Lista de Exercicios Matematica-51

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Quest
Q.63 Area bounded by the curve y = min {sin2x, cos2x}and x-axis between the ordinates x = 0 and x =
4
5π
 is
(A) 
2
5π
 square units (B) 
4
)2(5 −π
 square units
(C) 
8
)2(5 −π
 square units (D) 





−
π
2
1
8
 square units
Q.64 The equation to the orthogonal trajectories of the system of parabolas y = ax2 is
(A) 
2
2
y
2
x
+ = c (B) 
2
y
x
2
2 + = c (C) 
2
2
y
2
x
− = c (D) 
2
y
x
2
2 − = c
Q.65 If ∫
x
a
dt)t(yt = x2 + y (x) then y as a function of x is
(A) y = 2 – (2 + a2) 2
ax 22
e
−
(B) y = 1 – (2 + a2) 2
ax 22
e
−
(C) y = 2 – (1 + a2) 2
ax 22
e
−
(D) none
Q.66 A curve y = f (x) passing through the point 





e
1
,1 satisfies the differential equation 
dx
dy
 + 2
x
2
ex
−
=0.
Then which of the following does not hold good?
(A) f (x) is differentiable at x = 0.
(B) f (x) is symmetric w.r.t. the origin.
(C) f (x) is increasing for x 0.
(D) f (x) has two inflection points.
Q.67 The substitution y = zα transforms the differential equation (x2y2 – 1)dy + 2xy3dx = 0 into a homogeneous
differential equation for
(A) α = – 1 (B) 0 (C) α = 1 (D) no value of α.
Q.68 A curve passing through (2, 3) and satisfying the differential equation ∫
x
0
dt)t(yt = x2y (x), (x >0) is
(A) x2 + y2 = 13 (B) y2 = 
2
9
x (C) 1
18
y
8
x 22
=+ (D) xy = 6
Q.69 Which one of the following curves represents the solution of the initial value problem
 Dy = 100 – y, where y (0) = 50
(A) (B) (C) (D) 
Q.70 Solution of the differential equation
dx
dy
yee
22 yx 



 + + )xxy(e 2x2
− = 0, is
(A) 
2xe (y2 – 1) + 
2ye = C (B) 
2ye (x2 – 1) + 
2xe = C
(C) 
2ye (y2 – 1) + 
2xe = C (D) 
2xe (y – 1) + 
2ye = C