Choose the correct answer:
(i) The surface area of the cone frustum generated by revolving the line segment y = x
2 +
1
2
, 1 ≤ x ≤ 3 about
the x axis is
(a) 4π
√
5
(b) 2π
(c) 3π
√
5
(d) 98π/81
(ii) The curve y =
x
3
9
, 0 ≤ x ≤ 2 is revolved about the x axis. The area of the resulting surfaces is
(a) 4π
√
5
(b) 2π
(c) 3π
√
5
(d) 98π/81
(iii) The volume of a solid generated by revolving a region between the y axis and the curve x = R(y), c ≤
y ≤ d about the x axis, is given by
(a) R d
c
π[R(y)]2dy
(b) R d
c
2π[R(y)]2dy
(c) R d
c
π[R(y)]dy
(d) R d
c
π
2
[R(y)]2dy
(iv) The volume of the solid generated by revolving the region bounded by x =
√y, x = −y and y = 2 about
the x axis is
(a) 16π
15 (3√
2 + 5)
(b) 40π
3
(c) 8π
3
(d) π
6
(v) limx→0
x−sin x
x3 =?
(a) 0/0
(b) 2
(c) 1/2
(d) 1/6
(vi) limx→0+
sin x
x2 =?
(a) 0/0
1
(b) ∞
(c) 2
(d) 1/2
(vii) limx→(π/2)−
sec x
1+tan x =?
(a) ∞/∞
(b) 0/0
(c) 1
(d) 0
(viii) limx→∞
ln x
2
√
x =?
(a) 0
(b) ∞
(c) 0/0
(d) ∞/∞
(ix) A parameterized curve x = f(t), y = g(t) is smooth if
(a) f and g are differentiable
(b) the curve is differentiable at every parameter value
(c) f
0 and g
0 are continuous and not simultaneously zero
(d) none of the above
(x) If a smooth curve x = f(t), y = g(t), a ≤ t ≤ b, is traversed exactly once as t increases from a to b, the
curve’s length is
(a) L = 2πy R b
a
q
(
dx
dt )
2 + ( dy
dt )
2dt
(b) L = 2πx R b
a
q
(
dx
dt )
2 + ( dy
dt )
2dt
(c) L =
R b
a
q
(
dx
dt )
2 + ( dy
dt )
2dt
(d) none of the above
(xi) The tangent line to the curve x = 2 cost, y = 2 sin t at the point where t = π/4 is
(a) y = −x + 2√
2
(b) y =
√
3x −
π
√
3
3 + 2
(c) y = x + 2√
3
(d) y = − √
1
3+2x
(xii) The tangent line to the curve x = 4 sin t, y = 2 cost at the point where t = π/4 is
(a) y = x +
√
2
(b) y = −
1
2
x + 2√
2
(c) y = x +
1
4
(d) y = 2x −
√
3
2
(xiii) Which of the following describes a second order homogeneous linear differential equation?
(a) P(x)
d
2y
dx2 + Q(x)
dy
dx + R(x)y = G(x)
(b) P(x)
d
2y
dx2 + Q(x)
dy
dx + R(x)y = 0
(c) dy
dx + P(x)y = Q(x)
(d) none of the above
(xiv) A differential equation of the form ay00 + by0 + cy = 0 has the characteristic equation
(a) y = e
rx
(b) x
2 + y
2 = 1
(c) x
2
a2 +
y
2
b
2 = 1
(d) ar2 + br + c = 0
(xv) If the auxiliary equation has only one real root r, then the general solution of ay00 + by0 + cy = 0 is
given by
(a) y = c1x + c2
(b) y = c1e
r1x + c2e
r2x
Sample Solution