Mathematics

UNIT-1 [Matrices & Determinants]

1. Define Rank of a matrix.
 Answer: The rank of a matrix is the maximum
number of linearly independent row or column
vectors in the matrix. It indicates the dimension of
the vector space generated by its rows or columns.
2. State Cayley Hamilton theorem.
 Answer: The Cayley-Hamilton theorem states that
every square matrix satisfies its own characteristic
equation. If ( A ) is a square matrix and ( p(\lambda)
) is its characteristic polynomial, then ( p(A) = 0 ).
3. What is Hermitian and Skew-Hermitian matrix?
 Answer: A Hermitian matrix is a square matrix that
is equal to its own conjugate transpose, i.e., ( A =
A^* ). A skew-Hermitian matrix is a square matrix
that is equal to the negative of its conjugate
transpose, i.e., ( A = -A^* ).
4. Define symmetric and skew symmetric matrix.
 Answer: A symmetric matrix is a square matrix that
is equal to its transpose, i.e., ( A = A^T ). A skew-
symmetric matrix is a square matrix that is equal to
the negative of its transpose, i.e., ( A = -A^T ).
5. Evaluate the following determinants:
 (a) ( \begin{vmatrix} 3 & 1 \ 5 & 6 \end{vmatrix} )
  Answer: ( (3)(6) - (1)(5) = 18 - 5 = 13 )
 (b) ( \begin{vmatrix} A & A \ A & A \end{vmatrix} )
 Answer: The determinant of a matrix with two
identical rows is 0.
6. If ( A= \begin{pmatrix} 4 & 5 \ 6 & 7 \end{pmatrix} ),
find ( 4A + 2I ).
 Answer:
 ( 4A = \begin{pmatrix} 16 & 20 \ 24 & 28 \
end{pmatrix} )
 ( 2I = \begin{pmatrix} 2 & 0 \ 0 & 2 \
end{pmatrix} )
 ( 4A + 2I = \begin{pmatrix} 16 + 2 & 20 + 0 \ 24
+ 0 & 28 + 2 \end{pmatrix} = \begin{pmatrix}
18 & 20 \ 24 & 30 \end{pmatrix} )
7. Find the minor and Cofactor of the given matrix: ( A= \
begin{pmatrix} 4 & -7 \ -3 & 2 \end{pmatrix} ).
 Answer:
 Minor of element ( a_{11} = 4 ) is ( 2 )
(determinant of ( \begin{pmatrix} 2 \
end{pmatrix} )).
 Cofactor of ( a_{11} = 2 ).
 Minor of element ( a_{12} = -7 ) is ( -3 )
(determinant of ( \begin{pmatrix} -3 \
end{pmatrix} )).
 Cofactor of ( a_{12} = 3 ).
  Minor of element ( a_{21} = -3 ) is ( -7 ).
 Cofactor of ( a_{21} = 7 ).
 Minor of element ( a_{22} = 2 ) is ( 4 ).
 Cofactor of ( a_{22} = 4 ).
8. Show that in the matrices ( (AB)^{-1} = B^{-1} A^{-1} ).
 Answer: This can be shown using the property of
inverses:
 ( (AB)(B^{-1}A^{-1}) = A(BA^{-1}) = AIA = A )
and similarly for the other side, confirming the
equality.
9. Show that ( A = \begin{pmatrix} 1 & 1 \ -A & 2 \ 1 + A &
3 & A \ -A & 0 \end{pmatrix} ) is Hermitian.
 Answer: To show that ( A ) is Hermitian, we need to
verify that ( A = A^* ) (the conjugate transpose).
This involves checking that each element satisfies
the Hermitian property.
10. Determine the eigenvalues of matrix ( A= \
begin{pmatrix} A & h & A \ 0 & A & 0 \ 0 & A & A \
end{pmatrix} ).
 Answer: The eigenvalues can be found by solving
the characteristic polynomial ( \det(A - \lambda I) =
0 ).
This concludes the answers for Section A questions. If you
need further details or explanations, feel free to ask! UNIT-2
[Limits and Continuity]
1. Evaluate:
 Answer: The specific expression to evaluate is not
provided, please specify the limit to evaluate.
2. Evaluate:
 Answer: Similar to the previous question, please
provide the specific limit expression.
3. Define the term Continuity.
 Answer: A function is said to be continuous at a
point if the limit of the function as it approaches
that point is equal to the function's value at that
point.
4. Write the properties of a continuous function.
 Answer:
 A continuous function has no breaks, jumps, or
holes.
 The function is continuous on an interval if it is
continuous at every point in that interval.
 The composition of continuous functions is
also continuous.
5. Evaluate:
 Answer: Please provide the specific limit expression
for evaluation.
6. Define the term limit.
  Answer: The limit of a function at a certain point is
the value that the function approaches as the input
approaches that point.
7. Evaluate:
 Answer: Again, please specify the limit expression
to evaluate.
8. Explain continuity in an open interval.
 Answer: A function is continuous in an open
interval if it is continuous at every point within that
interval, meaning for every point in the interval, the
limit of the function as it approaches that point
equals the function's value at that point.
9. Evaluate:
 Answer: Please provide the specific limit expression
for evaluation.
10. If the given limit has value ( k/2 ). Find the value of
( k ).
 Answer: The specific limit expression is needed to
determine the value of ( k ). Please provide the limit
for evaluation. UNIT-3 [Differentiation]
11. Find ( \frac{dy}{dx} ); if ( \sin(x+y) = \log(x+y) ).
 Answer: To find ( \frac{dy}{dx} ), we can use implicit
differentiation:
 Differentiate both sides: ( \cos(x+y)(1 + \
frac{dy}{dx}) = \frac{1}{x+y}(1 + \frac{dy}{dx}) ).
  Rearranging gives ( \frac{dy}{dx} = \frac{\
cos(x+y) - \frac{1}{x+y}}{1 - \frac{1}{x+y}} ).
12. Find the ( n )th differential coefficient of ( x^3 \cos
x ).
 Answer: The ( n )th differential coefficient can be
found using the product rule and differentiating
multiple times, leading to a general formula based
on the Leibniz rule.
13. If ( y = \log[\log(\log x)] ), find ( \frac{dy}{dx} ).
 Answer: Using the chain rule:
 ( \frac{dy}{dx} = \frac{1}{\log(\log x)} \cdot \
frac{1}{\log x} \cdot \frac{1}{x} ).
14. Find the derivative of the function ( (x^n + a)(x^m
+ b) ).
 Answer: Using the product rule:
 ( \frac{d}{dx}[(x^n + a)(x^m + b)] = (x^n + a) \
cdot \frac{d}{dx}(x^m + b) + (x^m + b) \cdot \
frac{d}{dx}(x^n + a) ).
15. Differentiate ( \sin(2x) \sin(4x) ) with respect to ( x
).
 Answer: Using the product rule:
 ( \frac{d}{dx}[\sin(2x) \sin(4x)] = \cos(2x) \cdot
2 \sin(4x) + \sin(2x) \cdot \cos(4x) \cdot 4 ).
16. Find ( \frac{dy}{dx} ) when ( y = e^x \log(\sin^2
x) ).
  Answer: Using the product rule:
 ( \frac{dy}{dx} = e^x \log(\sin^2 x) + e^x \
cdot \frac{2 \cos x}{\sin x} ).
17. Find ( \frac{dy}{dx} ) when ( y = x^2 + y^2 =
4ax^2 ).
 Answer: Implicit differentiation gives:
 ( 2y \frac{dy}{dx} = 8ax ) leading to ( \frac{dy}
{dx} = \frac{8ax}{2y} = \frac{4ax}{y} ).
18. Find the derivative of the function: ( x^2 + y^2 =
4ax^2 ).
 Answer: Implicit differentiation yields:
 ( 2x + 2y \frac{dy}{dx} = 8ax ) leading to ( \
frac{dy}{dx} = \frac{8ax - 2x}{2y} = \frac{4a - 1}
{y} ).
19. If ( y = \log(\tan(A^4 + A^2)) ), find ( \frac{dy}
{dx} ).
 Answer: Using the chain rule:
 ( \frac{dy}{dx} = \frac{1}{\tan(A^4 + A^2)} \
cdot \sec^2(A^4 + A^2) \cdot \frac{d}{dx}(A^4
+ A^2) ).
20. Differentiate the following with respect to ( x ):
( 3x + 2 ).
 Answer: The derivative is simply ( 3 ).
UNIT-4 [Application of Differentiation]
1. State Maclaurin’s theorem with its expression.
 Answer: Maclaurin's theorem states that a function
( f(x) ) can be expressed as:
 ( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \
frac{f'''(0)}{3!}x^3 + \ldots ) for all ( x ) in the
neighborhood of 0.
2. State Taylor’s theorem with its expression.
 Answer: Taylor's theorem states that a function
( f(x) ) can be expressed as:
 ( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 +
\frac{f'''(a)}{3!}(x-a)^3 + \ldots ) for all ( x ) in
the neighborhood of ( a ).
3. Verify Rolle’s theorem for ( f(x) = x^4 - 1 ) in ( [-1, 1] ).
 Answer:
 Check that ( f(-1) = f(1) = 0 ).
 Since ( f(x) ) is continuous on ( [-1, 1] ) and
differentiable on ( (-1, 1) ), by Rolle's theorem,
there exists at least one ( c \in (-1, 1) ) such
that ( f'(c) = 0 ).
4. Verify Lagrange’s mean-value theorem for the function
( f(x) = x(2 - A) ) in the interval ( [0, 1] ).
 Answer:
 Calculate ( f(0) ) and ( f(1) ).
 The average rate of change is ( \frac{f(1) - f(0)}
{1 - 0} ).
  Find ( c ) such that ( f'(c) = ) average rate of
change.
5. Verify Rolle’s theorem for the function ( f(x) = x^3 -
6x^2 + 11x - 6 ) in the interval ( [1, 3] ).
 Answer:
 Check that ( f(1) = f(3) = 0 ).
 Since ( f(x) ) is continuous and differentiable,
there exists at least one ( c \in (1, 3) ) such that
( f'(c) = 0 ).
6. Find the maximum and minimum values of ( 3x^4 -
8x^3 + 12x^2 - 48x + 25 ) on ( [0, 3] ).
 Answer:
 Find the critical points by setting ( f'(x) = 0 ).
 Evaluate ( f(x) ) at the endpoints and critical
points to determine maximum and minimum
values.
7. State Leibnitz theorem with its expression.
 Answer: Leibnitz theorem states that if ( f(x) ) is a
product of two functions, then:
 ( \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) ).
8. Verify Rolle’s theorem for the function ( f(x) = \sin(2x) )
in the interval ( [0, \frac{\pi}{2}] ).
 Answer:
 Check that ( f(0) = f(\frac{\pi}{2}) = 0 ).
  Since ( f(x) ) is continuous and differentiable,
there exists at least one ( c \in (0, \frac{\pi}
{2}) ) such that ( f'(c) = 0 ).
9. Find the value of ( c ) by the mean value theorem when
( f(x) = x^3 - 3x - 2 ) on ( [-2, 3] ).
 Answer:
 Calculate ( f(-2) ) and ( f(3) ).
 Find the average rate of change and solve
( f'(c) = ) average rate of change for ( c ).
10. Evaluate:
 Answer: Please provide the specific expression to
evaluate.
UNIT-5 [Integration]
1. What do you understand by definite integral?
 Answer: A definite integral represents the signed
area under the curve of a function over a specified
interval ([a, b]) and is denoted as ( \int_a^b f(x) ,
dx ).
2. Integrate: ( \log x ).
 Answer: The integral is given by:
 ( \int \log x , dx = x \log x - x + C ).
3. Integrate: ( ax^5 + bx^3 + cx + d ).
 Answer: The integral is:
 ( \int (ax^5 + bx^3 + cx + d) , dx = \frac{a}{6}x^6 + \
frac{b}{4}x^4 + \frac{c}{2}x^2 + dx + C ).
4. Evaluate: ( \int \sin^3 x \cos x , dx ).
 Answer: Using substitution, let ( u = \sin x ), then
( du = \cos x , dx ):
 ( \int \sin^3 x \cos x , dx = \int u^3 , du = \
frac{u^4}{4} + C = \frac{\sin^4 x}{4} + C ).
5. Evaluate: ( \int x \cos x , dx ).
 Answer: Using integration by parts:
 Let ( u = x ) and ( dv = \cos x , dx ), then ( du =
dx ) and ( v = \sin x ):
 ( \int x \cos x , dx = x \sin x - \int \sin x , dx = x \
sin x + \cos x + C ).
6. Evaluate:
 Answer: Please provide the specific integral
expression for evaluation.
7. Prove that ( \int_0^{\frac{A}{2}} \log \tan A , dA = 0 ).
 Answer: This can be shown using properties of
definite integrals and symmetry.
8. Prove that ( -\frac{A}{2} \int_{\frac{A}{2}}^{A} A^3 A
A^2 A , dA = 0 ).
 Answer: This requires evaluating the integral and
showing that it simplifies to zero.
9. What do you understand by indefinite integral?
  Answer: An indefinite integral represents a family
of functions whose derivative is the integrand,
denoted as ( \int f(x) , dx = F(x) + C ), where ( F'(x) =
f(x) ).
10. Evaluate: ( \int \sec x \log(\sec x + \tan x) , dx ).
 Answer: This integral can be solved using
integration techniques, often involving substitution
or integration by parts.