MASTERSHEET: Differential Equation EXERCISE # 1

Function BY Om Sirwww.upliftmaths.com Connect with Om Sir Just Click or Touch
MASTERSHEET: Differential Equation
equation y = x dx
are-
(A) 1, 2 (B) 2, 1 (C) 1, 1 (D) 2, 2
Q.2 The order and degree of the differential
equation
yd are-
(A) 2, 2 (B) 3, 3 (C) 2, 3 (D) 3, 2
Q.3 The degree of the differential equation 3/2
3
3
dx
yd
Q.4 The order O and degree D of the differential
equation y = 1 + x
(C) D = 0 (D) D is not defined
Q.5 If p and q are order and degree of differential
equation y2
p =
2
Q.6 The differential equation of the family of
curves represented by the equation x2 + y2 = a2
is-
Q.7 The differential equation of the family of
curves y2 = 4a (x + a), where a is an arbitrary
constant, is-
(A) y
Q.8 The differential equation whose solution is
(x – h)2 + (y – k)2 = a2 is (where a is a constant)-
(A)
which are at a unit distance from the origin is
(A)
2
dx
www.upliftmaths.com Connect with Om Sir Just Click or Touch
Q.10 The differential equation of all parabolas that
have origin as vertex and y-axis as axis of
symmetry is-
(C) yy' = 2x (D) y" + y = 2x
Q.11 Differential equation whose general solution is
y = c1x + c2 /x for all values of c1 and c2 is-
(A) 2
dx
dy +
2
2
x1
y1
(D) None of these
dy = – sin x sin y is-
(A) sin y + cos x = c (B) sin y – cos x = c
(C) sin y. cos x = c (D) sin y = c cos x
Q.14 The solution of the equation dx
dy = cos (x – y) is
(A) y + cot

Q.15 The solution of the differential equation
x2
dx
(A) tan–1
x dx
(A) y = xecx (B) y + xecx = 0
(C) y + ex = 0 (D) None of these
Q.17 Solution of differential equation
(3y –7x + 7) dx + (7y –3x + 3) dy = 0 is-
(A) C = (2x + y –1)5 (y –3x + 1)2
(B) C = (x + y – 1)5 (y – x + 1)2
(C) C = (x + 2y –1)5 (2y –x + 1)2
(D) None of these
dy =
3y4x3
2y4x3
x log x dx
(A) y = log x + c
(B) y = log x2 + c
(C) y log x = (log x)2 + c
(D) y = x log x + c
Q.20 The solution of differential equation
dx
(A) ey =ex/2 + ce–x (B) ey =ex + ce–x
(C) 3ey =ex/2 + ce–x (D) None of these
sin y (dy/dx) = cos y (1 – x cos y) is-
(A) cos y = (x+1) – cex (B) sec y = (x + 1) – cex
(C) sec y = (x–1) – cex (D) None of these
Q.22 The value of k such that the family of parabola
y = cx2 + k is the orthogonal trajectory of the
family of ellipses x2 + 2y2 – y = c.
(A) 2
1 (B)
y
x –
)yx(
Q.25 The solution of
(A) exy + ex/y + c = 0 (B) exy –ex/y+ c = 0
(C) exy + ey/x + c = 0 (D) exy – ey/x + c = 0
Q.26 The degree and order of the differential
equation of the family of all parabolas whose
axis is x- axis are respectively-
(A) 2, 1 (B) 1, 2 (C) 3, 2 (D) 2, 3
Q.27 Number of values of m N for which y = emx
is a solution of the differential equation
D3y – 3D2y – 4Dy + 12y = 0 is-
(A) 0 (B) 1
Q.28 The solution to the differential equation
y ny + xy = 0, where y(1) = e, is-
(A) x(ny) = 1 (B) xy (ny) = 1
(C) (ny)2 = 2 (D) ny +

the differential equation x
9 x
(C) 8
Q.30 Family y = Ax + A3 of curve represented by the
differential equation of degree-
(A) three (B) two
Q.31 If dx
dy = 1 + x + y + xy and y(–1) = 0, then
function y is-
(A) 2/)x1( 2

Q.32 Solution of differential equation xdy – ydx = 0
represents-
(C) parabola whose vertex is at origin
(D) circle whose centre is at origin
Q.33 The differential equation y3dy + (x + y2) dx = 0
becomes homogeneous if we put y2 = t.
Q.34 The degree of the differential equation
2
2
2
2
dx
yd
order 3 & degree 1.
dy =
1y2
3ax
Q.37 Solution of differential equation
x2=1 +
is…………….
Q.39 If gradient of a curve at any point P(x, y) is
1x2y2
1yx
curve is……………………
dx
dy3 +
1x
y2
EXERCISE # 2
Questions
general solution is
e where
(A) 5 (B) 4 (C) 3 (D) none
Q.2 The order and the degree of the differential
equation whose general solution is, y = c(x – c)2,
are respectively -
(A) 1, 1 (B) 1, 2 (C) 1, 3 (D) 2, 1
Q.3 The order of the differential equation formed
by differentiating and eliminating the constants
from y = a sin2x + b cos2x + c sin 2x + d cos2x.
Where a, b, c, d are arbitrary constants; is -
(A) 1 (B) 2 (C) 3 (D) 4
Q.4 The degree of differential equation satisfying
the relation
(A) 1 (B) 2 (C) 3 (D) none
Q.5 The solution curves of the given differential
equation xdx – dy = 0 are given by a family of-
(A) parabola (B) hyperbola
(C) circles (D) ellipses
(2, 5) and having the area of, triangle formed
by the x-axis, the ordinate of a point on the
curve and the tangent at the point as 5 sq. units-
(A) xy = 10 (B) x2 = 10y
(C) y2 = 10x (D) xy1/2 = 10
log
dy = 4x – 2y – 2, y = 1 when x = 1 is -
(A) 2e2y+2 = e4x + e2 (B) 2e2y–2 = e4x + e4
(C) 2e2y+2 = e4x + e4 (D) 3e2y+2 = e3x + e4
Q. 8 Solution of the differential equation
(x2 + 1)y1 + 2xy = 4x2 is-
(A) y (1 + x2) = 3
x4 3
(C) y(1 – x2) = 2
dx
dy – y = cos x – sin x with the condition that y
is bounded when x + the longest interval
in which f(x) is increasing in the interval [0, ] is-
(A)
2x2y dx
2
is-
(A) sin x2 y2 = ex–1 (B) sin (x2y2) = x
(C) cos2xy2 + x = 0 (D) sin (x2y2) = ex
dx
dy =
x
y +
Q.12 The equation of curve through point (1, 0) and
whose slope is xx
(C) y = x1
xdy + ydx – 22yx1 dx = 0 is-
(A) sin–1 xy = C – x
(B) xy = sin (x + C)
(C) log (1 – x2y2) = x + C
(D) y = x sin x + C
Q.14 A curve passes through the point

y – cos2
(A) y = x tan–1

(C) y = x
1 tan–1
e n (D) none of these
Q.15 If x intercept of any tangent is 3 times the
x-coordinate of the point of tangent, then the
equation of the curve given that it passes
through (1, 1) is-
1 (D) none of these
Q.16 A particle moves in a straight path such that its
velocity is always 4 times its acceleration. If its
velocity at time t = 0 is 2m/sec, what is its
velocity at t = 2 sec?
(A) e2 m/sec (B) 2 e m/sec
(C) 2
2 m/sec
Q.17 A normal is drawn at a point P(x, y) of a curve.
It meets the x-axis and y-axis in the points A
and B respectively such that OA
1 +
OB
where 'O' is the origin. The equation of such a
curve passing through (5, 4) denotes
(A) a line (B) a circle
(C) a parabola (D) pair of straight line
Q.18 The equation of a curve for which the product
of the abscissa of a point P and the intercept
made by a normal at P on the x-axis equals
twice the square of the radius vector of the
point P is (curve passes through (1, 0))-
(A) x2 + y2 = x4 (B) x2 + y2 = 2x4
(C) x2 + y2 = 4x4 (D) none of these
Q.19 The latus-rectum of the conic passing through
the origin and having the property that normal
at each point (x, y) intersects the x-axis at
((x + 1), 0) is -
differential equation y2(x2 + 1) = 2xy1 passing
through the point (0, 1) and having slope of
tangent at x = 0 as 3 is -
(A) y = x2 + 3x + 2 (B) y2 = x2 + 3x + 1
(C) y = x3 + 3x + 1 (D) none of these
Q.21 If y = |cx|n
x
is the general solution of the differential equation
dx
dy =
x
y +
dy = 0 is
(C) (x/y)5 + x4/4 = C
(D) (xy)4 + x5/5 = C
xdy 2222
dx is
(A) y = x cot (c – x) (B) cos–1 y/x = –x + c
(C) y = x tan (c – x) (D) y2/x2 = x tan (c –x)
Q.24 if A =
dy is a differential equation
of order ‘m’ and degree ‘n’ then (m + n) is equal to
(A) 2 (B) 3 (C) 4 (D) 5
Part-B One or more than one correct
answer type Questions
the initial ordinate of any tangent is equal to the
corresponding subnormal -
(C) has separable variables
(D) is second order
plane must be
2 = 0
(D) y3 2 (1 – y1
2) – 3y1 y1 2 = 0
Q.27 The differential equation
(A) linear (B) homogeneous
y2
c
dv +
2
du
dv
(C) xy = cy + c2 (D) x2y = cy + c2
Q.30 The solution of dx
dy + x = xe(n – 1)y is-
(A) 1n
Ce + 1
(C) log
Ce + 1
dy =
4x2y3
5y6x4
transformed into one which is with separated
variable; by the substitution -
Q.32 The function f(x) satisfying the equation,
f 2(x) + 4f (x). f(x) + [f (x)]2 = 0, is-
(A) f(x) = c. x)32(e
(B) f(x) = c. x)32(e
(C) f(x) = c. x)23(e
(D) f(x) = c. x)32(e
through the point (0, 1) and satisfying the
differential equation dx
such that
(B) it is periodic
(C) it is neither an even nor an odd function
(D) it is continuous and differentiable for all x
Q.34 Water is drained from a vertical cylindrical tank
by opening a valve at the base of the tank. It is
known that the rate at which the water level
drops is proportional to the square root of water
depth y, where the constant of proportionality k
> 0 depends on the acceleration due to gravity
and the geometry of the hole. If t is measured in
minutes and k = 1/15 then the time to drain the
tank if the water is 4 meter deep to start with is-
(A) 30 min (B) 45 min
(C) 60 min (D) 80 min
Q.35 The solution of the differential equation,
x2
dx
x , is-
Q.36 If x
x is
ax 22
ax 22
ax 22
0
Part-C Assertion-Reason type Questions
Statement-2. While answering these questions
you are to choose any one of the following
four responses.
and the Statement-2 is correct explanation of
the Statement-1.
but Statement-2 is not correct explanation of
the Statement-1.
false.
true.
Q.38 Statement-1: The area of the ellipse 2x2 + 3y2 = 6
will be more than the area of the circle
x2 + y2 – 2x + 4y + 4 = 0.
Statement-2: The length of the semi-major axis
of ellipse 2x2 + 3y2 = 6 is more than the radius
of the circle x2 + y2 – 2x + 4y + 4 = 0.
Q.39 Statement-1: The differential equation
y3dy + (x + y2)dx = 0 becomes homogeneous if
we put y2 = t.
order and first degree becomes homogeneous if
we put y = tx.
Q.40 Statement-1 : The orthogonal trajectory to the
curve (x – a)2 + (y – b)2 = r2 is y = mx + b – am
where a and b are fixed numbers and r & m are
parameters.
centre of circle is normal to the circle.
Q.41 Statement-1 : sin x 2
2
dx
Statement-2 : A differential equation is said to
be linear if dependent variable and its
differential coefficients occurs in first degree
and are not multiplied together.
Q.42 Statement-1 : The equation of the curve
passing through (3, 9) which satisfies differential
equation dx
dy = x +
Statement-2 : The solution of differential
equation
2
dx
dy
y = c1ex + c2e–x
Part-D Column Matching type Questions
Q. 43 Column I Column II (A) A curve passing through (P) straight line
(2, 3) having the property
that length of the radius
vector of any of its point P
is equal to the length of the
tangent drawn at this point, can be
(B) A curve passing through (Q) circle
(1, 1) having the property
that any tangent intersects the
y-axis at the point which is
equidistant from the point of
tangency and the origin, can be
(C) A curve passing through (R) parabola
(1, 0) for which the length of
normal is equal to the radius
vector, can be
point (2, 1) and having the
property that the segment of
any of its tangent between the
point of tangency and the
x- axis is bisected by the
y- axis, can be
Q.44 Match the following
Column I Column II
dx
(2x – 10y3) 0y dx
sec2y dy + tan y dx = dx
is
sin y dx
EXERCISE # 3
JEE Main PYQ
1. If for x 0, y = y(x) is the solution of the differential equation, (x + 1)dy = ((x + 1)2 + y – 3)dx, y(2) = 0, then y(3)
is equal to _________.
[IIT JEE MAINS 2020]
2. If y = y(x) is the solution of the differential equation, ey dy
–1 dx
= ex such that y(0) = 0, then y(1) is equal to :
(1) 2 + loge2 (2) 2e (3) loge2 (4) 1 + loge2
3. Let y = y(x) be the solution curve of the differential equation, (y2 – x) dy
dx = 1, satisfying y(0) = 1. This curve
intersects the x-axis at a point whose abscissa is : [IIT JEE MAINS 2020]
(1) 2 (2) 2 – e (3) – e (4) 2 + e
4. Let y = y(x) be solution of the differential equation 21– x
dy
dx +
If y 1
(1) – 3
2 (2)
1
2
5. The differential equation of the family of curves, x2 = 4b(y + b), b R, is : [IIT JEE MAINS 2020]
(1) x(y)2 = x – 2yy (2) x(y)2 = 2yy– x (3) x(y)2 = x + 2yy (4) xy = y
6. If dy
xy
x y ; y(1) = 1; then a value of x satisfying y(x) = e is : [IIT JEE MAINS 2020]
(1) e
2 (2)
2 e (3) 3 e (4) 2 e
7. Let y = y(x) be the solution of the differential equation, 2 sin x
y 1
dy
dx = – cos x, y > 0, y(0) = 1. If y() = a and
dy
dx at x = is b, then the ordered pair (a, b) is equal to : [IIT JEE MAINS 2020]
(A) (2, 1) (B) 3
2, 2
(C) (1, –1) (D) (1, 1)
8. If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation,
2x2dy = (2xy + y2)dx, then f 1
2
(A)
e
1
1 log 2 (D) 1 + loge2
9. If x3dy + xy dx = x2 dy + 2y dx : y(2) = e and x > 1, then y(4) is equal to : [IIT JEE MAINS 2020]
(A) 3
10. The solution curve of the differential equation, (1 + e–x) (1 + y2) dy
dx = y2, which passes through the point (0, 1), is
x1 e
– x1 e
dx –
e
+ 3 = 0 is (where C is a constant of integration)
(A) x – 2 loge(y + 3x) = C (B) x – loge(y + 3x) = C
(C) x – 1
1
2 (logex)2 = C
12. Let y = y(x) be the solution of the differential equation, xy – y = x2(x cos x + sin x). x > 0. If y() = , then
y 2
(A) 2 + 2

13. Let y = y(x) be the solution of the differential equation cos x dy
dx + 2y sin x = sin 2x, x 0,
2
(A) 2 – 2 (B) 1
2 – 1 (C) 2 – 2 (D) 2 + 2
14. If y = y(x) is the solution of the differential equation
x5 e
2 y
dx + ex = 0 satisfying y(0) = 1, then a value of
y(loge 13) is : [IIT JEE MAINS 2020]
(A) 1 (B) –1 (C) 2 (D) 0
15. If y = 2
cosec x is the solution of the differential equation, dy
dx + p(x)y =
function p(x) is equal to [IIT JEE MAINS 2020]
(A) cot x (B) tan x (C) cosec x (D) sec x
16. The general solution of the differential equation 2 2 2 21 x y x y + xy
dy
(where C is constant of integration) [IIT JEE MAINS 2020]
(A) 21 y +
21 x = 1
17. If y = y(x) is the solution of the differential equation, x dy
dx + 2y = x2 satisfying y(1) = 1, then y
1
2
13
16
18. Let f : [0, 1] R be such that f(xy) = f(x).f(y), for all x, y [0, 1], and f(0) 0. If y = y(x) satisfies the
differential equation, dy
1
4
(1) 2 (2) 3 (3) 5 (4) 4
19. If dy
(1) 1
4
3
20. The curve amongst the family of curves represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which
passes through (1, 1) is [IIT JEE MAINS 2019]
(1) a circle with centre on the y-axis (2) an ellipse with major axis along the y-axis
(3) a circle with centre on the x-axis (4) a hyperbola with transverse axis along the x-axis
21. Let f be a differentiable function such that f (x) = 7 – 3
4
x 0 lim
(1) does not exist (2) exists and equals 4
7 (3) exists and equals 4 (4) exists and equals 0
22. If y(x) is the solution of the differential equation dy
dx +
2 e–2, then :
[IIT JEE MAINS 2019] (1) y(loge2) = loge4 (2) y(x) is decreasing in (0, 1)
(3) y(loge2) = elog 2
1 ,1
23. The solution of the differential equation, dy
dx = (x – y)2, when y(1) = 1, is : [IIT JEE MAINS 2019]
(1) –loge 1 x – y
1– x y
2 – x
1– x y
1 x – y
= 2(x – 1)
24. Let y = y(x) be the solution of the differential equation, x dy
dx + y = x loge x, (x > 1). If 2y(2) = loge 4 – 1, then y(e)
is equal to : [IIT JEE MAINS 2019]
(1) – e
2 (2) –
e
4
25. If a curve passes through the point (1, –2) and has slope of the tangent at any point (x, y) on it as 2x – 2y
x , then
the curve also passes through the point : [IIT JEE MAINS 2019]
(1) (–1, 2) (2) (– 2 , 1) (3) ( 3 , 0) (4) (3, 0)
26. Let y = y(x) be the solution of the differential equation, (x2 + 1)2 dy
dx + 2x(x2 + 1)y = 1 such that y(0) = 0. If
a y(1) = 32
, then the value of ‘a’ is - [IIT JEE MAINS 2019]
(1) 1
2 (2)
16 (4) 1
27. Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is 2
2y
x . If the curve passes through the
centre of the circle x2 + y2 – 2x – 2y = 0, then is equation is - [IIT JEE MAINS 2019]
(1) x loge |y| = 2(x – 1) (2) x loge |y| = 2(x + 1)
(3) x2 loge |y| = –2(x – 1) (4) x loge |y| = x – 1
28. The solution of the differential equation x dy
dx + 2y = x2(x 0) with y(1) = 1, is : [IIT JEE MAINS 2019]
(1) y = 3x
2
(1) – 2
4 3

30. If y = y(x) is the solution of the differential equation dy
dx = (tan x – y) sec2 x, x – ,
2 2
(1) 1
1
e – 2
31. Let y = y(x) be the solution of the differential equation, dy
dx + y tan x = 2x + x2 tan x, x – ,
2 2
(1) y 4
x – y

dy = 0, If value of y is 1 when x = 1, then the value of x for
which y = 2, is :
5
2 +
1
e
33. The general solution of the differential equation (y2 – x3)dx – xydy = 0 (x 0) is :
(where c is a constant of integration) [IIT JEE MAINS 2019]
(1) y2 + 2x2 + cx3 = 0 (2) y2 + 2x3 + cx2 = 0
(3) y2 – 2x + cx3 = 0 (4) y2 – 2x3 + cx2 = 0
34. Let y – y(x) be the solution of the differential equation sin x dy
dx + y cos x = 4x, x (0, ). If y 0
2
(1) 2–8
(y 1)cos x 0 and y(0) 1, dx
then y 2
(1) 4
3 (2)
1 –
3
36. If a curve y = f(x) passes through the point (1 , –1) and satisfies the differential equation,
y(1 + xy) dx = xdy then f 1
– 2
(1) 4
5 (2)
2
5
37. Let y(x) be the solution of the differential equation (x log x) dy
y 2x dx
(1) 2 (2) 2e (3) e (4) 0
38. Let the population of rabbits surviving at a time t be governed by the differential equation
dp(t) 1 p(t) – 200.
dt 2 and P(0) = 100 then p(t) equals : [JEE-Main - 2014]
(1) 400 – 300 et/2 (2) 300 – 200 e–t/2
(3) 600 – 500 et/2 (4) 400 – 300 e–t/2
39. At present a firm is manufacturing 2000 items. It is estimated that the rate of change of production P
w.r.t. additional number of workers x is given by dP
100 –12 x dx
. If the firm employs 25 more workers,
then the new level of production of items is : [AIEEE-2013] (1) 2500 (2) 3000 (3) 3500 (4) 4500
40. If the surface area of a sphere of radius r is increasing uniformly at the rate 8cm2/s, then the rate of
change of its volume is : [JEE-Main (On line) 2013]
(1) proportional to r2 (2) constant (3) proportional to r (4) proportional to r
41. Consider the differential equation 3
2 2
dy y
dx 2(xy – x ) [JEE-Main (On line)-2013]
Statement 1 : The substitution z = y2 transforms the above equation into a first order homogenous differential equation.
Statement 2 : The solution of this differential equation is

2y 2 xy e C
(1) Statement 1,is false and statement 2 is true. (2) Both statements are true. (3) Statement 1 is true and statement 2 is false. (4) Both statements are false.
42. If a curve passes through the point 7
2, 2
at any point (x, y) on it, then the
ordinate of the point on the curve whose abscissa is – 2 is : [JEE-Main (On line) 2013]
(1) 5
3
2
43. The equation of the curve passing through the origin and satisfying the differential equation
(1 + x2) dy
dx +2 xy = 4x2 is : [JEE-Main (On line) 2013]
(1) (1 + x2) y = x3 (2) 3(1 + x2) y = 4x3 (3) 3(1 + x2) y =2x3 (4) (1 + x2) y = 3x3
44. The population p(t) at time t of a certain mouse species satisfics the differential equation dp(t)
dt = 0.5
p(t) – 450. If p(0) = 850, then the time at which the population becomes zero is – [AIEEE-2012]
(1) In18 (2) 2 ln 18 (3) ln 9 (4) 1
2 In18
45. If dy
dx = y + 3 > 0 and y(0) = 2, then y(ln 2) is equal to :- [AIEEE-2011]
(1) 13 (2) –2 (3) 7 (4) 5 46. Let I be the purchase value of an equipment and V(t) be the value after it has been used for t years. The
value V(t) depreciates at a rate given by differential equation dV(t)
–k(T – t), dt
where k > 0 is a
constant and T is the total life in years of the equipment. Then the scrap value V(T) of the equipment is [AIEEE-2011]
(1) 2k(T – t)
T – k
(4) 2kT
I – 2
47. The curve that passes through the point (2, 3), and has the property that the segment of any tangent to
it lying between the coordinate axes is bisected by the point of contact, is given by : [AIEEE-2011]
(1)
6 y
48. Consider the differential equation y2dx + 1
x – y

dy = 0. If y(1) = 1, then x is given by : [AIEEE-2011]
(1)
1
y e
49. Solution of the differential equation cos x dy = y(sin x – y)dx, 0 < x < 2
is : [AIEEE-2010]
(1) sec x = (tan x + c) y (2) y sec x = tan x + c
(3) y tan x = sec x + c (4) tan x = (sec x + c) y
50. The differential equation which represents the family of curves y = 2c x
1c e where c1 and c2 arc arbitrary
constants, is :- [AIEEE-2009]
(1) yy ′′ = y ′ (2) yy ′′ = (y ′)2 (3) y ′ = y2 (4) y ′′ = y ′ y
51. The solution of the differential equation dy x y
dx x
satisfying the condition y(1) = 1 is – [AIEEE-2008]
(1) y - nx + x (2) y = x nx + x2 (3) y = xe(x–1) (4) y = x nx + x
52. The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is-
[AIEEE-2008]
(1) (x – 2)y′ 2 = 25 – (y – 2)2 (2) (y – 2)y′ 2 = 25 – (y – 2)2
(3) (y – 2)2 y′ 2 = 25 – (y – 2)2 (4) (x – 2)2 y′ 2 = 25 – (y – 2)2
53. The differential equation of all circles passing through the origin and having their centres on the x-axis
is- [AIEEE-2007]
dx (2) 2 2 dy
x y 3xy dx
dx (4) 2 2 dy
y x – 2xy dx

54. The differential equation whose solution is Ax2 + By2 = 1, where A and B are arbitrary constant is of –
[AIEEE-2006]
(1) first order and second degree (2) first order and first degree
(3) second order and first degree (4) second order and second degree
JEE Advanced PYQ
1. Let denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to at a
point P intersect the y-axis at YP. If PYP has length 1 for each point P on , then which of the following option
is/are correct? [JEE Adv 2019]
(A) 2
2 e
x
(C) 2
2 e
x
(D) 2xy '– 1– x 0
2. Let f. R R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation
dy
f(x) is _________. [JEE Adv 2018]
3. Let f : R R and g : R R be two non-constant differentiable functions. If
f (x) = (e(f(x)–g(x)))g(x) for all x R, and f(1) = g(2) = 1, then which of the following statement (s) is (are) TRUE ?
[JEE Adv 2018]
(A) f(2) < 1 – loge 2 (B) f(2) > 1 – loge 2
(C) g(1) > 1 – loge 2 (D) g(1) < 1 – loge 2
4. If y = y(x) satisfies the differential equation –1

(A) 3 (B) 9 (C) 16 (D) 80
5. A solution curve of the differential equation (x2 + xy + 4x + 2y + 4) 2dy
y 0, x 0 dx
, passes through the point
(1,3). Then the solution curve [JEE Adv 2016]
(A)intersects y = x + 2 exactly at one point (B) intersects y = x + 2 exactly at tow point
(C)intersects y = (x + 2)2 (D) does NOT intersect y = (x + 3)2
6. Let f : (0, )R be a differentiable function such that f (x) = 2 f (x)
x for all x (0, ) and f(1)1. Then
[JEE Adv 2016]
(A) x 0
(D) | f (x) | 2 for all x (0,2)
7. Consider the family of all circles whose centers lie on the straight line y = x. If this family of circles is represented
by the differential equation Py + Qy + 1 = 0, where P, Q are functions of x, y and y 2
2
dx dx

,
then which of the following statements is (are) true? [JEE Adv 2015]
(A) P = y +x (B) P = y –x
(C) P + Q = 1 – x + y + y + (y)2 (D) P – Q = x + y – y – (y)2
8. The function = f() is the solution of the differential equation 4
2 2

3 /2
3 /2
f (x)dx

. Let the slope of the curve at each point (x, y) be y y
sec , x 0 x x

(A) y 1

x
x
x 2

10. If y(x) satisfies the differential equation y ' – ytanx = 2x sec x and y(0) = 0, then [JEE 2012]
(A) 2

11.(a) Let f : [1,) [2, ) be a differentiable function such that f(1) = 2. If 6
x
3
I
f(t)dt 3xf(x) – x
for all x 1, then the value of f(2) is : [JEE 2011]
(b) Let y ' (x) + y(x)g '(x) = g(x)g '(x), y(0) = 0, x R, where f ′ (x) denotes df(x)
dx and g(x) is a given non-
constant differentiable function on R with g(0) = g(2) = 0. Then the value of y(2) [JEE 2011] 12. Let f be a real valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y-intercept
of the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P, then the value of f(– 3) is equal to : [JEE 2010]
13.(i) Match the statements/expressions in Column-I with the open intervals in Column-ll. [JEE 2009] Column-I Column-II

solutions of the differential equation (x – 3)2 y ' + y = 0

(x – 1)(x – 2)(x – 3)(x – 4)(x – 5)dx
(C) Interval in which at least one of the points of local maximum (r) 5
, 8 4

of cos2 x + sin x lies (D) Interval in which tan–1 (sin x + cos x) is increasing (s) (–,)
(ii) Match the statements/expressions in Column-I with the values given in Column-ll. Column-I Column-II (A) The number of solutions of the equation (p) 1
sinxxe – cosx 0 in the interval 0,
2

(B) Value(s) of k for which the planes (q) 2 kx + 4y + z = 0, 4x + ky + 2z = 0 and 2x + 2y + z = 0 intersect in a straight line (C) Value(s) of k for which |x –1| + |x –2| + |x +1| + |x +2| = 4k (r) 3 has integer solution(s) (D) If y ' = y + 1 and y(0) = 1, then value(s) of y(ln2) (s) 4 (t) 5
14. Let a solution y = y(x) of the differential equation, 2 2x x – 1dy – y y – 1dx 0 satisfy y(2) = 2
3 .

1 2 3 1 – 1–
y x x [JEE 2008]
(A) Statement-1 is True, Statement-2 is True ; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1. (C) Statement-1 is True, Statement-2 is False. (D) Statement-1 is False, Statement-2 is True.
15.(a) Let f(x) be differentiable on the interval (0, ) such that f(1) = 1 and 2 2
t x
t – x for
(A) 21 2x
3x 3 (B)
2–1 4x
3x 3 (C)
(b) The differential equation 21– ydy
dx y determines a family of circles with -
(A) variable radii and a fixed centre at (0, 1) (B) variable radii and a fixed centre at (0, –1) (C) fixed radius I and variable centres along the x-axis (D) fixed radius I and variable centres along the y-axis 16. Let C be a curve such that the tangent at any point P on it meets x-axis and y-axis at A and B
respectively. If BP : PA = 3 : 1 and the curve passes through the point (1, 1), then [JEE 2006]
(A) The curve passes through 1
2, 8

(B) Equation of normal to the curve at (1, 1) is 3y – x = 2 (C) The differential equation for the curve is 3y ' + xy = 0 (D) The differential equation for the curve is xy ' + 3y = 0
ANSWER KEY
EXERCISE # 1
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ans. A C B A D A B A C A D B D B A A B D C A
Q.No. 21 22 23 24 25 26 27 28 29 30 31 32
Ans. B B A D A B C A D A B B
33. True 34. False 35. True 36. False 37. y2 = x2 (2nx –1) + c 38. Ellipse
39. 6y –3x = n 2
2y3x3 40. y3 (x + 1)2 =
6
x6
+ 5
2x5
+ 4
EXERCISE # 2
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Ans. C C C A A A C A B A B A B A C
Q.No. 16 17 18 19 20 21 22 23 24
Ans. B B A B C D B C C
Q.No. 25 26 27 28 29 30 31 32 33 34 35 36 37
Ans. A,B A,B C,D A,B,C,D A,B,C A,B A,B,C C,D A,B,D C A A A
38. (B) 39. (C) 40. (A) 41. (A) 42. (C)
43. A P, S ; B Q ; C Q, S; D R 44. A R ; B P ; C S ; D Q EXERCISE # 3
JEE MAINS
1 2 3 4 5 6 7 8 9 10
3.00 4 3 2 3 3 D B B A
11 12 13 14 15 16 17 18 19 20
C A A B A A 2 2 1 3
21 22 23 24 25 26 27 28 29 30
3 4 4 4 3 3 1 4 4 2
31 32 33 34 35 36 37 38 39 40
2 1 2 2 2 1 1 2 3 3
41 42 43 44 45 46 47 48 49 50
2 3 2 2 3 4 3 4 1 2
51 52 53 54
4 3 3 3
1 2 3 4 5 6 7 8 9 10
A,B 0.4 B,C A A,D A B,C B A A,D
11 12
BONUS,0 9
13.((i) (A) p,q,s ; (B) p,t ; (C) p,q,r,t ; (D) s ; (ii) (A) p; (B) q,s ‘ (C) q,r,s,t ; (D) r)
14.(C) 15.((a) (A), (b) (C)) 16.(ABD)

Date post:	28-Mar-2022
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

MASTERSHEET: Differential Equation EXERCISE # 1

Documents