Select the correct alternatives : (More than one are correct)
Q.71 Familyofcurveswhosetangentatapointwithitsintersectionwiththecurvexy=c2 formanangleof

4
is:
(A) y2  2xy  x2 = k (B) y2 + 2xy  x2 = k
(C) y = x - 2 c tan1
x
c





 + k (D) y = c ln
c x
c x


 x + k
where k is an arbitraryconstant .
Q.72 The general solutionof the differential equation, x
dy
dx





 = y. ln
y
x





 is :
(A) y = xe1  cx (B) y = xe1 + cx(C) y = ex . ecx (D) y = xecx
where c is an arbitraryconstant.
Q.73 Whichofthefollowingequation(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.74 The function f(x)satisfyingthe 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 arbitraryconstant .
Q.75 Theequation ofthecurvepassingthrough (3 ,4)&satisfyingthedifferential 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.76 Theareaboundedbyacurve,theaxis ofco-ordinates&the ordinateofsomepoint ofthecurve is equal
tothelengthofthecorrespondingarcofthecurve. Ifthecurvepasses throughthepoint 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 = x
x
e
e
2


Q.77 Identifythestatement(s)whichis/areTrue.
(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 differential equation
(C) f(x , y) = x2 + sin x . cosy is not homogeneous
(D) (x2 + y2) dx - (xy2  y3) dy= 0 is a homegeneous differential equation .
Some Additional Problems on DE & AUC
NUCLEUS

Q.78 Thegraphofthefunction y=f(x)passingthroughthepoint(0,1)andsatisfyingthedifferentialequation
dx
dy
+ y cos x = cos x is such that
(A) itis aconstant function (B) it is periodic
(C) it is neither an even nor an odd function (D) itis continuous &differentiableforall x .
Q.79 A function y= f (x) satisfyingthe differential equation
dx
dy
·sin x – y cos x + 2
2
x
x
sin
= 0 is such that,
y  0 as x  then the statement which is correct is
(A)
0
x
Lim

f(x) = 1 (B)
0
2
/
 f(x) dx is less than

2
(C)
0
2
/
 f(x) dx is greater than unity (D) f(x)is an odd function
Select the correct alternative : (Only one alternative is correct)
Direction for Q.80 to Q.82 (3 question together)
Considerthefunction
f (x) = x3 – 8x2 + 20x – 13
Q.80 Numberof positiveintegers x forwhichf (x)is aprimenumber, is
(A) 1 (B) 2 (C) 3 (D) 4
Q.81 Thefunction f(x)definedfor R R
(A) is one one onto
(B) is manyone onto
(C) has 3 real roots
(D) is such that f (x1) · f(x2) < 0 where x1 and x2 are the roots of f ' (x) = 0
Q.82 Area enclosed byy= f (x) and the co-ordinate axes is
(A)
12
65
(B)
12
13
(C)
12
71
(D) none
Answers
Select the correct alternatives : (More than one are correct)
Q.71 ABCD Q.72 ABC Q.73 ACD Q.74 CD Q.75 AB Q.76 BC
Q.77 ABC Q.78 ABD Q.79 ABC
Select the correct alternative : (Only one alternative is correct)
Q.80 C Q.81 B Q.82 A