JNTUK B.Tech R23 M1 Important Questions

LINEAR ALGEBRA & CALCULUS

Unit-wise Important Questions (R23 Regulation)
Common to All Branches

UNIT I

2 Marks Questions

1. Define linear system of equations.

2. What is the normal form of a matrix?

3. State the Cauchy-Binet formulae for matrix multiplication.

5 Marks Questions

1. Find the rank of the matrix using echelon form:

[1 2 3 -2]
[2 -2 1 3]
[3 0 4 1]

2. Solve the system of equations using Gauss elimination method:
10x + y + z = 12, 2x + 10y + z = 13, x + y + 5z = 7

3. Find the inverse using Gauss-Jordan method:

[2 -1 3]
[1 1 1]
[1 -1 1]

4. Solve the system of equations using Gauss Seidel iteration method:
10x + y + z = 12, x + 10y - z = 10, x - 2y + 10z = 9

5. Reduce a matrix to normal form and find its rank.

6. Apply Jacobi iteration method to solve a system of linear equations.

UNIT II

2 Marks Questions

1. Find the sum of the Eigen values of matrix [1 2; 2 4]

2. State the Cayley-Hamilton theorem and its significance.

3. Define quadratic forms and classify their nature.

5 Marks Questions

1. Determine the eigen values of A^(-1) where A =

[-3 -7 -5]
[ 2 4 3]
[ 1 2 2]

2. Verify Cayley-Hamilton theorem for A =

[2 1 1]
[0 1 0]
[1 1 2]

3. Diagonalize the matrix A =

[-3 -1 1]
[ 1 5 -1]
[ 1 -1 3]

and find A^4 using modal matrix 'P'

4. Reduce the quadratic form 2x² + 2y² + 2z² - 2yz - 2zx - 2xy to the canonical form by orthogonal reduction. Hence find nature, rank, index, and signature.

5. Find eigenvalues and eigenvectors of a given matrix and diagonalize it.

6. Determine the nature of a quadratic form and classify it as positive definite, negative definite, or indefinite.

UNIT III

2 Marks Questions

1. State Cauchy's mean value theorem.

2. Write the geometrical interpretation for Lagrange's mean value theorem.

3. State Taylor's and Maclaurin's theorems with remainder terms.

5 Marks Questions

1. Verify Rolle's mean value theorem f(x) = sin²x/2 in [0, π]

2. Write the Taylor's series expansion for f(x) = log(1 + x) about x = 0

3. Prove that π/8 + 1/3√3 < sin⁻¹(3/5) < π/8 + 1/6

4. Show that for any 0 < x < 1, x < sin⁻¹x < x/√(1-x²)

5. Apply Rolle's theorem to find the roots of equations and verify the conditions.

6. Expand functions using Taylor's series about a given point and find approximations.

UNIT IV

2 Marks Questions

1. Find ∂f/∂x, ∂f/∂y for f(x, y) = xy + x² + 2y

2. Define continuity and differentiability for functions of several variables.

3. State Euler's theorem for homogeneous functions.

5 Marks Questions

1. If x = r cos θ, y = r sin θ then prove that ∂²θ/∂x² + ∂²θ/∂y² = 0

2. Determine whether the functions are functionally dependent: u = x√(1-y²) + y√(1-x²), v = sin⁻¹(x) + sin⁻¹(y)

3. Find extreme values f(x, y) = 1 - x² - y²

4. Find the maximum and minimum distance of the point (3,4,12) from the sphere x² + y² + z² = 1 using Lagrange's multiplier method

5. Find directional derivatives and apply the chain rule for functions of several variables.

6. Expand functions of two variables using Taylor's and Maclaurin's series.

UNIT V

2 Marks Questions

1. Evaluate ∫₀¹ ∫₀¹ xy dx dy

2. Write the formula for changing Cartesian coordinates to polar coordinates in double integrals.

3. State the conditions for changing the order of integration in double integrals.

5 Marks Questions

1. By change of Integration Evaluate ∬ xy dx dy over region x²+y² ≤ 1 10M

2. Find the volume of the sphere x² + y² + z² = a² using triple integration. 10M

3. Evaluate by change of order of Integration ∫₀^(2a) ∫₀^(4a²-y²) dx dy 10M

4. Evaluate ∭ z(x² + y²) dx dy dz where R is the Region bounded by the cylinder x² + y² = 1 and the planes z = 2 and z = 3 by changing it to cylindrical coordinates. 10M

5. Find areas using double integrals for regions bounded by given curves. 10M

6. Change variables to spherical coordinates in triple integrals and find volumes. 10M

Key Concepts kuda add chesanu

Unit I

• Matrix rank calculation using echelon and normal forms
• Gauss elimination and Gauss-Jordan methods
• Jacobi and Gauss-Seidel iteration methods
• Consistency of system of equations

Unit II

• Eigenvalue and eigenvector calculations
• Matrix diagonalization techniques
• Cayley-Hamilton theorem verification and applications
• Quadratic forms and orthogonal transformations

Unit III

• Mean value theorem applications and verifications
• Taylor's and Maclaurin's series expansions
• Rolle's theorem geometric interpretation
• Lagrange's and Cauchy's mean value theorems

Unit IV

• Partial derivative calculations and chain rule
• Jacobian computations and functional dependence
• Extrema problems with Lagrange multipliers
• Taylor series expansion of functions of two variables

Unit V

• Change of order of integration techniques
• Coordinate transformations (polar, cylindrical, spherical)
• Area and volume calculations using multiple integrals
• Applications to geometric problems