal
So (C) is incorrect
Statement (D):
If X is diagonal and it commutes with every matrix Y , then X must be a multiple of the identity matrix
If X is diagonal but not a multiple of I , there will be some matrices Y for which 
Only scalar multiples of identity X = λI commute with all matrices
So (D) is correct
Correct statements are (A), (B), and (D)
Hence Option (1) is the correct answer

Concept:

A potential function φ(x, y, z) for a vector field F(x, y, z) is a scalar function such that:

F(x, y, z) = ∇φ(x, y, z)

where ∇ is the gradient operator:

∇ = ∂/∂x î + ∂/∂y ĵ + ∂/∂z k̂

Explanation:

To find the vector field F, we need to calculate the gradient of the potential function φ(x, y, z):

∇φ(x, y, z) = (∂φ/∂x) î + (∂φ/∂y) ĵ + (∂φ/∂z) k̂

∂φ/∂x = y + z

∂φ/∂y = x + z

∂φ/∂z = x + y

Therefore, the vector field F is:

F(x, y, z) = (y + z) î + (x + z) ĵ + (x + y) k̂

⇒ A is correct

⇒ Option(4) is the correct answer.

Solution:
A^' = where, A' is the transpose of the matrix
A =
A + A' = =
By comparing 2cosα = 1
⇒ cosα = 1/2
⇒ α = cos^-1(1/2)
⇒ α = π/3

Concept:
First Order Linear Differential Equation:
A differential equation of the from dy/dx + P × y = Q, where P and Q are constants or functions of x only, is known as a first-order linear differential equation.
Steps to solve a First Order Linear Differential Equation:

• Convert into the standard form dy/dx + P × y = Q, where P and Q are constants or functions of x only.
• Find the Integrating Factor (F) by using the formula: F = e∫ P dx..
• Write the solution using the formula: y × F = ∫ (Q × F) dx + C where C is the constant of integration.

Calculation:
2x (dy/dx) - y = 3
⇒ dy/dx + (-1/2x) y = 3/2x
⇒ P = -1/2x and Q = 3/2x.
Integrating factor F:
e∫ P dx = e∫ (-1/2x) dx = e^{(ln (1/√x))} = 1/√x
The solution of the given differential equation is:
y × (1/√x) = ∫ ( (3/2x) × (1/√x) ) dx + C
y × (1/√x) = ∫ ( 3/2 (x^(3/2)) ) dx + C
y × (1/√x) = (3/2) ∫ (x^(-3/2)) dx + C
y × (1/√x) = (3/2) ∫ (x^(-3/2)) dx + C
y × (1/√x) = (3/2) (x^(-3/2 + 1) / (-3/2 + 1)) + C
y × (1/√x) = (-3) x^(-1/2) + C
⇒ y / √x = (-3 / √x) + C
⇒ This is an equation of a parabola.

Given equations are y = x² and y = 1/x
Intersection point of these two curves will be (1, 1).

Let consider a vertical strip as shown in the figure, then the limits will be
X varies from 0.5 to 1 and y varies from 1/x to x²
Now the area will be,



= ln 2 - 7/24

Concept:
If  and  are two non-zero vectors then scalar product will be-
a⃗ . b⃗ = |a⃗ | |b⃗ | cos θ ... (1)
Where,
θ is angle between a⃗ & b⃗
and, |a⃗ | = √(a₁² + b₁² + c₁²) and |b⃗ | = √(a₂² + b₂² + c₂²) …2)
Calculation:
Given,
a⃗ = 2î + ĵ + 3k̂
b⃗ = 3î - 2ĵ + k̂
Then,
a⃗ ⋅ b⃗ = |a⃗| |b⃗| cos θ
⇒ cos θ = a⃗ ⋅ b⃗ / |a⃗| |b⃗|
⇒ cos θ = (2î + ĵ + 3k̂) ⋅ (3î - 2ĵ + k̂) / √(2² + 1² + 3²) √(3² + (-2)² + 1²)
⇒ cos θ = (6 - 2 + 3) / √14 * √14
⇒ cos θ = 7/14 = 1/2
⇒ cos θ = cos 60° ⇒ θ = 60°

Concept:
Let f(x) = p(x) / q(x)
There are three conditions that need to be met by a function f(x) in order to be continuous at a number a. These are:

• f(a) is defined [you can’t have a hole in the function]
• lim x→a f(x) exists
• lim x→a f(x) = f(a)

Note:
if any of the three conditions of continuity is violated, the function is said to be discontinuous.
If sin x = 0 then x = nπ, n ∈ Z
Calculation:
Given, f(x) = 1 / (x - [x])where [x] is greatest integer function
∴ f(x) = 1 / (x - [x]) ( (x) is fractional part of x. )
∴ Domain of (x) ∈ R except integers ( ∵ at integer value of x, (x) = 0 )
∴ f(x) is discontinuous at all integers and we know that there infinitely many integers
Thus f(x) is discontinuous at infinite points.
i.e., The greatest integer function becomes discontinuous at every integer value of x. So, the whole function will become discontinuous at each
integer value of x.
It has infinite number of points of discontinuity.
Hence, the correct answer is option 4)
Additional Information
1. Greatest Integer Function: Greatest Integer Function [x] indicates an integral part of the real number x which is nearest and smaller integer to x. It is also known as floor of x

• In general, If n ≤ x ≤ n + 1 Then [x] = n (n ∈ Integer)
• This means if x lies in [n, n + 1) then the Greatest Integer Function of x will be n.

Concept -

(i) n³ + 1 = (n+1)(n² + 1 - n)

(ii) and

(iii)

Explanation -

We have the series

=

=

=

=

=

= e + (e - 1) = 2e - 1

Hence the given series is convergent and cgs to finite limit 2e - 1.

Hence option(iii) is correct.