Lista de Exercicios Matematica-53

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[11]Quest Tutorials
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Quest
Q.79 Which of the following equation(s) is/are linear.
(A) 
dy
dx
 + 
y
x
 = ln x (B) y 
dy
dx





 + 4x = 0 (C) dx + dy = 0 (D) 
d y
dx
2
2 = cos x
Q.80 The function f(x) satisfying the equation, f2(x) + 4 f ′ (x) . f(x) + [f ′ (x)]2 = 0 .
(A) f(x) = c . 
( )
e
2 3- x
(B) f(x) = c . 
( )
e
2+ 3 x
(C) f(x) = c . 
( )
e
3 − 2 x
(D) f(x) = c . 
( )
e
2+ 3− x
where c is an arbitrary constant.
Q.81 The equation of the curve passing through (3 , 4) & satisfying the differential equation,
y
2
dx
dy






 + (x − y)
dx
dy
 – x = 0 can be
(A) x − y + 1 = 0 (B) x2 + y2 = 25 (C) x2 + y2 − 5x − 10 = 0 (D) x + y − 7 = 0
Q.82 The area bounded by a curve, the axis of co-ordinates & the ordinate of some point of the curve is equal
to the length of the corresponding arc of the curve. If the curve passes through the point P (0, 1) then the
equation of this curve can be
(A) y = 
2
1
(ex − e – x + 2) (B) y = 
2
1
(ex + e−x)
(C) y = 1 (D) y = xx ee
2
−+
Q.83 Identify the statement(s) which is/are True.
(A) f(x , y) = ey/x + tan 
y
x
 is homogeneous of degree zero
(B) x . ln 
y
x
 dx + 
y
x
2
 sin−1
y
x
 dy = 0 is homogeneous of degree one
(C) f(x , y) = x2 + sin x . cos y is not homogeneous
(D) (x2 + y2) dx - (xy2 − y3) dy = 0 is a homegeneous differential equation .
Q.84 The graph of the function y = f (x) passing through the point (0 , 1) and satisfying the differential equation
dx
dy
 + y cos x = cos x is such that
(A) it is a constant function (B) it is periodic
(C) it is neither an even nor an odd function (D) it is continuous & differentiable for all x .
Q.85 A function y = f (x) satisfying the differential equation 
dx
dy
·sin x – y cos x + 2
2
x
xsin
= 0 is such that,
y → 0 as x → ∞ then the statement which is correct is
(A) 
0x
Lim
→
f(x) = 1 (B) 
0
2π /
∫ f(x) dx is less than 
π
2
(C) 
0
2π /
∫ f(x) dx is greater than unity (D) f(x) is an odd function