B.A. / B.Sc. 1st Semester Examination
BS-IS/12
229944 (S)
Mathematics
Course No. : 101
f(x, y) = { xy sin(1/x), if x ≠ 0
0, if x = 0
at origin.
b) Determine a and b so that the curve y = ax³ + 3bx² has a point of inflexion at (-1,2).
2. a) If Z = log(x² + xy + y²), then prove that:
x(∂Z/∂x) + y(∂Z/∂y) = 2.
b) Find the position and nature of double point(s) on the curve:
x³ - y² - 7x² + 4y + 15x - 13 = 0.
b) Evaluate lim(x→0) (sin x)^(tan/x).
4. a) Find the envelope of the family of lines (x/a) + (y/b) = 1 where a and b are connected by the relation ab = c².
b) Show that x + y + a = 0 is the only asymptote of the curve:
x³ + y³ - 3axy = 0.
6. a) If F→ = (2x² + y²) î + (3y - 4x) ĵ, evaluate Δ∫F→.dȓ around the triangle ΔABC whose vertices are A(0,0), B(2,0), and C(2,1).
b) Trace the curve r = a(1 + cos θ).
8. a) Trace the curve r = a sin 2θ.
b) i) Transform xy³ + x³y = a² to polar form.
b) Obtain a reduction formula for ∫ secⁿ x dx, n being a positive integer. Hence evaluate ∫sec⁴ x dx.
10. a) Evaluate 0∫a (x⁴/√(a² - x²)) dx.
b) A segment is cut off from a sphere of radius a by a plane at a distance (1/2)a from the center. Show that the volume of the segment is (5/32) of the volume of the sphere.
Time Allowed: 3 hours Maximum Marks: 80
Note: Attempt five questions in all, selecting one question from each unit. All questions carry equal marks.
Unit
- I
1. a) Discuss the continuity of the
function f(x, y) defined by:f(x, y) = { xy sin(1/x), if x ≠ 0
0, if x = 0
at origin.
b) Determine a and b so that the curve y = ax³ + 3bx² has a point of inflexion at (-1,2).
2. a) If Z = log(x² + xy + y²), then prove that:
x(∂Z/∂x) + y(∂Z/∂y) = 2.
b) Find the position and nature of double point(s) on the curve:
x³ - y² - 7x² + 4y + 15x - 13 = 0.
Unit
- II
3. a) Trace the curve y = x²(x -
3a), a > 0.b) Evaluate lim(x→0) (sin x)^(tan/x).
4. a) Find the envelope of the family of lines (x/a) + (y/b) = 1 where a and b are connected by the relation ab = c².
b) Show that x + y + a = 0 is the only asymptote of the curve:
x³ + y³ - 3axy = 0.
Unit
- III
5. a) Find the value of ȓ satisfying
the equation d²ȓ/dt² = ȃ,
where ȃ is a constant vector.
Also, it is given that ȓ = ô and dȓ/dt = ȗ, at t = 0.
b) Find the directional derivative of
the function 2xy + z² at (1, -1, 3) in the direction of the vector î + 2ĵ + 2ќ.
6. a) If F→ = (2x² + y²) î + (3y - 4x) ĵ, evaluate Δ∫F→.dȓ around the triangle ΔABC whose vertices are A(0,0), B(2,0), and C(2,1).
b) Prove that div ȓ = 2/r.
Unit
- IV
7. a) With usual notations, prove
that tan φ = (r dθ/dr).b) Trace the curve r = a(1 + cos θ).
8. a) Trace the curve r = a sin 2θ.
b) i) Transform xy³ + x³y = a² to polar form.
ii) Transform r² = sin 2θ to
Cartesian form.
Unit
- V
9. a) Find the curved surface of the
solid generated by the revolution about the x-axis of the area bounded by the
parabola y² = 4ax, the ordinate x = 3a, and the x-axis.b) Obtain a reduction formula for ∫ secⁿ x dx, n being a positive integer. Hence evaluate ∫sec⁴ x dx.
10. a) Evaluate 0∫a (x⁴/√(a² - x²)) dx.
b) A segment is cut off from a sphere of radius a by a plane at a distance (1/2)a from the center. Show that the volume of the segment is (5/32) of the volume of the sphere.
