Q.2. (a) Find the volume ∬R xy dA where R is the region bounded by the line y = x − 1 and the parabola
y² = 6x + 2. (10)
(b) Evaluate the following line integral:
∫C (y² dx + x dy)
where C = C₂, is the line segment joining the points (−5, −3) to (0, 2), and C₂ is the arc of the parabola
x = 4 − y². (10)
Q.3. (a) Three forces P, Q and R act at a point parallel to the sides of a triangle ABC taken in the same order. Show that the magnitude of the resultant is
(b) A hemispherical shell rests on a rough inclined plane whose angle of friction is λ. Show that the inclination of the plane base to the horizontal cannot be greater than
sin⁻¹(2 sin λ). (10)
Q.4. (a) A uniform square lamina of side 2a rests in a vertical plane with two of its sides in contact with two smooth pegs distant b apart and in the same horizontal line. Show that if
θ√2 < b < a
a non-symmetric position of equilibrium is possible in which
b(sinθ + cosθ) = a. (10)
(b) Find the centre of mass of a semicircular lamina of radius a whose density varies as the square of the distance from the centre. (10)
Q.5. (a) Evaluate the integral
∫₀¹ ∫x² (x² + y²) dy dx
also show that the order of integration is immaterial. (10)
(b) Find the directional derivative of the function
f(x, y, z) = 4xz³ − x²y²,
at the point
P = (2, 1, −2),
along the z-axis. (10)
SECTION-B
Q.6. (a) A particle is moving along the parabola
x² = 4ay
with constant speed v. Determine the tangential and the normal components of its acceleration when it reaches the point whose abscissa is √5a. (10)
(b) Find the distance travelled and the velocity attained by a particle moving in a straight line at any time t, if it starts from rest at t = 0 and is subject to an acceleration
a = t² + sin t + e²ᵗ. (10)
Q.7. (a) A particle moves in the xy-plane under the influence of a force field which is parallel to the y-axis and varies as the distance from the x-axis. Show that, if the force is repulsive and the path of the particle is not straight, then
y = a cosh(nx) + b sinh(nx),
where a and b are constants. (10)
(b) Discuss the motion of a particle moving in a straight line with an acceleration
x³, where x is the distance of the particle from a fixed point O on the line, if it starts at t = 0 from a point x = c with the velocity c²/√2. (10)
Q.8. (a) A battleship is steaming ahead with speed V and a gun is mounted on the battleship so as to point straight backwards and is set at angle of elevation α. If v₀ is the speed of projection (relative to the gun), show that the range is
(2v₀ sinα (v₀ cosα − V)) / g. (10)
(b) Show that the law of force towards the pole of a particle describing the curve
rⁿ = aⁿ cos(nθ)
is given by
F = [(n +1) h² a²ⁿ] / r²ⁿ⁺³, where h is a constant. (10)
APPLIED MATHEMATICS, PAPER-II
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-A and THREE Questions from SECTION-B, All questions carry EQUAL marks.
SECTION-A
Q.1. (a) Solve by method of variation of parameter
d²y/dx² − dy/dx + y = x eⁿˣ (10)
(b) Solve first order non-linear differential equation
