(a) Find eigenvectors for the matrix 

Solve the following system of equations using Cramer's rule. 

x + y - z = 6

3x - 2y + z = -5

x + 3y - 2z = 14

Verify that the following matrix A is orthogonal 

 (a) If A =  then show that 

(i) 1/2 (A + A') is symmetric, and

(ii) 1/2 (A - A') is skew symmetric.

Solve the following differential equation: 

(1/y²) (dy/dt) = 1 - e⁻³ᵗ

(c) Show that the equation 

(y - 2x³)dx = x(1 - xy)dy

becomes exact on multiplication by x⁻² and solve it.

Using the ideal gas equation, estimate the change in the pressure of 1.0 mol of an ideal gas at 0°C when its volume is changed from 22.414 L to 21.414 L. 

 (a) Find the differential equation whose solution is given by 

y = eˣ(A cos x + B sin x)

where, A and B are arbitrary constants.

∫₁³ x²eˣ³ dx

 ∫((1+ln x)³)/x dx

 ∫(2eˣ - 3√x)dx

 (d²y/dx²)¹ᐟ⁵ = k[1+(dy/dx)²]⁵ᐟ²

 (dy/dx)³ = √(1+(dy/dx)²)

√(d³y/dx³) = dy/dx + x⁴

(dy/dx)² = (7x)/(4y²)

Find the asymptotes of the following functions: 

(i) (3x-4)/(2x+6)

(ii) (x²-2x-8)/(x-1)

Find the equation of tangent and normal to the curve f(x) = x³ - 3x² + 6x - 1 at x = 2. 

(a) Find all the second order partial derivatives of the following function. 

f(x, y) = x² - 8xy + y²

Find the derivative of the following functions with respect to x: (2+2)

(i) (x⁻¹ᐟ² - x¹ᐟ²)/(x⁻¹ᐟ² + x¹ᐟ²)

(ii) 3x/√(5+2x²)

(a) Evaluate the following limit. 

(b) If the law of motion of a particle is given as: s = -t³ + 3t² + 25, then (2+1)

i) find its velocity and acceleration.

ii) find the distance covered by the particle in time t = 5 units

Prove, with the help of vectors, that the diagonals of a parallelogram bisect each other.